Control and observer design for nonlinear finite and infinite dimensional systems [1 ed.] 3540279385, 9783540279389

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Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

322

Thomas Meurer
Knut Graichen
Ernst Dieter Gilles (Eds.)

Control and Observer Design for Nonlinear Finite and Inˇnite Dimensional Systems With 150 Figures

Series Advisory Board F. Allg¨ower · P. Fleming · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · A. Rantzer · J.N. Tsitsiklis

Editors Dipl.- Ing. Thomas Meurer Dipl.- Ing. Knut Graichen Prof. Dr.- Ing. Dr. h.c. mult. Ernst Dieter Gilles Universit¨at Stuttgart Insitut für Systemdynamik und Regelungstechnik (ISR) Pfaffenwaldring 9 70569 Stuttgart Germany

ISSN 0170-8643 ISBN-10 ISBN-13

3-540-27938-5 Springer Berlin Heidelberg New York 978-3-540-27938-9 Springer Berlin Heidelberg New York

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Preface

The field of nonlinear control has gathered increasing importance during the last decades and is still — or rather more than ever — a very active field of research. Thereby, the development of nonlinear control theory and design is directly linked to the progress in theory and design of nonlinear observers. For instance, the differential geometric viewpoint had impact in both research areas. Another important aspect linking both fields is the fact that modern nonlinear control laws usually require full or at least partial state information, such that the classical control–loop has to be amended by an observer. Besides research, nonlinear control and observer design and especially their realtime realization and implementation enters more and more into industrial applications. Particularly in the fields of automotive engineering and mechatronics, higher control performance is demanded with the simultaneous demand of actuator and sensor size reduction. Additionally, market pressure increases the demand to reduce costly sensors, such that signals have to be reconstructed mathematically. Although most of the results on nonlinear controller and observer design are intended for finite–dimensional systems, their application to infinitedimensional systems described by partial differential equations gathers more and more importance. Recently, this area experienced a revival due to the demands e.g. in the control of highly nonlinear flexible structures from mechatronics, or bio–chemical systems, where spatial effects can not be neglected. The intension of this volume is to present a well balanced combination of state-of-the-art theoretical results in the field of nonlinear controller and observer design, combined with industrial applications stemming from mechatronics, electrical, (bio–) chemical engineering, and fluid dynamics. The table of contents provides an impression of the scope of contributions from leading experts in system theory and control engineers from universities and industry. The unique combination of results for finite as well as infinite–dimensional systems makes this book a remarkable contribution addressing postgraduates, researchers, and engineers both at universities and in industry.

VI

Preface

The contributions to this book were presented at the Symposium on Nonlinear Control and Observer Design: From Theory to Applications (SYNCOD). The symposium was held September 15–16, 2005, at the University of Stuttgart, Germany, and was dedicated to the 65th birthday of Prof. Dr.–Ing. Dr.h.c. Michael Zeitz to honor his life–long research and contributions on the fields of nonlinear control and observer design. All of the contributions were subject to strict reviewing to guarantee the high quality of the book. The editors would like to thank all authors and reviewers for the careful work and their participation at the symposium as the basis for this book. Furthermore, the editors are grateful to Beate Witteler–Neul for the significant help and advice with the organization of the symposium.

Stuttgart, Germany June 2005

Thomas Meurer Knut Graichen Ernst Dieter Gilles

Contents

Part I Nonlinear Observer Design – from Theory to Industrial Applications Observers as Internal Models for Remote Tracking via Encoded Information Alberto Isidori, Lorenzo Marconi, Claudio De Persis . . . . . . . . . . . . . . . . . .

3

Extended Luenberger Observer for Nonuniformly Observable Nonlinear Systems Klaus R¨ obenack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Approximate Observer Error Linearization by Dissipativity Methods Jaime A. Moreno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 On Invariant Observers Silv`ere Bonnabel, Pierre Rouchon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Remarks on Moving Horizon State Estimation with Guaranteed Convergence Tobias Raff, Christian Ebenbauer, Rolf Findeisen, Frank Allg¨ ower . . . . . . 67 Least Squares Smoothing of Nonlinear Systems Arthur J. Krener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 State Estimation of a Molten Carbonate Fuel Cell by an Extended Kalman Filter Michael Mangold, Markus Gr¨ otsch, Min Sheng, Achim Kienle . . . . . . . . . 93 Bioprocess State Estimation: Some Classical and Less Classical Approaches Guillaume Goffaux, Alain Vande Wouwer . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

VIII

Contents

Part II Nonlinear Control of Finite–Dimensional Systems – Geometric Methods, Differential Flatness, Optimal Control and their Applications Convergent Systems: Analysis and Synthesis Alexey Pavlov, Nathan van de Wouw, Henk Nijmeijer . . . . . . . . . . . . . . . . 131 Smooth and Analytic Normal Forms: A Special Class of Strict Feedforward Forms Issa A. Tall, Witold Respondek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Constructive Nonlinear Dynamics – Foundations and Application to Robust Nonlinear Control Johannes Gerhard, Martin M¨ onnigmann, Wolfgang Marquardt . . . . . . . . . 165 Optimal Control of Piecewise Affine Systems: A Dynamic Programming Approach Frank J. Christophersen, Mato Baoti´c, Manfred Morari . . . . . . . . . . . . . . . 183 Hierarchical Hybrid Control Synthesis and its Application to a Multiproduct Batch Plant J¨ org Raisch, Thomas Moor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Closed-Loop Fault-Tolerant Control for Uncertain Nonlinear Systems Michel Fliess, C´edric Join, Hebertt Sira-Ram´ırez . . . . . . . . . . . . . . . . . . . . . 217 Feedforward Control Design for Nonlinear Systems under Input Constraints Knut Graichen, Michael Zeitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 System Inversion and Feedforward Control via Formal Power Series and Summation Methods Marc Oliver Wagner, Thomas Meurer, Michael Zeitz . . . . . . . . . . . . . . . . . 253 Flatness-Based Improved Relative Guidance Maneuvers for Commercial Aircraft Thierry Miquel, Jean L´evine, F´elix Mora-Camino . . . . . . . . . . . . . . . . . . . . 271 Vehicle Path-Following with a GPS-aided Inertial Navigation System Steffen Kehl, Wolf-Dieter P¨ olsler, Michael Zeitz . . . . . . . . . . . . . . . . . . . . . 285 Control of Switched Reluctance Servo-Drives Achim A.R. Fehn, Ralf Rothfuß . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Contents

IX

Flatness-Based Two-Degree-of-Freedom Control of Industrial Semi-Batch Reactors Veit Hagenmeyer, Marcus Nohr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Part III Nonlinear Control of Infinite–Dimensional Systems – Applications in Mechatronics, Fluid Flow and Chemical Engineering Controlled Friction Damping using Optimally Located Structural Joints Lothar Gaul, Stefan Hurlebaus, Hans Albrecht, Jan Wirnitzer . . . . . . . . . . 335 Infinite-Dimensional Decoupling Control of the Tip Position and the Tip Angle of a Composite Piezoelectric Beam with Tip Mass Andreas Kugi, Daniel Thull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Nonlinear Flow Control Based on a Low Dimensional Model of Fluid Flow Rudibert King, Meline Seibold, Oliver Lehmann, Bernd. R. Noack, Marek Morzy´ nski, Gilead Tadmor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Flatness Based Approach to a Heat Conduction Problem in a Crystal Growth Process Joachim Rudolph, Jan Winkler, Frank Woittennek . . . . . . . . . . . . . . . . . . . 387 Model–based Nonlinear Tracking Control of Pressure Swing Adsorption Plants Matthias Bitzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Observers฀as฀Internal฀Models฀for฀Remote Tracking฀via฀Encoded฀Information Alberto฀Isidori1,2,3 ,฀Lorenzo฀Marconi1 ,฀and฀Claudio฀De฀Persis2 1฀

C.A.SY.฀–฀Dipartimento฀di฀Elettronica,฀Informatica฀e฀Sistemistica, University฀of฀Bologna,฀40136฀Bologna,฀Italy. 2฀ Dipartimento di Informatica฀e฀Sistemistica,฀Universit` a฀di฀Roma฀“La฀Sapienza”, 00184฀Rome,฀Italy. 3฀ Department฀of฀Electrical฀and฀Systems฀Engineering,฀Washington฀University, St.฀Louis,฀MO฀63130,฀USA. [email protected],฀[email protected] Summary.฀ In฀this฀paper,฀we฀consider฀a฀servomechanism฀problem฀in฀which฀the฀command฀and฀control฀functions฀are฀distributed฀in฀space,฀and฀hence฀the฀system฀consists of฀different฀components฀linked฀by฀a฀communication฀channel฀of฀finite฀capacity.฀The desired฀ control฀ goal฀ is฀ achieved฀ by฀ designing฀ appropriate฀ encoders,฀ decoders฀ and internal฀models฀of฀the฀exogenous฀signals.฀As฀an฀application,฀we฀describe฀a฀how฀the output฀of฀a฀system฀can฀be฀forced฀to฀track฀a฀reference฀signal฀generated฀by฀a฀remotely located฀nonlinear฀oscillator.

Keywords: Nonlinear฀tracking,฀internal฀model,฀encoding,฀remote฀control.

1 Introduction Generally speaking, the problem of tracking and asymptotic disturbance rejection (sometimes also referred to as the generalized servomechanism problem or the output regulation problem) is to design a controller so as to obtain a closed-loop system in which all trajectories are bounded, and a regulated output asymptotically decays to zero as time tends to infinity. The peculiar aspect of this design problem is the characterization of the class of all possible exogenous inputs (disturbances, commands, uncertain constant parameters) as the set of all possible solutions of a fixed (finite-dimensional) differential equation. In this setting, any source of uncertainty (about actual disturbances affecting the system, about actual trajectories that are required to be tracked, about any uncertain constant parameters) is treated as uncertainty in the initial condition of a fixed autonomous finite dimensional dynamical system, known as the exosystem.

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 3–18, 2005. © Springer-Verlag Berlin Heidelberg 2005

4

A. Isidori, L. Marconi, and C. De Persis

The body of theoretical results that was developed in this domain over about three decades has scored numerous important successes and has now reached a stage of full maturity. The design of a controller that solves a generalized servomechanism problem is centered around the design of an internal model, which is an autonomous dynamical system capable of generating all possible “feed-forward inputs” capable of securing perfect tracking. Even though several different approaches to the design of internal models have been pursued in the literature (see e.g. [13, 3, 19, 8]) it was only recently that the design in question was understood to be based on the very same principles underlying the design of state observers. And, of course, in the design of regulators for nonlinear systems, it is the design on nonlinear observers that plays a crucial role. The theory of nonlinear observers has been used, in the design of nonlinear regulators, at different levels of generality. The earliest contribution of this kind is directly related to the pioneering work of Michael Zeitz on the design of nonlinear observers [2]. Professor Zeitz investigated the problem of determining when and how the dynamics of the observation error can be made diffeomorphic to a linear dynamics, a problem that later became known as problem of linearization by output injection. As a matter of fact, the same principles inspiring the method of linearization by output injection have been used in [9] for the design of a (special class of) nonlinear internal models. An adaptive version of the method, based on the works of [1, 18], was used later in [10] for the design of a class of adaptive nonlinear internal models. Finally, the theory of high-gain nonlinear observers as developed by [12] was used in [6] for the design of a fully general (though not adaptive) nonlinear internal model. In this paper, we consider a servomechanism problem in which the command and control functions are distributed in space, and hence the system consists of different components linked by communication networks. The simplest case in which this situation may occur is when generation of reference signals and control functions take place at distant locations. The problem addressed is the control of a plant so as to have its output tracking (a family of) reference commands generated at a remote location and transmitted through a communication channel of finite capacity. What renders the problem in question different from a conventional tracking problem is that the tracking error, that is the difference between the command input and the controlled output, is not available as a physical entity, as it is defined as difference between two quantities residing at different (and possibly distant) physical locations. Therefore the tracking error as such cannot be used to drive a feedback controller, as it is the case in a standard tracking problem. As a simple example of application of our method, we describe a how the output of a system can be forced to track a reference signal generated by a remotely located Van der Pol oscillator.

Remote Tracking via Encoded Information

5

2 Problem Statement The problem outlined in the introduction can be defined in the following terms. Consider a single-input single-output nonlinear system modeled by equations of the form x˙ = f (x) + g(x)u (1) y = h(x) and suppose its output y is required to asymptotically track the output ydes of a remotely located exosystem w˙ = s(w) ydes = yr (w) .

w ∈ Rr

(2)

The problem is to design a control law of the form ξ˙ = ϕ(ξ, y, wq ) u = θ(ξ, y, wq )

(3)

in which wq represents a sampled and quantized version of the remote exogenous input w, so as to have the tracking error e(t) = y(t) − yr (w(t))

(4)

asymptotically converging to zero as time tends to ∞. Note that the controller in question does not have access to e, which is not physically available, but only to the controlled output and to a sampled and quantized version of the remotely generated command. We will show in what follows how the theory of output tracking can be enhanced so as to address this interesting design problem. In particular, we will show how, by incorporating in the controller two (appropriate) internal models of the exogenous signals, the desired control goal can be achieved. One internal model is meant to asymptotically reproduce, at the location of the controlled plant, the behavior of the remote command input. The other internal model, as in any tracking scheme, is meant to generate the “feedforward” input which keeps the tracking error identically at zero. We begin by describing, in the following section, the role of the first internal model.

3 The Encoder-Decoder Pair In order to overcome the limitation due to the finite capacity of the communication channel, the control structure proposed here has a decentralized structure consisting of two separate units: one unit, co-located with the command generator, consists of an encoder which extracts from the the reference signal the data which are transmitted through the communication channel;

6

A. Isidori, L. Marconi, and C. De Persis

the other unit, co-located with the controlled plant, consists of a decoder which processes the encoded received information and of a regulator which generates appropriate control input. The problem at issue will be solved under a number of assumptions most of which are inherited by the literature of output regulation and/or control under quantization. The first assumption, which is a customary condition in the problem of output regulation, is formulated as follows. (A0) The vector field s(·) in (2) is locally Lipschitz and the initial conditions for (2) are taken in a fixed compact invariant set W0 . The next assumption is, on the contrary, newer and motivated by the specific problem addressed in this paper. In order to formulate rigorously the assumption in question, we need to introduce some notation. In particular let |x|S denote the distance at a point x ∈ Rn from a compact subset S ⊂ Rn , i.e. the number |x|S := max |x − y| y∈S

and let

L0 =

max

i∈[1,...,r] (x,y)∈W0 ×W0

|xi − yi | .

(5)

Furthermore, having denoted by Nb the number of bits characterizing the communication channel constraint, let N be the largest positive integer such that (6) Nb ≥ r log2 N

where υ , υ ∈ R, denotes the lowest integer such that υ ≥ υ. With this notation in mind, the second assumption can be precisely formulated as follows.

(A1) There exists a compact set W ⊃ W0 which is invariant for w˙ = s(w) and such that √ L0 w ¯∈W ⇒ |w| ¯ W0 ≥ r . 2N W being compact and s(·) being locally Lipschitz, it is readily seen that there exists a non decreasing and bounded function M (·) : R≥0 → R>0 , with M (0) = 1, such that for all w10 ∈ W and w20 ∈ W and for all t ≥ 0 |w1 (t) − w2 (t)| ≤ M (t)|w10 − w20 |

(7)

where w1 (t) and w2 (t) denote the solutions of (2) at time t passing through w10 and, respectively, w20 at time t = 0. This function, the sampling interval T , the number L0 defined in (5) and the number N fulfilling (6), determine the parameters of the encoder-decoder pair, which are defined as follows (see [20], [17], [11] for more details). Encoder dynamics. The encoder dynamics consist of a copy of the exosystem dynamics, whose state is updated at each sampling time kT , k ∈ N and

Remote Tracking via Encoded Information

7

determines (depending on the actual state of the exosystem) the centroid of the quantization region, and of an additional discrete-time dynamics which determines the size of the quantization region. Specifically, the encoder is characterized by w˙ e = s(we )

L(k + 1) =

√ M (T ) L(k) r N

we (kT ) = we (kT − ) + wq (k) we (0− ) ∈ W0

L(k) N

L(0) = L0

in which wq represents the encoded information given by, for i = 1, . . . , r, ⎧ N |wi (kT )−we,i (kT − )| 1 ⎪ ⎪ N even − ⎪ ⎨ L(k) 2 wq,i (k) = sgn(wi (kT )−we,i (kT − ))· − ⎪ ⎪ ⎪ N |wi (kT )−we,i (kT )| − 1 N odd . ⎩ L(k) 2

At each sampling time kT , the vector wq (k) is transmitted to the controlled plant through the communication channel and then used to update the state of the decoder unit as described in the following. To this regard note that each component of the vector wq (k) can be described by log2 N bits and thus the communication channel constraint is fulfilled. Decoder dynamics The decoder dynamics is a replica of the encoder dynamics and it is given by w˙ d = s(wd )

wd (kT ) = wd (kT − ) + wq (k) wd (0− ) = we (0− )

√ M (T ) L(k) L(k + 1) = r N

L(k) N

(8)

L(0) = L0

If, ideally, the communication channel does not introduce delays, it turns out that wd (t) ≡ we (t) for all t ≥ 0. Furthermore, it can be proved that the set W characterized in Assumption (A1) in invariant for the encoder (decoder) dynamics and that the asymptotic behavior of we (t) (wd (t)) converges uniformly to the true exosystem state w(t), provided that T is properly chosen with respect to the number N and the function M (·). This is formalized in the next proposition (see [17], [11] for details). Proposition 1. Suppose Assumptions (A0)-(A1) hold and that the sampling time T and the number N satisfy √ (9) N > r M (T ) .

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A. Isidori, L. Marconi, and C. De Persis

Then: (i) for any wd (0− ) ∈ W0 and w(0) ∈ W0 , wd (t) ∈ W for all t ≥ 0; (ii) for any wd (0− ) ∈ W0 and w(0) ∈ W0 , lim |w(t) − wd (t)| = 0

t→∞

with uniform convergence rate, namely for every > 0 there exists T ∗ > 0 such that for all initial states wd (0− ) ∈ W0 , w(0) ∈ W0 , and for all t ≥ T ∗ , |w(t) − wd (t)| ≤ . Proof. As W is an invariant set for w˙ = s(w), the proof of the first item reduces to show that, for all k ≥ 0, if wd (kT − ) ∈ W then necessarily wd (kT ) ∈ W . For, note that this is true for k = 0. As a matter of fact, since wd (0− ) ∈ W0 ⊂ W and by√bearing in mind the definition of wq , it turns out that |wd (0) − w(0)| ≤ rL0 /2N which implies, by definition of W in Assumption 0 note that, again by definition of (A1), that wd (0) ∈ W . For a generic k > √ wq , it turns out that |wd (kT ) − w(kT )| ≤ rL(k)/2N . But, by the second of (8) √ and by condition (9), L(k) < L(k − 1) ≤ L0 yielding |wd (kT ) − w(kT )| ≤ rL0 /2N which implies wd (kT ) ∈ W . This completes the proof of the first item. The second item has been proved in [17], [11]. Remark 1. By composing (6) with (9) it is easy to realize that the number of bits Nb and the sampling interval T are required to satisfy the constraint √ r M (T ) (10) Nb ≥ r log2 in order to have the encoder-decoder trajectories asymptotically converging to the exosystem trajectories. Since the function M (·) depends on the exosystem dynamics and on the set W0 of initial conditions for (2), equation (10) can be interpreted as a relation between the bit-rate of the communication channel and the exosystem dynamics which must be satisfied in order to remotely reconstruct the reference signal.

4 The Design of the Regulator 4.1 Standing Hypotheses As in most of the literature on regulation of nonlinear system, we assume in what follows that the controlled plant has well defined relative degree and normal form. If this is the case and if the initial conditions of the plant are allowed to vary on a fixed (though arbitrarily large) compact set, there is no loss of generality in considering the case in which the controlled plant has relative degree 1 (see for instance [4]). We henceforth suppose that system (1) is expressed in the form

Remote Tracking via Encoded Information

z˙ = f (z, y, μ) y˙ = q(z, y, μ) + u

z ∈ Rn y∈R

9

(11)

in which μ is a vector of uncertain parameters ranging in a known compact set P . Initial conditions (z(0), y(0)) of (11) are allowed to range on a fixed (but otherwise arbitrary) compact set Z × Y ⊂ Rn × R. It is well known that, if the regulation goal is achieved, in steady-state (i.e when the tracking error e(t) is identically zero) the controller must necessarily provide an input of the form uss = Ls yr (w) − q(z, yr (w), μ)

(12)

(where Ls yr (·) stands for the derivative of yr (·) along the vector field s(·)) in which w and z obey μ˙ = 0 (13) w˙ = s(w) z˙ = f (z, yr (w), μ) . As in [5], we assume in what follows that system (13) has a compact attractor, which is also locally exponentially stable. To express this assumption in a concise form, it is convenient to group the components μ, w, z of the state vector of (13) into a single vector z = col(μ, w, z) and rewrite the latter as z˙ = f0 (z) . Consistently, the map (12) is rewritten as uss = q0 (z) , and it is set Z = P × W × Z. The assumption in question is the following one:

(A2) there exists a compact subset Z of P × W × Rn which contains the positive orbit of the set Z under the flow of (13) and ω(Z) is a differential submanifold (with boundary) of P × W × Rn . Moreover there exists a number d1 > 0 such that z ∈ P × W × Rn ,

|z|ω(Z) ≤ d1

⇒

z ∈ Z.

Finally, there exist m ≥ 1, a > 0 and d2 ≤ d1 such that z0 ∈ P × W × Rn ,

|z0 |ω(Z) ≤ d2

⇒

|z(t)|ω(Z) ≤ me−at |z0 |ω(Z) ,

in which z(t) denotes the solution of (13) passing through z0 at time t = 0. In what follows, the set ω(Z) will be simply denoted as A0 . The final assumption is an assumption that allows us to construct an internal model of all inputs of the form uss (t) = q0 (z(t)), with z(t) solution of (13) with initial condition in A0 . This assumption, which can be referred to as assumption of immersion into a nonlinear uniformly observable system, is the following one:

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A. Isidori, L. Marconi, and C. De Persis

(A3) There exists an integer d > 0 and a locally Lipschitz map ϕ : Rd → R such that, for all z ∈ A0 , the solution z(t) of (13) passing through z0 at t = 0 is such that the function u(t) = q0 (z(t)) satisfies u(d) (t) + ϕ(u(t), u(1) (t), . . . , u(d−1) (t)) = 0 .

(14)

4.2 A Nonlinear Observer as Nonlinear Internal Model As mentioned in the introduction, nonlinear observers play a fundamental role in the design of nonlinear regulators. To see why this is the case consider a candidate controller having the following structure ξ˙ = Φ(ξ) + Ψ (ξ)v u = γ(ξ) + v

(15)

in which ξ ∈ Rν and v is an additional control, to be determined at a later stage. Controlling the plant (11) by means of (15) yields a system μ˙ = 0 w˙ = s(w) z˙ = f (z, y, μ) y˙ = q(z, y, μ) + γ(ξ) + v ξ˙ = Φ(ξ) + Ψ (ξ)v e = y − yr (w) which, regarded as a system with input v and output e, has a well-defined relative degree, equal to one. If the vector field Ψ (ξ) is complete, this system has a globally-defined normal form (see e.g. [14, pages 427-432]). Its zero dynamics are those of μ˙ w˙ z˙ ξ˙

=0 = s(w) = f (z, yr (w), μ) = Φ(ξ) + Ψ (ξ)[Ls yr (w) − q(z, yr (w), μ) − γ(ξ)] ,

and these equations, using the concise notation z = col(μ, w, z) introduced above, can be rewritten as z˙ = f0 (z) ξ˙ = Φ(ξ) + Ψ (ξ)[q0 (z) − γ(ξ)] .

(16)

It is known that if a system has relative degree one, a globally defined normal form, and a zero dynamics whose trajectories asymptotically converge to a compact attractor, control by means of high-gain output feedback has the effect of keeping trajectories bounded and steering the output itself to zero. In view of this fact, it is reasonable to expected that if the trajectories of (16) converge to a compact attractor, the choice of the additional control v in

Remote Tracking via Encoded Information

11

(15) as a high-gain feedback on e can be used to solve the problem of output regulation. Leaving aside, for the time being, the fact that the variable e is not – in the present setting – available for feedback, we describe in what follows how the desired asymptotic properties of (16) can be achieved. Note that the dynamics in question can be viewed as the cascade connection of two subsystems, the upper of which has trajectories which are bounded and attracted by the compact invariant set A0 (see Assumption (A2)). Thus, the idea is to design Φ(ξ), Ψ (ξ), γ(ξ) so that also in the full system (16) the trajectories are bounded and attracted by a compact invariant set. Looking at how the upper and the lower subsystem of (16) are coupled, it is seen that the coupling takes places through the function uss (t) = q0 (z(t)), which is seen as “output” of the upper subsystem and “input” of the lower subsystem. In view of Assumption (A3), as long as z0 ∈ A0 , the function in question can be regarded also as output of the autonomous nonlinear system ζ˙1 = ζ2 ζ˙2 = ζ3 ··· ζ˙d−1 = ζd ζ˙d = −ϕ(ζ1 , ζ2 , . . . , ζd ) uss = ζ1 which is trivially uniformly completely observable, in the sense of [12]. Taking advantage of this property, it seems quite natural at this point to choose the lower subsystem of (16) as an observer for the set of variables ζ1 , . . . , ζd (which is indeed always possible, because the latter possesses the required observability properties). In this way, one is guaranteed that the components of the vector ξ are attracted by a compact set, and the required asymptotic property of (16) is obtained. The nonlinear observer will be designed according to the so-called “highgain” construction proposed in [12]. To this end, consider the sequence of functions recursively defined as τ1 (z) = q0 (z) ,

...,

τi+1 (z) =

∂τi f0 (z) ∂z

for i = 1, . . . , d − 1, with d as introduced in assumption (A3), and consider the map τ : P × W × Rn → Rd (μ, w, z) → col(τ1 (z), τ2 (z), . . . , τd (z)) .

If k, the degree of continuous differentiability of the functions in (11), is large enough, the map τ is well defined and C 1 . In particular τ (A0 ), the image of A0 under τ is a compact subset of Rd , because A0 is a compact subset of P × W × Rn . Let ϕc : Rd → R be any locally Lipschitz function of compact support which agrees on τ (A0 ) with the function ϕ defined in (A3), i.e. a function

12

A. Isidori, L. Marconi, and C. De Persis

such that, for some compact superset S of τ (A0 ) satisfies ϕc (η) = 0 for all η ∈ S ϕc (η) = ϕ(η) for all η ∈ τ (A0 ). With this in mind, consider the system ξ˙ = Φc (ξ) + G(uss − Γ ξ)

(17)

in which ⎛

⎞ ξ2 ⎜ ⎟ ξ3 ⎜ ⎟ ⎟, Φc (ξ) = ⎜ · · · ⎜ ⎟ ⎝ ⎠ ξd −ϕc (ξ1 , ξ2 , . . . , ξd )

⎛

⎞ κcd−1 ⎜κ2 cd−2 ⎟ ⎜ ⎟ ··· ⎟ G=⎜ ⎜ d−1 ⎟, ⎝ κ c1 ⎠ κ d co

Γ = 1 0 ··· 0 ,

the ci ’s are such that the polynomial λd + c0 λd−1 + · · · + cd−1 = 0 is Hurwitz and κ is a positive number. As shown in [6], if κ is large enough, the state ξ(t) of (17) asymptotically tracks τ (z(t)), in which z(t) is the state of system (13). Therefore Γ ξ(t) asymptotically reproduces its output (12), i.e. the steady state control uss (t). As a matter of fact, the following result holds. Lemma 1. Suppose assumptions (A1) and (A2) hold. Consider the triangular system z˙ = f0 (z) (18) ξ˙ = Φc (ξ) + G(q0 (z) − Γ ξ) . Let the initial conditions for z range in the set Z and let Ξ be an arbitrarily large compact set of initial condition for ξ. There is a number κ∗ such that, if κ ≥ κ∗ , the trajectories of (18) are bounded and graph(τ |A0 ) = ω(Z × Ξ) . In particular graph(τ |A0 ) is a compact invariant set which uniformly attracts Z × Ξ. Moreover, graph(τ |A0 ) is also locally exponentially attractive. 4.3 The Remote Regulator and its Properties In view of Lemma 1, it would be natural – if the true error variable e were available for feedback purposes – to choose for (11) a control of the form ξ˙ = Φc (ξ) − Gke u = Γ ξ − ke ,

(19)

with k a large number. This control, in fact, would solve the problem of output regulation (see [5]). The true error e not being available, we choose instead

Remote Tracking via Encoded Information

13

eˆ = y − yr (wd )

(20)

e ξ˙ = Φc (ξ) − Gkˆ u = Γ ξ − kˆ e.

(21)

and the controller accordingly as

The main result which can be established is that there exists k ∗ > 0 such that if k ≥ k ∗ the regulator designed above solves the problem in question (provided that N and T satisfy the condition of Proposition 1). To this end, it is shown first of all the trajectories of the controlled system, namely those of the system w˙ d = s(wd )

wd (kT ) = wd (kT − ) + wq (k)

z˙ = f (z, y, μ)

L(k) 2N

y˙ = q(z, y, μ) + Γ ξ − k(y − yr (wd )) ξ˙ = Φc (ξ) − Gk(y − yr (wd ))

(22) are bounded. To study trajectories of (22) it is convenient to replace the coordinate y by eˆ = y − yr (wd ) to obtain the system

w˙ d = s(wd ) z˙ = f (z, eˆ + yr (wd ), μ) ξ˙ = Φc (ξ) − Gkˆ e eˆ˙ = q(z, eˆ + yr (wd ), μ) − Ls yr (wd ) + Γ ξ − kˆ e.

(23)

This system can be further simplified by changing the state variable ξ into ˜ so as to obtain a system of the ξ˜ = ξ − Gˆ e and setting p = col(μ, wd , z, ξ), form p˙ = F0 (p) + F1 (p, eˆ)ˆ e (24) eˆ˙ = H0 (p) + H1 (p, eˆ)ˆ e − kˆ e, in which

⎛

⎞ 0 ⎜ ⎟ s(wd ) ⎟ F0 (p) = ⎜ ⎝ ⎠ f (z, yr (wd ), μ) ˜ + G(−q(z, yr (wd ), μ) + Ls yr (wd ) − Γ ξ) ˜ Φ(ξ) H0 (p) = q(z, yr (wd ), μ) − Ls yr (wd ) + Γ ξ˜

and F1 (p, eˆ), H1 (p, eˆ) are suitable continuous functions. With this notation at hand, it is possible to show that a large value of k succeeds in rendering bounded the trajectories of the switched nonlinear system (24) provided that the sampling interval T is sufficiently large.

14

A. Isidori, L. Marconi, and C. De Persis

Proposition 2. Consider system (22) with initial conditions in P × W × Z × Y × Ξ. Suppose assumptions (A0)-(A3) hold. Let κ be chosen as indicated in Lemma 1. Then there exist T ∗ > 0 and k ∗ > 0 such that for all sampling intervals T > T ∗ and all k ≥ k ∗ the trajectories are bounded in positive time. Proof. See [16]. Proposition 2 shows that trajectories of the controlled system remain bounded if the time interval T exceeds a minimum number T ∗ (minimal “dwell-time”) which depends on the parameters of the controlled system and on the sets of initial conditions. This, in view of (10), requires Nb to exceed a suitable minimum number Nb∗ . 4 To prove that the tracking error converges to zero, it is useful to observe that, if the coordinate y of (22) is replaced by e = y − yr (w) the system in question can be also rewritten as w˙ = s(w) z˙ = f (z, e + yr (w), μ) ξ˙ = Φc (ξ) + G(−ke) + G(−k˜ e) e˙ = q(z, e + yr (w), μ) − Ls yr (w) + Γ ξ − ke − k˜ e having set

(25)

e˜ = eˆ − e .

The same change of variables used to put (23) in the form (24) yields now a system of the form p˙ = F0 (p) + F1 (p, e)e e, e˙ = H0 (p) + H1 (p, e)e − ke − k˜

(26)

˜ and F0 (p), F1 (p, e), H0 (p), H1 (p, e) are the same as in which p = col(μ, w, z, ξ) in (24). This system can be viewed as system p˙ = F0 (p) + F1 (p, e)e e˙ = H0 (p) + H1 (p, e)e − ke

(27)

forced by a perturbation e˜ = yr (w) − yr (wd ) which, since yr (w) is continuous, is asymptotically vanishing because of Proposition 1. 4

If the number Nb is fixed and not compatible with the minimal dwell time T ∗ determined in the proof of Proposition 2, a more elaborate control structure has to be used, as suggested in [16].

Remote Tracking via Encoded Information

15

The asymptotic properties of (24) have been investigated in [7]. In particular, the results presented in that paper show that if k is large enough, system (24) is input-to-state stable, with restrictions, with respect to a compact subset which is entirely contained in the set {(p, e) : e = 0}. This property can be exploited to prove the main result of the paper. Proposition 3. Consider system (22) with initial conditions in P × W × Z × Ξ × Y . Suppose assumptions (A0)-(A3) hold. Let κ be chosen as indicated in Lemma 1. Then there exist T ∗ > 0 and k ∗ > 0 such that for all sampling intervals T > T ∗ and all k ≥ k ∗ , trajectories are bounded in positive time and lim e(t) = 0 .

t→∞

Proof. See [16].

5 Simulation Results We consider the problem of synchronizing two oscillators located at remote places through a constrained communication channel. The master oscillator (playing the role of exosystem) is a Van der Pol oscillator described by w˙ 1 = w2 + (w1 − aw13 ) w˙ 2 = −w1

(28)

whose output yr = w2 must be replied by the output y of a remote system of the form y˙ = u . (29) Simple computations show that, in this specific case, the steady state control input uss coincides with uss = −w1 and the assumption (A3) is satisfied by ξ˙1 = ξ2 ξ˙2 = −ϕ(ξ1 , ξ2 )

uss = ξ1

where ϕ(ξ1 , ξ2 ) = ξ1 − (ξ2 − 3aξ12 ξ2 ) through the map τ (w) = −w1 −w2 + (w1 − aw13 )

T

We consider a Van der Pol oscillator with = 1.5 and a = 1. The regulator T (21) is tuned choosing κ = 3, G = (12 36) and k = 8. We consider two different simulative scenarios which differ for the severity of the communication channel constraint. In the first case we suppose that the number of available bits is Nb = 2 yielding, according to (6) and to the fact that r = 2, N = 2. In this case, for a certain set of initial conditions, condition (9) is fulfilled with

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A. Isidori, L. Marconi, and C. De Persis

T = 0.15 s. In the second case the available number of bits is assumed Nb = 4 from which (6) and (9) yield a bigger N and T respectively equal to N = 4 and T = 0.5 s. The simulation results, obtained assuming the exosystem (28) and the system (29) respectively at the initial conditions w(0) = (1, 0) and y(0) = 5, are shown in the figure 1 for the first scenario and figure 2 for the second one. In particular figure 1 (respectively 2) shows in the left-half side the phase portrait of the Van der Pol oscillator with overlapped the actual state trajectory of the encoder (decoder) and, in the right-half side, the time behavior of the reference trajectory yr (t) (dotted line) and of the controlled output y(t) (solid line).

1.5

y (t), y(t) r

3

1 2

0.5 1

w2

0

−0.5

0

−1 −1

−1.5 −2

−2

−2.5 −2

−1

0

1 w

1

2

3

−3

0

2

4 6 Time (s)

8

10

Fig. 1. First control scenario (N = 2, T = 0.15 s). Left: phase portrait of the exosystem (dotted line) and trajectory (we1 , we2 ) (solid line). Right: time behavior of the reference trajectory yr (t) (dotted line) and of the controlled output y(t) (solid line).

Acknowledgement. This work was partially supported by NSF under grant ECS0314004, by ONR under grant N00014-03-1-0314, and by MIUR.

References 1. G. Bastin and M. R. Gevers, Stable adaptive observers for non-linear time varying systems, IEEE Trans. Autom. Contr., AC-33, pp. 650–657, 1988. 2. D. Bestle and M. Zeitz, Canonical form observer design for nonlinear time variable systems, Int. J. Control, 38, pp. 419-431, 1983. 3. C.I. Byrnes, F. Delli Priscoli, A. Isidori and W. Kang, Structurally stable output regulation of nonlinear systems. Automatica, 33, pp. 369–385, 1997.

Remote Tracking via Encoded Information

17

y(t), y (t) 2

r

3

1.5

2 1

1

w

2

0.5

0

0

−0.5

−1 −1

−2 −1.5

−2 −2

−1

0 w1

1

2

−3

0

5

15 10 Time (s)

20

25

Fig. 2. Second control scenario (N = 4, T = 0.5 s). Left: phase portrait of the exosystem (dotted line) and trajectory (we1 , we2 ) (solid line). Right: time behavior of the reference trajectory yr (t) (dotted line) and of the controlled output y(t) (solid line). 4. C.I. Byrnes, A. Isidori, L. Marconi, Further results on output regulation by pure error feedback, to be presented at 16th IFAC World Congress, July 2005. 5. C.I. Byrnes, A. Isidori, Limit sets, zero dynamics and internal models in the problem of nonlinear output regulation, IEEE Trans. on Automatic Control, AC-48, pp. 1712–1723, 2003. 6. C.I. Byrnes, A. Isidori, Nonlinear Internal Models for Output Regulation, IEEE Trans. on Automatic Control, AC-49, pp. 2244–2247, 2004. 7. C.I. Byrnes, A. Isidori, L. Praly, On the Asymptotic Properties of a System Arising in Non-equilibrium Theory of Output Regulation, Preprint of the MittagLeffler Institute, Stockholm, 18, 2002-2003, spring. 8. Z. Chen, J. Huang, Global robust servomechanism problem of lower triangular systems in the general case, Systems and Control Letters, 52, pp. 209–220, 2004. 9. F. Delli Priscoli, Output regulation with nonlinear internal models, Systems and Control Letters, 53, pp. 177-185, 2004. 10. F. Delli Priscoli, L. Marconi, A. Isidori, A new approach to adaptive nonlinear regulation, preprint, arXiv.math.OC/0404511v1, 28 April 2004. 11. C. De Persis, A. Isidori. Stabilizability by state feedback implies stabilizability by encoded state feedback, Systems and Control Letters, 53, pp. 249–258, 2004. 12. J.P. Gauthier, I. Kupka, Deterministic Observation Theory and Applications, Cambridge Univeraity Press, 2001. 13. J. Huang and C.F. Lin. On a robust nonlinear multivariable servomechanism problem. IEEE Trans. Autom. Contr., AC-39, pp. 1510–1513, 1994. 14. A. Isidori. Nonlinear Control Systems. Springer Verlag (New York, NY),3rd edition, 1995. 15. A. Isidori. Nonlinear Control Systems II. Springer Verlag (New York, NY), 1st edition, 1999. 16. A. Isidori, L. Marconi, C. De Persis, Remote tracking via encoded information for nonlinear systems, arXiv:math.OC/0501351, Jan. 2005.

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17. D. Liberzon, J. Hespanha, Stabilization of nonlinear systems with limited information feedback, IEEE Trans. on Automatic Control, to appear 2005. Also in Proc. of the 42nd Conf. Dec. Contr., pp. 182–186, 2003. 18. R. Marino and P. Tomei, Global adaptive observers for nonlinear systems via filtered transformations, IEEE Trans. on Automatic Control, AC-37, pp. 1239– 1245, 1992. 19. A. Serrani, A. Isidori and L. Marconi, Semiglobal nonlinear output regulation with adaptive internal model, IEEE Trans. Autom. Contr., AC-46, pp. 11781194, 2001. 20. S. Tatikonda, S. Mitter, Control under Communication Constraints,IEEE Trans. on Automatic Control, AC-49, pp. 1056–1068, 2004.

Extended Luenberger Observer for Nonuniformly Observable Nonlinear Systems Klaus R¨ obenack Institut f¨ ur฀Regelungs-฀und฀Steuerungstheorie,฀Fakult¨ at฀Elektrotechnik฀und Informationstechnik,฀Technische฀Universit¨ at฀Dresden,฀Mommsenstr.฀13,฀01099 Dresden,฀Germany. [email protected] Summary.฀ We฀consider฀ the฀observer฀ design฀ problem฀ for฀ nonlinear฀ single฀output systems.฀In฀contrast฀to฀most฀well-known฀design฀procedures฀the฀observability฀matrix may฀have฀a฀rank฀deficiency.฀The฀observer฀design฀is฀based฀on฀a฀partial฀observer฀form. The฀proposed฀observer฀is฀a฀generalization฀of฀the฀extended฀Luenberger฀observer.฀The observer฀gain฀can฀easily฀be฀computed.

Keywords: Nonlinear฀observer,฀normal฀form,฀detectability.

1 Introduction We consider a single output system x˙ = f (x) + g(x, u),

y = h(x)

(1)

with smooth maps f : Rn → Rn , g : Rn ×Rm → Rn , h : Rn → R, and g(x, 0) = 0 for all x ∈ Rn . The problem of observer design for nonlinear systems (1) has attracted the attention of many researchers [25]. One particularly interesting approach is the design based on differential-geometric concepts. These design methods are based on various normal forms. For example, the approaches developed in [21, 3] are based on the observer canonical form consisting of a linear output map and linear observable dynamics driven by a nonlinear output injection. The observer design method suggested in [18, 20] relies on a similar normal form, where the output map is allowed to be nonlinear. In [12, 9, 10] the observability canonical form is used. These normal forms require that the nonlinear matrix is regular. More recently, new approaches have been developed for systems which are not uniformly observable. Several approaches are based on appropriate decompositions of system (1), see [1, 16, 17, 30]. In [16] the Byrnes-Isidori normal form [7] is used for observer design in almost the same way as the observability

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 19–34, 2005. © Springer-Verlag Berlin Heidelberg 2005

20

K. R¨ obenack

canonical form in [12, 9]. Similarly, the work [21] is generalized in [17], where a certain partial observer canonical form is used. These approaches decompose the original system in an observable and an unobservable part. For the design procedure, only the observable part is taken into consideration. The unobservable part must satisfy a steady state (stability) property. Systems with this property are called detectable [1, 34]. In this paper we present a new observer design method for systems of the form (1) with a singular observability matrix. We use a similar partial observer form as in [17]. However, the form described in Sect. 2 has significantly weaker existence conditions compared to [17]. Our observer design method explained in Sect. 3 is based on an observer error linearization similar to [3, 35]. The resulting observer can be considered as a generalization of the extended Luenberger observer (see [4, 32]). The feasibility of the design method is demonstrated on three examples in Sect. 4.

2 Partial Observer Form Consider system (1) with h(x0 ) = 0 for some x0 ∈ Rn . We will present sufficient existence conditions for a local diffeomorphism z = T (x),

x = S(z)

(2)

defined in an open neighbourhood of x0 transforming (1) into a partial nonlinear observer form z˙ = A z + α(zr , zr+1 , . . . , zn ; u) y = cT z + β(zr+1 , . . . , zn )

(3a) (3b)

for r ∈ {1, . . . , n} with the maps α : Rn−r+1 × Rm → Rn and β : Rn−r → R. The matrix A ∈ Rn×n and the vector c ∈ Rn are given by A=

A0 0 0 0

,

cT = cT0 0 ,

where A0 ∈ Rr×r and c0 ∈ Rr are in dual Brunovsky form [6]: ⎞ ⎛ 0 0 ··· 0 0 ⎜1 0 ··· 0 0⎟ ⎟ ⎜ A0 = ⎜ . . . . . ⎟ , cT0 = 0 · · · 0 1 . ⎝ .. .. . . .. .. ⎠ 0 0 ··· 1 0

For r = n the output (3b) is linear (i.e., y = zn ) and the map α is an input-output injection. In this case, Eq. (3) is called observer canonical form [21, 22]. For r < n, system (3a) appears decomposed into a first subsystem of dimension r and a possibly nonlinear second subsystem of dimension

Extended Luenberger Observer for Nonuniformly Observable Systems

21

n − r. The first subsystem has linear dynamics driven by a nonlinear injection α of the input u and the last n − r + 1 coordinates zr , . . . , zn . Replacing zr by zr = y − β(zr+1 , . . . , zn ) according to (3b), the resulting injection α(y −β(zr+1 , . . . , zn ), zr+1 , . . . , zn ; u) is an input-output injection additionally driven by the coordinates zr+1 , . . . , zn of the second subsystem. The first subsystem is locally weakly observable (see [14, 11]) if the coordinates zr+1 , . . . , zn of the second subsystem are known. For the observer design we will later impose a detectability-like condition on the second subsystem. The following notation is well-established in nonlinear control theory [15]. The Lie derivative of a scalar field h along the vector field f is defined by the scalar product Lf h(x) = dh(x), f (x) , where dh denotes the gradient of h. h(x)) with L0f h(x) = Iterated Lie derivatives are given by Lkf h(x) = Lf (Lk−1 f h(x). If v is a second vector field, the Lie bracket is defined by [f, v] = v∗ f −f∗ v, where f∗ = f denotes the Jacobian matrix of f . Iterated Lie brackets are given v] with ad0f v = v. For a positive integer r < n the matrix by adkf v = [f, adk−1 f ⎛

⎜ Qr (x) = ⎝

dh(x) .. . dLfr−1 h(x)

⎞ ⎟ ⎠

(4)

is called reduced observability matrix. In case of n = r we call (4) observability matrix. An arbitrary smooth solution v : Rn → Rn of Qr (x) · v(x) = er ∈ Rr

(5)

is called starting vector, where er denotes the rth column of the identity matrix. Sufficient conditions for the existence of form (3) are given by the following theorem: Theorem 1. There exists a local diffeomorphism (2) in a neighbourhood of x0 ∈ Rn with T (x0 ) = 0, transforming (1) into (3) if

C1 rank Qr = r, C2 [adif v, adjf v] = 0, 0 ≤ i, j ≤ r − 1, C3 [g, adif v] = 0, 0 ≤ i ≤ r − 2, in some neighbourhood of x0 .

The case r = n is treated in [21, 22]. Note that the positive integer r is usually not uniquely determined by system (1). The upper bound for r is the rank of the observability matrix. The choice of the integer r offers some degrees of freedom. In particular, smaller values of r lead to weaker existence conditions C2 and C3. Before we prove Theorem 1 we recall the following theorem on simultaneous rectification [26, p. 54]: Theorem 2. Let X1 , . . . , Xr be linearly independent vector fields in a neighbourhood of p, satisfying

22

K. R¨ obenack

[Xi , Xj ] = 0

for

1 ≤ i, j ≤ r .

Then there exists a coordinate system (U, x1 , . . . , xr ) in a neighborhood of p such that on U ∂ for 1 ≤ i ≤ r . Xi = ∂xi Now, we prove Theorem 1: Proof. Assume that the conditions C1-C3 of Theorem 1 are fulfilled. Due to the rank condition C1 the system (5) has a solution v in some neighbourhood of x0 . Equation (5) can equivalently be written as Lv Lif h(x) = dLif h(x), v(x) =

0 for i = 0, . . . , r − 2 , 1 for i = r − 1 .

This implies ⎛

⎞ 0 ··· 0 1 dh(x0 ) ⎟ ⎜ .. ⎜. ∗⎟ ⎜ ⎟ .. r−1 ⎜ ⎟ ⎠ v(x0 ) ad−f v(x0 ) · · · ad−f v(x0 ) = ⎜ ⎝ . .. ⎟ , (6) ⎝0 r−1 .⎠ dLf h(x0 ) 1 ∗ ··· ∗ ⎛

⎞

see [15, Lemma 4.1.2]. Therefore, the vector fields

v(x), adf v(x), . . . , adr−1 v(x) f are linearly independent for all x in neighbourhood of x0 . Now, we additionally make use of condition C2. According to Theorem 2 there exists a local diffeomorphism (2) such that T∗ adi−f v =

∂ ∂zi+1

for i = 0, . . . , r − 1 .

(7)

This diffeomorphism transforms the maps occurring in (1) into f¯(z) = T∗ f (x)|x=S(z) g¯(z, u) = T∗ g(x, u)|x=S(z) ¯ h(z) = h(x)|x=S(z) . Because of (6) we have ∂ ¯ h = dh, adi−f v = ∂zi+1

0 for i = 0, . . . , r − 2 , 1 for i = r − 1 .

¯ has the form given Hence, in transformed coordinates the output map h in (3b).

Extended Luenberger Observer for Nonuniformly Observable Systems

23

Next, we turn our attention to the drift vector field ∂ ∂ f¯ = f¯1 + · · · + f¯n . ∂z1 ∂zn Due to (7) we have ∂ i−1 = T∗ adi−f v = T∗ [−f, adi−1 −f v] = [−T∗ f, T∗ ad−f v] ∂zi+1 n ∂ ∂ ∂ ¯ ∂ = [−T∗ f, ∂z ] = [−f¯, ∂z ] = j=1 ∂zi fj ∂zj i i

(8)

for i = 1, . . . , r − 1. Comparing both ends of (8) yields ∂ ¯ ∂zi fj ∂ ¯ ∂zi fi+1

= 0 for 1 ≤ j ≤ n, j = i + 1, 1 ≤ i ≤ r − 1 = 1 for 1 ≤ i ≤ r − 1 .

(9)

Finally, we consider the input-dependent vector field g¯. Because of condition C3 and (7) we have 0 = T∗ [g, adi−f v] = [T∗ g, T∗ adi−f v] = [¯ g , ∂z∂i+1 ] =

n j=1

∂ ¯j ∂zi+1 g

∂ ∂zj

(10)

for i = 0, . . . , r − 2. This implies ∂ g¯j = 0 ∂zi+1

for 1 ≤ j ≤ n, 0 ≤ i ≤ r − 2 .

(11)

Form (9) and (11) we can conclude that the vector field (f¯ + g¯) of the transformed system has the form (3a). The system (5) of linear equations is solvable if rank Qr = r. Then, every solution of (5) can be written as v(x) = Q− r (x) er ,

(12)

− where Q− r is a generalized inverse of Qr , i.e., Qr Qr Qr = Qr , see [2, 8]. The particular choice of the generalized inverse offers additional degrees of freedom which can be exploited to fulfill the existence conditions C2 and C3. One uniquely defined generalized inverse of Qr is the Moore-Penrose inverse Q+ r (see [23, 27]). If the rows of the matrix Qr are linearly independent we have T −1 T . If we use the Moore-Penrose inverse, Eq. (12) becomes Q+ r = Qr (Qr Qr )

v(x) = Q+ r (x) er .

(13)

3 Extended Luenberger Observer We will derive an observer with a Luenberger structure by an extended Taylor linearization technique in the coordinates of the partial observer form (3). The observer gain can be expressed in the original coordinates. The section also contains a local convergence analysis.

24

K. R¨ obenack

3.1 Observer Error Linearization We proceed similar as in case of the extended Luenberger observer [3, 35]. We use an observer of the structure xˆ˙ = f (ˆ x) + g(ˆ x, u) + k(ˆ x, u)(y − h(ˆ x)),

(14)

where we have to determine the observer gain k : Rn × Rm → Rn . We assume that the conditions of Theorem 1 are fulfilled. Applying the transformation (2) to system (14) yields zˆ˙ =Aˆ z + α(ˆ zr , zˆr+1 , . . . , zˆn ; u) z )) + (S (ˆ

−1

k(S(ˆ z ), u) · (zr + β(zr+1 , . . . , zn ) − zˆr − β(ˆ zr+1 , . . . , zˆn )) .

The observation error z˜ = z − zˆ is governed by

z˜˙ = A˜ z + α(zr , zr+1 , . . . , zn ; u) − α(ˆ zr , zˆr+1 , . . . , zˆn ; u) −1 − (S (ˆ z )) k(S(ˆ z ), u) · (˜ zr + β(zr+1 , . . . , zn ) − β(ˆ zr+1 , . . . , zˆn )) .

(15)

For the following considerations we assume that zˆi (t) → zi (t) for t → ∞ and i = r + 1, . . . , n. This is a steady-state property for the subsystem defined by the last (n−r) coordinates [1]. We now consider the limit case characterized by (16) zi = zˆi for i = r + 1, . . . , n. In the 2n-dimensional state-space of system (1) with observer (14), Eq. (16) defines a (n − r)-dimensional subspace. The error dynamics (15) restricted to this subspace has the form z˜˙ = A˜ z + α(zr , zˆr+1 , . . . , zˆn ; u) − α(ˆ zr , zˆr+1 , . . . , zˆn ; u) − (S (ˆ z ))−1 k(S(ˆ z ), u) z˜r .

(17)

Linearizing α along the estimated trajectory on this subspace yields α(zr , zˆr+1 , . . . , zˆn ; u) = α(ˆ zr , zˆr+1 , . . . , zˆn ; u) ∂ α(ˆ zr , zˆr+1 , . . . , zˆn ; u) z˜r + O(|˜ zr |2 ). + ∂ zˆr

(18)

Substituting (18) into (17) results in ∂ −1 z˜˙ =A˜ z+ α(ˆ zr , zˆr+1 , . . . , zˆn ; u) z˜r − (S (ˆ z )) k(S(ˆ z ), u) z˜r + O(|˜ z r |2 ) ∂ zˆr ∂ −1 =A˜ z − (S (ˆ α(ˆ zr , zˆr+1 , . . . , zˆn ; u) cT z˜ + O(|˜ z r |2 ) . z )) k(S(ˆ z ), u) − ∂ zˆr Next, we want to choose the observer gain k such that −1

l = (S (ˆ z ))

k(S(ˆ z ), u) −

∂ α(ˆ zr , zˆr+1 , . . . , zˆn ; u) ∂ zˆr

Extended Luenberger Observer for Nonuniformly Observable Systems

25

is a constant vector l=

l0 0

∈ Rn

⎛

⎞ p0 ⎟ ⎜ l0 = ⎝ ... ⎠ ∈ Rr . pr−1

with

This is equivalent to k(S(ˆ z ), u) = S (ˆ z ) l + S (ˆ z)

∂ α(ˆ zr , zˆr+1 , . . . , zˆn ; u) . ∂ zˆr

(19)

(20)

Using (20) yields the error dynamics z˜˙ = (A − lcT )˜ z + O( z˜ 2 ) for the limit case (16). The error dynamics of the first r-dimensional subsystem on the subspace defined by (16) is governed by ⎛ ⎞ 0 · · · 0 −p0 ⎛ ⎞ ⎛ ⎞ z˜1 ⎜ . . .. ⎟ z˜1 d ⎜ . ⎟ ⎜ 1 . . −p1 ⎟ . ⎟ 2 ⎟⎜ (21) ⎝ .. ⎠ = ⎜ ⎜. . ⎟ ⎝ .. ⎠ + O( z˜ ) , . dt .. ⎠ ⎝ .. . . 0 z˜r z˜r 0 · · · 1 −pn−1 A0 − l0 cT0

where the linear part has the characteristic polynomial det(λI − (A0 − l0 cT0 )) = p0 + p1 λ + p2 λ2 + · · · + pr−1 λr−1 + λr .

(22)

By choosing the coefficients p0 , . . . , pr−1 of (22) we can assign arbitrary eigenvalues to the subsystem (21). 3.2 Computation of the Observer Gain Now we want to express the observer gain (20) in x-coordinates. The diffeomorphisms T and S from (2) are inverse maps to each other, i.e., S(T (x)) ≡ x and T (S(z)) ≡ z. This implies T∗ S∗ = I and S∗ T∗ = I. From (7) we get adi−f v = S∗

∂ ∂zi+1

for i = 0, . . . , r − 1 .

With (19) we obtain S (ˆ z )l = p0 v(ˆ x) + p1 ad−f v(ˆ x) + · · · + pr−1 adr−1 x) . −f v(ˆ The next part is similar to (8). We have

(23)

26

K. R¨ obenack

T∗ (adr−f v + ad−g adr−1 −f v) = = = =

T∗ adr−f v + T∗ ad−g adr−1 −f v T∗ [−f, adr−1 v] + T [−g, adr−1 ∗ −f v] −f r−1 [−T∗ (f + g), T∗ ad−f v] [−(f¯ + g¯), ∂z∂ r ] n = j=1 ∂z∂ r f¯j + g¯j ∂z∂ j = =

n j=1

∂ ∂zr αj

∂ ∂zj

∂ α, ∂zr

or equivalently z) S (ˆ

∂ x) . α(ˆ zr , zˆr+1 , . . . , zˆn ; u) = adr−f v(ˆ x) + ad−g adr−1 −f v(ˆ ∂ zˆr

(24)

Using (23) and (24) we can express (20) in the original coordinates: x) k(ˆ x, u) = p0 v(ˆ x) + p1 ad−f v(ˆ x) + · · · + pr−1 adr−1 −f v(ˆ + adr−f v(ˆ v(ˆ x , u) . x) + ad−g adr−1 −f

(25)

The dependence of the observer gain k on u is due to the input-dependent vector field g occurring in the last summand of (25). Equation (25) can be regarded as a version of the nonlinear Ackermann formula for systems that are not uniformly observable (see [3, 4, 32, 35]). In order to obtain the observer gain we assumed in Sect. 3.1 that the existence conditions of Theorem 1 are fulfilled. However, for the computation of observer gain (25) only condition C1 is required. This phenomenon has already been observed in connection with the classical extended Luenberger observer [32, Th. 9.4]. The observer gain (25) can be computed symbolically with computer algebra packages such as Mathematica, Maple, or MuPAD. In case of a complicated system (1) the multiple differentiations required to obtain (25) may result in very large symbolic expressions. Then, it is recommendable to use an alternative differentiation technique called automatic differentiation [13]. The computation of the Lie derivatives and Lie brackets required here is discussed in [31, 29]. 3.3 Local Convergence Analysis To derive a formula for the observer gain via an extended Taylor linearization technique we restricted the error system (15) to the subspace defined by (16). Now, we will analyse the error dynamics of the resulting observer in the whole state-space. Inserting the observer gain (20) into (15) results in error dynamics governed by zr , . . . , zˆn ; u) z˜˙ = A˜ z + α(zr , . . . , zn ; u) − α(ˆ

− l+ ∂∂zˆr α(ˆ zr , . . . , zˆn ; u) (˜ zr + β(zr+1 , . . . , zn )−β(ˆ zr+1 , . . . , zˆn )) .

(26)

Extended Luenberger Observer for Nonuniformly Observable Systems

27

The linearization n

zr , . . . , zˆn ; u) + α(zr , . . . , zn ; u) = α(ˆ i=r n

β(zr+1 , . . . , zn ) = β(ˆ zr+1 , . . . , zˆn ) +

∂ α(ˆ zr , . . . , zˆn ; u) z˜i + O( z˜ 2 ) ∂ zˆi

∂ β(ˆ zr+1 , . . . , zˆn ) z˜i + O( z˜ 2 ) ∂ z ˆ i i=r+1

of α and β along the estimated trajectory zˆ yields n

z˜˙ = (A − lcT ) z˜ + − l+

∂ zr , . . . , zˆn ; u) ∂ zˆi α(ˆ

i=r+1 ∂ α(ˆ z ˆn ; u) r, . . . , z ∂ zˆr

∂ zr+1 , . . . , zˆn ) ∂ zˆi β(ˆ

z˜i + O( z˜ 2 ) .

The linear part of the error system has a block triangular structure z˜˙ =

A0 − l0 cT0 0

∗ ∗

z˜ + O( z˜ 2 ) .

(27)

For given bounded trajectories z and u of (1) the error system (26) is decomposed according to Eq. (27) into a r-dimensional first subsystem and an (n − r)-dimensional second subsystem, i.e, ˙ ξ˜ = (A0 − l0 cT0 ) ξ˜ + ϕ(ξ˜r , η˜, t) , η˜˙ = ψ(ξ˜r , η˜, t) ,

(28a) (28b)

where ξ˜ = (˜ z1 , . . . , z˜r )T and η˜ = (˜ zr+1 , . . . , z˜n )T . By choosing l0 via the coefficients p0 , . . . , pr−1 we place all eigenvalues of A0 − l0 cT0 in the open left half-plane. Since ϕ is smooth, it is locally Lipschitz. Therefore, there exist a smooth positive definite function V1 and two constants γ1 , γ2 > 0 such that ˜ V˙ 1 (ξ)

(28a)

≤ − γ1 ξ˜

2

+ γ2 η˜

2

for all sufficiently small ξ˜ and η˜, where · denotes the Euclidean norm. If we consider η˜ as an input to subsystem (28a), the function V1 is a local exponential-decay input-to-state stable (ISS) Lyapunov function [33]. The choice of l0 and V1 offers some degrees of freedom. It can be shown by similar techniques as in [16, 17, 30] that through appropriate selection of l0 and V1 the constant γ1 > 0 can be made arbitrary large and the constant γ2 > 0 arbitrary small. For the subsystem (28b) we require a steady state property (see [1, 34]). More precisely, we assume that there exist a smooth positive definite function V2 and two constants γ3 , γ4 > 0 such that V˙ 2 (˜ η)

(28b)

≤ γ3 |ξ˜r |2 − γ4 η˜

2

(29)

28

K. R¨ obenack

˜ The function V2 is a local exponential-decay for all sufficiently small η˜ and ξ. ISS Lyapunov function for subsystem (28b) if we consider ξ˜r as an input. For β ≡ 0 we have ξ˜r = y˜ with y˜ = y−h(ˆ x). In this special case, the function V2 is a local exponential-decay output-to-state stable (OSS) Lyapunov function [34]. Now we take the whole error system (28) into consideration. The smooth ˜ + V2 (˜ function V (˜ z ) = V1 (ξ) η ) is positive definite. The derivative of V along the observer error dynamics (26) has the form V˙ (˜ z)

(26)

= V˙ (˜ z)

(28)

≤ −(γ1 − γ3 ) ξ˜

2

− (γ4 − γ2 ) η˜

≤ − max{γ1 − γ3 , γ4 − γ2 } × z˜

2 2

(30)

˜ The form (30) is negative definite if γ1 is for all sufficiently small η˜ and ξ. sufficiently large and γ2 sufficiently small (i.e., γ1 > γ3 and 0 < γ2 < γ4 ). Then, the equilibrium z˜ = 0 of (26) is locally asymptotically stable. 3.4 Remarks on the Design Method The observer design presented in this paper uses a decomposition of system (1) into two interconnected subsystems. The design procedure is basically carried out for the first subsystem. This is possible since the first subsystem is locally weakly observable if we ignore the interconnection from the second subsystem. The observer converges if the second subsystem, which may be unobservable, satisfies the steady state property (29). This local steady state property of the unobserved dynamics means that system (1) is locally detectable [1]. The partition of system (1) into two subsystems is based on Theorem 1. As mentioned above, the integer r, which is the dimension of the first subsystem, may offer some degrees of freedom that can be exploited during the design process. Clearly, the value of r can not exceed the rank of the observability matrix because of condition C1 in Theorem 1. For large r it may be difficult to satisfy the conditions C2 and C3. For small values of r, the unobserved dynamics governed by the second subsystem of dimension n − r may not posses the steady state property (29).

4 Examples The design of extended Luenberger observers introduced in Sect. 3 is illustrated on three examples.

Extended Luenberger Observer for Nonuniformly Observable Systems

29

4.1 Synchronous Motor Consider the model of a synchronous motor [24, 19, 5]: x˙ 1 x˙ 2 x˙ 3 y

= = = =

x2 B1 − A1 x2 − A2 x3 sin x1 − 12 B2 sin(2x1 ) u − D1 x3 + D2 cos x1 x1 .

The observability matrix ⎛

⎞ 1 0 0 ⎠ Q3 (x) = ⎝ 0 1 0 −A2 x3 cos(x1 ) − B2 cos(2x1 ) −A1 −A2 sin(x1 )

(31)

(32)

is singular for x1 ∈ πZ, where Z denotes the set of integers. Each singularity of (32) results in a pole of the associated starting vector v. The integrability condition C2 is violated since [ad1f v, ad2f v] = 0. Note that one can still design a classical extended Luenberger observer. However, the observer gain is a very large expression and has singularities for x1 ∈ πZ (see [28]). Now, we will consider the new approach. Let us try r = 2. The reduced observability matrix has the form Q2 =

100 010

.

The staring vector v is computed by (13). We obtain v = (0, 1, 0)T and ad−f v = (1, −A1 , 0)T . There holds [v, adf v] ≡ 0 because these vector fields are constant. The observer gain (25) given by ⎞ ⎛ p1 − A1 x1 ) ⎠ ˆ3 cos xˆ1 + A21 − B2 cos(2ˆ (33) k(ˆ x) = ⎝ p0 − A1 p1 − A2 x ˆ1 −D2 sin x

is well-defined for all x ∈ R3 . In this case, the observer gain (33) does not depend on the input u since [g, adf v] ≡ 0. For the simulation we used the parameters A1 = 0.2703, A2 = 12.01, B1 = 39.19, B2 = −48.04, D1 = 0.3222, D2 = 1.9 and u ≡ 1.933 as in [24]. ˆ(0) = (0.5, 0.1, 5)T . All obThe initial values are x(0) = (0.8, 0, 10)T and x server eigenvalues where placed at −10. The trajectories of system (31) are drawn with solid lines (see Fig. 1). We used dashed lines for the observer with the gain (33). In addition we plotted the trajectories of the classical extended Luenberger observer using dash-dotted lines. It can be seen that the classical extended Luenberger observer converges faster than the new observer with the gain vector (33). In fact, the slow rate of convergence is due to the internal dynamics of (31). On the other hand, the observer gain (33) has no singularities and is easier to compute.

30

K. R¨ obenack

Fig. 1. Trajectories of the synchronous motor model (31) with two versions of the extended Luenberger observer

4.2 Example from Jo and Seo (2002) The next example is taken from [17, Sect. 5.1]: x˙ 1 x˙ 2 x˙ 3 y

= x2 + x31 + u = x21 x2 + x1 u sin x1 = −x3 − x33 + x1 x2 + x1 u = x1 .

(34)

The observability matrix Q3 (x) is singular with rank Q3 (x) = 2 for all x ∈ R3 . For r = 2 one obtains the reduced observability matrix Q2 (x) =

dh(x) dLf h(x)

=

1 00 3x21 1 0

.

The starting vector is chosen by ⎞ 1 0 ⎝ −3x21 1 ⎠ v(x) = Q+ 2 (x) e2 = 0 0 ⎛

0 1

⎛ ⎞ 0 = ⎝1⎠. 0

Moreover, we have [v, adf v] ≡ 0. and [g, v] ≡ 0, i.e., the assumptions of Theorem 1 are fulfilled. The observer gain (25) has the form ⎞ ⎛ x21 p1 + 4ˆ x1 + u sin x ˆ1 + uˆ x1 cos xˆ1 − xˆ41 + p1 x ˆ21 ⎠ . k(ˆ x, u) = ⎝ p0 − 2uˆ 2 −ˆ x1 + p1 x ˆ1 − 3ˆ x1 x ˆ3

Extended Luenberger Observer for Nonuniformly Observable Systems

31

In contrast to [17], the observer is obtained without explicit computation of the associated normal form. For the simulation we stabilize system (34) with a high-gain controller u = −Ky using K = 10 (see [15, Sect. 4.7]). The two observer eigenvalues are placed at −5, i.e., p0 = 25 and p1 = 10. The initial values were x(0) = (1, 2, 3)T and xˆ(0) = (0, 0, 0)T . Figure 2 shows the simulation results.

Fig. 2. Simulation of System (34) with controller and observer

4.3 Mini-Example with Limit Cycle The following 2-dimensional system has an unstable equilibrium point in the origin and an asymptotically stable limit cycle around the origin: x˙ 1 = −x2 + x1 (1 − x21 ) x˙ 2 = x1 + x2 (1 − x22 ) .

(35)

For the output we use the four-quadrant arctangent y = arctan(x2 , x1 ) ,

(36)

which is basically the same as arctan(x2 /x1 ) but x1 is allowed to be zero. In several programming languages such as C, Fortran, and Java this function is denoted by atan2. The 2 × 2 observability matrix Q2 (x) =

2 − x2x+x 2 1

2

−x52 +x41 x2 +4x21 x32 (x21 +x22 )2

x1 x21 +x22 x51 −x1 x42 −4x31 x22 (x21 +x22 )2

(37)

32

K. R¨ obenack

has the determinant det(Q2 (x)) =

2x1 x2 (x1 + x2 )(x2 − x1 ) , (x21 + x22 )2

which is zero if x1 = 0 ∨ x2 = 0 ∨ x1 = x2 ∨ x1 = −x2 . Following the limit cycle for one period, the observability matrix (37) has already 8 singularities. From the reduced observability matrix Q1 (x) =

2 − x2x+x 2 1

2

x1 x21 +x22

we compute v(x) = Q+ 1 (x) e1 =

−x2 x1

−x32 + 3x21 x2 x31 − 3x1 x22

and ad−f v(x) =

.

The observer gain is k(ˆ x) =

x ˆ2 (3ˆ x21 − xˆ22 − p0 ) x22 + p0 ) x ˆ1 (ˆ x21 − 3ˆ

.

(38)

The simulation is carried out with the initial values x(0) = (0, 0.5)T and x ˆ(0) = (1, 0)T . The observer eigenvalue is placed at −5, i.e., p0 = −5. Figure 3 shows the trajectories. The observer works as expected. With (36) we measure the angular part of the trajectories converging to the limit cycle. The radial part is not observed, but moves to the limit cycle due to the asymptotic stability of the periodic solution.

5 Summary We developed a method for observer design based on a partial observer form similar as in [17]. However, we rely on weaker existence conditions. Moreover, the observer gain can be obtained without an explicit computation of this form. In fact, only the Jacobian matrix of the transformation is needed. The proposed observer can be regarded as a version of the extended Luenberger observer for systems whose observability matrix is singular.

References 1. G. L. Amicucci and S. Monaco. On nonlinear detectability. J. Franklin Inst., 335B(6):1105–1123, 1998. 2. A. Ben-Israel and T. N. E. Greville. Generalized Inverses: Theory and Applications. Wiley-Interscience, 1974. 3. D. Bestle and M. Zeitz. Canonical form observer design for non-linear timevariable systems. Int. J. Control, 38(2):419–431, 1983.

Extended Luenberger Observer for Nonuniformly Observable Systems

33

Fig. 3. Simulation of system (35) with the observer using (38) 4. J. Birk. Rechnergest¨ utzte Analyse und Synthese nichtlinearer Beobachtungsaufgaben, volume 294 of VDI-Fortschrittsberichte, Reihe 8: Meß-, Steuerungs- und Regelungstechnik. VDI-Verlag, D¨ usseldorf, 1992. 5. J. Birk and M. Zeitz. Extended Luenberger observer for non-linear multivariable systems. Int. J. Control, 47(6):1823–1836, 1988. 6. P. Brunovsky. A classification of linear controllable systems. Kybernetica, 6(3):173–188, 1970. 7. C. I. Byrnes and A. Isidori. Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. on Automatic Control, 36(10):1122–1137, 1991. 8. S. L. Campbell and C. D. Meyer. Generalized Inverses of Linear Transformations. Dover Publications, New York, 1979. 9. G. Ciccarella, M. Dalla Mora, and A. Germani. A Luenberger-like observer for nonlinear systems. Int. J. Control, 57(3):537–556, 1993. 10. M. Dalla Mora, A. Germani, and C. Manes. A state observer for nonlinear dynamical systems. Nonlinear Analysis, Theory, Methods & Applications, 30(7):4485–4496, 1997. 11. J. P. Gauthier and G. Bornard. Observability for any u(t) of a class of nonlinear systems. IEEE Trans. on Automatic Control, 26(4):922–926, 1981. 12. J. P. Gauthier, H. Hammouri, and S. Othman. A simple observer for nonlinear systems — application to bioreactors. IEEE Trans. on Automatic Control, 37(6):875–880, 1992. 13. A. Griewank. Evaluating Derivatives — Principles and Techniques of Algorithmic Differentiation, volume 19 of Frontiers in Applied Mathematics. SIAM, Philadelphia, 2000. 14. R. Hermann and A. J. Krener. Nonlinear controllability and observability. IEEE Trans. on Automatic Control, AC-22(5):728–740, 1977. 15. A. Isidori. Nonlinear Control Systems: An Introduction. Springer-Verlag, London, 3rd edition, 1995.

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16. N. H. Jo and J. H. Seo. Input output linearization approach to state observer design for nonlinear system. IEEE Trans. on Automatic Control, 45(12):2388– 2393, 2000. 17. N. H. Jo and J. H. Seo. Observer design for non-linear systems that are not uniformly observable. Int. J. Control, 75(5):369–380, 2002. 18. N. Kazantzis and C. Kravaris. Nonlinear observer design using Lyapunov’s auxiliary theorem. Systems & Control Letters, 34:241–247, 1998. 19. H. Keller. Entwurf nichtlinearer Beobachter mittels Normalformen. VDI-Verlag, 1986. 20. A. Krener and M. Xiao. Nonlinear observer design in the Siegel domain. SIAM J. Control and Optimization, 41(3):932–953, 2002. 21. A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems & Control Letters, 3:47–52, 1983. 22. R. Marino. Adaptive observers for single output nonlinear systems. IEEE Trans. on Automatic Control, 35(9):1054–1058, 1990. 23. E. H. Moore. On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc., 26:394–395, 1920. 24. B. K. Mukhopadhyay and O. P. Malik. Optimal control of synchronous-machine excitation by quasilinearisation techniques. Proc. IEE, 119(1):91–98, 1972. 25. H. Nijmeijer and T. I. Fossen, editors. New Directions in Nonlinear Observer Design, volume 244 of Lecture Notes in Control and Information Science. SpringerVerlag, 1999. 26. H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control systems. Springer, 1990. 27. R. Penrose. A generalized inverse for matrices. Proc. Cambridge Philosophical Society, 51:406–413, 1955. 28. K. R¨ obenack. Beobachterentwurf f¨ ur nichtlineare Zustandssysteme mit Hilfe des Automatischen Differenzierens. Shaker-Verlag, Aachen, 2003. 29. K. R¨ obenack. Computation of the observer gain for extended Luenberger observers using automatic differentiation. IMA Journal of Mathematical Control and Information, 21(1):33–47, 2004. 30. K. R¨ obenack. Zum High-Gain-Beobachterentwurf f¨ ur eingangs-ausgangslinearisierbare SISO-Systeme. Automatisierungstechnik, 52(10):481–488, 2004. 31. K. R¨ obenack and K. J. Reinschke. The computation of Lie derivatives and Lie brackets based on automatic differentiation. Z. Angew. Math. Mech., 84(2):114– 123, 2004. 32. J. Schaffner and M. Zeitz. Variants of nonlinear normal form observer design. In Hijmeijer and Fossen [25], pages 161–180. 33. E. D. Sontag and Y. Wang. On characterizations of the input-to-state stability. Systems & Control Letters, 24:351–359, 1995. 34. E. D. Sontag and Y. Wang. Output-to-state stability and detectability of nonlinear system. Systems & Control Letters, 29(5):279–290, 1997. 35. M. Zeitz. The extended Luenberger observer for nonlinear systems. Systems & Control Letters, 9:149–156, 1987.

Approximate฀Observer฀Error฀Linearization฀by Dissipativity฀Methods Jaime฀A.฀Moreno Automatizacin,฀Instituto฀de฀Ingenier´ıa,฀Universidad฀Nacional฀Aut´ onoma฀de฀M´exico (UNAM).฀Ciudad฀Universitaria,฀Ed.12,฀Circuito฀Exterior฀S/N.฀Coyoac´ an,฀04510. M´exico,฀D.F.,฀Mexico. [email protected] Summary.฀ In฀ this฀ paper฀ a฀ method฀ to฀ design฀ observers฀ for฀ nonlinear฀ systems฀ is proposed.฀The฀basic฀idea฀is฀to฀decompose฀the฀system฀in฀a฀nonlinear฀part,฀that฀can be฀transformed฀into฀a฀nonlinear฀observer฀form,฀and฀a฀perturbation฀term฀connected in฀the฀feedback฀loop.฀In฀transformed฀coordinates฀the฀observer฀error฀becomes฀linear with฀ a฀ feedback฀ perturbation.฀ By฀ using฀ the฀ dissipativity฀ theory฀ it฀ is฀ possible฀ to design฀the฀observer฀gains฀so,฀that฀the฀closed฀loop฀is฀internally฀stable฀if฀some฀LMIlike฀conditions฀are฀satisfied.฀This฀method฀is฀very฀flexible฀and฀allows฀the฀design฀of nonlinear฀ observers฀ using฀ exact฀ linearization฀ methods฀ to฀ a฀ much฀ bigger฀ class฀ of systems.฀The฀method฀is฀shown฀to฀be฀very฀general,฀since฀it฀includes฀(and฀generalizes) as฀ special฀ cases฀ several฀ observer฀ design฀ methods,฀ as฀ for฀ example฀ the฀ exact฀ error linearization฀method,฀the฀high-gain฀method,฀the฀circle฀criterion฀design฀method,฀and the฀design฀ for฀ Lipschitz฀ nonlinear฀ systems.฀ The฀ design฀ is฀ computationally฀ simple in฀ many฀ cases,฀ since฀ it฀ reduces฀ to฀ the฀ solution฀ of฀ a฀ feasibility฀ LMI฀ problem,฀ for which฀ highly฀ efficient฀ numerical฀ methods฀ are฀ available.฀ The฀ method฀ offers฀ great flexibility฀in฀the฀design,฀since฀the฀particular฀properties฀of฀the฀nonlinearities฀can฀be characterized฀ by฀ means฀ of฀ one฀ or฀ several฀ quadratic฀ forms,฀ i.e.฀ supply฀ rates.฀ This feature฀can฀be฀used฀to฀design฀observers฀for฀systems฀with฀special฀properties฀in฀the nonlinearities.

Keywords:฀Dissipativity,฀absolute฀stability,฀nonlinear฀observer฀design,฀approximate฀error฀linearization,฀LMI.

1 Introduction The design of observers for nonlinear systems is a very important and challenging task in control and has attracted the interest of many researchers. Since the observer design for linear systems is well understood it is attractive to try to reduce the nonlinear case to the linear one. This idea has been proposed for the first time independently by Bestle and Zeitz [4] and Krener and Isidori [18], where conditions for the possibility of transforming a nonlinear system into a

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 35–51, 2005. © Springer-Verlag Berlin Heidelberg 2005

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nonlinear observer form are given, for which the dynamics of the state estimation error for a proposed observer is exactly linear, so that the design of the observer is reduced to a linear one. This method, called nonlinear normal form observer design or exact error linearization, has been extended and refined since then in many different directions [19, 39, 37, 27, 25, 31, 15, 12, 16, 1]. Despite of its elegance and beauty the main drawback of the error linearization method is that it is applicable to a very small class of nonlinear systems, since it imposes very restrictive and generically not satisfied conditions. Several attempts to enlarge the class of systems, for which an exact or an approximate error linearization can be achieved, have been done. Approximate methods [3, 21], in which the nonlinear system is decomposed into a nonlinear part, that is exactly linearizable, and a perturbation term, are probably the ones that consider the largest class of systems. They aim at selecting the non unique decomposition such that the perturbation term is minimized in some sense. After transforming the system into an approximate nonlinear observer form (ANOF), the observer design is done for the linear part and, since the perturbation term has been minimized, it is expected that the convergence is assured. However, there is no guarantee that this is the case. For nonlinear systems, that can be decomposed as a linear time invariant part and a nonlinear feedback element, the classical absolute stability problem [17] deals with the question on when the stability of the equilibrium point is maintained for the whole class of nonlinearities contained in a sector. The circle criterion gives a sufficient condition for absolute stability, and has been used in [2, 7] to design observers for a special class of nonlinear system in Lure form and with square and monotonic nonlinearities. The dissipativity theory [35] extends the absolute problem and the circle criterion to a much wider class of systems, and it can be used for the design of nonlinear observers, as shown in [23] for the class of Lur’e systems. In [28] the same idea of designing observers with absolute error dynamics were used for the robust synchronization of dynamical systems. In particular for systems with linearizable error dynamics they give frequency conditions using the Kalman-Yakubovich Lemma for synchronization when the output injection term of the observer is not realized exactly. Similar ideas have been also used for robust synchronization (see for example [32] and the references therein). The objective of this paper is to propose a method to design observers for the class of systems that can be brought to an ANOF. Instead of minimizing the perturbation term it is shown that if the transformation into the ANOF can be performed so, that the perturbation term satisfies a sector condition for which the conditions of a dissipative design as in [23] are satisfied, then the observer convergence is assured. This reflects the fact that not only the magnitude but also its ”direction” is of importance. This distinction is in the basis of the classical small gain and the passivity theorems. It is shown that the proposed method encompasses and generalizes several known methodologies, improving and eliminating some restrictions imposed by them. Compared to

Approximate Observer Error Linearization by Dissipativity Methods

37

[28] our objective and the structure of our observer are more general, and we consider explicitly the case of multiple sector conditions. In this paper only sector conditions or local quadratic constrains are considered but, as it is done in [28], integral quadratic constrains can also be used, giving less conservative conditions. Our results show that several apparently different observer design methods, as for example the Exact error linearization and the High-gain designs, can be put in the same perspective and further generalized. The rest of the paper is organized as follows. In Section 2 the exact error linearization method is reviewed. In Section 3 the dissipative approach of the observer design from [23] is reviewed and extended. In Section 4 a new design method is proposed that combines both methods. In Section 5 the use and some advantages of the proposed methodology are illustrated with some examples. For simplicity of the presentation only the Single Output (SO) case will be considered.

2 Exact Error Linearization Method Consider single output nonlinear systems of the form Σ : x˙ = f (x) + g (x, u) ,

x (0) = x0 , y = h (x) ,

(1)

where x ∈ M ⊂ Rn is the state, u ∈ Rm is the input, y ∈ R is the measured output, f and g are smooth vector fields with f (0) = 0 and g (x, 0) = 0, ∀x ∈ M , and h is a smooth function with h (0) = 0. The observer error linearization problem [4, 18, 19, 37](see also [22, 31]) consists in finding (if possible) a neighborhood U0 of x = 0, and a coordinates transformation (i.e. a local diffeomorphism) z = T (x) ,

T (0) = 0 ,

z ∈ Rn ,

(2)

transforming system (1) into the observer normal form (ONF) Σonf : z˙ = Ao z + γ (y, u) ,

z (0) = z0 , y = Co z ,

(3)

where (Co , Ao ) is an observable pair, and they are in the dual Brunovsky j canonical form, i.e. Co = 1 0 · · · 0 , and Ao = [aij ], with aij = δi+1 , where j δi is the Kronecker δ symbol. The solution of this problem requires certain observability properties of the system (1). We will discuss only those requirements for the system without inputs, i.e. (1) with u = 0. For further discussion of this topic, including the properties required by the system with inputs, see [37, 26, 8, 9, 13]. For the system without inputs the observability map is given by T , (4) q : M → Rn : q (x) = h (x) , Lf h (x) , · · · , Lfn−1 h (x) and its Jacobian at x ∈ M defines the observability matrix

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Q (x) =

⎡

∂q (x) ⎢ ⎢ =⎢ ⎣ ∂x

dh (x) dLf h (x) .. . dLfn−1 h (x)

⎤

⎥ ⎥ ⎥ ∈ M n×n . ⎦

The injectivity (or invertibility) of q in M assures the global observability, whereas the regularity of Q (x) at x implies the local observability around the point x. The observer error linearization problem requires the regularity of the observability map for every x ∈ U0 , and is completely characterized by Theorem 1. There exists a local diffeomorphism (2) in a neighborhood U0 of the origin transforming system (1) into the observer normal form (3) if, and only if, in U0 : (i) rank dh (x) , dLf h (x) , · · · , dLfn−1 h (x) = n ,

(ii) adif r , adjf r = 0 , 0 ≤ i , j ≤ n − 1 ,

(iii) g , adjf r = 0 , 0 ≤ j ≤ n − 2 , ∀u ∈ Rm with r being the vector field solution of Lr h (x) , Lr Lf h (x) , · · · Lr Lfn−1 h (x) = 0 0 · · · 1

.

(5)

There exists a global diffeomorphism, i.e. U0 = Rn , if, and only if, conditions (i)-(iii) hold in Rn and, in addition, (iv) adif r , 0 ≤ i ≤ n − 1 , are complete vector fields. Remark 1. Lf h (x) =

∂h(x) ∂x f

(x) denotes the Lie derivative of the function

h along the vector field f , and Lif h (x) = Lf Li−1 f h (x) . The differential dh of a smooth function h : U ⊂ Rn → R may be denoted in local coordinates ∂h(x) as a row gradient vector ∂h(x) . For two vector fields f and g a ∂x1 , · · · , ∂xn ∂g new vector field is defined by the Lie bracket [f , g] = ∂x f − ∂f ∂x g, in local coordinates. For repeated Lie brackets the ad operator is defined as

g ad0f g = g , adif g = f , adi−1 f

.

A vector field f is said to be complete if the solutions to the differential equation x˙ = f (x) may be defined for all the time t ∈ R. Remark 2. The conditions of Theorem 1 are very strong and are not generically satisfied. Condition (i) corresponds to the local observability of the system (1) without inputs for all x ∈ U0 , and is satisfied in most cases. The commutativity Condition (ii) is very restrictive and it amounts to the possibility of transforming the system (1) without inputs to the observability normal form. Condition (iii) corresponds to the fact that the vector field g

Approximate Observer Error Linearization by Dissipativity Methods

39

is a function of y and u in the new coordinates. Condition (iv) is required if the transformation is valid globally, and is satisfied if, for example, the vector fields are globally Lipschitz. The motivation for finding a transformation of system (1) to the normal form (3) comes from the fact that the design of a (local or global) observer for the plant becomes a linear problem, i.e. the observer is given by the system ·

Ωonf : zˆ = Ao zˆ + γ (y, u) + L (y − Co zˆ) ,

zˆ (0) = zˆ0 , x ˆ = T −1 (ˆ z) ,

(6)

where L is selected so that the matrix (Ao − LCo ) is Hurwitz, and the equation for the error z˜ = zˆ − z is linear ·

Ξonf : z˜ = (Ao − LCo ) z˜ ,

z˜ (0) = z˜0 , x˜ = T −1 (ˆ z ) − T −1 (z) .

(7)

Remark 3. This is the simplest version of the observer error linearization problem. The class of systems for which the observer design problem can be reduced to a linear one can be enlarged loosening the conditions of the Theorem 1. This can be achieved extending the observer error linearization problem in different manners. For example, allowing the map (2) to be a semi-diffeomorphism (i.e. the inverse is not required to be differentiable) continuous observers can be designed when Condition (i) in Theorem 1 is not satisfied [38, 31]. The inclusion of an output transformation [19] or/and time-scale transformations [12, 30] or the use of immersion techniques, for which the transformed system (3) is allowed to be of higher dimension than the original one, lead to further enlargement of the class of systems [20, 15]. Furthermore, it is possible to consider nonlinear observer forms with derivatives of the input and the output [40, 11]. Example 1. [19, 27] For systems in observability normal form x˙ = x2 , xn · · · ϕn (x)

T

,

x (0) = x0 , y = x1 ,

there exist a state transformation z = T (x) and an output transformation y¯ = λ (y) that bring the system in the ONF (3) if and only if If n = 2: ϕ2 (x) has the polynomial form ϕ2 (x) = k0 (y)+k1 (y) x2 +k2 (y) x22 . In this case the output transformation is calculated from λ (y) = −k2 (y)λ (y), with λ (0) = 0, and where λ (y)
dλ(y) dy . The output injection terms in (3) are derived from γ1 (y) = k1 (y) λ (y), and γ2 (y) = k0 (y) λ (y), with γ1 (0) = γ2 (0) = 0. If n = 3: ϕ3 (x) has the polynomial form ϕ3 (x) = k0 (y)+k1 (y) x2 +k2 (y) x3 + k3 (y) x22 + k4 (y) x2 x3 + k5 (y) x32 , and the coefficients satisfy the relations: 3k5 (y) = k4 (y) − k42 (y) /3, and k3 (y) = k2 (y) − k2 (y) k4 (y) /3. If this is the case the output transformation is calculated from λ (y) = − 13 k4 (y) λ (y), with λ (0) = 0. The output injection terms results from γ1 (y) = k2 (y) λ (y), γ2 (y) = k1 (y) λ (y), γ3 (y) = k0 (y) λ (y), with γ1 (0) = γ2 (0) = γ3 (0) = 0.

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If no output transformation is allowed, then further restrictions of the coefficients are necessary, that can be derived by setting λ (y) = 1 in the previous relations.

3 Dissipative Design of Nonlinear Observers Motivated by the circle criterion design of nonlinear observers in [2] the author has proposed in [23] a methodology for designing nonlinear observers for a class of nonlinear systems. This method will be briefly reviewed in this section. 3.1 Preliminaries In this work the stability properties of dissipative systems will be used for the design of observers for systems that can be represented as the feedback interconnection of a dynamical linear time invariant (LTI) system in the forward loop and a memoryless nonlinearity in the feedback loop. From the general dissipativity theory [34, 35, 36, 14, 17] the following results are of relevance here. Consider the feedback interconnection x˙ = Ax+Bu , x (0) = x0 , y = Cx , u = −ψ (t, y) , x ∈ Rn , u ∈ Rq , y ∈ Rm , (8) and quadratic supply rates ω (v, w) = v T Qv + 2v T Sw + wT Rw =

v w

T

Q S ST R

v w

, v ∈ Rr , w ∈ Rs ,

where Q ∈ Rr×r , S ∈ Rr×s , R ∈ Rs×s , and Q, R symmetric.

(9)

Definition 1. The linear part (A, B, C) of system (8) is said to be state strictly dissipative (SSD) with respect to the supply rate ω (y, u), or for short (Q, S, R)-SSD, if there exist a matrix P = P T > 0, and > 0 such that P A + AT P + P , P B C T QC C T S − T B P 0 ST C R

≤0 .

(10)

For quadratic systems, i.e. m = q, passivity corresponds to the supply rate ω (y, u) = y T u. If (A, B) controllable, (A, C) observable, then condition (10) is equivalent, by the Kalman-Yakubovich-Popov Lemma [17], to the fact that the transfer matrix of Σ, i.e. G (s) = C (sI − A)−1 B, is strictly positive real (SPR). Note that this definition assures the existence of a quadratic positive definite storage function V (x) = xT P x, and a positive definite loss T function Z (x, u) = K T x + W T u K T x + W T u + xT P x, such that along any trajectory of the system V˙ (x (t)) = ω (y (t) , u (t)) − Z (x (t) , u (t)).

Approximate Observer Error Linearization by Dissipativity Methods

41

Definition 2. The nonlinear part of system (8), a time-varying memoryless nonlinearity ψ : [0, ∞) × Rm → Rq , u = ψ (t, y), piecewise continuous in t and locally Lipschitz in y, such that ψ (t, 0) = 0, is said to satisfy a dissipative condition in Γ with respect to the supply rate ω (u, y) (9), or for short (Q, S, R)-D in Γ , if ω (u, y) = ω (ψ (t, y) , y) ≥ 0 , ∀t ≥ 0 , ∀y ∈ Γ ⊆ Rm , where Γ is a subset of Rm whose interior is connected and contains the origin. If Γ = Rm , then ψ satisfies the dissipativity condition globally, in which case it is said that ψ is dissipative with respect to ω, or for short, (Q, S, R)-D. Remark 4. Note that the classical sector conditions [17] for square nonlinearities, i.e. m = q, can be represented in this form. If ψ is in the secT tor [K1 , K2 ], i.e. (y − K1 u) (K2 u − y) ≥ 0, then it is (Q, S, R)-D, with 1 (Q, S, R) = −I, 2 (K1 + K2 ) , − 12 K1T K2 + K2T K1 . If ψ is in the sector [K1 , ∞], i.e. (y − K1 u)T u ≥ 0, then it is 0, 12 I, − 21 K1 + K1T -D. Remark 5. Classically the concept of dissipativity has been defined globally. However, it is of interest to consider the local (or non global) case, for which the local version of dissipativity of a nonlinearity has been introduced. For the interconnected system (8) a generalization of the passivity and of the small gain theorems for non square systems can be easily obtained, and it will be used in the sequel. Lemma 1. Consider the system (8). If the linear system (C, A, B) is (−R, S T , −Q)-SSD, then the equilibrium point x = 0 of (8) is globally (locally) exponentially stable for every (Q, S, R)-D (in Γ for some Γ ⊆ Rm ) nonlinearity. Proof. By hypothesis (10) is satisfied with −R, S T , −Q . Take V (x) = xT P x as Lyapunov function candidate for the closed loop system. The time derivative of V (x) along the solutions of (8) is V˙ = (Ax + Bu)T P x + xT P (Ax + Bu), or, because of (10) and (8) x V˙ = u =−

T

ψ y

P A + AT P P B BT P 0 T

Q S ST R

x x ≤ u −ψ

T

−C T RC C T S T SC −Q

x − xT P x −ψ

ψ − V (x) ≤ − V (x) , y

since ψ is (Q, S, R)-dissipative. If ψ satisfies the dissipativity condition only for Γ ⊆ Rm , then the foregoing analysis shows that the origin is (locally) exponentially stable in some finite domain.

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3.2 Dissipative Observer Design Consider the class of systems described by a LTI subsystem with a nonlinear perturbation term, connected in feedback, i.e. Σ : z˙ = Az + Gψ (σ, y, u) + γ (y, u) ,

z (0) = z0 , y = Cz , σ = Hz , (11)

where z ∈ Rn is the state, y ∈ Rp is the measured output, u ∈ Rm is the input, and σ ∈ Rr is a (not necessarily measured) linear function of the state. γ (y, u) is an arbitrary nonlinear function of the input and the output. ψ (σ, y, u) is a q-dimensional vector that depends on σ, y, u. ψ and γ are assumed to be locally Lipschitz in σ, y, u, so that existence and uniqueness of solutions is guaranteed. It will be assumed that the trajectories of interest of Σ are defined for all the time, i.e. there are no finite escape times. An observer for system (11) is a dynamical system Ω that has as inputs the input u and the output y of Σ, and its output zˆ is an estimation of the state z of Σ. A full order observer for Σ of the form ·

Ω : zˆ = Aˆ z + L (ˆ y − y) + Gψ (ˆ σ +N (ˆ y − y) , y, u)+γ (y, u) , yˆ = C zˆ, σ ˆ = H zˆ (12) is proposed, where matrices L ∈ Rn×p , and N ∈ Rr×p have to be designed. Defining the state estimation error by z˜
zˆ−z, the output estimation error by y˜
yˆ − y, and the function estimation error by σ ˜
σ ˆ − σ, ξ
(H + N C) z˜ = σ ˜ + N y˜, and a new nonlinearity φ (ξ, σ, y, u)
ψ (σ, y, u) − ψ (σ + ξ, y, u) ,

(13)

the dynamics of the error can be written as ·

Ξ : z˜ = AL z˜ + Gν ,

z˜ (0) = z˜0 , ξ = HN z˜ , ν = −φ (ξ, σ, y, u) ,

(14)

where AL
A + LC, and HN
H + N C. Note that φ (0, σ; y, u) = 0 for all σ, y, u. Remark 6. Note that when the plant Σ is LTI (at least up to an output injection term), then φ = 0, and the error dynamics Ξ is LTI and autonomous, i.e. it does not depend on the plant state. The same is true if σ is dependent on the output y, since in this case there exists a matrix such that σ = F y, and there exists an N such that HN = H + N C = 0. In these both cases detectability of the pair (A, C) is a necessary and sufficient condition to construct an observer. However, in general, the error dynamics (14) is not autonomous, but it is driven by the system (11) through the linear function of the state σ = Hx. φ is therefore a time varying nonlinearity, whose time variation depends on the state trajectory of the plant. The observer design consists in finding matrices L and N , if they exist, so that Ξ satisfies the conditions of Lemma 1. For this it is necessary to assume that the nonlinear part of (14) belongs to one or several sectors.

Approximate Observer Error Linearization by Dissipativity Methods

43

Assumption 1 φ in (14) is (Qi , Si , Ri )-dissipative (in Γ ) for some finite set of non positive semidefinite quadratic forms ωi (φ, z) = φT Qi φ + 2φT Si z + z T Ri z ≥ 0, for all σ, y, u, for i = 1, 2, · · · , M .

It is clear that it is necessary that the quadratic forms be independent. M It is also easy to see that then φ is i=1 θi (Qi , Si , Ri )-dissipative (in Γ ) for every θi ≥ 0, i.e. φ is dissipative with respect to the supply rate ω (φ, z) = M i=1 θi ωi (φ, z). Example 2. Consider a lower triangular nonlinearity

ψ (x, u) = [ψ1 (x1 , u) , · · · , ψn−1 (x1 , · · · , xn−1 , u) , ψn (x, u)]T : Rn ×Rm → Rn (15) with ψ (0, u) = 0 for all u. Assume that each component is (globally) Lipschitz, uniformly in u (or for u in a compact set). i.e. ψi xi , u − ψi y i , u ≤ ki xi − y i , i = 1, · · · , n, where ki > 0 is the Lipschitz constant of ψi , and T

xi = x1 · · · xi . Defining φ (z, x, u) = ψ (x, u) − ψ (x + z, u) the Lipschitz condition on ψ implies for each component of phi that φi z i , xi , u ≤ ki z i , i = 1, · · · , n. Considering the Euclidean norm this implies φ2i (z1 , · · · , zi , x1 , · · · , xi , u) ≤ ki2 z12 + · · · + zi2 , i = 1, · · · , n.

These inequalities show that φ is (Qi , Si , Ri )-dissipative for every i = 1, · · · , n, with (Qi , Si , Ri ) = −bi bTi , 0, ki Ii , where bi are the basis vectors of Rn , Ii = diag (Ii , 0n−i ), and Ip is the identity matrix of dimension p. In this case the design is as follows Theorem 2. Suppose that Assumption 1 is satisfied. If there are matrices L and N , and a vector θ = (θ1 , · · · , θM ), θi ≥ 0, such that the linear subsystem M of Ξ is −Rθ , SθT , −Qθ -SSD, with (Qθ , Sθ , Rθ ) = i=1 θi (Qi , Si , Ri ), then Ω is a global (local) exponential observer for Σ, i.e. there exist constants κ, μ > 0 such that for all z˜ (0) (in a vicinity of z˜ = 0) z˜ (t) ≤ κ z˜ (0) exp (−μt). Proof. By definition of −Rθ , SθT , −Qθ -SSD there exist a θ = (θ1 , · · · , θM ), θi ≥ 0, a matrix P = P T > 0, and > 0 such that T T T Rθ HN , P G − HN Sθ P AL + ATL P + P + HN T G P − S θ HN Qθ

≤0.

vector (16)

The application of Lemma 1 leads immediately to V˙ ≤ − V (e), with V (e) =

eT P e. Convergence follows with κ =

λmax (P ) λmin (P ) ,

and μ =

2λmax (P ) .

Remark 7. In general (16) is a nonlinear matrix inequality feasibility problem. Under some conditions it becomes a Linear Matrix Inequality (LMI) feasibility problem, for which a solution can be effectively found by several algorithms in the literature [6]. For a further discussion of the computational issues see [23]. Note also that when the inequality (16) is feasible, there exist in general several solutions for L and N .

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4 Approximate Error Linearization using Dissipativity The major drawback of the observer error linearization method is that it can be applied only to a very small class of nonlinear systems. To deal with a much bigger set of systems several approximate techniques have been proposed (see [21] and its references, [3, 24]). The basic idea of these methods is to decompose the nonlinear system (1) in two parts Σ : x˙ = f (x, u) + g (x, u) ,

x (0) = x0 , y = h (x) ,

(17)

where the system (f, h) can be transformed into the observer normal form (3) after a state transformation (2), and the full system is transformed into an Approximate Nonlinear Observer Form (ANOF) Σaonf : z˙ = Ao z + Gψ (σ, y, u) + γ (y, u) ,

z (0) = z0 , y = Co z , σ = Hz , (18) where Gψ (σ, y, u) = ∂T∂x(x) g (x, u), with x = T −1 (z). Note that (18) is a special form of (11), and that in general G = H = I. This procedure can be applied to a very big class of systems. Since the decomposition in (17) can be made in many different forms, the transformation and the perturbation term ψ in (18) are not uniquely determined. In [21, 3] an observer in the new coordinates (18) is proposed as (12), where A = Ao , C = Co , N = 0, and L is selected so that the matrix (Ao − LCo ) is Hurwitz. The error equation, z˜ = zˆ − z, is then approximately linear (14). In these references T , γ and ψ are selected to reduce the size of the error perturbation term φ in a compact region of the state space. Doing so it is hoped that the error equation becomes more linear, so that it has an asymptotically stable equilibrium point at z˜ = 0 with a possible big attraction region. Although intuitively appealing, in the previous method it is not justified that the minimization does indeed reach its objective for the observer design. Moreover, the approach is restricted to local observers, since the state space has to be restricted to a compact set, and the minimization has to be done numerically. Instead of relying on reducing the magnitude of the perturbation term φ, what in general can be only achieved on a compact set, it is proposed in this paper to select the transformation (if possible) so that the transformed system (18) satisfies the conditions of Theorem 2 (globally or locally). It is possible to state the following result of the combination of both methods, that is simple to prove. Theorem 3. Suppose that a nonlinear system (1) can be decomposed as in (17), such that after a diffeomorphic state transformation z = T (x) the approximate observer normal form (18) is obtained. Suppose furthermore that the conditions of Theorem 2 are satisfied (locally or globally) by the approximate observer normal form (18). Then the system (12) with the output mapz ) is a (local or global) exponential observer for the system, if ping xˆ = T −1 (ˆ T −1 is uniformly continuous.

Approximate Observer Error Linearization by Dissipativity Methods

45

This result includes the exact error linearization method (when ψ = 0) but it offers a greater flexibility: Since the decomposition (17) can be made in many different ways, one can try different possibilities. Note that the same idea can be used for the extensions of the exact linearization method described in Remark 3.

5 Examples In this section some examples are given to illustrate how this method can be applied and the advantages and flexibility it offers in the design of nonlinear observers. 5.1 Example 1 Consider the system x˙ 1 = x2 − x1 u , x˙ 2 = −x1 − x2 − x21 − x22 − αx32 + x1 x2 u , α > 0 , y = x1 , for which an observer in a compact set of the state space was proposed in [21]. For u = 0 and from Example 1 it is seen that the system is not transformable into the observer form because of the cubic term. Decomposing the system as x˙ =

x2 0 −x1 u+ + −x1 − x2 − x21 − x22 x1 x2 −αx32

, y = x1 ,

it can be readily seen (see Example 1) that the first term can be linearized by a state and an output transformation [19], given by y¯ = θ (y) = exp (y) − 1 , y = θ−1 (¯ y) = ln |¯ y + 1| , θ : R → (−1, ∞) , z = T (x) =

exp (x1 ) − 1 exp (x1 ) (x2 + 1) − 1

x = T −1 (z) =

ln |z1 + 1| z2 +1 z1 +1 − 1

, T : R2 → (−1, ∞) × R

.

In the new coordinates the system has the approximate observer form (18) 0 01 −¯ y − u (¯ y + 1) ln |¯ y + 1| , z+ + α (¯ y − z2 )3 00 − ln |¯ y + 1| (1 + y¯) (1 + ln |¯ y + 1| + u) (1+¯ y )2 y¯ = z1 .

z˙ =

This system can be written in the form (11), and the error equation (14) is given with

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α (¯ y − σ)3 01 0 1 , G = HT = , CT = , ψ (σ, y¯) = 2 , 00 1 0 (1 + y¯) α 3 3 ¯) − (σ − y¯) , σ = z2 . φ (ξ, σ, y¯) = 2 (ξ + σ − y (1 + y¯) A=

It is easy to check that ω (φ, ξ, y¯) = Qφ2 + 2Sφξ + Rξ 2 ≥ 0 for all σ with

α . If y¯ is bounded there is a minimum value of (Q, S, R) = 0, 1, − 2(1+¯ y )2 ¯ ¯ R so that φ is 0, 1, R -dissipative. The observer design can be done if the design inequality (16) is feasible. The inequality is satisfied, for example, for ¯ l1 = 3R, ¯ l2 = −2R ¯ 2 . This observer is not global, because z1 has to N = 2R, be bounded, but there is no restriction on z2 , and u.

5.2 Example 2: High-Gain Observer Design It will be shown that the well-known high-gain observer (HGO) design [9, 10] for non linear systems is covered by Theorem 3, and its solution represents a particular one. Consider the class of systems (1) (affine in the control) that are uniformly observable for any input, defined in [9]. These systems can be transformed by the diffeomorphic observability map (4), z = q (x), into the approximate observer normal form (18), where the nonlinearity has the triangular structure given by (15) and, in general, G = H = I and γ (y, u) = 0 [8, 9]. The HGO design is done for this plant representation, and this design can be derived from the dissipative design in the previous section. Lemma 2. Under the same assumptions as in [9] the high-gain observer designed in that reference is a particular solution of Theorem 2. Proof. Note first that the high-gain observer in [9] corresponds to Ω (12), with A = Ao , C = Co , H = G = I, and N = 0. Next it will be shown that the high-gain solution satisfies theorem 2. By Hypothesis each component of ψ (x, u) is globally Lipschitz, uniformly in u. Following Example 2 it is easy to show that φ (e, x, u) = ψ (x, u) − ψ (e + x, u) is low triangular and globally Lipschitz. This implies that φ is (Qi , Si , Ri )dissipative, i = 1, 2, · · · , n, with (Qi , Si , Ri ) = −bi bTi , 0, ki Ii , where ki > 0 is the Lipschitz constant of ψi , and the other symbols are as in Example 2. It is required to show that the high-gain solution [9], given by L = −Xg−1 C T , and P = Xg , that satisfies for some (high) gain g > 0 gXg + AT Xg + Xg A = C T C ,

(19)

> 0 satisfies Theorem 2, there exist θ = (θ1 , · · · , θn ), θi > 0, and such that (16) is satisfied, where Qθ = − ni=1 θi bi bTi = − diag (θi ), Rθ = diag

n−i j=0 θn−j kn−j

, Sθ = 0, A = Ao , C = Co , H = G = I, and N = 0.

Pre- and postmultiplying (19) with Λ = diag g, g 2 , · · · , g n it is obtained ΛXg Λ + AT ΛXg Λ + ΛXg ΛA = gC T C, since, as it is easily checked, CΛ = gC,

Approximate Observer Error Linearization by Dissipativity Methods

47

and Λ−1 AΛ = gA. If Π is the symmetric and positive definite solution of Π + AT Π + ΠA = C T C, then it follows that ΛXg Λ = gΠ. It will be shown that this solution satisfies (16). Replacing L and P in (16) and using (19) −gXg − C T C + Xg + Rθ Xg Xg Qθ

≤0.

Pre- and post-multiplying this inequality with the matrix Λ = diag g, g 2 , · · · , g n , dividing by g and simplifying the expression one obtains −gΠ − gC T C + Π + g1 ΛRθ Λ Π ≤0. 1 Π g ΛQθ Λ

(20)

. 1 2i−1 θi = −I ,then θi = g ΛQθ Λ = − diag g n−i −2(n−(i+j)) kn−j . Note that, since in the g −(2i−1) and g1 ΛRθ Λ = diag j=0 g last expression (i + j) ≤ n there is a finite limit limg→∞ g1 ΛRθ Λ = diag (ki ).

Selecting θ (positive) such that

It follows then easily that for any > 0, and for g big enough the inequality (20) is satisfied. This can be seen using the Schur complement [5, Appendix A]: (20) is satisfied iff 1 −gΠ − gC T C + Π + ΛRθ Λ + Π 2 ≤ 0 . g Since Π is positive definite, i.e. Π > δI for some δ > 0, the inequality is satisfied for g big enough.

This result is interesting for different reasons: 1) It shows that the proposed method is always applicable when the high-gain methodology is. 2) Moreover, it represents a generalization of the HG method, and can deliver a solution when the HG does not. 3) Since the HG represents only one possible solution, the proposed method can find better solutions, and is not constrained to highgain ones. Moreover, other optimization and design criteria can be included to find a better solution. 4) On the other side, the HG method and the strong theoretical characterization of when it is feasible, provides a class of systems for which the proposed method is assured to deliver a solution. Since Theorem 2 does not give explicit conditions on the system for a solution to be feasible, the HG case shows that it is possible to assure existence of solutions for a large class of systems. 5) A simple and immediate generalization of HG is to use the additional injection depending on matrix N . 6) Since in the proposed method the particular characteristics of the nonlinearities can be taken into account using several supply rates, and not only the triangular and Lipschitzness characteristics used by the HG method, it is expected that the system class for which this method is applicable is much bigger, the conservativeness of the results is reduced, and the quality of the results increased. It can be shown [23] that the proposed method includes several other design strategies proposed in the literature as the Circle Criterion Design [2, 7] and the Lipschitz Observer Design [29, 33].

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5.3 Example 3: Observer Design for Detectable Systems Most design methods require the system to be uniformly observable for every input, as for example the HGO and the exact error linearization. Our method does not require this property but it is able to construct observers for detectable systems, as the following example shows. Consider the planar system x˙ 1 = g (x2 ) u , x˙ 2 = x1 − x2 , y = x1 ,

where g (x2 ) is any globally Lipschitz nonlinearity, with Lipschitz constant kg . This system is not observable for every input, since for u = 0 it is detectable but not observable. This system can be written in the form (11), and the error equation (14) is given with A=

1 0 1 , H = 0 1 , ψ (σ, u) = g (σ) u − σ , , G = CT = 0 1 −1

φ (ξ, σ, u) = z + (g (σ) − g (σ + ξ)) u , σ = x2

φ (ξ, σ, u) is globally Lipschitz in ξ with constant k = kg |u| + 1, whenever u is bounded. φ is therefore dissipative for (Q, S, R) = −1, 0, k 2 . It is easy (but tedious) to check that, for example, with N = 0, l1 = − (k + 1), l2 = −2 the design inequality (16), which is an LMI, is feasible. This means that a global observer can be designed for any Lipschitz g (x2 ) for bounded inputs, despite of the fact that the system is not observable.

6 Conclusions A new method to design observers for nonlinear systems has been proposed. The basic idea is to decompose the system in a nonlinear part, that can be transformed into a nonlinear observer form, and a perturbation term connected in the feedback loop. In transformed coordinates the observer error becomes linear with a feedback perturbation. By using the dissipativity theory it is possible to design the observer gains so, that the closed loop is internally stable if some LMI-like conditions are satisfied. This method is very flexible and allows the design of nonlinear observers using exact linearization methods to a much bigger class of systems. The method has been shown to be very general, since it includes (and generalizes) as special cases several observer design methods, as for example the exact error linearization method, the high-gain method, the circle criterion design method, and the design for Lipschitz nonlinear systems. The design is computationally simple in many cases, since it reduces to the solution of a feasibility LMI problem, for which highly efficient numerical methods are available. The method offers great flexibility in the design, since the particular properties of the nonlinearities can be characterized

Approximate Observer Error Linearization by Dissipativity Methods

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by means of one or several quadratic forms, i.e. supply rates. This feature can be used to design observers for systems with special properties in the nonlinearities. Further generalization can be achieved combining the ideas presented in this paper and the results of [16, 1]. As pointed out in [28], the use of integral quadratic constrains as generalization of the (local) quadratic supply rates used in the present paper leads to less restrictive designs. Acknowledgement. This work has been done with the financial aid of DGAPAUNAM under project PAPIIT IN111905-2.

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16. N. Kazantzis and C. Kravaris. Nonlinear observer design using Lyapunov’s auxiliary theorem. Systems & Control Letters, 34:241–247, 1998. 17. H.K. Khalil. Nonlinear Systems. Prentice–Hall, Upsaddle River, New Jersey, third edition, 2002. 18. A.J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Syst. Control Lett., 3:47–52, 1983. 19. A.J. Krener and W. Respondek. Nonlinear observers with linear error dynamics. SIAM J. Control and Optimization, 23:197–216, 1985. 20. J. Levine and R. Marino. Nonlinear system immersion, observers and finite dimensional filters. Syst. Control Lett., 7:133–142, 1986. 21. A. Lynch and S. Bortoff. Nonlinear observers with approximated linear error dynamics: the multivariable case. IEEE Trans. Automatic Control, 46(6):927– 932, 2001. 22. R. Marino and P. Tomei. Nonlinear Control Design; Geometric, Adaptive & Robust. Prentice Hall, London, 1995. 23. J. A. Moreno. Observer design for nonlinear systems: A dissipative approach. In Proceedings of the 2nd IFAC Symposium on System, Structure and Control SSSC2004, pages 735–740, Oaxaca, Mexico, Dec. 8-10, 2004, 2004. IFAC. 24. S. Nicosia, P. Tomei, and A. Tornamb´e. A nonlinear observer for elastic robots. IEEE Journal of Robotics and Automation, 4(1):45–52, February 1988. 25. H. Nijmeijer and T.I. Fossen, editors. New Directions in Nonlinear Observer Design. Number 244 in Lecture notes in control and information sciences. Springer– Verlag, London, 1999. 26. H. Nijmeijer and A. van der Schaft. Nonlinear Dynamical Control Systems. Springer–Verlag, New York, 1990. 27. A.R. Phelps. On constructing nonlinear observers. SIAM J. Control Optimization, 29:516–534, 1991. 28. Alexander Pogromsky and Henk Nijmeijer. Observer-based robust synchronization of dynamical systems. International Journal of Bifurcations and Chaos, 8(11):2243–2255, 1998. 29. R. Rajamani. Observers for Lipschitz nonlinear systems. IEEE Trans. Automatic Control, 43:397–401, 1998. 30. W. Respondek, H. Nijmeijer, and P. Gromoski. Observer linearization by output-dependent time-scaling transformation. In 5th IFAC Symposium Nonlinear Control Systems (NOLCOS), Saint Petersburg, Russia, July 2001. 31. J. Schaffner and M. Zeitz. Variants of nonlinear normal form observer design. In Nijmeijer and Fossen [25], pages 161–180. 32. J.A.K. Suykens, P.F. Curran, J. Vandewalle, and L.O. Chua. Robust nonlinear H∞ synchronization of chaotic lur’e systems. IEEE Trans. on Circuits and Systems–I: Fundamental theory and applications, 44(10):891–904, 1997. 33. F.E. Thau. Observing the state of nonlinear dynamic systems. Int. J. Control, 17:471–479, 1973. 34. A. Van der Schaft. L2 -Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag, London, 2nd edition, 2000. 35. J.C. Willems. Dissipative dynamical systems, part I: General theory. Archive for Rational Mechanics and Analysis, 45:321–351, 1972. 36. J.C. Willems. Dissipative dynamical systems, part II: Linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis, 45:352– 393, 1972.

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37. X. Xia and W. Gao. Nonlinear observer design by observer error linearization. SIAM Journal of Control and Optimization, 27(1):199–216, 1989. 38. X. Xia and M. Zeitz. On nonlinear continuous observers. Int. J. Control, 66(6):943–954, 1997. 39. M. Zeitz. The extended Luenberger observer for nonlinear systems. Syst. Control Lett., 9:149–156, 1987. 40. M. Zeitz. Canonical normal forms for nonlinear systems. In A. Isidori, editor, Nonlinear Control Systems Design - Selected Papers from IFAC-Symposium, pages 33–38. Pergamon Press, Oxford, 1989.

On Invariant Observers Silv`ere Bonnabel and Pierre Rouchon Ecole des Mines de Paris, Centre Automatique et Syst`emes, 60 Bd Saint-Michel, 75272 Paris cedex 06, France. {silvere.bonnabel,pierre.rouchon}@ensmp.fr

Summary.฀ A฀ definition฀ of฀ invariant฀ observer฀ and฀ compatible฀ output฀ function฀ is proposed฀ and฀ motivated.฀ For฀ systems฀ admitting฀ a฀ Lie฀ symmetry-group฀ G฀ of฀ dimension฀less฀or฀equal฀to฀the฀state฀dimension฀and฀with฀a฀G-compatible฀output,฀an explicit฀ procedure฀ based฀ on฀ the฀ moving฀ frame฀ method฀ is฀ proposed฀ to฀ construct such฀invariant฀observers.฀It฀relies฀on฀an฀invariant฀frame฀and฀a฀complete฀set฀of฀invariant฀estimation฀errors.฀Two฀examples฀of฀engineering฀interest฀are฀considered:฀an exothermic฀chemical฀reactor฀and฀an฀inertial฀navigation฀problem.฀For฀both฀examples we฀ show฀ how฀ invariance฀ and฀ the฀ proposed฀ construction฀ can฀ be฀ a฀ useful฀ guide฀ to design฀non-linear฀convergent฀observers,฀although฀the฀part฀of฀the฀design฀procedure which฀achieves฀asymptotic฀stability฀is฀not฀systematic฀and฀must฀take฀into฀account the฀specific฀nonlinearities฀of฀the฀case฀under฀study.

Keywords:฀Observers,฀symmetries,฀invariance,฀moving฀frame,฀chemical฀reactor,฀inertial฀navigation.

1 Introduction Symmetries are important in physics and mathematics, see, e.g., [13, 15]. In control theory, they also play a fundamental role, especially in feedback design and optimal control, see, e.g., [7, 6, 11, 9, 10, 16, 18, 14]. To our knowledge only very few results exploiting symmetries are available for observer design. Most of the results are based on special structure once a proper set of state coordinates has been chosen [12, 5, 17, 19, 8]. Thus most design techniques are not coordinates free. For mechanical system with position measures, an intrinsic design is proposed in [3]: it relies on the metric derived from the kinetics energy; it is invariant with respect to change of configuration coordinates. In [2, 1], it is shown how to exploit symmetry for the design of asymptotic observer via the notion of invariant estimation errors. The observer dynamics remains unchanged up to any transformation of state coordinates belonging

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 53–65, 2005. © Springer-Verlag Berlin Heidelberg 2005

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to a Lie-group of symmetries. This paper prolonges and completes theses results and can be seen also the counter-part on the observer side of invariant tracking [14]. For clarity’s sake, we consider here only the local and regular case. Global and singular cases are much more difficult and require additional assumptions. However, as shown by the analytical examples of the exothermic reactor and inertial navigation, such local analysis are in fact sufficient to get invariant observer and formulae that are also valid on the entire state manifold. In section 2 we propose a natural definition of an invariant observer. In section 3, we characterize in terms of invariant frame and invariant estimation errors such invariant observers. In section 4, we show that the existence of invariant observers imposes some strong property on the output map: it has to be compatible with the group action on the state space. In section 5, we adapt the moving frame method and propose a procedure to construct explicitly (i.e., up to inversion of non-linear map) the general form of invariant observers. Sections 6 and 7 are devoted to two engineering examples for which we propose nonlinear and convergent invariant observers. The authors thank Philippe Martin for interesting discussion and comments relative to the inertial navigation problem.

2 Invariance Consider the following dynamics d x = f (x, t) dt

(1)

where the state x lies in an open subset X of Rn and f is a smooth function of its arguments. Let G be a r-dimensional Lie group acting locally on X : ϕg is a diffeomorphism on X close to identity when g is close to e, the identity element of G. We assume that the action of G is locally free: the dimension of the orbit passing through x ∈ X , i.e., the set of all ϕg (x) for g close to e is r, the dimension of G; for any x ∈ X and any y ∈ X close to x and belonging to the orbit of x, exists a unique g close to e such that ϕg (x) = y and such a g depends smoothly of y on the orbit of x. All along the paper, when we consider g ∈ G, we always implicitly assume that g is in a small neighborhood of e. Definition 1. Dynamics (1) is called G-invariant if for all g ∈ G, x ∈ X f (ϕg (x), t) =

∂ϕg (x) · f (x, t). ∂x

(2)

G is thus a symmetry group: for each g ∈ G, the change of variables X = ϕg (x) d X = f (X, t). leaves the dynamics unchanged: dt Let h : x → y = h(x) be a regular output map from X to Y, an open subset of Rp (p ≤ n).

On Invariant Observers

Definition 2. Take a G-invariant dynamics put y = h(x). The dynamical system

d dt x

55

= f (x, t) with a smooth out-

d x ˆ = fˆ(ˆ x, h(x), t) dt is called a G-invariant observer if, and only if, for all g ∈ G and for all x and x ˆ in X we have fˆ(x, h(x), t) = f (x, t),

∂ϕg (ˆ x) · fˆ(ˆ x, h(x), t) = fˆ(ϕg (ˆ x), h(ϕg (x)), t). ∂x

This definition means that the observer dynamics is unchanged under transˆ = ϕg (ˆ formations of the form X x), X = ϕg (x), g being an arbitrary element of G: d ˆ ˆ h(X), t) X = fˆ(X, dt Notice that this definition does not deal with convergence issues. We separate intentionally invariance from convergence and robustness. We have kept the terminology observer because, with this definition, the system trajectory t → x(t) is solution of the observer dynamics: x ˆ(t) = x(t) when x ˆ(0) = x(0). Clearly such definition must be completed by convergence and robustness properties. Since we do not have general result relative to convergence and robustness we will just complete the definition simply by the following one : d x ˆ = fˆ(ˆ x, h(x), t) is called asymptotic if x ˆ(t) converges to x(t) the observer dt when t tends to +∞.

3 Characterization Assumptions relative to the action of G on X , imply that exist n point-wise linearly independent G-invariant vector fields w1 , ...., wn , forming a frame on X , i.e. a basis of the tangent space at x (see, theorem 2.84 of [15]). d x ˆ = fˆ(ˆ x, y, t) as described in definition 2. For Take an invariant observer dt ˆ each y and t, the vector field f can be decomposed according to such invariant frame (wi )i=1,...,n : fˆ(ˆ x, y, t) =

n

Fi (ˆ x, y, t)wi (ˆ x) i=1

where the Fi ’s are smooth scalar functions of their arguments. Definition 2 means that n

∀x ∈ X ,

f (x, t) =

Fi (x, h(x), t)wi (x) i=1

and for any i ∈ {1, ..., n}, any g ∈ G and x, x ˆ in X , x), h(ϕg (x)), t) = Fi (ˆ x, h(x), t) Fi (ϕg (ˆ

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Thus we have fˆ(ˆ x, y, t) = f (ˆ x, t) +

n

Ei (ˆ x, y, t)wi (ˆ x) i=1

x, y, t) = Fi (ˆ x, y, t) − Fi (ˆ x, h(ˆ x), t) and satisfies for all x, xˆ ∈ X where each Ei (ˆ and g ∈ G: Ei (x, h(x), t) = 0,

Ei (ϕg (ˆ x), h(ϕg (x)), t) = Ei (ˆ x, h(x), t).

These Ei can be interpreted thus as scalar invariant errors. This motivates the following definition. Definition 3. A scalar smooth function E(ˆ x, y, t) is called an invariant error if, and only if, it satisfies the following equation E(x, h(x), t) = 0,

E(ϕg (ˆ x), h(ϕg (x)), t) = E(ˆ x, h(x), t).

for any g ∈ G, x, xˆ ∈ X .

We have thus the following lemma Lemma 1. Any G-invariant observer

d ˆ dt x

= fˆ(ˆ x, y, t) (see definition 2) reads: n

d x ˆ = fˆ(ˆ x, y, t) = f (ˆ x, t) + Ei (ˆ x, y, t)wi (ˆ x) dt i=1

(3)

where the Ei ’s are invariant errors and (w1 , ..., wn ) is an invariant frame. We will see in section 5 how to build such Ei and wi with the knowledge, in local coordinates, of the transformation ϕg .

4 G-Compatible Output The existence of G-invariant observer in the sense of definition 2 implies a compatibility condition on the output map y = h(x). Under rank conditions, the action of G on state-space X can be transported via the output map h on the output space Y in the sense of the following definition: d x = f (x, t) be a G-invariant dynamics. The smooth outDefinition 4. Let dt put map X x → y = h(x) ∈ Y is said G-compatible if, and only if, for any g ∈ G, there exists a smooth invertible transformation ρg on Y such that

∀x ∈ X ,

h(ϕg (x)) =

g (h(x)).

This definition implies that G acts also on the output space Y, via the transformations g .

On Invariant Observers

57

d Theorem 1. Consider a G-invariant dynamics dt x = f (x, t) with output y = d h(x) and assume there exists a G-invariant observer dt x ˆ = fˆ(ˆ x, y, t) in the sense of definition 2. Assume that the rank of the jacobian ∂f (ˆ x, y, t)/∂y is maximum and equal to the dimension of y. Then, necessarily, the output map is G-compatible in the sense of definition 4.

Proof. By lemma 1, we have fˆ(ˆ x, y, t) = f (ˆ x, t) +

n

Ei (ˆ x, y, t)wi (ˆ x) i=1

where the Ei are invariant errors and wi invariant vector fields. Since the rank ˆ of ∂∂yf is equal to p = dim(y), there exist p invariant errors, E = (Ei1 , ..., Eip ), 1 ≤ i1 < i2 ... < ip ≤ n such that the Jacobian ∂E ∂y is invertible. Denote by F the inverse map versus y (locally defined by the implicit function theorem), then we have F (ˆ x, E(ˆ x, y, t), t) ≡ y. Moreover we have, for all x ˆ, x ∈ X and g ∈ G,

x), h(ϕg (x)), t) = E(ˆ x, h(x), t). E(ϕg (ˆ Thus

h(ϕg (x), t) = F (ϕg (ˆ x), E(ˆ x, h(x), t), t).

Let us fix x ˆ and t to some nominal values, said x¯ and t¯. The above identity means that for any g ∈ G, exists ρg (depending also on x ¯ and t¯, but these dependencies are omitted here) a map from Y to Y such that, for all x ∈ X , h(ϕg (x), t) = ρg (h(x)). Thus the output map is G-compatible and ρg is a transformation on Y with inverse ρg−1 .

5 Construction The above discussion and results justify the following question. Assume that we have explicitly the action of G on the state space X with a G-compatible output map y = h(x). This means that we have at our disposal r scalar parameters a = (a1 , ..., ar ) corresponding to a particular parametrization of G, n smooth scalar functions ϕ = (ϕ1 , ..., ϕn ) of a and x = (x1 , ..., xn ) corresponding to the action of G in local coordinates on X , Xi = ϕi (a, x),

i = 1, ..., n

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We have also p output scalar maps of x, h = (h1 , ..., hp ), and p scalar functions of a and y, = ( 1 , ..., p ), corresponding to action of G on the output-space Y such that for any group parameter a and state x h(ϕ(a, x)) = (a, h(x)). These transformations ϕ and are derived in general from obvious and physical symmetries. According to lemma 1, invariant observers are built with invariant errors Ei and invariant frame (w1 , ..., wn ). More precisely, we will see that, under standard regularity conditions, we have an explicit parametrization of such invariant observers based on a complete set (E1 , ..., Ep ) of invariant errors. Let us first explain how to build an invariant frame from the knowledge of ϕ. This construction is standard (see [15]) and as follows. Since the orbits are of dimension r, the dimension of G, we can decompose the components of ϕ into two sets: ϕ = (ϕ, ¯ ϕ) ˜ with dim(ϕ) ¯ = r and dim(ϕ) ˜ = n − r ≥ 0, such that for each x, the map a → ϕ(a, ¯ x) is invertible. This decomposition x = (¯ x, x ˜) is clearly not unique. Take a normalization of the first set of components, x ¯, ¯ x) = x ¯0 versus a to obtain a = α(¯ x0 , x) denoted by x ¯0 . We can solve ϕ(a, where α is a smooth function. Thus α is characterized by the identity: ϕ(α(¯ ¯ x0 , x), x) ≡ x¯0 . ∂ , ..., ∂x∂n ). For each i ∈ {1, ..., n} and x, set Consider the canonical frame ( ∂x 1

wi (x) =

∂ϕ (α(¯ x0 , x), x) ∂x

−1

·

∂ |ϕ(α(¯x0 ,x),x). ∂xi

Then wi is invariant because for any group parameter b and x we have "

wi (ϕ(b, x)) = ∂ϕ (b, x) ∂x

"

−1

∂ϕ x0 , ϕ(b, x)), ϕ(b, x)) ∂x (α(¯

·wi (ϕ(b, x)) =

−1

·

∂ ∂xi

and thus

∂ϕ ∂ϕ (α(¯ x0 , ϕ(b, x)), ϕ(b, x)) (b, x) ∂x ∂x

−1

·

∂ ∂xi

the group structure implies that, for any group parameters c, d, we have ϕ(d, ϕ(c, x)) = ϕ(d · c, x); thus ϕ(α(¯ x0 , ϕ(b, x)), ϕ(b, x)) = ϕ(α(¯ x0 , ϕ(b, x)) · b, x) and also ∂ϕ ∂ϕ ∂ϕ (α(¯ x0 , ϕ(b, x)), ϕ(b, x)) (b, x) = (α(¯ x0 , ϕ(b, x)) · b, x); ∂x ∂x ∂x where · corresponds to the composition law on G.

On Invariant Observers

"

59

since α(¯ x0 , ϕ(b, x)) · b = α(¯ x0 , x), we have thus ∂ϕ (b, x) ∂x

−1

· wi (ϕ(b, x)) =

∂ϕ (α(¯ x0 , x), x) ∂x

−1

·

∂ = wi (x). ∂xi

The computation of a complete set of invariants Ei is obtained via similar manipulations based on the normalization function α. We recall here the procedure presented in [2] based on the moving frame method (see also [14] for closely related computations of invariant tracking errors). We consider here the action of G on the product space X × Y defined by X ×Y

(ˆ x, y) → (ϕ(a, xˆ), (a, y)) ∈ X × Y.

Consider once again the normalization function α(¯ x0 , x ˆ). Then each component of ϕ(α(¯ ˜ x0 , x ˆ), x ˆ) and of (α(¯ x0 , x ˆ), y) are invariant scalar function of (ˆ x, y). Moreover, they form a complete set of invariant relative to the action of G on X × Y: this means that any invariant function I of x ˆ and y is a function of these p + n − r fundamental invariants. Thus every invariant errors x, y, t) admits the following expression: Ei (ˆ Ei (ˆ x, y, t) = Fi (ϕ(α(¯ ˜ x0 , x ˆ), x ˆ), (α(¯ x0 , x ˆ), y), t) where Fi is a smooth function of its arguments. To be a little more explicit, x), ..., In−r (ˆ x) the components of ϕ(α(¯ ˜ x0 , x ˆ), x ˆ) and by ε1 (ˆ x, y), denote by I1 (ˆ ..., εp (ˆ x, y) the components of (α(¯ x0 , x ˆ), y) − (α(¯ x0 , x ˆ), h(ˆ x)). Then the invariant errors Ei have the form Ei (ˆ x, y, t) = Ei (I, ε, t) where Ei is a smooth function of its arguments I, ε and t, with Ei (I, 0, t) ≡ 0.

6 An Exothermic Reactor Let us consider the classical exothermic reactor of [4]. With slightly different notations, the dynamics reads d E X = D(t)(X in − X) − k exp − X dt RT E d T = D(t)(T in (t) − T ) + c exp − X + u(t) dt RT where (E, R, k, c) are positive and known constant parameters, D(t), T in (t) and u(t) are known time functions. The parameter X in > 0, the inlet composition is unknown. The available online measure is T , the temperature inside the reactor. The reactor composition X is not measured.

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These two differential equations correspond to material and energy balances. Their structure is independent of the units. Let us formalize such independence in terms of invariance. We just consider change of material unit corresponding to the following scaling X → aX and X in → aX in with a > 0. The group G will be the multiplicative group R∗+ .Take x = (X in , X, T, c) as state and the action of G is defined for each a > 0 via the transformation ⎛ ⎞ ⎛ in ⎞ aX in X ⎜ aX ⎟ ⎜ X ⎟ ⎜ ⎟ ⎟ ⎜ ⎝ T ⎠ → ϕ(a, x) = ⎝ T ⎠ . c/a c

Then the dynamics

d in X =0 dt d E X = D(t)(X in − X) − k exp − X dt RT d E T = D(t)(T in (t) − T ) + c exp − X + u(t) dt RT d c=0 dt is invariant and y = (T, c) is a G-compatible output. Notice that we have added to the original state (X, T ), the inlet composition X in and the parameter c. Since c it is known, it has been added to the output function. ∂ ∂ ∂ , ∂T , c ∂c ). Invariant errors An obvious invariant frame is (X in ∂X∂in , X ∂X are built with the following complete set of invariants ˆ X , ˆ in X

Tˆ,

ˆ cˆX,

Tˆ − T,

ˆ in . (ˆ c − c)X

Since c is known (ˆ c = c), we do not care for the estimation of c and we thus consider invariant observers for (X in , X, T ). X in and X must be estimated since they are unknown. Since X in can only be constructed as an integral of the estimation error Tˆ − T , Tˆ must be calculated anyway although measured. Moreover the temperature measure can be noisy and thus noiseless estimation Tˆ of T could be useful for feedback. According to lemma 1, invariant observers have the following form d ˆ in X =A dt

ˆ X ˆ Tˆ, Tˆ − T , cX, ˆ X in

!

ˆ in X

! « „ ˆ d ˆ X ˆ ˆ − k exp − E X ˆ Tˆ, Tˆ − T X ˆ +B ˆ in − X) , cX, X = D(t)(X ˆ in dt RTˆ X ! „ « ˆ E d ˆ X in ˆ Tˆ , Tˆ − T ˆ + u(t) + C , cX, T = D(t)(T (t) − Tˆ ) + c exp − X ˆ in dt RTˆ X

On Invariant Observers

61

where A ,B and C are smooth functions such that A

ˆ X ˆ Tˆ, 0 , cX, ˆ in X

=B

ˆ X ˆ Tˆ, 0 , cX, ˆ in X

=C

ˆ X ˆ Tˆ, 0 , cX, ˆ in X

= 0.

ˆ and X ˆ in are positive quantities. Such observers preserve the fact that X Although we have no general method for choosing A, B and C to ensure convergence, we propose here a stabilizing design. First, up to a change of A, B and C, we can replace the Arrhenius term exp − RETˆ by exp − RTE(t) where T (t) is the measure (kind of output in-

jection). Thus the invariant observer reads (without changing the notations for A, B and C): d ˆ in X =A dt

ˆ X ˆ Tˆ, Tˆ − T , cX, ˆ X in

!

ˆ in X

! „ « ˆ d ˆ X ˆ − k exp − E ˆ Tˆ, Tˆ − T X ˆ ˆ in − X) ˆ +B , cX, X = D(t)(X X ˆ in dt RT (t) X ! „ « ˆ d ˆ E X in ˆ Tˆ , Tˆ −T . ˆ + u(t) + C T = D(t)(T (t)−T (t)) + c exp − , cX, X ˆ in dt RT (t) X

Let us choose ˆ exp − A = B = −βcX and ˆ exp − C = −γcX

E RT (t)

E RT (t)

(Tˆ − T (t))

(Tˆ − T (t))

with β and γ strictly positive design parameters. Take the variables ξˆ = ˆ instead of X ˆ in and X, ˆ since we have homogeˆ X ˆ in ) and Zˆ = log(X) log(X/ in ˆ ˆ neous equations in X and X. Then we have the following triangular structure „ « d ˆ ˆ − D(t) − k exp − E ξ = D(t) exp(−ξ) dt RT (t) « „ « „ E d ˆ E ˆ (Tˆ − T (t))−D(t)−k exp − Z = D(t) exp(−ξ)−βc exp Zˆ exp − dt RT (t) RT (t) „ « d ˆ E T = D(t)(T in (t) − T (t)) + c exp − exp Zˆ (1 − γ(Tˆ − T (t))) + u(t). dt RT (t)

Take Z˜ = Zˆ − log(X(t)) and T˜ = Tˆ − T (t) instead of Zˆ and Tˆ. We get

„ « d ˆ ˆ − D(t) − k exp − E ξ = D(t) exp(−ξ) dt RT (t) « „ d ˜ ˆ − exp(−ξ(t))) − βc exp − E + Z(t) exp Z˜ T˜ Z = D(t)(exp(−ξ) dt RT (t) „ « „ « d ˜ E E T = c exp − + Z(t) (exp Z˜ − 1) − γc exp − + Z(t) exp Z˜ T˜ dt RT (t) RT (t)

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where ξ = log(X/X in ) and Z = log X. Assume that exist M and η > 0 ˆ Z, ˜ T˜ ) remain such that for all t ≥ 0, M ≥ X in , D(t), X(t), T (t) ≥ η . Then (ξ, ˆ bounded for all t ≥ 0 and limt→+∞ (ξ(t) − ξ(t)) = 0. It is thus enough to ˜ T˜ ): analyze the convergence to 0 of the following reduced system in (Z, « „ d ˜ E Z = −βc exp − + Z(t) exp Z˜ T˜ dt RT (t) « „ « „ E E d ˜ + Z(t) (exp Z˜ − 1) − γc exp − + Z(t) exp Z˜ T˜. T = c exp − dt RT (t) RT (t)

Consider the regular change of time scale τ = Then:

t 0

c exp − RTE(s) + Z(s)

ds.

dZ˜ = −β exp Z˜ T˜ dτ dT˜ = (exp Z˜ − 1) − γ exp Z˜ T˜ dτ ˜ + β T˜ 2 as Lyapounov function. A standard This system admits Z˜ + exp(−Z) 2 application of Lasalle invariance principle shows that 0 is globally asymptotically stable. Guided by invariance considerations, we have obtained the following globally converging non-linear observer: d ˆ in E ˆ X ˆ in X = −β exp − (Tˆ − T (t)) cX dt RT (t) d ˆ ˆ ˆ in − X) ˆ X ˆ − exp − E X = D(t)(X k + β(Tˆ − T (t))cX dt RT (t) E d ˆ ˆ + D(t)(T in (t)−T (t)) + u(t) T = exp − 1 − γ(Tˆ − T (t)) cX dt RT (t) where the design parameters β and γ have to be chosen strictly positive.

7 Inertial Navigation Consider a flying body carrying an Inertial Measurement Unit which measures the earth magnetic field (denoted B in the earth frame) and the instantaneous rotation vector ω. Let b denote the magnetic field in the body frame (measured by the IMU). Let R ∈ SO(3) denote the rotation matrix which maps the mobile body frame to the earth frame. The kinematic relations between R and ω reads, d Ra = R(ω(t) ∧ a) dt for any vector a. Omitting a, we have the following dynamics for R in SO(3):

On Invariant Observers

63

d R = R(ω(t) ∧ ·) dt

(4)

y = b = R−1 B

(5)

with output

If the earth frame rotates via g ∈ SO(3), then R becomes gR, B becomes gB and the above equations remain unchanged. This means that the dynamics d R = R(ω(t) ∧ ·), dt

d B=0 dt

is invariant under the action of the group G = SO(3) via left multiplication: ϕg (R, B) = (gR, gB),

g ∈ SO(3)

and the output map y = R−1 B is clearly G-compatible since invariant ( g = Id ). Let e1 ,e2 ,e3 be a basis of R3 . Then, an invariant frame on SO(3), the state-space for R, is as follows: SO(3)

R → wi (R) = R(ei ∧ ·)

i = 1, 2, 3

Notice that wi (R) belongs to the tangent space at R. Since each component of b = R−1 B is invariant, they form a complete set of scalar invariants relative to the action of G = SO(3) on SO(3) × R3 , the space of (R, B). Every invariant observer reads 3

d ˆ ˆ −1 B, R ˆ −1 B − b ˆ R = R(ω(t) ∧ ·) + Ei R dt i=1

ˆ i ∧ ·) R(e

(6)

ˆ −1 B, 0) ≡ 0. where the scalar functions Ei are such that Ei (R We can adjust the function Ei in order to have the convergence of the ˆ −1 B − R−1 B to 0 when t tends to +∞. Assume that the estimation error R Ei correspond to the coordinates of the vector b ∧ ˆ(R−1 B). More precisely we set 3

i=1

ˆ −1 B) Ei ei = Kb ∧ (R

where K > 0 is some design parameter. Then we get the following invariant observer d ˆ ˆ ([ω(t) + Ke] ∧ ·) R=R dt ˆ −1 B). Standard computations (using R−1 = RT ) show that with e = b(t) ∧ (R d ˆ −1 ˆ −1 B − b) − Ke ∧ (R ˆ −1 B) (R B − b) = −ω ∧ (R dt

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and 1 d ˆ −1 R B−b 2 dt

2

= −K e

2

ˆ −1 B Thus e tends to zeros when t tends to +∞. This implies that b and R tend to be co-linear. Since their modules are equal, this means the convergence of the estimated output to the measured output (a direct analysis shows that ˆ −1 B is unstable and thus will not be obtained in the situation when b = −R practice). ˆ to R is only partial since, if t → R(t) ˆ The convergence of R is solution of ˆ the invariant observer (6), then t → Oθ R(t) is also solution of (6) where Oθ is the rotation around B by an arbitrary angle θ. In a certain sense, we cannot improve the convergence since with the output b = R−1 B, the dynamics on R is not observable: the trajectories t → R(t) and t → Oθ R(t) lead to exactly the same output trajectory t → b(t).

8 Conclusion We have proposed a systematic method to design observers preserving the symmetries of the original system. We do not have, up to now, a similar systematic procedure to tackle convergence. Nevertheless, the two previous examples indicate that invariance could be a useful guide for the design of convergent non-linear observers.

References 1. N. Aghannan. Contrˆ ole de R´eacteurs de Polym´erisation, observateur et invariance. PhD thesis, Ecole des Mines de Paris, November 2003. 2. N. Aghannan and P. Rouchon. On invariant asymptotic observers. In Proceedings of the 41st IEEE Conference on Decision and Control, volume 2, pages 1479– 1484, 2002. 3. N. Aghannan and P. Rouchon. An intrinsic observer for a class of lagrangian systems. IEEE AC, 48(6):936–945, 2003. 4. R. Aris and N.R. Amundson. An analysis of chemical reactor stability and control- i,ii,iii. Chem. Engng. Sci., 7:121–155, 1958. 5. J. Birk and M. Zeitz. Extended Luenberger observers for nonlinear multivariable systems. Int. J. Control, 47:1823–1836, 1988. 6. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and R. Murray. Nonholonomic mechanical systems with symmetry. Arch. Rational Mech. Anal., 136:21–99, 1996. 7. F. Fagnani and J. Willems. Representations of symmetric linear dynamical systems. SIAM J. Control and Optim., 31:1267–1293, 1993. 8. J.P. Gauthier and I. Kupka. Deterministic Observation Theory and Applications. Cambridge University Press, 2001.

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9. J.W. Grizzle and S.I. Marcus. The structure of nonlinear systems possessing symmetries. IEEE Trans. Automat. Control, 30:248–258, 1985. 10. B. Jakubczyk. Symmetries of nonlinear control systems and their symbols. In Canadian Math. Conf. Proceed., volume 25, pages 183–198, 1998. 11. W. S. Koon and J. E. Marsden. Optimal control for holonomic and nonholonomic mechanical systems with symmetry and lagrangian reduction. SIAM J. Control and Optim., 35:901–929, 1997. 12. A.J. Krener and W. Respondek. Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim., 23:197–216, 1985. 13. J.E. Marsden and T.S. Ratiu. Introduction to Mechanics and Symmetry. Springer-Verlag, New York, 1994. 14. Ph. Martin, P. Rouchon, and J. Rudolph. Invariant tracking. ESAIM: Control, Optimisation and Calculus of Variations, 10:1–13, 2004. 15. P. J. Olver. Equivalence, Invariants and Symmetry. Cambridge University Press, 1995. 16. W. Respondek and I.A. Tall. Nonlinearizable single-input control systems do not admit stationary symmetries. Systems and Control Letters, 46:1–16, 2002. 17. J. Rudolph and M. Zeitz. A block triangular nonlinear observer normal form. Systems and Control Letters, 23:1–8, 1994. 18. A.J. van der Schaft. Symmetries in optimal control. SIAM J. Control Optim., 25:245259, 1987. 19. M. Zeitz and X. Xia. On nonlinear continuous observers. Int. J. Control, 66(6):943–954, 1997.

Remarks on Moving Horizon State Estimation with Guaranteed Convergence Tobias Raff, Christian Ebenbauer, Rolf Findeisen, and Frank Allg¨ ower 1

Institute for Systems Theory in Engineering (IST), University of Stuttgart, Germany. {raff,ce,findeise,allgower}@ist.uni-stuttgart.de

Summary. In this paper, a moving horizon state estimation scheme is proposed. The scheme is inspired by combining the system-theoretic concept of observability maps with optimization-based estimators. As a result, a simple moving horizon state estimator scheme for nonlinear discrete time control systems is established which guarantees global convergence under certain observability conditions.

Key words: Moving horizon state estimator, optimization-based estimator, nonlinear discrete time control system, inverse problem, observability.

1 Introduction The problem of state estimation is basically an inverse problem [4], i.e., given a set of external measurements, one would like to estimate the internal states of a given control system. The inverse nature of this problem is one reason why state estimation for nonlinear control systems is still a hard and challenging problem. Despite its difficulty, state estimation or state observation has gained constantly high interest in the control literature since its advent for more than 60 years ago. Clearly, this is because state estimation is of fundamental importance in control engineering with an incredible variety of applications. Ranging from standard control applications, like output feedback design, fault detection, softsensors, to applications in computer vision, digital image processing, and even cryptography. Basically, there are two classes of state estimators in systems and control theory. The first class of state estimators is the class of estimators to which belong for example the Kalman filter and the Luenberger observer. In this traditional class, state estimators are realized in a typically system-theoretic fashion, namely as control systems where the available measurements are feed into the input of the estimator and the output delivers the estimated state. The second class of state estimators is the class of optimization-based state estimators to which belong moving horizon state estimators and optimization-based observers. In this class, state

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 67–80, 2005. © Springer-Verlag Berlin Heidelberg 2005

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estimators are realized in a typically optimization-based fashion, roughly spoken, that often means as a least square minimizer. That this two classes are related, is clearly demonstrated by the Kalman filter, which can be perfectly interpreted as an optimal recursive least square minimizer but also as a system controlled by an optimally designed estimator gain. The purpose of this paper is to study an estimation approach from the second class, namely a moving horizon estimation scheme. Moving horizon estimators gained an increasing interest over the last two decades and were explicitly addressed first in the papers [11, 23, 12]. Moving horizon estimators are optimization-based estimators, which explicitly use a set of output measurements measured over a past time horizon with a certain length. This measurement horizon or measurement window is moving forward in time. That means the estimator uses the last, lets say N , measurements to estimate the state. The use of a certain (finite) number of measurements is simply motivated by the fact of limited amount of memory and of keeping the online-optimization load tractable. In contrast to moving horizon estimators and optimization-based estimators, traditional estimators do not use a set of measurements over a certain past time horizon for estimation. Instead they use only the current measurement for state estimation. However, in this paper a moving horizon estimator for nonlinear discrete time control systems is proposed which, under certain observability conditions, guarantees global convergence of the estimated state to the actual state. This is achieved with the help of the system-theoretic concept of observability maps. The idea used in this paper is in the line with the approaches presented in [6, 13, 2] and in [11, 23, 12]. The papers [13, 2] put more emphasis on the design of traditional estimators with the help of observability maps and the papers [11, 23, 12] considered the design from a purely optimization-based point of view without an explicit use of observability maps to ensure a guaranteed feasible contraction. This paper emphasis on the use of observability maps to combine them into a moving horizon scheme. As a result, a simple moving horizon estimator scheme for nonlinear discrete time control systems is proposed which guarantees global convergence under certain observability conditions. The remainder of the paper is organized as follows: In Section 2 the problem formulation and a detailed problem description is given as well as related results on optimization-based estimation are summarized. In Section 3, the main result, namely a moving horizon estimator scheme with guaranteed convergence for nonlinear discrete time systems is proposed. Finally, Section 4 concludes with a discussion and summary.

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Notation. A function V : Rn → R is called positive definite, if V (0) = 0, V (x) > 0, ∀x ∈ Rn \ {0} and positive semidefinite if V (x) ≥ 0, ∀x ∈ Rn . 1 n The Euclidian norm of x ∈ Rn is denoted by x = ( i=1 x2i ) 2 . A function n V : R → R is called proper or radially unbounded, if V (x) → ∞, whenever x → ∞. Let R+ denote the set of positive real numbers, then K is the class of functions from R+ to R+ which are zero at zero, strictly increasing, and continuous.

2 State Estimation Problem The control system for which the states have to be estimated from output measurements is of the form x(i + 1) = f (x(i), u(i))

x0

(1)

y(i) = h(x(i), u(i)), where x ∈ Rn is the state, u ∈ Rp is the control input, y ∈ Rm is the measured output and x0 is the initial condition. For simplicity, it is assumed that the functions f, h and all admissible controls u = u(i) are appropriately defined such that the solution x of (1) is forward complete in its domain of interest, i.e., the solution exists for all time steps i and for all admissible initial conditions x0 . In a first step, no state and input constraints are considered. Later in the paper, state and input constraints will be discussed shortly (Remark 5). The reason why discrete time control systems are considered here will became clear later. However, one can always think about (1) as a sampled-data representation of a continuous time control system: x(t) ˙ = a(x(t), u(t)) y(t) = c(x(t), u(t)),

x0

(2)

where a, c are assumed to be sufficiently smooth, with a sampling interval T such that the output measurements of (1) and (2) are equal at time instances t = kT , i.e., y(k; x0 ) = y(t = kT ; x0 ), with piecewise constant control inputs over the sampling interval [(k − 1)T, kT ]. The notation y(k; xk−N ) means the output at time step i = k of (1), where (1) was initialized at time step k − N with the initial condition x0 = x(k − N ). A state estimation problem could now be formulated as follows. Given the set of N past output measurements y(k − N ; xk−N ), ..., y(k − 1; xk−N ) with x0 = x(k − N ), as well as N past control inputs u(k − N ), ..., u(k − 1). Find ˆ(k − N ) of the state x0 = x(k − N ) based on the given an estimate x ˆ0 = x information. One possible optimization-based approach to solve this problem is via a least square minimization:

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min x ˆ0

y(k − j; xk−N ) − yˆ(k − j; x ˆk−N )

j=1

2

s.t.ˆ x(i + 1) = f (ˆ x(i), u(i))

(3)

yˆ(i) = h(ˆ x(i), u(i)) x ˆ0 = xˆ(k − N ), where the optimization has to be carried out over all admissible “initial” states x ˆ0 = x ˆ(k − N ). In case of no state constraints and no a priori knowledge of the state x(k − N ), the optimization problem has to be carried out over the whole space Rn . Additionally, this type of dynamic optimization problem is usually non-convex, and hence it is very hard to obtain a global minimum, without getting stuck in a local minimum. In the following, some results from the literature concerning optimizationbased estimation and observability maps are summarized, which will be utilized in Section 3. Looking at optimization problem (3), a question arises, namely, what does it mean if the objective takes the value zero? One expects that it is reasonable to assume that in this case the states are estimated correctly, i.e., x ˆ(k − N ) = x(k − N ), ..., xˆ(k − 1) = x(k − 1). This expectation is a kind of observability assumption, which can be defined as follows [17]: Definition 1. The control system (1) is said to be uniformly observable if there exists a positive integer N and a K-function ϕ such that for any control ˆ0 and input u(0), ..., u(N − 1) and for any two admissible initial states x0 , x corresponding output measurements y(j; x0 ), yˆ(j; xˆ0 ), ϕ( x0 − xˆ0 ) ≤

N −1 j=0

y(j; x0 ) − yˆ(j; x ˆ0 )

(4)

holds. A similar idea of observability notion can be already found in [7]. Definition 1 looks quite different from traditional definitions of observability in terms of observability maps (cf. e.g. [9, 22, 15]). However, there is a close relation to traditional observability maps. Instead of formulating the state estimation problem as optimization problem, one could use the following approach. The solution of the following nonlinear equations h(ˆ x(k − N ), u(k − N )) = y(k − N ; xk−N )

h(f (ˆ x(k − N ), u(k − N )), u(k − N + 1)) = y(k − N + 1; xk−N ) x ˆ(k−N +1)

.. . h(f (ˆ x(k − 2), u(k − 2)), u(k − 2)) = y(k − 1; xk−N ), x ˆ(k−1)

(5)

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with respect to x ˆ(k − N ) gives an estimate of the state x(k − N ) and hence by forward simulation an estimate of the states x(k − N + 1), ..., x(k − 1). Indeed, f (ˆ x(k − 2), u(k − 2)) just represents a forward simulation from the initial condition x ˆ(k − N ). Notice that the given data are u(k − N ), ..., u(k − 1) as well as y(k − N ; xk−N ), ..., y(k − 1; xk−N ), hence the unknown quantity in ˆ(k − N ). Therefore, to find the solution with the nonlinear equation (5) is x respect to x ˆ(k − N ) is an inverse problem . It is convenient to write (5) in a more compact form, namely as φ(ˆ x(k − N ), ν(k, N )) = η(k, N )

(6)

with ⎤ u(k − N ) ⎥ ⎢ .. ν(k, N ) = ⎣ ⎦ . ⎡

⎤ y(k − N ; xk−N ) ⎥ ⎢ .. η(k, N ) = ⎣ ⎦ . ⎡

y(k − 1; xk−N ) ⎤ h(ˆ x(k − N ), u(k − N )) ⎢ ⎥ .. φ(ˆ x(k − N ), ν(k, N )) = ⎣ ⎦. . u(k − 1)

⎡

(7)

(8)

h(ˆ x(k − 1), u(k − 1))

One can show now that the nonlinear equation (6) has a unique solution for any admissible data u(k−N ), ..., u(k−1) and y(k−N, xk−N ), ..., y(k−1, xk−N ), if the control system (1) is observable in the sense of Definition 1. Hence, φ plays the role of an observability map, which was also utilized in [13, 7]. This desirable property of an unique solution of (6) is summarized in the following stronger assumption: Assumption 2 The map φ = φ(ξ, ν) defined in (8) is assumed to be diffeomorph, that is φ must be a proper map and the Jacobian matrix of φ, i.e., ∂φ ∂ξ , must have full column rank for any admissible control ν. First, notice that in case of N = n, Assumption 2 is nothing else then Hadamard’s characterization of a global diffeomorphism [8]. Notice also that in case of a compact domain of admissible states, properness of φ can be skipped [21]. However, in the following properness (radially unboundedness) of φ is assumed due to simplicity of explanation and for the reason of emphasis on global convergence in Section 3. Finally, one may keep in mind that Assumption 2 is nothing else then a uniform global observability assumption. Therefore, the following points should be summarized up to now: optimization problem (3) is a straight forward approach to the state estimation problem. This is underlined by Definition 1, which defines observability in terms of the distance between output measurements. On the other hand, the nonlinear equation (6) gives an inverse problem perspective on the state estimation problem which is directly connected to the notion of observability maps. This is underlined by Assumption 2, which ensures observability similar to traditional observability conditions. Whether one prefers to state observability in

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a traditional way by using observability maps and invertible (injective) maps or just as an optimization problem is basically a matter of taste. However, although both formulations have their advantages with respect to generality and utilizability, the explicit use of observability maps like φ has one advantage, namely they allow a guaranteed estimation of the state. This was exploited in [6, 13, 2] and this is also exploited in the present paper to establish global convergence in a constructive way. The idea how to solve the nonlinear equation (6) under the Assumption 2 is simple, elegant, and powerful at once, and it is basically a Lyapunov-type argument. In the following, this procedure to solve nonlinear equations of type (6) under the Assumption 2 is outlined. Basically, it goes along Lyapunov-type arguments of a possible proof of Hadamard’s global theorem on diffeomorphisms, as presented in the enlighting paper [8]. Due to reasons of readability, the arguments in (6) are skipped: φ(ˆ x, ν) = η. Notice also that the solution x ˆ = x(k − N ) of φ(ˆ x, ν) = η is an exact estimate: x(k − N ) = xˆ(k − N ). The arguments used in [8] to solve φ(ˆ x, ν) = η, with respect to x ˆ, for a given ν, η, are the following: Consider the (proper) Lyapunov function V (ξ) =

1 φ(ξ, ν) − η 2

2

(9)

with the gradient vector field Newton flow ∂φ (ξ, ν) ξ˙ = − ∂ξ

T

φ(ξ, ν) − η .

(10)

Then, by differentiation of V with respect to the vector field (10), one can easily establish convergence of ξ to x, such that φ(x, ν) = η is satisfied, by using a Lyapunov-type argument and by Assumption 2. Therefore, basically, a simple forward simulation of the vector field (10), where the solution ξ of (10) is sometimes called Newton flow solution, gives a constructive procedure to estimate xˆ for any initial guess ξ0 . The outlined procedure above is sometimes also referred to as dynamic inversion. The Newton flow as stated in (10) converges to x for t → ∞. One can, however, easily modify the vector field in such a way that convergence is archived in finite time, for example by establishing V˙ (ξ) ≤ − V (ξ). A similar type of argument was already exploited in the 50’s to solve inverse problems. In the literature one can find it often under the so called Showalter or Landweber method [5]. Furthermore, notice that a discretization of the Newton flow (10) leads to an ordinary Newton iteration as used in standard optimization techniques. This is also the reason why optimization-based estimators using this solution approach are sometimes called Newton estimators. Hence, one possible interpretation is to obtain an estimate of the state via minimizing V , which closes the chain of reasoning, as started in the beginning of this section. Therefore, a constructive and numerical feasible solution of the state estimation problem is possible, by utilizing observability maps.

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Summarizing, observability can be seen from an optimization-based point of view, i.e., by the optimization problem (3), or from a more traditional point of view, i.e., by observability maps like (8). The explicit use of an observability map allows a guaranteed estimation of the state with the help of the arguments outlined above, in particular, via the Newton flow (10). Therefore, given a set of output measurements and the corresponding control sequence, the state estimation problem can be solved in a constructive and reliable fashion under observability conditions as stated in Assumption 2. However, several issues concerning the solution of the state estimation via an inverse problem are not addressed up to now. Most importantly, the problem of measurement noise and uncertainty has to be treated. Fortunately, the problem of solving nonlinear equations is quite mature and hence there are powerful solution approaches which can handle noisy and uncertainty data. This mathematical discipline is called regularization theory, in which so called ill-posed problems [20], these are problems with have no (unique) solution or depend discontinuously on the optimization parameters, are regularized. The most famous regularization approach is Tikhonov’s regularization approach [20]. Another regularization method is the so called continuous or dynamical regularization approach [16, 19], which is based upon contraction and Lyapunov-type arguments. Notice that the Newton flow can be seen as a continuous regularization approach, in which the regularization parameter is the time t. Finally, some further remarks are necessary. First, it should be remarked that in [10] (p. 205) was stated that the Newton flow (10) is often not suitable from a numerical point of view for a simple forward simulation. However, alternative approaches exists. More information can be found in the literature and in [10] as well as references therein. Second, nothing has been said about the number of measurement samples N needed to estimate the state. There are some generically results available in the literature [1, 18] which proof that N = 2n + 1 is enough for almost all control systems. However, it is reasonable to have at least as many measurements as states, i.e., N ≥ n. Such a state estimation approach as discussed above may be appropriate for very slow dynamics and for particular offline state estimation problems by solving the nonlinear equation (6) at each sampling instance. However, in online state estimation one has to take into account the evolution of the state and to exploit the fact that new measurements become available. The incorporation of this in an optimization-based state estimator is the objective of the next section.

3 Moving Horizon State Estimation A traditional observer is usually composed of two terms. The first term is the so called simulation term, that is a copy of the control systems which takes the evolution of the control system into account. The second term is the error control term or corrector term which asymptotically controls the estimated

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state to the actual state of the control systems. In an optimization-based state estimator, this structure often does not appear any more. Nevertheless, these two tasks, error control and simulation, have to be incorporated. For example, one could solve the state estimation problem as follows. Take the first N output measurements and then, under observability assumption, the state can be estimated exactly and by a simple forward simulation future states can be determined. However, due to the presence of noise and uncertainty in the model of the control system, this strategy will not lead to a satisfactory state estimation. Hence, it is desirable to reestimate the state whenever a new output measurement becomes available. It is also needless to say that a forward simulation of equation (10) would only lead to an approximate state estimate, since a forward simulation for t → ∞ is computationally intractable. Furthermore, from a computational point of view, it may be desirable to not estimate the state exactly after the first N output measurements, but instead, estimate the state in an asymptotic fashion, like traditional estimators. This may significantly reduce the computational load. In the literature several approaches to these issues exist. The asymptotic convergence of the moving horizon estimators is usually ensured by a decreasing argument or Lyapunovtype argument [12, 23]. Often, decreasing is ensured in an ad hoc manner by solving a nonlinear (non-convex) optimization problem. Moreover, to guarantee convergence, the evolution of the state has to be taken into account. This is often realized via Lipschitz assumptions or worst case (min-max) bounds or high-gain arguments. In the following, a simple moving horizon state estimator scheme is proposed which handles the error control and simulation task in an alternative way. The observability map (8) is used to ensure that the state estimation error converges globally to zero. In particular, this is done by a decreasing argument which is always feasible due to the use of the Newton flow (10). This was also used in [13, 2], but for designing Luenberger-type estimators. The evolution of the state is taken into account by a forward simulation of the control system over the last sampling interval. All these concepts and ideas are summarized in the next theorem, which establishes a simple moving horizon state estimator scheme: Theorem 1. Given the control system (2) and its sampled-data representation (1). Under Assumption 2 with N ≥ n, the decreasing condition V (k; x ˆ(k − N )) ≤ ε · V (k − 1; x ˆ(k − N − 1)) with 0 < ε < 1, V (k − 1; x ˆ(k − N − 1)) > 0, and with

(11)

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N

V (k; x ˆ(k − N )) =

j=1

y(k − j; xk−N ) − yˆ(k − j; x ˆk−N )

s.t.ˆ x(i + 1) = f (ˆ x(i), u(i))

2

(12)

yˆ(i) = h(ˆ x(i), u(i)) x ˆ0 = xˆ(k − N ), is guaranteed feasible via numerical algorithms at any time step k ≥ N and provides a moving horizon state estimation scheme with guaranteed global convergence, i.e., x ˆ(k − N ) → x(k − N ) xˆ(k) → x(k)

(13) (14)

for k → ∞. Proof. First notice that V is a positive, radially unbounded function which is zero if and only if x ˆ(k − N ) = x(k − N ). This follows from Assumption 2. Since ε is strictly less than one, the decreasing condition (11) guarantees that V (k; x ˆ(k − N )) converges to zero. The feasibility of the decreasing condition (11) via numerical algorithms is guaranteed by the Newton flow argument as outlined in Section 2. In particular, a forward simulation of (10) at any time step k with the data y(k−N ; xk−N )...y(k−1; xk−N ),u(k−N )...u(k−1) leads to an estimate of x ˆ(k−N ) (ˆ x(k−N )...ˆ x(k−1)). And for any > 0, there the exists a (finite) simulation time tk−N , such that V (k; ξ(tk−N ; ξk−N )) < , where ξ = ξ(t; ξk−N ) is a solution of the Newton flow (10) and ξ0 = ξ(k − N ) is the initial guess for x ˆ(k − N ). A reasonable guess is to use the previous estimated x(k − N − 1), u(k − N − 1)). Hence, state x ˆ(k − N − 1), i.e., ξ0 = ξ(k − N ) = f (ˆ ξ(tk−N ; ξk−N ) is an estimate of x ˆ(k − N ) which converges to x(k − N ) for tk−N → ∞. Finally, the estimate of the state at time k is obtained recursively via x ˆ(k) = f (ˆ x(k − 1), u(k − 1)). Note that the argument of forward simulation of the Newton flow is valid in the presence of numerical errors caused by discretization, since, this discretization error can be made sufficiently small by a sufficiently small step size and by an Euler polygon approximation. Note also that in contrast to x ˆ(k − N ), which is obtained by the Newton flow argument, the state estimate x ˆ(k) at time k is obtained by a forward simulation of the control system (1). That convergence of x ˆ(k−N ) → x(k−N ) implies convergence of x ˆ(k) → x(k), follows from the fact of the continuous dependence of the solution of (2) on the initial conditions. This is satisfied, since the control system (2) is assumed to be sufficiently smooth and the forward simulation is carried out over a finite time horizon.  From a practical perspective, following remarks to Theorem 1 may be helpful to improve and to implement a practical moving horizon state estimation scheme.

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Remark 1. It might happen that V (k; ξ(k − N )) > V (k − 1; ξ(k − N − 1)) (see Fig. 1). One can avoid this effect by using a delayed measurement window. Note that up to now, at time step k only the measurements y(k − N ; xk−N ), ..., y(k − 1; xk−N ) without y(k; xk−N ) was used. This delayed measurement window can be used to guarantee V (k; ξ(k−N )) ≤ V (k−1; ξ(k−N − 1)). This follows from the fact that ξ(tk−N −1 ; ξk−N −1 ) converges to x(k − N − x(k−N −1), u(k−N −1)) 1) for tk−N −1 → ∞ and therefore also ξ(k−N ) = f (ˆ converges to x(k − N ), i.e., V (k; ξ(k − N )) converges to zero. In other words, one can use the old measurements y(k − N ; xk−N )...y(k − 1; xk−N ) and perform a forward simulation with the Newton flow until V (k; ξ(k − N )) ≤ V (k − 1; ξ(k − N − 1)) holds, where for the evaluation of V (k − 1; ξ(k − N − 1)) the new measurements y(k − N + 1; xk−N +1 )...y(k; xk−N +1 ) are used. V (k; ξ(k − N))

V

Newton flow simulation

V (k − 1; ξ(k − N − 1))

V (k − 1; ξ(tk−N −1 ; ξk−N −1 )) = V (k − 1; x ˆ(k − N − 1))

V (k; ξ(tk−N ; ξk−N )) = V (k; x ˆ(k − N))

k−1

k

k+1

k‘s

Fig. 1. Convergence behavior.

Remark 2. Although convergence of x ˆ(k) is guaranteed, oscillating behavior of the estimate xˆ(k) might occur since x ˆ(k) is obtained by a forward simulation with the control system (1), which may accumulate the estimate error of x ˆ(k −N ). In case this problem appears, one possible solution is to not perform a forward simulation with the control system but instead to perform a forward simulation with a local estimator of the control system which will be initialized by xˆ(k − N ). For example, such an local estimator may be the extended Kalman filter. Another practical problem which might occur is in case that V (k − 1; x ˆ(k − N − 1)) is already small, i.e., x ˆ(k − 1 − N ) ≈ x(k − 1 − N ). Then instead of forcing a further decrease in V , one may just keep V below a certain small upper bound, i.e., to achieve practical convergence.

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Remark 3. An immediate consequence of Remark 2 is the question how to choose this upper bound which defines a kind of stopping criteria for the contraction? This upper bound is definitively not easy to obtain. Basically, one could argue to define this upper bound in such a way, that whenever V is below this upper bound, a local estimator, e.g., an extended Kalman filter, converges when initialized by the currently available estimate x ˆ(k − N ). Unfortunately, this would assume an explicit knowledge of the region of convergence of a local estimator which is usually not available. Again, one could argue to perform a forward simulation and when the distance in the sense of an l2 -norm between the estimated output trajectory from the local estimator and the measured output trajectory is bounded, i.e., ∞ ˆ(k − N + j; x ˆk−N ) ≤ M , then by the obj=0 y(k − N + j; xk−N ) − y servability assumption, the local estimator converges. Such an idea was used for stabilization in [14]. But in estimation this cannot be realized and even not approximated, since the future measurement are not available. However, one can use the following strategy: First, perform a forward simulation from k − N − 1....k − 1 with a local estimator and then a backward simulation from k − 1....k − N − 1 with a local estimator for the reverse control system, i.e., the system with a reversed time flow (see also Fig. 2). The two distances of the output trajectories generated by a forward and backward simulation with respect to the measured output trajectory is a measure for the quality of the estimate, since the distance is zero if and only if the estimate is exact, which follows from the observability assumption. Depending on the computational available power, these forward and backward simulations can be repeated a few times and a possible convergence of the obtained distances may indicate that the estimate is good enough for the use of the local estimator. Finally assume that if the control system (1) is considered as a discretization of (2) and that a local estimator for x(t) ˙ = ν · a(x(t), u(t)) y(t) = c(x(t), u(t)),

(15)

with ν ∈ {−1, 1} as additional input, is known. Then one can conclude the following: If these forward and backward simulations are repeated infinitely often and the distances of the output trajectories generated by these forward and backward simulations with respect to the measured output trajectory converges to zero, then the initial estimate xˆ(k − N ) was in the region of convergence of the local estimator. Remark 4. There are different possible strategies in the case k < N . One can use for example only the available data and assume zero for the rest, or just wait until N output measurements are available. Note that the quick generation of enough output measurements is often no problem in practice, since the sampling rate is no the limiting factor in applications where moving horizon control strategies are used, like process industries.

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y(.; ˆ x ˆk−N )

x ˆ(k − 1)

x ˆ(1) (k − N )

(1) y ˆ(1) (.; x ˆ k−N )

x ˆ(1) (k − 1)

.......

x ˆ(2) (k − N )

forward simulation → ← backward simulation

x(k − N )

y(.; x(k − N ))

x(k − 1)

Fig. 2. Forward and backward simulations.

Remark 5. If some information about uncertainty and noise in the measurement is available, e.g., the frequency spectrum, then the output measurements can be regularized for example by a prefiltering, which can be easily incorporated in such a moving horizon state estimation framework. However, the proposed moving horizon state estimator is purely deterministic and does not explicitly take uncertainty and noise into account, as it is done, for example, in [17]. Also state constraints are not explicitly taken into account, although one strategy could be to project the estimated state in an admissible region. Remark 6. Finally, as already outlined in Section 2, the Newton flow argument was used to proof global convergence. However, from a numerical point of view, other approaches may be numerically better conditioned. Furthermore, it is reasonable to use previous estimates at time step k − 1 for initial guests at time step k and to control the speed of convergence by the parameter ε. To sum up, Theorem 1 and the above remarks provide a simple moving horizon estimator scheme. The basic ideas used in this paper are not new. Observability maps and contraction arguments were already used, explicitly or at least implicitly, in [6, 13, 12] and in [23, 12] respectively. However, Theorem 1 emphasis explicitly on the usefulness to combine observability maps and contraction arguments in a moving horizon estimator setup without going into details how to do a concrete realization of the moving horizon estimator. Furthermore, several aspects, in particular, the use of a delayed measurement window or the combination of moving horizon estimators with locally convergent estimators (Remark 1 and 2) may be exploited in a concrete moving horizon estimator.

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4 Summary and Discussion In this paper, a moving horizon state estimation scheme with global guaranteed convergence was proposed. The scheme was inspired by combining the ideas as used for example in [6, 13, 12], to integrate optimization-based estimators and observability maps into a moving horizon scheme. The global convergence was established with the help of an observability assumption and by solving the resulting inverse problem. Basically no optimization is necessary, or in other words, optimization is implicitly done by the Newton flow argument, which can be seen as a curved-line search strategy with guaranteed decreasing property. Alternatively, this forward simulation can be replaced by a regularized forward simulation with the help of a local estimator, like the extended Kalman filter. Possible extension are an explicit incorporation of disturbances, uncertainty, and constraints in the scheme. Another challenging issue is the real-time solution of the resulting inverse problem. In particular, for control systems with many states (n large), efficient numerical methods are necessary which exploit the special nature of the problem. Moreover, instead of estimation, the same approach could be used in steering and tracking by replacing the observability map by the controllability map. Further of interest is the development of a practical moving horizon estimator strategy combined with an efficient numerical implementation as well as practical applications and relaxed observability assumption, e.g., nonglobal observability assumptions. Acknowledgement. This paper is dedicated to Professor Michael Zeitz, a friend, teacher and most valued colleague, on the occasion of his 65th birthday.

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8. W. Gordon. On the diffeomorphisms of euclidian space. American Math. Monthly, 79:755–759, 1972. 9. Y. Inouye. On the observability of autonomous nonlinear systems. J. Math. Anal. Appl., 60:236–247, 1977. 10. D.H. Jacobson. Extensions of linear-quadratic control, optimization and matrix theory. Academic Press, 1977. 11. S. Jang, B. Joseph, and H. Mukai. Comparison of two approaches to on-line parameter and state estimation of nonlinear systems. Ind. Eng. Chem. Proc. Des. Dev., 25:809–814, 1986. 12. H. Michalska and D.Q. Mayne. Moving horizon observers and observer-based control. IEEE Trans. of Automatic Control, 40:995–1006, 1995. 13. P. E. Moraal and J. W. Grizzle. Observer design for nonlinear systems with discrete-time measurements. IEEE Trans. of Automatic Control, 40:395–404, 1995. 14. G. De Nicolao, L. Magni, and R. Scattolini. Stabilizing receding-horizon control of nonlinear time varying systems. IEEE Trans. of Automatic Control, 43:1030– 1036, 1998. 15. J. Nijmeijer and A. van der Schaft. Nonlinear Dynamical Control Systems. Springer, London, 1990. 16. A. G. Ramm. Dynamical systems method for solving nonlinear operator equations. Communications in Nonlinear Science and Numerical Simulation, 9:383– 402, 2004. 17. C. V. Rao, J. B. Rawlings, and D. Q. Mayne. Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations. IEEE Trans. of Automatic Control, 48:246–258, 2003. 18. J. Stark, D. S. Broomhead, M. E. Davies, and J. Huke. Takens embedding theorems for forced and stochastic systems. Nonlinear Analysis, Theory, Methods and Applications, 30:5303–5314, 1997. 19. U. Tautenhahn. On the asymptotic regularization of nonlinear ill-posed problems. Inverse Problems, 10:1405–1418, 1994. 20. A. N. Tikhonov, V. Y. Arsenin, and F. John. Solutions of ill-posed problems. John Wiley and Sons, 1977. 21. G. Zampieri. Finding domains of invertibility for smooth functions by means of attraction basins. Journal of Differential Equations, 104:11–19, 1993. 22. M. Zeitz. Observability canonical (phase-variable) form for nonlinear timevariable systems. Int. J. Systems Science, 15:949–958, 1984. 23. G. Zimmer. State observation by on-line minimization. Int. Journal of Control, 60:595–606, 1994.

Least Squares Smoothing of Nonlinear Systems Arthur J. Krener Department of Mathematics, University of California, Davis, CA, 95616, USA. [email protected]

Summary.฀ We฀consider฀the฀fixed฀interval฀smoothing฀problem฀for฀data฀from฀linear or฀nonlinear฀models฀where฀there฀is฀a฀priori฀information฀about฀the฀boundary฀values of฀the฀state฀process.฀The฀nonlinearities฀and฀boundary฀values฀preclude฀a฀stochastic approach฀so฀instead฀we฀use฀a฀least฀squares฀methodology.฀The฀resulting฀variational equations฀are฀a฀coupled฀system฀of฀ordinary฀differential฀equations฀for฀the฀state฀and costate฀involving฀boundary฀conditions.฀If฀the฀model฀is฀linear฀and฀the฀a฀priori฀information฀is฀only฀about฀the฀initial฀state฀then฀several฀authors฀have฀given฀methods฀for solving฀the฀resulting฀equations฀in฀two฀sweeps.฀If฀the฀model฀is฀linear฀but฀the฀a฀priori information฀is฀about฀both฀the฀initial฀and฀final฀states฀then฀direct฀methods฀have฀been proposed.฀If฀the฀state฀dimension฀is฀large฀these฀methods฀can฀be฀very฀expensive฀and moreover฀they฀don’t฀readily฀generalize฀to฀nonlinear฀models.฀Therefore฀we฀present฀an iterative฀method฀for฀solving฀both฀linear฀and฀nonlinear฀problems. Keywords:฀Nonlinear฀smoothing,฀boundary฀value฀processes,฀least฀squares฀smoothing.

1 Introduction We consider the problem of smoothing a boundary value process, i.e. a nonlinear system of the form x˙ = f (t, x, u) y = h(t, x, u)

(1) (2)

0 = b(x(0), x(T ))

(3)

where x(t) ∈ Rn is the unknown state of the process, u(t) ∈ Rm is the known input or control, y(t) ∈ Rp is the known output or observation and the boundary condition (3) is k dimensional. The notation x(0 : T ) denotes the curve t → x(t), for t ∈ [0, T ]. Research supported in part by NSF DMS-0204390.

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 81–92, 2005. © Springer-Verlag Berlin Heidelberg 2005

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The fixed interval smoothing problem [5] is to estimate the state x(0 : T ) from the knowledge of the functions f, h, b and the inputs and outputs u(0 : T ), y(0 : T ). This problem is called fixed interval smoothing because to estimate x(t) at any t ∈ [0, T ], we use all the inputs u(t) and outputs y(t) on the interval [0, T ]. But the above formulation is different from the standard fixed interval smoothing problem in two ways. The first is that the model is allowed to be nonlinear. The second is that the a priori partial information that we have available can be about both x(0) and x(T ). The usual approach is to assume that only a priori information is known about x(0) so the boundary conditions (3) are replaced by initial conditions, 0 = b(x(0))

(4)

Both of these differences add substantial difficulty to the smoothing problem. There are several closely related problems which we briefly describe. For a more complete discussion of these problems we refer the reader to [5]. The filtering problem is to estimate x(t) for t > 0 given the past inputs and outputs u(0 : t), y(0 : t). In filtering problems it is usually assumed a priori information is only available about x(0). The fixed point smoothing problem is to estimate x(t) given the inputs u(0 : T ) and outputs y(0 : T ) as T → ∞. Again it is usually assumed a priori information is only available about x(0). There are extensions of the above model to the situation where a priori information is available at times interior to [0, T ]. Then the boundary condition is replaced by a multipoint condition 0 = b(x(0), x(t1 ), . . . , x(tl ), x(T ))

(5)

where 0 < t1 < . . . < tl < T . For simplicity of exposition we shall not discuss such problems but the methods that we will present readily generalize. In some estimation problems, the input and/or output data may be censored. For example in fixed interval smoothing the output may be unknown on some subset [t0 , t1 ] ⊂ [0, T ]. The model can be adjusted to handle this by setting y(t) = 0 and h(t, x, u) = 0 for t ∈ [t0 , t1 ]. If the input is unknown on [t0 , t1 ] then we can treat it as a driving noise, see below. The approach that we take is an extension of Bryson and Frazier [3]. The above model may not be known exactly or there may be driving noise, observation noise or boundary noise so we don’t expect to be able to estimate x(t) exactly. To account for this, we add noises β, w(t), v(t) to the model, x˙ = f (t, x, u) + g(t, x)w y = h(t, x, u) + k(x, t)v β = b(x(0), x(T ))

(6)

We could take w(t), v(t) to be independent white Gaussian noises and β to be an independent random vector but then we would have to deal with

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the mathematical technicalities of a stochastic ODE with random boundary conditions. Not much is known about this except for linear ODEs with wellposed linear boundary conditions. In this case x(t) is a reciprocal process [8], [9] and there are well-developed linear smoothers [1], [14], [15], [4]. Instead we assume that w(t) ∈ L2 ([0, T ], Rl ), v(t) ∈ L2 ([0, T ], Rq ) are unknown functions and β is an unknown vector in Rk . The model (6) is well-posed if for every u(0 : T ), y(0 : T ), β, w(0 : T ) there exists an unique solution x(0 : T ), v(0 : T ) to (6). We do not assume that the model is well-posed. If it is not well-posed then for a given u(0 : T ), y(0 : T ) we restrict our attention to those noise triples β, w(0 : T ), v(0 : T ) for which there exists a unique x(0 : T ) satisfying (6). Such noise triples are said to be admissible. To simplify the discussion we shall restrict our attention to the case where k(x, t) = I so that v(t) = y(t) − h(x(t), t). We define an ”energy” associated with the three noises, a simple choice is 1 2

T

β Pβ +

w (t)Q(t)w(t) + v (t)R(t)v(t) dt

(7)

0

where P is positive definite and Q(t), R(t) are positive definite for any t ∈ [0, T ]. The larger P is relative to Q(t), R(t) the more we assume that the boundary condition is satisfied exactly. The larger Q(t) is relative to P, R(t) the more we assume that the dynamics is satisfied exactly. The larger R(t) is relative to P, Q(t) the more we assume that the observation is exact. One could generalize the definition (7) of the energy by allowing Q, R to depend on x, u or even more general nonlinear functions but for simplicity we do not do so. We postulate that optimal state estimate x ˆ(t), t ∈ [0, T ] is generated by the admissible noise triple of minimum energy consistent with the known functions u(0 : T ), y(0 : T ). Therefore for each known u(0 : T ), y(0 : T ), we must solve the optimal control problem of minimizing the energy (7) over all admissible noise triples. If the minimum energy is achieved by the noise triple ˆ w(0 ˆ(0 : T ) satisfies β, ˆ : T ), vˆ(0 : T ) then the optimal estimate x x ˆ˙ = f (t, x ˆ, u) + g(t, x ˆ)wˆ y = h(t, x ˆ, u) + v βˆ = b(ˆ x(0), xˆ(T )).

(8)

This is an application of the least squares approach to estimation that has been widely used since Gauss. Generally speaking for linear systems the least squares and stochastic approaches give the same estimates when P, Q(t), R(t) of the least squares approach are the inverses of the covariances of the boundary, driving and observation noises of the stochastic approach. The advantage of the least squares approach is that it readily generalizes to nonlinear systems with a priori boundary information while the stochastic approach can lead to considerable technical difficulties.

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2 The Variational Approach We apply the Pontryagin minimum principle to the above problem. This yields the first order necessary that must be satisfied by an optimal solution. For ˆ w(0 fixed, known u(0 : T ), y(0 : T ), the noise triple β, ˆ : T ), vˆ(0 : T ) that ˆ : T) minimizes the energy (7) generates the state x ˆ(0 : T ) and adjoint λ(0 trajectories that satisfy the following conditions. Define the Hamiltonian 1 H(t, λ, x, w) = λ (f (t, x, u(t)) + g(t, x)w) + w (t)Q(t)w(t) 2 1 + (y(t) − h(t, x, u(t))) R(t)(y(t) − h(t, x, u(t))). 2

(9)

Then x ˆ˙ (t) =

∂H ˆ (t, λ(t), x ˆ(t), w(t)) ˆ ∂λ

(10)

ˆ˙ λ(t) =−

∂H ˆ (t, λ(t), xˆ(t), w(t)) ˆ ∂x ˆ w(t) ˆ = argmin H(t, λ(t), x ˆ(t), w)

(11) (12)

w

ˆ λ(0) =

∂b (ˆ x(0), x ˆ(T )) ∂x0

P b(ˆ x(0), xˆ(T ))

(13)

ˆ )= λ(T

∂b (ˆ x(0), x ˆ(T )) ∂xT

P b(ˆ x(0), x ˆ(T ))

(14)

Because of the form of the Hamiltonian we have that ˆ w(t) ˆ = −Q−1 (t)g (t, x ˆ(t))λ(t)

(15)

and so ˆ x ˆ˙ (t) = f (t, x ˆ(t), u(t)) − g(t, x ˆ(t))Q−1 (t)g (t, xˆ(t))λ(t) ∂f ∂g ˆ ˆ˙ (t, xˆ(t), u(t)) − (t, xˆ(t))Q−1 (t)g (t, x ˆ(t))λ(t)) λ(t) =− ∂x ∂x ∂h + (t, x ˆ(t), u(t)) R(t) (y(t) − h(t, xˆ(t), u(t))) ∂x ∂b ˆ λ(0) = (ˆ x(0), x ˆ(T )) P b(ˆ x(0), x ˆ(T )) ∂x0 ∂b ˆ )= λ(T (ˆ x(0), x ˆ(T )) P b(ˆ x(0), x ˆ(T )). ∂xT

ˆ λ(t) (16)

This is a nonlinear two point boundary value problem in 2n variables that can be solved by direct methods, shooting methods or iterative methods [6], [7], [12]. Below we shall introduce iterative methods that takes advantage of the variational nature of the problem.

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3 Smoothing of Linear Boundary Value Processes Let’s look at the linear case where the noisy system takes the form x˙ = F (t)x + B(t)u + G(t)w y = H(t)x + D(t)u + v β = V 0 x(0) + V T x(T )

(17)

then the two point boundary value problem becomes x ˆ˙ (t) = ˆ˙ λ(t) = ˆ λ(0) = ˆ )= λ(T

ˆ F (t)ˆ x(t) + B(t)u(t) − G(t)Q−1 (t)G (t)λ(t) ˆ + H (t)R(t) (y(t) − H(t)ˆ −F (t)λ(t) x(t) − D(t)u(t)) (V 0 ) P V 0 x ˆ(0) + (V 0 ) P V T x ˆ(T ) T 0 T T ˆ(0) + (V ) P V x ˆ(T ). (V ) P V x

(18)

We define x ˆ ξ= ˆ λ μ= A(t) = B(t) =

u y F (t) −G(t)Q−1 (t)G (t) −H (t)R(t)H(t) −F (t)

B(t) 0 −H (t)R(t)D(t) H (t)R(t)

C(t) = I 0 V0 =

(V 0 ) P V 0 −I (V T ) P V 0 0

VT =

(V T ) P V 0 0 (V T ) P V T −I

then the two point boundary value problem becomes ξ˙ = A(t)ξ + B(t)μ

0 = V ξ(0) + V ξ(T ) 0

T

(19) (20)

This problem is well-posed if F = V 0 + V T Φ(T, 0) is invertible where Φ(t, s) is the 2n × 2n matrix solution to ∂Φ (t, s) = A(t)Φ(t, s) ∂t Φ(s, s) = I.

(21) (22)

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Then the Green’s matrix is Φ(t, 0)F −1 V 0 Φ(0, s) t > s −Φ(t, 0)F −1 V T Φ(T, s) t < s

G(t, s) = and

(23)

T

ξ(t) = 0

G(t, s)B(s)μ(s) ds

so the least squares smoothed estimate is T

x ˆ(t) = 0

C(t)G(t, s)B(s)μ(s) ds.

We say that F (t), G(t) is controllable on [0, T ] if the gramian T

Φ(T, s)G(s)G (s)Φ (T, s) ds

0

is positive definite. The pair H(t), F (t) is observable on [0, T ] if the gramian T

Φ (t, 0)H (t)H(t)Φ(t, 0) ds

0

is positive definite It can be shown that if F (t), G(t) is controllable on [0, T ], H(t), F (t) is observable on [0, T ] and the eigenvalues of Q(t), R(t) are bounded away from zero then there exists a unique solution to the optimization problem and this solution must satisfy (18) so it is well-posed.

4 Solution of the Linear Two Point Boundary Value Problem by Direct Methods The most direct way of solving the two point boundary value problem is to compute the Green’s matrix (23). This requires finding the fundamental matrix which satisfies an 2n × 2n linear differential equation (21). If n is large then this is a nontrivial task. Moreover since the dynamics is Hamiltonian (18) we expect it to have modes that are unstable in forward time and also modes that are unstable in backward time. This is a difficulty for any numerical scheme Several authors [13], [1], [11], [2], [14] have proposed other direct methods based on diagonalization or triangularization of the dynamics (18). We shall describe the triangularization method due to Weinert [14]. Our use of notation is different from his. He assumes that the input u(t) = 0. The first step is to make a change of variables in x, λ space so as to triangularize the dynamics (18). Let N (t) be the n×n matrix that satisfies the backward Riccati equation

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87

N˙ = −N F − F N + N GQ−1 G N − H RH

N (T ) = 0 Define

ρ(t) = λ(t) − N (t)x(t)

(24)

then the dynamics is triangular F − GQ−1 G N x˙ −GQ−1 G = ρ˙ 0 −F + GQ−1 G N

x 0 + y. ρ HR

(25)

The boundary conditions become V0

I 0 N (0) I

x(0) x(T ) 0 + VT = . 0 ρ(0) ρ(T )

(26)

Moreover Weinert asserts that the dynamics of x is stable in the forward direction when ρ = 0 and the dynamics of ρ is stable in the backward direction. If there is only a priori data about the initial condition (V 0 = I, V T = 0) then V0 =

P −I 0 0

VT =

0 0 0 −I

and the boundary conditions also triangularize (P − N (0))x(0) − ρ(0) = 0 ρ(T ) = 0.

(27)

We can find the solution of (25) and (27) by first integrating the differential equation for ρ backward from the final condition ρ(T ) = 0 and then integrating the differential equation for x forward from the initial condition (P − N (0))x(0) − ρ(0) = 0. But if there is a priori data about both the initial and final conditions then the boundary conditions do not triangularize. We can still find a particular solution by first integrating the lower half of (25) backward from the final condition ρ(T ) = 0 and then integrating the upper half forward from the initial condition x(0) = 0. Denote this solution by x ¯(t), ρ¯(t), it will probably not satisfy the correct boundary conditions (26). Therefore we need to compute the 2n × 2n matrix solution of d F − GQ−1 G N −GQ−1 G Σ(t) = Σ(t) 0 −F + GQ−1 G N dt Σ(0) = I.

(28)

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This can be reduced to the solution of two n × n matrix differential equations. Let Ψ (t, s) be the solution of ∂ Ψ (t, s) = F − GQ−1 G N Ψ (t, s) ∂t

(29)

Ψ (s, s) = I and let M (t) be the solution of the Lyapunov equation M˙ = F − GQ−1 G N M + M F − GQ−1 G N

− GQ−1 G

(30)

M (0) = 0 then Σ(t) =

Ψ (t, 0) M (t)Ψ (T, t) . 0 Ψ (T, t)

(31)

The solution to (25) that we seek is of the form x(t) x¯(t) = + Σ(t)ξ ρ(t) ρ¯(t)

(32)

where ξ is determined by the boundary conditions (26). Weinert’s method is more efficient than the direct computation of the Green’s matrix (23). The latter requires finding the fundamental solution of a 2n × 2n linear differential equation. Weinert’s method requires finding the fundamental solution of a n × n linear differential equation and the n × n solutions of a Riccati differential equation and a Lyapunov differential equation. But neither method is practical if n is large.

5 Solution of the Linear Smoothing Problem by an Iterative Method We present an iterative method to solve the linear smoothing problem and generalize it to the nonlinear problem in the next section. We are given the input u(0 : T ) and observation y(0 : T ) and we wish to find the solution of (17) that minimizes the least square criterion (7). We do not assume that the boundary value problem (17) is well-posed so for given β and w(0 : T ) there may not be a solution and/or it may not be unique. Therefore to parametrize the solutions of (17) we must choose a well-posed boundary condition like x(0) = x0 . We fix the observations y(0 : t), then given x0 and w(0 : T ) there is a unique solution x(0 : T ), v(0 : T ) to

Least Squares Smoothing of Nonlinear Systems

x˙ = F (t)x + B(t)u + G(t)w y = H(t)x + D(t)u + v x(0) = x0

89

(33)

to which we associate the cost π(x0 , w) =

1 2

T

β Pβ +

w (t)Q(t)w(t) + v (t)R(t)v(t) dt

(34)

0

We compute the first variation in π due to variations in x0 , w(0 : T ), δπ = π(x0 + δx0 , w + δw) − π(x0 , w) = β P V 0 + V T Φ(T, 0) δx0 T

−

v (s)R(s)H(s)Φ(s, 0) ds δx0

0 T

+β P V T

Φ(T, s)G(s)δw(s) ds 0

T

+

w (s)Q(s)δw(s) ds

0 T

−

0

T

v (t)R(t)H(t)Φ(t, s)dtG(s)δw(s) ds

s 0

+O(δx , δw)2 where Φ(t, s) is the n × n matrix solution of ∂ Φ(t, s) = F (t)Φ(t, s) ∂t Φ(s, s) = I. If n is large it is expensive to compute Φ(t, s) so instead we define μ (s) = β P V T Φ(T, s) −

T

v (t)R(t)H(t)Φ(t, 0) dt.

s

Then μ(s) satisfies the final value problem d μ(s) = −F (s)μ(s) + H (s)R(s)v(s) ds μ(T ) = (V ) P β T

and the first variation is π(x0 + δx0 , w + δw) = π(x0 , w) + β P V 0 + μ (0) δx0 T

+

(w (s)Q(s) + μ (s)G(s)) δw(s) ds

0

+O(δx0 , δw)2 .

(35)

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We solve the problem by gradient descent. Given u(0 : T ) and y(0 : T ), choose any x0 and w(0 : T ), solve the initial value problem (33) and compute the cost (34). Then compute the gradient of the cost with respect to x0 , w(0 : T ) by solving the final value problem (35). Choose a step size and define δx0 = − (V 0 ) P β + μ(0) δw(s) = − (Q(s)w(s) + G (s)μ(s)) .

(36)

Replace x0 , w(0 : T ) by x0 + δx0 , w(0 : T ) + δw(0 : T ), solve the new initial value problem (33) and compute the new cost (34). If the new cost is not sufficiently less than the previous cost then we change the step size and recompute (33) and (34). We repeat until we find a variation that reduces the cost sufficiently. Standard step size rules such as Armijo’s can be used [10]. Once we find a variation that sufficiently lowers the cost, we accept it. Then we again compute the gradient of the cost with respect to x0 , w(0 : T ) by solving the final value problem (35) and repeat until it is too difficult to find a variation that reduces the cost any further. The last solution to the initial value problem (33) is the smoothed estimate x ˆ(0 : T ).

6 Solution of the Nonlinear Smoothing Problem by an Iterative Method We return to the nonlinear smoothing problem and seek to minimize (7) subject to (6). Again we use gradient descent. Given u(0 : T ) and y(0 : T ) choose any x0 and w(0 : T ), solve the initial value problem, x˙ = f (t, x, u) + g(t, x(t))w

(37) (38)

x(0) = x

0

and compute the cost (34). Then compute the gradient of the cost with respect to x0 , w(0 : T ) by solving the final value problem (35) where ∂f (t, x(t), u(t)) ∂x G(t) = g(t, x(t)) ∂h H(t) = (t, x(t), u(t)) ∂x F (t) =

We continue as in the linear case. Choose a step size δx0 = −

and define

(V 0 ) P β + μ(0)

δw(s) = − (Q(s)w(s) + G (s)μ(s)) .

(39) (40)

Replace x0 , w(0 : T ) by x0 + δx0 , w(0 : T ) + δw(0 : T ) and solve the new initial value problem (37) and compute the new cost (34). If the new cost is

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not sufficiently less than the previous cost then we change the step size and recompute (37) and (34). We repeat until we find a variation that reduces the cost sufficiently. Once we find a variation that sufficiently lowers the cost, we accept it. Then we again compute the gradient of the cost with respect to x0 , w(0 : T ) by solving the final value problem (35) and repeat until it is too difficult to find a variation that reduces the cost any further. The last solution to the initial value problem (37) is the smoothed estimate x ˆ(0 : T ).

7 Conclusion We have discussed the problem of obtaining smoothed estimates of a nonlinear process from continuous observations and boundary information which we call boundary value processes. Because of technical difficulties associated with the stochastic approach, we have opted for a least squares approach. The latter is applicable even when the description of the boundary value process is not well-posed. We reviewed an existing direct method for smoothing linear boundary value processes and presented new iterative methods for smoothing both linear and nonlinear boundary value process. For low dimensional linear processes direct methods can be used but for high dimensional and/or nonlinear processes the iterative method is preferable. Acknowledgement. It is a pleasure and honor to dedicate this paper to my esteemed colleague, Michael Zeitz on the occasion of his sixty fifth birthday.

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7. Keller H B (1976) Numerical Solution of two point boundary value problems. SIAM, Philadelphia 8. Krener A J (1986) Reciprocal processes and the stochastic realization problem for acausal systems. In: Byrnes C B, Lindquist A (eds) Modeling, identification and robust control. Elsevier Science. 9. Krener A J, Frezza R, Levy B C (1991) Gaussian reciprocal processes and selfadjoint stochastic differential equations of second order. Stochastics 34:29-56. 10. McCormick G P (1983) Nonlinear programming, theory, algorithms and applications. Wiley Interscience, New York 11. Nikoukhah R, Adams M B ,Willsky A S, Levy B C (1989) Estimation for boundary value descriptor systems. Circuits, Systems and Signal Processing 8:25–48 12. Roberts S M, Shipman J S (1972) Two-point boundary value problems: shooting methods. Blaisdell, Waltham, MA. 13. Weinert H L, Desai U (1981) On complementary models and fixed interval smoothing. IEEE Transactions on Automatic Control 26:863–867 14. Weinert H L (1991) A note on efficient smoothing for boundary value models. Int. J. Control, 53:503–507 15. Weinert H L (2001) Fixed interval smoothing for state space models. Kluwer Academic Publishers, Norwell MA

State฀Estimation฀of฀a฀Molten฀Carbonate฀Fuel Cell฀by฀an฀Extended฀Kalman฀Filter Michael฀Mangold1 ,฀Markus฀Gr¨ otsch1 ,฀Min฀Sheng1 ,฀and฀Achim฀Kienle1,2 1฀

Max-Planck-Institut฀f¨ ur฀Dynamik฀komplexer฀technischer฀Systeme,฀Sandtorstraße 1,฀39106฀Magdeburg,฀Germany. {mangold,groetsch,sheng,kienle}@mpi-magdeburg.mpg.de 2฀ Otto-von-Guericke-Universit¨ at฀Magdeburg,฀Lehrstuhl฀f¨ ur Automatisierungstechnik฀/฀Modellbildung,฀Universit¨ atsplatz฀2,฀39106 Magdeburg,฀Germany. Summary.฀ Industrial฀ fuel฀cell฀ stacks฀ only฀ provide฀very฀limited฀ measurement฀ information.฀To฀overcome฀this฀deficit,฀a฀state฀estimator฀for฀a฀molten฀carbonate฀fuel cell฀is฀developed฀in฀this฀contribution.฀The฀starting฀point฀of฀the฀work฀is฀a฀rigorous spatially฀distributed฀model฀of฀the฀system.฀Due฀to฀its฀complexity,฀this฀model฀is฀hardly suitable฀for฀the฀design฀of฀a฀state฀estimator.฀Therefore,฀a฀reduced฀model฀is฀derived฀by using฀a฀Galerkin฀method฀and฀the฀Karhunen฀Lo`eve฀decomposition฀technique.฀A฀low order฀system฀of฀ordinary฀differential฀equations฀and฀algebraic฀equations฀results.฀The reduced฀model฀is฀used฀to฀study฀the฀observability฀of฀the฀system฀for฀different฀sensor configurations.฀ An฀extended฀Kalman฀ filter฀ with฀a฀continuous฀time฀simulator฀ part and฀a฀discrete฀time฀corrector฀part฀is฀designed฀on฀the฀basis฀of฀the฀reduced฀model. The฀filter฀is฀tested฀in฀simulations.

Keywords:฀Fuel฀cell,฀MCFC,฀partial฀differential฀equations,฀model฀reduction, Karhunen฀Lo`eve฀decomposition,฀state฀estimation,฀Kalman฀filter.

1฀Introduction Fuel฀cells฀are฀a฀promising฀technology฀for฀the฀efficient฀generation฀of฀electrical energy฀[3].฀They฀are฀able฀to฀convert฀chemical฀energy฀directly฀into฀electrical energy,฀avoiding฀an฀intermediate฀step฀of฀producing฀mechanical฀energy.฀Therefore,฀the฀electrical฀efficiency฀of฀fuel฀cells฀is฀considerably฀higher฀than฀that฀of most฀conventional฀processes฀for฀electrical฀power฀generation. For฀stationary฀applications,฀high฀temperature฀fuel฀cells฀like฀solid฀oxide฀fuel cells฀ or฀ molten฀ carbonate฀ fuel฀ cells฀ are฀ most฀ suitable.฀ These฀ cell฀ types฀ are very฀flexible฀with฀respect฀to฀the฀fuel.฀They฀can฀be฀operated฀with฀most฀kinds of฀gaseous฀hydrocarbons฀like฀natural฀gas,฀bio฀gas฀from฀fermentation,฀waste gas฀from฀chemical฀processes,฀or฀coal฀gas.฀Due฀to฀their฀high฀operation฀temperature,฀they฀are฀very฀attractive฀for฀the฀co-generation฀of฀electricity฀and฀heat.

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 93–109, 2005. © Springer-Verlag Berlin Heidelberg 2005

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However, it requires further research efforts to make high temperature fuel cells economically competitive. A better process understanding and an improved process operation can help to exploit the potential of the cells to a higher degree than is done today. Process operation is challenging, because industrial high temperature fuel cell stacks are highly integrated processes, which combine the internal generation of hydrogen by steam reforming, the mass coupling between anode and cathode gas channels, and the heat integration of endothermic reforming reactions and exothermic electrochemical reactions in one apparatus. The process operation is further complicated by the limited measurement information available online. Currently, most high temperature fuel cell stacks are operated manually, based on experience and heuristic knowledge. This approach requires large safety factors and therefore is unsatisfactory. The purpose of this work is to support the process operation of high temperature fuel cells by developing a model based measuring system. As an example, the molten carbonate fuel cell (MCFC) stack HotModule by MTU CFC Solutions [2] is considered. A prototype of this industrial cell stack has been installed at the Magdeburg university hospital. The modelling, optimisation, and control of the stack is subject of a joint research project carried out by researchers at the University of Magdeburg, the Max Planck Institute Magdeburg, and the University of Bayreuth in cooperation with industrial partners from IPF Heizkraftwerk Betriebsgesellschaft mbH. The results reported here are a part of this project. Section 2 presents a detailed first principle model of the MCFC and a reduced model for process control purposes. Section 3 discusses the design of an extended Kalman filter based on the reduced model. The filter is tested in simulation studies.

2 Modelling of the MCFC In the following section, the working principle of the MCFC is introduced briefly. Section 2.2 outlines the key features of a detailed physical model used as a starting point for this work. In Section 2.3, a reduced model of low order is derived from the detailed model. 2.1 Working Principle of the MCFC In the following, a MCFC system is considered whose scheme is shown in Figure 1 (a). The anode feed to the system consists of methane and steam. Before entering the anode gas channels, the feed goes through a pre-reformer. In the pre-reformer, the methane steam reforming reaction and the water-gas shift reaction take place. The hydrogen and carbon monoxide generated by the pre-reformer are oxidised at the anode of the fuel cell. The gas flow that

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leaves the anode channels enters a burner, where unconverted methane, hydrogen, and CO are oxidised to CO2 and water. The outlet of the burner is connected to the inlet of the cathode gas channels. The cathode gas channels run perpendicular to the anode gas channels establishing a cross-flow in the fuel cell. The cathode exhaust is partly recycled to the burner. Figure 1 (b) shows the internal structure of the fuel cell in more detail. Because the anode gas channels are coated with reforming catalyst, an internal reforming step can produce additional hydrogen and CO. Both react electrochemically at the anode, consuming carbonate ionsat the anode and releasing free electrons. The carbonate ions are generated in the cathodic reaction, where oxygen and CO2 react under consumption of two free electrons. The free electrons produced at the anode are transported to the cathode by an external electrical circuit, performing electrical work at an external load. The electrical circuit is closed internally by the migration of carbonate ions from the cathode to the anode through the electrolyte. The cell stack of the HotModule consists of several hundred cells. However, if all the cells in the cell stack behave in a similar fashion, the single cell shown in Figure 1 (a) can be considered as an average cell representative for the whole stack, and the model of the depicted MCFC system can serve as a model for the HotModule. 2.2 Detailed Model In [7, 8] a detailed model was developed for the described MCFC system. This model serves as a reference and as a basis for the state estimator presented in this contribution. The main assumptions of the model are: " " " " " " "

"

The gas phases are isobaric and behave like ideal gases. In all channels, plug flow is assumed. In the fuel cell, spatial gradients in the direction of the ζ1 and the ζ2 axis, i.e. in the direction of the anodic and cathodic gas fluxes are taken into account. A pseudo-homogeneous energy balance is formulated for the solid parts of the cell, assuming an identical temperature in the anode, the electrolyte and the cathode; Reversible potential kinetics are used to describe the internal reforming in the cell; for the electrochemical reactions, Butler-Volmer kinetics are used. The electrodes possess infinite electrical conductivity. Charge transport in the electrolyte occurs only perpendicular to the ζ1 and ζ2 axis; the electrolyte is modelled as a ohmic resistance, the concentration of carbonate ions is assumed to be constant; a spatially distributed potential field and a spatially distributed current density field are computed. The pre-reformer has a negligible hold-up; all reactions in the pre-reformer occur instantaneously; CO is converted completely to CO2 ; a constant rate of conversion for CH4 is assumed.

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The burner has a negligible hold-up and works with total conversion.

Under the met assumptions, one obtains from the component mass balances, the total mass balance, and the energy balance for the anode gas channels the following set of hyperbolic partial differential equations: ∂xA ∂xA i = −γ A ϑA i + σiA ∂τ ∂ζ1 ∂ 0=− γ A ϑA + σγA ∂ζ1 ∂ϑA ∂ϑA V A cA = −γ A ϑA cA + σϑA P P ∂τ ∂ζ1 VA

(1) (2) (3)

In the above equations, the superscript A stands for the anode, i is a component index for the seven gas components CH4 , H2 O, H2 , CO, CO2 , O2 , N2 ; A is a dimensionless molar flux density, ϑA is a xA i are the molar fractions, γ A dimensionless temperature, σi , σϑA , σγA are nonlinear source terms caused by the chemical reactions in the gas phase as well as by heat and mass transfer; the other variables are constant model parameters. An analogous set of equations describes the cathode gas channels. An energy balance of the solid phase leads to the following parabolic partial differential equation for the dimensionless solid temperature ϑS : cSP

1 ∂ 2 ϑS ∂ϑS 1 ∂ 2 ϑS = + + σϑS ∂τ P eS1 ∂ζ12 P eS2 ∂ζ22

(4)

The nonlinear source term σϑS is due to heat transfer and due to the exothermic electrochemical reactions, cSP , P eS1 , and P eS2 are constants. The model is completed by ordinary differential equations for the electric potentials and by algebraic equations for the reactions kinetics, the mass and energy balances in the pre-reformer and the burner [7, 8]. In total, the described spatially distributed model consists of 17 hyperbolic and parabolic partial differential equations, 2 ordinary differential equations in space, 8 ordinary differential equations in time, and additional algebraic equations. Its numerical solution by the method of lines is quite expensive. The model is discretised on finite volumes with a fixed equidistant grid. The resulting discretised system consists of 1901 differential equations and 2858 algebraic equations. The simulation of 40 h real time in Figure 2 requires 23 900 s of CPU time on a PC. Furthermore, the design of a state estimator based directly on the detailed model is very difficult. Therefore, a reduced model of low order is desirable that approximates the detailed model with a reasonable accuracy but is more suitable for process control purposes. Such a reduced model will be derived in the next section.

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2.3 Reduced Model The model reduction is done in two steps. A first slight simplification of the detailed model is achieved by reducing the number of dynamic degrees of freedom. The time constants of the electric potential equations and the time constants in the gas phases are much smaller than the time constant of the energy balance of the solid. As the slow dynamics of the solid temperature dominates many process control problems, the electric potential fields as well as the anode and cathode gas channels can be considered as quasi-stationary. However, the use of a dynamic equation for the cathode gas temperature turns out to be advantageous for the numerical solution. The reason is that a change of the cell current causes a jump of the cathode gas temperature if quasi-stationarity is assumed. A much stronger reduction of the system order is possible if the spatially distributed model can be approximated by a lumped system. For such a model reduction of parabolic partial differential equations, orthogonal projection methods have become a frequently used technique [1, 9]. One of these methods, the Karhunen-Lo`eve-Galerkin procedure, is used in this work. A detailed description of the method was given in [12] for a simpler fuel cell model. The procedure is briefly illustrated here for the example of the solid temperature equation (4). The basic idea of the projection method is to approximate the unknown variable ϑS by a finite sum of products ϑ˜S : ϑ˜S (ζ1 , ζ2 , τ ) =

N

ϑSi (τ )ϕi (ζ1 , ζ2 ).

(5)

i=1

The functions ϕi in (5) are space dependent orthogonal basis functions, the ϑSi are time dependent amplitude functions. The approximative solution ϑ˜S will not solve equation (4) exactly, but a nonzero residual Res := cSP

1 ∂ 2 ϑ˜S 1 ∂ 2 ϑ˜S ∂ ϑ˜S − − − σϑS ∂τ P eS1 ∂ζ12 P eS2 ∂ζ22

(6)

will remain. Galerkin’s method of weighted residuals [4] requires that this residual must vanish if weighted by a basis function, i.e. !

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(7)

where Ω is the space domain of the system. Inserting (5) into (7) leads to N ordinary differential equations for the time dependent amplitude functions ϑSi (τ ). In order to obtain a reduced model of the MCFC system, not only the solid temperature profile has to be approximated by a set of basis functions, but also the space profiles of the gas temperatures, the total molar flow rates, the gas compositions, and the electric potential fields. The resulting reduced model is a differential algebraic system with a differential index of one that has the following structure:

Kalman Filter for a Molten Carbonate Fuel Cell

dϑS = f S ϑS , ϑA , ϑC , xi , γ A , γ C , ΔφA , ΔφC dt dϑC = f C ϑS , ϑA , ϑC , xi , γ A , γ C , ΔφA , ΔφC dt 0 = g ϑS , ϑA , ϑC , xi , γ A , γ C , ΔφA , ΔφC

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(8) (9) (10)

In (8)-(10), ϑS = (ϑS1 , ϑS2 , . . . , ϑSN ) is the vector of the amplitude functions for the solid temperature, ϑA , ϑC , xi , γ A , γ C , ΔφA , ΔφC are corresponding vectors of the time dependent amplitude functions for the anode and cathode gas temperatures, the gas compositions, the anode and cathode molar flux densities, and the electrical potential fields at the anode and at the cathode, respectively. The accuracy of the reduced model, i.e. its deviation from the detailed model, mainly depends on two factors. The first one is the number of terms considered in the series approximations. The second is the choice of the basis functions. Obviously, the best approximation with a low order model is achievable, if basis functions are used that are tailored to the specific problem. The Karhunen-Lo`eve (KL) decomposition method allows to compute such problem-specific basis functions numerically [10, 14]. The KL decomposition method uses numerical simulation results at discrete time points, socalled snapshots, obtained from dynamic simulations with the detailed model. The basis functions are taken as linear superpositions of these snapshots in the following form: T

ϕi (ζ1 , ζ2 ) = j=1

αij · vj (ζ1 , ζ2 ),

i = 1, . . . , N,

(11)

where vj is the snapshot taken at the jth time point, T denotes the total number of time points, and αij are unknown weighting factors. The basis functions are required to be orthogonal and to minimise the time averaged quadratic deviation from the snapshots. The weighting factors αij can then be determined by solving an eigenvalue problem [14]. The corresponding eigenvalues can be interpreted as a measure of how well a basis function is able to approximate the time average of the snapshots. In this sense, the basis function belonging to the largest eigenvalue can be considered as the most typical structure of the snapshots. In order to determine suitable basis functions for the MCFC model, the response of the detailed model to an increase of the cell current and to a subsequent decrease to the original value is computed numerically. The necessary number of basis functions for each variable is found by comparing the simulation results of the reduced model with the results of the detailed model for this test simulation. Good results are obtained when using 8 basis functions for the temperatures, 10 basis functions for the total anode flow rate and the anode potential, 9 basis functions for cathode flow rate and the cathode potential, and between 2 and 10 basis functions for the molar fractions of the gas

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components. The resulting reduced model consists of 16 ordinary differential equations and 115 algebraic equations, thereby reducing the dynamic order of the system by a factor of more than 100 compared to the detailed model. The KL decomposition generates the basis functions not in the form of analytical expressions, but only as numerical data arrays. Therefore, the evaluation of the right-hand sides of (8)-(10) requires a numerical quadrature. However, it should be noted that the reduced model still depends on the physical and operation parameters of the detailed model, i.e. a change of a physical parameter value directly affects the behaviour of the reduced model. Simulations show that the extrapolation qualities of the reduced model with respect to various model parameters are quite good [12]. As a validation experiment for the reduced model the cell current is varied randomly. Figure 2 shows a comparison between the reduced and the detailed model for this case. The deviation of the reduced model from the detailed model with respect to the temperatures is always below 1 %. The cell voltages computed by both models agree very well at most times. The only exception are instants immediately after a change of the cell current. Because the electric potential fields are considered as quasi-stationary in the reduced model, but as dynamic in the reference model, a jump of the cell current causes a jump of the cell voltage in the reduced model, but a continuous variation of the cell voltage in the reference model. This causes a difference of the cell voltage of up to 5 % at the moment of a discontinuous change of the cell current. However, as the potential dynamics are extremely fast, this mismatch vanishes after a very short time span and is of no importance for the applications. The CPU time on a PC is for this simulation 190 s in the case of the reduced model, compared to 23 900 s in the case of the detailed model. The reliability of the reduced model was confirmed by further simulations, where the anode feed composition was varied. From these tests it is concluded that the reduced model describes the behaviour of the MCFC system with reasonable accuracy and can be used as a basis for a state estimator.

3 Development of an Extended Kalman Filter The development of a state estimator for the MCFC starts with observability studies for different measurement configurations in Section 3.1. The design of a Kalman filter is described in Section 3.2. Section 3.3 presents some simulation results obtained with the developed Kalman filter. 3.1 Investigation of Observability In the HotModule, a certain amount of on-line measurement information is available: Apart from electrical quantities like the total cell voltage, various temperature and concentration measurements are made. The temperatures of

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Fig. 2. Validation of the reduced model by a random variation of the cell current; top diagram: dimensionless cell current used as input signal; middle diagram: maximum deviation in % of the reduced from the detailed model with respect to the solid temperature (solid line), the anode temperature (dashed line), and the cathode temperature (dotted line); bottom diagram: deviation in % of the reduced from the detailed model with respect to the cell voltage. All diagrams use a dimensionless time coordinate, 1 dimensionless time unit corresponding to about 6 s of real time.

the gases at the outlet of several anode gas channels are taken. However, as the anode gases are mixed strongly between the anode outlet and the burner, these temperature measurements can only give information on a middle outlet temperature averaged over all anode gas channels. Further temperature measurements are made at the outlets of several cathode gas channels. As the mixing is not that strong at the cathode outlet, those measurements also can give information on the spatial temperature profile at the cathode outlet in ζ1 direction. Some additional temperature sensors are placed inside the cell stack and can give information on the local cell temperature. They are extremely sensitive and are intended only for use during the very first start-up of the HotModule. After a longer operation time, most of these sensors break down, and a direct solid temperature measurement is no longer possible. Finally, the

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compositions of the anode exhaust gases and of the cathode exhaust gases are analysed by a gas chromatograph in intervals of about one hour. In order to investigate the observability of the systems, three scenarios of different sensor configurations are considered: Scenario 1: The average outlet temperature of the anode gases, the outlet temperature of three cathode gas channels, and the cell voltage are measured. Scenario 2: In addition to the measurement information given in Scenario 1, the temperature at one point in middle of the solid is measured. Scenario 3: In addition to the measurement information given in Scenario 1, the average gas composition at the outlet of the anode gas channels and at the outlet of the cathode gas channels is measured. For the three scenarios, the local observability is tested on the basis of a linearisation of the reduced model around an operation point. Algebraic variables are eliminated by solving the linearised algebraic equations and a linear time invariant dynamic system of the form x˙ = Ax,

y = Cx

(12)

with a state vector x ∈ R and an output vector y is obtained. The following observability tests are made: n

"

"

The classical Kalman criterion is used, which requires the observability matrix M := [C, CA, . . . , CAn−1 ] to have full rank. This criterion only allows a yes or no decision about observability, but does not give a quantitative statement about how well a certain state can be observed. The criterion by Hautus [6, 11] defines the following observability measure for a mode i belonging to an eigenvalue λi : κi :=

"

v ∗i C T Cv i . v ∗i v i

(13)

In (13), v i is the corresponding eigenvector. The value of κi ranges from 0 for an unobservable mode up to 1 for a directly measurable mode. The drawback of the Hautus criterion is that modes have no direct physical meaning but are linear combinations of the physical states. The convergence of a continuous Kalman filter is checked in simulations, assuming that the observed system behaves exactly like the linear system (12), that the measurement is continuous in time, and that there are no measurement errors and no measurement noise. Obviously, the Kalman filter will converge in the case of an observable system. The purpose of this test is to obtain information on the rate of convergence for the different variables, i.e. to assess how well observable the different states are.

For the linearised MCFC model, the Kalman criterion indicates a full rank of M and hence full observability for all three scenarios. By using the Hautus criterion, it is found that the modes closely connected to the amplitude

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functions of the cathode gas temperatures are well observable, whereas the observability of the modes connected to the amplitude functions of the solid temperature is worse. This result is confirmed by the simulations with the linear Kalman filter shown in Figure 3. All amplitude functions of the cathode temperatures converge quickly for all three scenarios. The convergence rate is also quite good for the leading amplitude functions of the solid temperature, but deteriorates for higher amplitude functions, especially in the case of scenario 1. The convergence behaviour improves for scenario 2, and becomes even better in the case of scenario 3 (see diagrams in the last row of Figure 3). However, it turns out that a poor estimate of the higher amplitude functions only has a minor effect on the estimate of the actual temperature profiles and the other physical states of the system. Considering the technical problems of measuring the solid temperature and the efforts needed to take concentration measurements, it is concluded that the best trade-off between observability and measurement costs is achieved by scenario 1. Therefore, the further development of the state estimator is carried out on the basis of the worst case scenario 1. 3.2 Design of the Kalman Filter As a state estimator for the HotModule, an extended Kalman filter with time discrete measurements and a time discrete correction term has been designed. In classical textbooks, e.g. [5], the extended Kalman filter is derived for systems of ordinary differential equations. However, an extension to differential algebraic systems with a differential index of one is straight forward [13]. The filter equations are briefly summarised in the following. Defining T

ˆ D := ϑS , ϑC x T

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(15)

the reduced model (8-10) can be written as ˆ A) x ˆ˙ D = f (ˆ xD , x ˆ A) 0 = g (ˆ xD , x

(16) (17)

Because the cell voltage depends nonlinearly on the state vector, a nonlinear ˆ: output equation results for the measurement vector y ˆ = h (ˆ ˆ A) y xD , x

(18)

The actions of the filter can be divided into two steps: In a prediction step, a time continuous simulation of the reduced model is used to estimate the

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states of the system. In a time discrete update step that is performed at measurement times the model states are corrected. The corrected states are used as initial conditions for the subsequent prediction step. The two steps are described in more detail in the following. During the prediction step, the model equations (16,17) and the equations for the error covariance matrix P P˙ = F P + P F T + Q

(19)

have to be solved. In (19), the Jacobian F is the total derivative of the righthand side vector of (16) with respect to xD after elimination of the algebraic states xA : xA ∂f dˆ ∂f + (20) F = ˆD ˆ A dˆ ∂x ∂x xD The matrix Q is a symmetric positive definite matrix, which should be equal the spectral density matrix of the process noise in the case of a linear system. In the update step, the difference between the predicted measurement values and the actually taken measurement values is weighted by a gain matrix K given by the following correlation: K HP (−)H T + R = P (−)H T

(21)

P (−) is the predicted value of the error covariance matrix at the time of measurement; R is the symmetric and positive definite covariance matrix of the measurement noise; H is the Jacobian of the output equation (18) after elimination of xA : ∂h dˆ xA ∂h + (22) H= ˆD ˆ A dˆ ∂x ∂x xD ˆ and measured The weighted difference between predicted output values y output values y is used to update the dynamic states of the system according to: ˆ D (−) + K(y − y ˆ) ˆ D (+) = x (23) x ˆ D (+) ˆ D (−) is the predicted state vector at the measurement time, x In (23), x is the corrected state vector used as an initial condition for the subsequent prediction step. Finally, the update step corrects the error covariance matrix according to: P (+) = [I − KH]P (−)

(24)

xD required in (20) and (22) is obtained by deriving (17) The gradient dˆ xA /dˆ ˆD: with respect to x xA ∂g ∂g dˆ + . (25) 0= ˆD ˆ A dˆ ∂x ∂x xD Because the reduced model has a differential index of one, (25) is regular and xD . can always be solved for dˆ xA /dˆ

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For the design of the filter, the matrices Q and R are used as design parameters. Due to the nonlinearity of the model, a systematic design approach is very difficult. Therefore, Q and R are assumed to be diagonal matrices; the elements on the main diagonal are chosen heuristically by simulations. In the implementation of the Kalman filter for the MCFC, the Jacobians F and H are computed numerically by finite differences. Because the measurement intervals are considerably smaller than the dominating time constants of the system, the variation of F is rather small between two measurement times. In order to increase the numerical efficiency, F is computed in every update step and is kept constant in the prediction steps. 3.3 Simulation Results The filter is tested with simulated measurement values generated either by the reduced model or by the detailed model. The initial condition of the filter is a steady state for a different value of the cell current. At time 0, the cell current is set to the correct value. A measurement interval of 10 time units is assumed. In a first test shown in Figure 4 a steady state of the reduced model is used as a reference. The filter states converge rapidly to the reference states. The convergence is much faster than the transient of the uncorrected model to the steady state. In a second test shown in Figure 5, the detailed model generates the measurement data and serves as a reference. Also in this case, the filter converges quickly. The mismatch between reduced and detailed model causes a small and tolerable remaining estimation error.

4 Conclusions and Outlook A low order model of a MCFC system has been derived from a detailed first principle model. The reduced model can be solved easily and fast on PC and is suitable for real-time applications. The reduced model is used as a basis for a state estimator. A local observability analysis shows that the measurement of the cell voltage and of the outlet temperatures of the anode and the cathode gas channels is sufficient to make the system completely observable. Based on these measurement information and on the reduced model, an extended Kalman filter has been designed and tested in simulations. The Kalman filter shows good convergence behaviour. At the moment, an experimental validation of the Kalman filter at the HotModule is under preparation. Acknowledgement. The authors thank Mykhaylo Krasnyk, M.Sc. for his assistance in implementing the Kalman filter. The financial support of the German Ministry of Education and Research under contract no. 03C0345B is gratefully acknowledged.

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3 2 1 0 0

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Fig. 4. Convergence of the extended Kalman filter, if the reduced model is used as a reference; deviations of the filter states from the reference states without a correction at measurement times (dashed lines) and with a correction at measurement times (solid lines); first row from left to right: maximum relative error of the estimated solid temperature, the estimated cathode gas temperature, and the estimated anode gas temperature; second row from left to right: relative error of the estimated cell voltage, maximum relative error of the H2 molar fraction in the anode gas, maximum relative error of the O2 molar fraction in the cathode gas.

References 1. J. Baker and P. Christofides. Finite-dimensional approximation and control of non-linear parabolic PDE systems. International Journal of Control, 73:439– 456, 2000. 2. M. Bischoff and G. Huppmann. Operating experience with a 250 kW molten carbonate fuel cell (MCFC) power plant. Journal of Power Sources, 105:216– 221, 2002. 3. L. Carrette, K.A. Friedrich, and U. Stimming. Fuel cells – fundamentals and applications. Fuel Cells, 1:5–39, 2001. 4. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. Academic Press, New York, 1972. 5. A. Gelb, editor. Applied Optimal Estimation. MIT Press, Cambridge, 1974.

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Fig. 5. Convergence of the extended Kalman filter, if the detailed model is used as a reference; first row from left to right: maximum relative error of the estimated solid temperature, the estimated cathode gas temperature, and the estimated anode gas temperature; second row from left to right: relative error of the estimated cell voltage, maximum relative error of the H2 molar fraction in the anode gas, maximum relative error of the O2 molar fraction in the cathode gas.

6. M.L.J. Hautus. Controllability and observability conditions of linear autonomous systems. Indagationes Mathematicae, 31:443–448, 1969. 7. P. Heidebrecht. Analysis and Optimisation of a Molten Carbonate Fuel Cell with Direct Internal Reforming (DIR-MCFC). PhD thesis, Otto-von-GuerickeUniversit¨ at Magdeburg, 2005. Fortschritt-Berichte VDI, Reihe 3, Nr. 826, VDI Verlag D¨ usseldorf. 8. P. Heidebrecht and K. Sundmacher. Dynamic modelling and simulation of a counter-current molten carbonate fuel cell (MCFC) with internal reforming. Chemical Engineering Science, 58:1029–1036, 2002. 9. K. Hoo and D. Zheng. Low-order control-relevant models for a class of distributed parameter systems. Chemical Engineering Science, 56:6683–6710, 2001. 10. M. Lo`eve. Probability Theory. Van Nostrand, Princeton, 1955. 11. J. L¨ uckel and P.C. M¨ uller. Analysis of controllability, observability and disturbability structures in linear time invariant systems. Regelungstechnik, 23:163–171, 1975.

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12. M. Mangold and M. Sheng. Nonlinear model reduction for a two-dimensional MCFC model with internal reforming. Fuel Cells, 4:68–77, 2004. ¨ 13. T. Obertopp. Modellierung, Identifikation und Uberwachung sicherheitstechnisch schwieriger Prozesse in mehrphasigen Reaktoren. PhD thesis, Universit¨ at Stuttgart, 2001. Fortschritt-Berichte VDI, Reihe 3, Nr. 701, VDI Verlag D¨ usseldorf. 14. H.M. Park and D.H. Cho. The use of the Karhunen-Lo`eve decomposition for the modeling of distributed parameter systems. Chemical Engineering Science, 51:81–98, 1996.

Bioprocess฀State฀Estimation:฀Some฀Classical and฀Less฀Classical฀Approaches Guillaume฀Goffaux฀and฀Alain฀Vande฀Wouwer Service฀d’Automatique,฀Facult´e฀Polytechnique฀de฀Mons,฀Boulevard฀Dolez฀31, B-7000฀Mons,฀Belgium. {guillaume.goffaux,alain.vandewouwer}@fpms.ac.be Summary.฀ This฀paper฀reviews฀some฀classical฀state฀estimation฀techniques฀for฀bioprocess฀applications,฀i.e.,฀the฀extended฀Kalman฀filter฀and฀the฀asymptotic฀observer, as฀well฀ as฀ a฀more฀recent฀ technique฀based฀on฀ particle฀ filtering.฀In฀ this฀application context,฀all฀ these฀techniques฀are฀based฀on฀ a฀continuous-time฀nonlinear฀ prediction model฀and฀discrete-time฀(low฀sampling฀rate)฀measurements.฀A฀hybrid฀asymptoticparticle฀filter฀is฀then฀developed,฀which฀blends฀the฀advantages฀of฀both฀techniques,฀i.e., robustness฀to฀model฀uncertainties฀(through฀a฀linear฀state฀transformation฀eliminating the฀reaction฀kinetics)฀and฀a฀rigorous฀consideration฀of฀the฀process฀and฀measurement noises.฀A฀simulation฀case-study฀is฀used฀throughout฀this฀paper฀to฀illustrate฀the฀performance฀of฀these฀state฀estimation฀techniques.

Keywords:฀Nonlinear฀estimation,฀particle฀filter,฀asymptotic฀observer,฀Kalman฀filter,฀biotechnology.

1 Introduction In standard bioprocess operation, on-line measurements are often limited to basic variables such as temperature and pH. Measurements of component concentrations, i.e. essential substrates, biomass and products of interest, are usually the results of sampling and off-line laboratory analysis. As such, they are available at discrete times only, and with relatively long sampling intervals (several hours up to 1-2 days). In recent years, on-line probes for measuring component concentrations, e.g. biomass probes based on capacitance measurements, have been developed, but their use is still very limited due to high costs. In this context, the design of “software sensors” based on state estimation techniques takes on particular importance. Software sensors allow the online reconstruction of non-measured variables (i.e. component concentrations) based on a process model and some available measurements (from “hardware sensors” or from on-line sampling and analysis). In bioprocess applications, two major techniques have emerged [4]:

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 111–128, 2005. © Springer-Verlag Berlin Heidelberg 2005

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Kalman filters, and in particular the continuous-discrete Extended Kalman Filter (EKF), which allows the use of a continuous-time dynamic model of the bioprocess together with discrete-time measurements, and which takes the process and measurement noises into account. The process nonlinearity is dealt with in an approximate way, through linearization along the state estimate trajectory. State and measurement noises are assumed to be normally distributed. Asymptotic observers, which are based on a state transformation eliminating the reaction kinetics from the model equations. This deterministic state estimation technique provides an asymptotic convergence whose rate is determined by the process operating conditions (i.e. the dilution rate in fed-batch and continuous processes).

In recent years, a new state estimation technique, called particle filtering, has been developed [1] and used mostly in navigation and tracking [2]. The particle filter builds a discrete estimation of the conditional probability density given the measurement information. The particle filter has several interesting features, which make it potentially attractive in the field of bioprocesses: " " " "

It is a nonlinear state estimation technique which does not require assumptions on the model equations. Discrete-time measurements are easily accommodated. It is a general method which handles non-Gaussian process and measurement noise. The particle filter is easily implemented and is almost insensitive to state dimension.

The objective of this study is to assess the performance of the particle filter in the context of bioprocess applications, as compared to traditional techniques such as the EKF and the asymptotic observer, and to propose a hybrid asymptotic-particle filter, which aims at blending the advantages of the asymptotic observer and the particle filter (robustness to uncertainties in the kinetic model and consideration of the noise statistics).

2 Bioprocess Modelling 2.1 Mass Balance Equations Consider a biological system described by a set of macroscopic components such as biomass, substrates and metabolic products. A reaction scheme and its associated pseudo-stoichiometry are defined by ϕk

i∈Rk

νi,k Si,k →

νj,k Pj,k j∈Pk

k ∈ 1, ..., M

(1)

where M is the number of reactions, ν•,k are the pseudo-stoichiometric coefficients of the k th reaction, Si,k is the ith consumed component in this

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reaction, Pj,k is the j th produced component in this reaction, and ϕk is the reaction rate. Assuming that the bioprocess takes place in a continuous stirred tank bioreactor (CSTBR), the system of mass balances for each component involved in the reaction scheme (1) leads to the general dynamic model [3] dξ(t) = Kϕ(ξ(t)) + (ξ in (t) − ξ(t)) D(t) − Gout (t) dt dV (t) = Qin (t) − Qout (t) dt

(2) (3)

where ξ ∈ RN is the vector of component concentrations, K ∈ RN ×M is the matrix of pseudo-stoichiometric coefficients, ϕ(ξ) ∈ RM is the vector of reaction rates, Qin ∈ R is the feed rate, Qout ∈ R is the outflow rate, V ∈ R is the reactor volume, D ∈ R is the dilution rate (D = QVin ), ξ in ∈ RN is the feed concentration vector, and Gout ∈ RN represents the vector of outflow rates in gaseous form. The system of mass balances (2-3) is a nonlinear differential equation system, which can be cast into the more general form ˙ x(t) = f (x(t), u(t)) + η(t)

x(t0 ) = x0

y k = y(tk ) = h(x(tk )) + (tk )

(4) (5)

where x(t) ∈ Rnx is the vector of state variables, u(t) ∈ Rnu is the vector of inputs, y(tk ) ∈ Rny is the vector of sampled measurements (at time tk ), η(t) ∈ Rnx is an additive process noise vector, and (tk ) ∈ Rny is an additive measurement noise vector. f and h are, in general, nonlinear vector functions. 2.2 System Observability A system is completely observable if it is possible to reconstruct the state vector completely from a finite number of measurements of the system outputs. Global observability analysis of nonlinear systems is a delicate task (observability generally depends on the system inputs), which can be simplified through the introduction of canonical forms ([12], [13], [6]). This way, a system is said to be globally observable if the nonlinear model can be set in the following form and if the following conditions are satisfied ⎤ ⎤ ⎡ x˙ 1 f 1 (x1 , x2 , u) ⎥ ⎢ ... ⎥ ⎢ ... ⎥ ⎢ i ⎥ ⎢ i ⎢ x˙ ⎥ ⎢ f (x1 , ..., xi+1 , u) ⎥ ⎥, ⎥=⎢ x˙ = ⎢ ⎥ ⎢ ... ⎥ ⎢ ⎥ ⎢ q−1 ⎥ ⎢ q−1 ... 1 q ⎦ ⎣f ⎣ x˙ (x , ..., x , u) ⎦ f q (x1 , ..., xq , u) x˙ q ⎡

⎤ h1 (x11 ) ⎢ h2 (x11 , x12 ) ⎥ ⎥ y=⎢ ⎦ ⎣ ... 1 1 hn1 (x1 , ..., xn1 ) ⎡

(6)

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where x i ∈ R ni ,

∀i ∈ {1, ..., q} ,

n1 ≥ n2 ≥ ... ≥ nq ,

ni = nx

(7)

1≤i≤q

and ∀x1 ∈ Rn1 ,

x1 = x11 , . . . , x1n1

∀i ∈ {1, ..., q − 1} ,

T

,

∂hj =0 ∂x1j

(8)

∂f i (x, u) = ni+1 ∂xi+1

(9)

∀j ∈ {1, ..., n1 } :

∀(x, u) ∈ Rnx × Rnu : rank

Conditions (8) imply that the first state subvector x1 can be directly inferred from the measurements, whereas (9) implies a pyramidal influence of the state subvector xi+1 on xi , so that any differences in the state trajectory can be detected in the measurements. 2.3 An Example In this section, a simple reaction scheme is considered, which describes the growth of biomass on glutamine and a maintenance process based on glucose and leading to the production of lactate. Growth: Maintenance:

ϕg ∧

νGln Gln −→ X ϕm

G + νX X −→ νX X + νL L

(10) (11)

where G, Gln, X and L denote glucose, glutamine, biomass and lactate, respectively. νGln , νX et νL are pseudo-stoichiometric coefficients. The symbol ”−→∧ ” means that the growth reaction is auto-catalysed by X and the presence of biomass on both sides of the maintenance reaction means that X catalyses this latter reaction. The growth rate ϕg and the maintenance rate ϕm are described by classical Monod laws and inhibition factors. Gln Kig X g + Gln Ki + G Kim G X ϕm (X, G) = μm max m m KM + G Ki + X

ϕg (X, G, Gln) = μgmax

g KM

(12) (13)

Simple mass balances allow the following dynamic model to be derived (see also (2))

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dX = ϕg (X, G, Gln) − D X (14) dt dG = −ϕm (X, G) + D (Gin − G) (15) dt dGln = −νGln ϕg (X, G, Gln) + D (Glnin − Gln) (16) dt dL = νL ϕm (X, G) − D L (17) dt Numerical values of the model parameters and operating conditions are listed in tables 1 and 2, respectively. The reactor is operated in continuous mode (Qin = Qout ) and samples are taken every 8 h during 20 days. Analysis gives measurements of lactate and glutamine, which are corrupted by normally distributed white noises with zero mean and variance matrix 2 , σL2 ). Simulation results are shown in figure 1. R = diag(σGln Global observability can be studied by using the canonical form (6), in which the dynamic equations (14-17) appear naturally. The state vector is subdivided into a measured vector x1 and a non-measured vector x2 . Table 2. Operating conditions

Table 1. Model parameters νL μgmax (h−1 ) g KM (mM) Kig (mM)

1.7 0.05 5 30

νGln (mM/(105 cell/ml)) −1 μm ) max (h m KM (mM) Kim (105 cell/ml)

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x˙ 1 x˙ 2

x˙ = with x1 = [Gln, L]T ,

=

f 1 (x1 , x2 , u) , f 2 (x1 , x2 , u)

x2 = [G, X]T ,

y = x1

(18)

u = [Gin , Glnin , D]T ,

f 1 (x1 , x2 , u) =

−νGln ϕg (X, G, Gln) + D (Glnin − Gln) , νL ϕm (X, G) − D L

f 2 (x1 , x2 , u) =

−ϕm (X, G) + D (Gin − G) . ϕg (X, G, Gln) − D X

The global observability condition becomes ∀(x, u) ∈ Rnx × Rnu : rank

∂f 1 (x, u) = n2 = 2 ∂x2

(19)

As ∂f 1 (x, u) = ∂x2

∂ϕ (X,G,Gln)

−νGln g ∂G (X,G) νL ∂ϕm∂G

∂ϕ (X,G,Gln)

−νGln g ∂X (X,G) νL ∂ϕm∂X

(20)

the global observability condition is satisfied if X, G and Gln do not vanish.

3 Kalman Filtering In this section, the extended Kalman filter is introduced according to the Bayesian approach. In this approach, the posterior probability density function (pdf) of the state based on the available measurements is evaluated recursively and approximated by a Gaussian distribution. 3.1 Bayesian State Estimation The process and measurement noises are both assumed independent identically distributed (i.i.d.) and mutually independent. The initial pdf p(x0 ) is assumed a priori known. The pdf p(xk |Yk ) = p(x(tk ) |[y(t0 ), . . . , y(tk )]) can be recursively calculated in two steps (see [11] for the discrete case): A prediction step which calculates the prior pdf p(xk |Yk−1 ) via the evolution equations (4) and the posterior pdf obtained at the previous time instant p(xk−1 |Yk−1 ). p(xk |Yk−1 ) =

Rn x

p(xk |xk−1 ) p(xk−1 |Yk−1 ) dxk−1

(21)

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where p(xk |xk−1 ) is the transitional density defined by the evolution equations and the process noise. A correction step which updates the prior pdf p(xk |Yk−1 ) using the knowledge of the measurements Yk and the measurement noise distribution p. p(xk |Yk ) = p(y k |Yk−1 ) =

p(y k |xk )p(xk |Yk−1 ) p(y k |Yk−1 ) Rn x

p(y k |xk )p(xk |Yk−1 ) dxk

(22)

where p(y k |xk ) = p (y k − h(xk )) is defined by the observation equations and the measurement noise. 3.2 The Extended Kalman Filter Consider the continuous-discrete case described by equations (4)-(5) where η(t) ∼ N (0, Q(t)), (tk ) ∼ N (0, R(tk )), and x(t0 ) ∼ N (x0 , P0 ). Moreover, pdfs in (21),(22) are approximated by Gaussian distributions and can therefore be specified by a mean and a covariance, i.e. p(x(t)|Yk−1 ) ≈ N (mt|tk−1 , Pt|tk−1 ) p(xk |Yk ) ≈ N (mtk |tk , Ptk |tk )

(23) (24)

The EKF uses a linearization along the state estimate trajectory to compute the evolution of the covariance matrix. The extended Kalman filter provides the evolution of the mean and covariance in two steps: "

Continuous prediction step for tk−1 ≤ t < tk ˆ (tk−1 ) = mtk−1 |tk−1 x P (tk−1 ) = Ptk−1 |tk−1 ˙x ˆ (t) = f (ˆ x(t), u(t)) P˙ (t) = F (ˆ x, u)P (t) + P (t)F T (ˆ x, u) + Q(t) ˆ (t) Pt|tk−1 = P (t) mt|tk−1 = x where F (ˆ x(t), u(t)) =

"

∂f (x(t),u(t)) . ∂x x(t)=ˆ x(t)

Discrete correction step at t = tk ˆ (tk ) Ptk |tk−1 = P (tk ) mtk |tk−1 = x

x(tk ))T (H(ˆ x(tk ))Ptk|k−1 H(ˆ x(tk ))T + R(tk ))−1 K(tk ) = Ptk |tk−1 H(ˆ x(tk ))) mtk |tk = mtk |tk−1 + K(tk )(y(tk ) − h(ˆ Ptk |tk = Ptk |tk−1 − K(tk )H(ˆ x(tk ))Ptk |tk−1

where H(ˆ x(tk )) =

∂h(x(t)) ∂x

x(t)=ˆ x(tk )

.

3.3 Example Returning to the simple application example of section 2.3, the performance of the EKF is examined in three different situations.

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Situation 1: A Perfect Model

25

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Figure 2 compares the evolution of the real and estimated concentrations with an EKF tuned with the parameters given in table 3. The measurement error 2 , σL2 ) whereas the process covariance covariance matrix R is equal to diag(σGln matrix Q is set to a small diagonal matrix because of the relative confidence in the model. State estimation is very satisfactory.

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Fig. 2. EKF state estimation with a perfect model.

Situation 2: Erroneous Model Parameters A more realistic case corresponds to the situation where the model parameters have been estimated with some errors, e.g. see table 4 (errors in the kinetic parameters). Figure 3 shows the estimation results starting with the same initial conditions as in table 3. The process covariance matrix is increased to take the higher model uncertainty into account. However, this precaution is not sufficient, and the estimation of the non-measured variables is not acceptable, particularly in the case of the biomass. Table 3. Initial filter parameters ˆ 0 Gln ˆ L0 ˆ0 G ˆ0 X

= = = =

yGln (t0 ) (mM) yL (t0 ) (mM) 24 (mM) 1 (105 cell/ml)

Table 4. Erroneous model parameters νL μgmax g KM Kig

= = = =

1, 7 0.045 (-10%) 5.6 (+12%) 27 (-10%)

νGln μm max m KM Kim

= = = =

0, 2 0.11 (+10%) 0.95 (-5%) 3.36 (+12%)

25

10

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Glucose [mM]

Bioprocess State Estimation

15 10 5 200 300 time (h)

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0 0

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Fig. 3. EKF state estimation based on erroneous model parameters (table 4).

Situation 3: Uniform Measurement Noise

25

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Glucose [mM]

Figure 4 shows the estimation results when the process model is perfectly known, but the measurement errors follow a uniform distribution on the interval [ξ• − Δy• , ξ• + Δy• ] where ξ• represents the lactate or glutamine concentration, and ΔyL = 0.2yL , ΔyGln = 0.2yGln . Despite the accurate model, the convergence speed is deteriorated.

15 10 5

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0 0

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200 300 time (h)

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400

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10 5 0 0

Fig. 4. EKF state estimation based on a perfect model (table 1) and uniform measurement noises.

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4 Asymptotic Observers 4.1 Description In contrast with the EKF, asymptotic observers are open-loop state estimation techniques, i.e. they do not have an adjustable rate of convergence. Dedicated to bioprocess systems (2)-(3), their major advantage is however that they do not require the knowledge of the reaction kinetics, which can be eliminated through a state transformation. This transformation is based on a partition of the state vector into a measured state subvector ξ1 ∈ RN1 and a non-measured state subvector ξ 2 ∈ RN2 (N1 + N2 = N ). The mass balance state equations (2) can therefore be rewritten as K1 ξ˙ ϕ(ξ) + D ξ˙ = ˙ 1 = K2 ξ2

ξ 1,in ξ − 1 ξ 2,in ξ2

−

G1,out G2,out

(25)

In this expression, it is assumed that the reaction rate vector ϕ(ξ) ∈ RM is unknown and the yield coefficients in K ∈ RN ×M are known. Moreover, rank(K) = M , i.e. the M reactions are linearly independent (otherwise it is possible to eliminate some dependent reactions) and the matrix K1 ∈ RN1 ×M has full rank, i.e. rank(K1 ) = M (the condition of asymptotic observability is that the number of measured state variables is equal to or greater than the number of independent reactions). Based on these assumptions, a linear transformation is defined z = ξ 2 − K2 K1−1 ξ 1 ,

(26)

z˙ = −Dz + Dξ 2,in − G2,out − K2 K1−1 (Dξ 1,in − G1,out )

(27)

which allows the unknown rate functions ϕ(ξ) to be eliminated from the evolution equations, i.e.

The asymptotic observer proceeds with a prediction and a correction step: Continuous prediction step for tk−1 ≤ t < tk ˆ (tk−1 )= ξˆ2 (tk−1 ) − K2 K1−1 ξˆ1 (tk−1 ) z z ˆ˙ (t) = −Dˆ z (t) + Dξ − G2,out − K2 K −1 (Dξ 2,in

1

Discrete correction step at t = tk

ξˆ1 (tk ) = y(tk )) ˆ (tk ) − K2 K1−1 ξˆ1 (tk ) ξˆ2 (tk ) = z

1,in

− G1,out )

(28)

(29)

The prediction step implies some form of extrapolation of the measurements between sampling instants. Zero- or first-order extrapolation can be used to this end. Alternatively, a full model, whenever available, can be used for extrapolation. However, this latter option increases the sensitivity to modelling errors (and can therefore be less robust in some instances).

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4.2 Example Continuing the example of sections 2.3 and 3.3, an asymptotic observer is now tested based on partition (18) and the erroneous kinetic parameters of table 4. The same initial conditions as for the EKF (table 3) are considered. The first-order and the full-model-based measurement extrapolation techniques are illustrated in figure 5. The estimation results are quite good, despite the error in the kinetic model. However a higher sensitivity to measurement noise (than with the EKF) is observed. st

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5 Particle Filters Recently, particle filters [10],[7] have been developed and applied mostly in the field of navigation and positioning. Particle filtering is based on an exploration of the state space by particles driven by a random process. A state vector xik and a weight ωki are associated to each particle. The weight of a particle is related to the likelihood of its state vector at time tk , taking the measurements up to tk into account. Therefore, the posterior pdf is approximated by N

p(xk |Yk ) ≈

ωki δxik (xk )

(30)

i=1

where δxik (xk ) is the Dirac function defined by

Rn x

δz (x)g(x)dx = g(z).

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Particle filters can be applied to the general model (4),(5) with noise statistics that can take arbitrary forms. The initial state statistics is assumed a priori known. 5.1 Algorithm Description Several particle filtering algorithms exist, which are described in [1], [2]. The algorithm presented here is known as the particle filter, the bootstrap filter or the sample importance resampling (SIR) filter. The algorithm can be subdivided into four steps : "

An initialization step which associates to each particle an initial state vector according to the known initial pdf N particles :

"

Drawing of xi0 according px0

i=1,...,N

Moreover, a uniform weighting is applied to the particles: ω0i = N1 . A propagation step in which particles are propagated according to the evolution equations. N process noises η i (t) are generated according to the noise statistics pη x˙ i (t) = f (xi (t), u(t)) + ηi (t)

"

A weighting step where the weights are updated based on the measurement noise statistics p ωki =

ρik , N P ρik

i ρik = p(y k |xik )ωk−1

where p(y k |xik ) = p (y k − h(xk )) .

i=1

"

An estimation step which provides state estimates using a weighted sum, which represents the mean of the posterior pdf p(xk |Yk ) N

ˆk = x

i=1

"

ωki xik

A resampling step which avoids the degeneracy problem where most of the particle weights vanish. In the limit, all the information is then concentrated in one particle. Resampling should occur if Nef f =

1 N i=1

< Nth

(31)

(ωki )2

which compares the number of particles that are really used to a threshold value (which is a design parameter). There are several methods for redistributing the weights [8]. For instance, one may choose the multinomial resampling where nik samples are drawn from each particle according to a multinomial rule (n1k , . . . , nN k ) ∼

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i M(N, ωk1 . . . , ωkN ) where N ∀k. This resampling procedure can i=1 nk = N, be realized in O(n) operations by sampling N ordered uniforms on [0, 1]. However, a major drawback is the introduction of a large variance of nik . To improve the variance, one may use the residual resampling which consists in ¯ N ) by taking the integer part N ω i for each ¯ 1, . . . , N estimating a first set (N k k k particle. Then, a multinomial procedure is derived from the residual set N

Nkres = N −

N ωki

(32)

i=1

An additional issue related to the degeneracy problem and introduced by these resampling methods is the so-called sample impoverishment. This problem occurs in the case of a weak process noise. In this case, after resampling, some particles will have more or less the same state vector. So, the amount of identical particles will increase with time and part of the calculations will become useless. Eventually, the particles will collapse to a single point, and this can happen within a few iterations. To alleviate this problem, it is possible to artificially add noises to the particles around the resampled state vectors, or to add artificial process noise in the evolution equations. However, this latter technique increases the estimation error covariance. Much more can be said on particle filters concerning convergence properties, and in particular the number of particles that are needed to achieve a given level of accuracy. Some information can be found in [5] and [9]. 5.2 Example This section parallels section 4.2, and three simulation case studies are considered again, in which either a perfect process model is at hand, or kinetic parameters have not been estimated accurately, or measurement noises follow uniform distributions. Two filter parameters can be tuned to achieve good performance, i.e. the number of particles N and the resampling threshold Nth . Obviously, a large number of particles is the best solution to explore the state space, but results in a higher computational load. The resampling threshold allows the degeneracy phenomenon to be avoided. The more the threshold number is close to the number of particles, the more often the particles are resampled. The resampling algorithm used here is based on residual resampling, and the particles are slightly noised after resampling in order to alleviate the lack of process noise. In each simulation case, the same initial conditions are considered (table 5) and it is assumed that each particle is drawn from a uniform distribution. Situation 1: A Perfect Model Figure 6 shows typical results obtained with a particle filter with N = 300, and Nth = 0.6N, and based on a perfect model (table 1). As the particle filter

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G. Goffaux and A. Vande Wouwer Table 5. Initial conditions drawn from uniform distributions Non-measured

Measured

Gi0 ∼ U[16,32] X0i ∼ U[0,2]

Glni0 ∼ U[yGln (t0 )−2.58σGln0 ,yGln (t0 )+2.58σGln0 ] Li0 ∼ U[yL (t0 )−2.58σL0 ,yL (t0 )+2.58σL0 ]

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is a random filter, the results can differ from one run to another. However, with a higher number of particles, the fluctuations decrease.

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Fig. 6. State estimation with a particle filter (N = 300, Nth = 0.6N) based on a perfect model (table 1).

Situation 2: Erroneous Model Parameters As with the EKF, state estimation deteriorates, especially for the biomass (figure 7). To improve on this situation, a new algorithm combining the particle filter and the asymptotic observer will be presented in the next chapter. Situation 3: Uniform Measurement Noise Finally, the particle filter is tested with uniform measurement noise like in section 3.3. Comparing the results in figure 8 with figure 4, it is apparent that the particle filter can easily handle non-Gaussian noise distributions.

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Fig. 7. State estimation with a particle filter (N = 300, Nth = 0.6N) with erroneous model parameters (table 4).

6 A Hybrid Asymptotic-Particle Filter 6.1 Description The objective of this section is to blend the advantages of the asymptotic observer and the particle filter, i.e. to improve robustness to uncertainties in the kinetic model and to take the noise statistics into account. The algorithm description closely follows those of the particle filter in section 5.1. The main difference is that the samples are propagated without using the kinetic model, which is eliminated through a linear transformation introduced in section 4. Specifically, the algorithm can be subdivided into four steps: "

An initialization step N initial state vector ξ it0 are generated according an initial pdf pξ0 and equal weights ω0i = N1 are associated to each particle, as in the classical particle filter. In analogy with the asymptotic observer, the state vector ξ is partitioned into a measured state vector ξ1 and a nonmeasured state vector ξ 2 and a linear transformation is applied to the initial condition z i0 = ξ i2,0 − K2 K1−1 ξi1,0 . The initial estimation is given by N ξˆt0 = i=1 ω0i ξit0 .

"

A propagation step in the time interval (tk−1 ≤ t < tk ) For each particle, the evolution of z is computed from ∀i,

z i (tk−1 ) = z ik−1 , z˙ i (t) = −Dz i (t) + Dξ 2,in − G2,out − K2 K1−1 (Dξ 1,in − G1,out ),

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Fig. 8. State estimation with a particle filter (N = 300, Nth = 0.4N) with a perfect model (table 1) and uniform measurement noises.

whereas the measured state vector ξ i1 is linearly extrapolated based on ξi1,k−2 and ξ i1,k−1 ξ i1 (t) = ξi1 (tk−1 ) +

ξi1 (tk−1 )−ξi1 (tk−2 ) tk−1 −tk−2

(t − tk−1 ),

and the non-measured state vector is obtained from ξ i2 (t) = z i (t) + K2 K1−1 ξ i1 (t).

i ˆ = Moreover, the weights ωk−1 are kept constant and ξ(t)

"

N i=1

i ωk−1 ξi (t).

A weighting step in tk As in the particle filter, the weighting step compares the measurements with the state estimates obtained from the propagation step ∀i

ωki =

ρik N i=1

ξ itk = ξi1,k , ξ i2,k

T

ξ i1,k = ξ i1 (tk ),

ξ i2,k = ξ i2 (tk )

z ik = z i (tk )

ρik

,

i ρik = p(y k |xik )ωk−1 where p(y k |xik ) = p (y k − h(xk ))

and the updated weights are used to estimate the state vector ξˆtk = N i i i=1 ωk ξ tk .

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A resampling step The effective number of particles Nef f is first evaluated (31). If Nef f is less than the threshold Nth , a resampling is operated, e.g. using the residual or the multinomial resampling. Then, the samples are slightly noised to avoid sample impoverishment.

6.2 Example

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To illustrate the performance of the hybrid asymptotic-particle filter, we return to the case where the kinetic model parameters have been poorly estimated (table 4). The initial filter conditions are the same as in table 5. The results shown in figure 9 demonstrate that the hybrid filter performs better than the particle filter (see figure 7), i.e. it is robust to uncertainties in the kinetic model, and performs also better than the asymptotic observer (see figure 5), i.e. it takes the noise statistics into account.

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7 Conclusions This paper reviews some classical state estimation techniques, e.g. the extended Kalman filter and the asymptotic observer, in the context of bioprocess applications where the process models are often uncertain and the measurement of component concentrations can only be achieved at low sampling rate.

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This paper also introduces particle filtering techniques, which have been recently proposed and applied in the field of navigation and positioning, and which are appealing solutions for bioprocess state estimation as they use nonlinear process models (without requiring linearization along the state estimate trajectory as in the EKF) and take the process and measurement noise statistics rigorously into account. Finally, a hybrid asymptotic-particle filter is developed, which combines the advantage of robustness to uncertainties in the kinetic model and consideration of the noise statistics. Preliminary numerical results show good performance. Investigations of the convergence properties of this filter are required and will be the subject of future work.

References 1. N. de Freitas A. Doucet and N. Gordon. Sequential Monte Carlo Methods in Practice. Springer-Verlag, New-York, January 2001. 2. S. Arulampalam B. Ristic and N. Gordon. Beyond the Kalman Filter – Particle Filters for Tracking Applications. Artech House, Boston, 2004. 3. G. Bastin and D. Dochain. On-Line Estimation and Adaptive Control of Bioreactors, volume 1 of Process Measurement and Control. Elsevier, Amsterdam, 1990. 4. P. Bogaerts and A. Vande Wouwer. Software sensors for bioprocesses. ISA Transactions, 42:547–558, 2003. 5. D. Crisan and A. Doucet. A survey of convergence results on particle filtering methods for practitioners. IEEE Transactions on Signal Processing, 50(3):736– 746, March 2002. 6. J.-P. Gauthier and I. Kupka. Observability and observers for nonlinear systems. SIAM Journal Control and Optimization, 32(4):975–994, 1994. 7. D.J. Salmon N.J. Gordon and A.F.M. Smith. Novel approach to nonlinear/nongaussian bayesian state estimation. IEE Proceedings, Part F, 140(2):107–113, April 1993. 8. P.-J. Nordlund. Sequential Monte Carlo filters and Integrated Navigation. Licentiate thesis 945, Department of Electrical Engineering, Link¨ oping University, 2002. 9. N. Oudjane. Stabilit´e et approximations particulaires en filtrage non-lin´ eaire – Application au filtrage. PhD thesis, University of Rennes I, December 2000. 10. G. Rigal P. Del Moral and G. Salut. Estimation et commande optimale nonlin´eaire. Convention D.R.E.T.-DIGILOG-LAAS/CNRS, SM.MCY/685.92/A, 89.34.553.00.470.75.01, 2, March 1992. 11. T. S¨ oderstr¨ om. Optimal estimation, chapter 5, pages 123–135. Advanced Textbooks in Control and Signal Processing. Springer-Verlag, London, 2002. 12. M. Zeitz. Observability canonical (phase-variable) form for nonlinear timevariable systems. International Journal of System Science, 15(9):949–958, 1984. 13. M. Zeitz. Canonical forms for nonlinear systems, chapter In: Nonlinear Control Systems Design (A. Isidori, Ed.), pages 33–38. Pergamon Press, 1989.

Convergent Systems: Analysis and Synthesis Alexey Pavlov, Nathan van de Wouw, and Henk Nijmeijer Eindhoven University of Technology, Department of Mechanical Engineering, P.O.Box. 513, 5600 MB, Eindhoven, The Netherlands. {A.Pavlov,N.v.d.Wouw,H.Nijmeijer}@tue.nl

Summary.฀ In฀this฀paper฀we฀extend฀the฀notion฀of฀convergent฀systems฀defined฀by B.P.฀Demidovich฀and฀introduce฀the฀notions฀of฀uniformly,฀exponentially฀convergent and฀ input-to-state฀convergent฀ systems.฀ Basic฀ (interconnection)฀ properties฀ of฀ such systems฀are฀established.฀Sufficient฀conditions฀for฀input-to-state฀convergence฀are฀presented.฀ For฀ a฀ class฀ of฀ nonlinear฀ systems฀ we฀ design฀ (output)฀ feedback฀ controllers that฀make฀the฀closed-loop฀system฀input-to-state฀convergent.฀The฀conditions฀for฀such controller฀design฀are฀formulated฀in฀terms฀of฀LMIs.

Keywords:฀Convergent฀systems,฀stability฀properties,฀asymptotic฀properties, interconnected฀systems,฀observers,฀output-feedback.

1 Introduction In many control problems it is required that controllers are designed in such a way that all solutions of the corresponding closed-loop system “forget” their initial conditions. Actually, this is one of the main tasks of a feedback to eliminate the dependency of solutions on initial conditions. In this case, all solutions converge to some steady-state solution which is determined only by the input of the closed-loop system. This input can be, for example, a command signal or a signal generated by a feedforward part of the controller or, as in the observer design problem, it can be the measured signal from the observed system. This “convergence” property of a system plays an important role in many nonlinear control problems including tracking, synchronization, observer design and the output regulation problem, see e.g. [15, 16, 18, 13] and references therein. For asymptotically stable linear time invariant systems with inputs, this is a natural property. Indeed, due to linearity of the system every solution is globally asymptotically stable and, therefore, for a given input, all solutions of such a system “forget” their initial conditions and converge to each other. After transients, the dynamics of the system are determined only by the input.

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 131–146, 2005. © Springer-Verlag Berlin Heidelberg 2005

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For nonlinear systems, in general, global asymptotic stability of a system with zero input does not guarantee that all solutions of this system with a non-zero input “forget” their initial conditions and converge to each other. There are many examples of nonlinear globally asymptotically stable systems which, being excited by a periodic input, have coexisting periodic solutions. Such periodic solutions do not converge to each other. This fact indicates that for nonlinear systems the convergent dynamics property requires additional conditions. The property that all solutions of a system “forget” their initial conditions and converge to some limit- or steady-state solution has been addressed in a number of papers. In [17] this property was investigated for systems with right-hand sides which are periodic in time. In that work systems with a unique periodic globally asymptotically stable solution were called convergent. Later, the definition of convergent systems given by V.A. Pliss in [17] was extended by B.P. Demidovich in [3] (see also [11]) to the case of systems which are not necessarily periodic in time. According to [3], a system is called convergent if there exists a unique globally asymptotically stable solution which is bounded on the whole time axis. Obviously, if such solution does exist, all other solutions, regardless of their initial conditions, converge to this solution, which can be considered as a limit- or steady-state solution. In [3, 2] (see also [11]) B.P. Demidovich presented a simple sufficient condition for such a convergence property. With the development of absolute stability theory, V.A. Yakubovich showed in [20] that for a linear system with one scalar nonlinearity satisfying some incremental sector condition, the circle criterion guarantees the convergence property for this system with any nonlinearity satisfying this incremental sector condition. In parallel with this Russian line of research, the property of solutions converging to each other was addressed in the works of T. Yoshizawa [21, 22] and J.P. LaSalle [9]. In [9] this property of a system was called extreme stability. In [21] T. Yoshizawa provided Lyapunov and converse Lyapunov theorems for extreme stability. Several decades after these publications, the interest in stability properties of solutions with respect to one another revived. In the mid-nineties, W. Lohmiller and J.-J.E. Slotine (see [10] and references therein) independently reobtained and extended the result of B.P. Demidovich. A different approach was pursued in the works by V. Fromion et al, [4, 6, 5]. In this approach a dynamical system is considered as an operator which maps some functional space of inputs to a functional space of outputs. If such operator is Lipschitz continuous (has a finite incremental gain), then, under certain observability and reachability conditions, all solutions of a state-space realization of this system converge to each other. In [1], D. Angeli developed a Lyapunov approach for studying both the global uniform asymptotic stability of all solutions of a system (in [1], this property is called incremental stability) and the so-called incremental input-to-state stability property, which is compatible with the input-to-state stability approach (see e.g. [19]). The

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drawback of the incremental stability and incremental input-to-state stability notions introduced in [1] is that they are not coordinate independent. In this paper we extend the notion of convergent systems defined by B.P. Demidovich. More specifically, we introduce the notions of (uniformly, exponentially) convergent systems and input-to-state convergent systems in Section 2. These notions are coordinate independent, which distinguishes them from the other approaches mentioned above. In Section 3 we present results on basic properties of (interconnected) convergent systems. Sufficient conditions for exponential and input-to-state convergence properties are presented in Section 4. In Section 5 we present (output) feedback controllers that make the corresponding closed-loop system input-to-state convergent. Section 6 contains the conclusions.

2 Convergent Systems In this section we give definitions of convergent systems. These definitions extend the definition given in [3] (see also [11]). Consider the system z˙ = F (z, t),

(1)

where z ∈ Rd , t ∈ R and F (z, t) is locally Lipschitz in z and piecewise continuous in t. Definition 1. System (1) is said to be

! convergent if there exists a solution z¯(t) satisfying the following conditions (i) z¯(t) is defined and bounded for all t ∈ R, (ii) z¯(t) is globally asymptotically stable. ! uniformly convergent if it is convergent and z¯(t) is globally uniformly asymptotically stable. ! exponentially convergent if it is convergent and z¯(t) is globally exponentially stable. The solution z¯(t) is called a limit solution. As follows from the definition of convergence, any solution of a convergent system “forgets” its initial condition and converges to some limit solution which is independent of the initial condition. In general, the limit solution z¯(t) may be non-unique. But for any two limit solutions z¯1 (t) and z¯2 (t) it holds that |¯ z1 (t) − z¯2 (t)| → 0 as t → +∞. At the same time, for uniformly convergent systems the limit solution is unique, as formulated below. Property 1. If system (1) is uniformly convergent, then the limit solution z¯(t) is the only solution defined and bounded for all t ∈ R. Proof. Suppose there exists another solution z˜(t) defined and bounded for all t ∈ R. Let R > 0 be such that |˜ z (t) − z¯(t)| < R for all t ∈ R. Such R exists

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since both z˜(t) and z¯(t) are bounded for all t ∈ R. Suppose at some instant t∗ ∈ R the solutions z¯(t) and z˜(t) satisfy |˜ z (t∗ ) − z¯(t∗ )| ≥ ε > 0 for some ε > 0. Since z¯(t) is globally uniformly asymptotically stable, there exists a number T (ε, R) > 0 such that if |˜ z (t0 ) − z¯(t0 )| < R for some t0 ∈ R then |˜ z (t) − z¯(t)| < ε,

∀ t ≥ t0 + T (ε, R).

(2)

Set t0 := t∗ −T (ε, R). Then for t = t∗ inequality (2) implies |˜ z (t∗ )− z¯(t∗ )| < ε. Thus, we obtain a contradiction. Since t∗ has been chosen arbitrarily, this implies z˜(t) ≡ z¯(t).  The convergence property is an extension of stability properties of asymptotically stable linear time-invariant (LTI) systems. Recall that for a piecewise continuous vector-function f (t), which is defined and bounded on t ∈ R, the system z˙ = Az + f (t) with a Hurwitz matrix A has a unique solution z¯(t) which is defined and bounded on t ∈ (−∞, +∞). It is given by the formula t z¯(t) := −∞ exp(A(t − s))f (s)ds. This solution is globally exponentially stable with the rate of convergence depending only on the matrix A. Thus, an asymptotically stable LTI system excited by a bounded piecewise-continuous function f (t) is globally exponentially convergent. In the scope of control problems, time dependency of the right-hand side of system (1) is usually due to some input. This input may represent, for example, a disturbance or a feedforward control signal. Below we will consider convergence properties for systems with inputs. So, instead of systems of the form (1), we consider systems z˙ = F (z, w)

(3)

with state z ∈ Rd and input w ∈ Rm . The function F (z, w) is locally Lipschitz in z and continuous in w. In the sequel we will consider the class PCm of piecewise continuous inputs w(t) : R → Rm which are bounded for all t ∈ R. Below we define the convergence property for systems with inputs. Definition 2. System (3) is said to be (uniformly, exponentially) convergent if for every input w ∈ PCm the system z˙ = F (z, w(t)) is (uniformly, exponentially) convergent. In order to emphasize the dependency on the input w(t), the limit solution is denoted by z¯w (t). The next property extends the uniform convergence property to the input-tostate stability (ISS) framework. Definition 3. System (3) is said to be input-to-state convergent if it is uniformly convergent and for every input w ∈ PCm system (3) is ISS with respect to the limit solution z¯w (t), i.e. there exist a KL-function β(r, s) and a K∞ function γ(r) such that any solution z(t) of system (3) corresponding to some input w(t) ˆ := w(t) + Δw(t) satisfies |z(t) − z¯w (t)| ≤ β(|z(t0 ) − z¯w (t0 )|, t − t0 ) + γ( sup |Δw(τ )|). t0 ≤τ ≤t

(4)

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In general, the functions β(r, s) and γ(r) may depend on the particular input w(t). If β(r, s) and γ(r) are independent of the input w(t), then such system is called uniformly input-to-state convergent. Similar to the conventional ISS property, the property of input-to-state convergence is especially useful for studying convergence properties of interconnected systems as will be illuminated in the next section.

3 Basic Properties of Convergent Systems As follows from the previous section, the (uniform) convergence property and the input-to-state convergence property are extensions of stability properties of asymptotically stable LTI systems. In this section we present certain properties of convergent systems that are inherited from asymptotically stable LTI systems. Since all ingredients of the (uniform) convergence and the input-to-state convergence properties are invariant under smooth coordinate transformations (see Definitions 1, 3), we can formulate the following property. Property 2. The uniform convergence property and input-to-state convergence are preserved under smooth coordinate transformations. The next statement summarizes some properties of uniformly convergent systems excited by periodic or constant inputs. Property 3 ([3]). Suppose system (3) with a given input w(t) is uniformly convergent. If the input w(t) is constant, the corresponding limit solution z¯w (t) is also constant; if the input w(t) is periodic with period T , then the corresponding limit solution z¯w (t) is also periodic with the same period T . Proof. Suppose the input w(t) is periodic with period T > 0. Denote z˜w (t) := z¯w (t + T ). Notice that z˜w (t) is a solution of system (3). Namely, by the definition of z˜w (t), it is a solution of the system z˙ = F (z, w(t + T )) ≡ F (z, w(t)). Moreover, z˜w (t) is bounded for all t ∈ R due to boundedness of the limit solution z¯w (t). Therefore, by Property 1 it holds that z˜w (t) ≡ z¯w (t), i.e. the limit solution z¯w (t) is T -periodic. A constant input w(t) ≡ w∗ is periodic for any period T > 0. Hence, the corresponding limit solution z¯w (t) is also periodic with any period T > 0. This implies that z¯w (t) is constant.  If two inputs converge to each other, so do the corresponding limit solutions, as follows from the next property. Property 4. Suppose system (3) is uniformly convergent and F (z, w) is C 1 . Then for any two inputs w1 (t) and w2 (t) satisfying w1 (t) − w2 (t) → 0 as t → +∞, the corresponding limit solutions z¯w1 (t) and z¯w2 (t) satisfy z¯w1 (t) − z¯w2 (t) → 0 as t → +∞.

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Proof. See Appendix. The next two properties relate to parallel and series connections of uniformly convergent systems. Property 5 (Parallel connection). Consider the system z˙ = F (z, w), y˙ = G(y, w),

z ∈ Rd y ∈ Rq .

(5)

Suppose the z- and y-subsystems are uniformly convergent (input-to-state convergent). Then system (5) is uniformly convergent (input-to-state convergent). Proof. The proof directly follows from the definitions of uniformly convergent and input-to-state convergent systems.  Property 6 (Series connection). Consider the system z˙ = F (z, y, w), z ∈ Rd y˙ = G(y, w), y ∈ Rq .

(6)

Suppose the z-subsystem with (y, w) as input is input-to-state convergent and the y-subsystem with w as input is input-to-state convergent. Then system (6) is input-to-state convergent. Proof. See Appendix. The next property deals with bidirectionally interconnected input-to-state convergent systems. Property 7. Consider the system z˙ = F (z, y, w), y˙ = G(z, y, w),

z ∈ Rd y ∈ Rq .

(7)

Suppose the z-subsystem with (y, w) as input is input-to-state convergent. Assume that there exists a class KL function βy (r, s) such that for any input (z, w) ∈ PCd+m any solution of the y-subsystem satisfies |y(t)| ≤ βy (|y(t0 )|, t − t0 ).

Then the interconnected system (7) is input-to-state convergent. Proof. Denote z¯w (t) to be the limit solution of the z-subsystem corresponding to y = 0 and to some w ∈ PCm . Then (¯ zw (t), 0) is a solution of the interconnected system (7) which is defined and bounded for all t ∈ R. Performing the change of coordinates z˜ = z − z¯w (t) and applying the small gain theorem for ISS systems from [7] we establish the property.  Remark 1. Property 7 can be used for establishing the separation principle for input-to-state convergent systems as it will be done in Section 5. In that context system (7) represents a system in closed loop with a state-feedback controller and an observer generating state estimates for this controller. The y-subsystem corresponds to the observer error dynamics.

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4 Sufficient Conditions for Convergence In the previous sections we presented the definitions and basic properties of convergent systems. The next question to be addressed is: how to check whether a system exhibits these convergence properties? In this section we provide sufficient conditions for convergence for smooth systems. A simple sufficient condition for the exponential convergence property for smooth systems was proposed in [2] (see also [11]). Here we give a different formulation of the result from [2] adapted for systems with inputs and extended to include the input-to-state convergence property. Theorem 1. Consider system (3) with the function F (z, w) being C 1 with respect to z and continuous with respect to w. Suppose there exist matrices P = P T > 0 and Q = QT > 0 such that P

∂F T ∂F (z, w) + (z, w)P ≤ −Q, ∂z ∂z

∀z ∈ Rd , w ∈ Rm .

(8)

Then system (3) is exponentially convergent and input-to-state convergent. Proof. A complete proof of this theorem is given in Appendix. It is based on the following technical lemma, which we will use in Section 5. Lemma 1 ([2, 11]). Condition (8) implies (z1 − z2 )T P (F (z1 , w) − F (z2 , w)) ≤ −a(z1 − z2 )T P (z1 − z2 ).

(9)

for all z1 , z2 ∈ Rd , w ∈ Rm , and for some a > 0. We will refer to condition (8) as the Demidovich condition, after B.P. Demidovich, who applied this condition for studying convergence properties of dynamical systems [2, 3, 11]. In the sequel, we say that a system satisfies the Demidovich condition if the right-hand side of this system satisfies condition (8) for some matrices P = P T > 0 and Q = QT > 0. Example 1. Let us illustrate Theorem 1 with a simple example. Consider the system z˙1 = −z1 + wz2 + w

(10)

z˙2 = −wz1 − z2 .

The Jacobian of the right-hand side of system (10) equals J(z1 , z2 , w) =

−1 w −w −1

.

Obviously, J + J T = −2I < 0. Thus, the Demidovich condition (8) is satisfied for all z1 , z2 and w (with P = I and Q = 2I). By Theorem 1, system (10) is input-to-state convergent.

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The next example illustrates the differences between the input-to-state convergence and incremental ISS (δISS) defined in [1]. Example 2. Consider the scalar system z˙ = −z + w3 . As follows from [1], this system is not δISS. At the same time, by Theorem 1 this system is input-tostate convergent. Remark 2. In some cases feasibility of the Demidovich condition (8) can be concluded from the feasibility of certain LMIs. Namely, suppose there exist matrices A1 , . . . , As such that ∂F (z, w) ∈ co{A1 , . . . , As }, ∂z

∀ z ∈ Rd , w ∈ Rm ,

where co{A1 , . . . , As } is the convex hull of matrices A1 , . . . , As . If the LMIs P Ai + ATi P < 0,

i = 1, . . . , s

(11)

admit a common positive definite solution P = P T > 0, then condition (8) is satisfied with this matrix P . Taking into account the existence of powerful LMI solvers, this is a useful tool for checking convergence properties. In some cases, feasibility of the LMI (11) can be checked using frequency domain methods following from the Kalman-Yakubovich lemma (see [20, 8]). For example, one can use the circle criterion, as follows from the next lemma. Lemma 2 ([8, 20]). Consider a Hurwitz matrix A ∈ Rd×d , matrices B ∈ Rd×1 , C ∈ R1×d and some number γ > 0. Denote A− γ := A − γBC and T := A + γBC. There exists P = P > 0 such that A+ γ − T P A− γ + (Aγ ) P < 0,

+ T P A+ γ + (Aγ ) P < 0

if and only if the inequality C(iωI − A)−1 B
0,

Ai Pc + Pc ATi + Bi Y + Y T BiT < 0, i = 1, . . . , s

(15)

is feasible. Then the system x˙ = f (x, K(x + v), w),

(16)

with K := YPc−1 and (v, w) as inputs is input-to-state convergent. Proof. Denote F (x, v, w) := f (x, K(x+v), w). The Jacobian of the right-hand side of system (16) equals ∂f ∂f ∂F (x, v, w) := (x, K(x + v), w) + (x, K(x + v), w)K. ∂x ∂x ∂u Due to assumption A1, ∂F ∂x (x, v, w) ∈ co{(Ai + Bi K), i = 1, . . . , s} for all (x, v, w) ∈ Rn+n+m . Since the LMI (15) is feasible, for the matrix K := YPc−1 it holds that Pc−1 (Ai + Bi K) + (Ai + Bi K)T Pc−1 < 0,

i = 1, . . . , s.

Therefore, by Remark 2 the closed-loop system (16) satisfies the Demidovich condition with the matrix P := Pc−1 > 0. By Theorem 1 system (16) with  (v, w) as inputs is input-to-state convergent. The next lemma shows how to design an observer based on the Demidovich condition. Lemma 4. Consider system (14). Suppose the LMI Po = PoT > 0, Po Ai + ATi Po + X Ci + CiT X T < 0, i = 1, . . . , s

(17)

is feasible. Then the system x ˆ˙ = f (ˆ x, u, w) + L(h(ˆ x, w) − y),

with L := Po−1 X

(18)

is an observer for system (14) with a globally exponentially stable error dynamics. Moreover, the error dynamics Δx˙ = G(x + Δx, u, w) − G(x, u, w),

(19)

where G(x, u, w) := f (x, u, w) + Lh(x, w) is such that for any bounded x(t) and w(t) and any feedback u = U (Δx, t) all solutions of system (19) satisfy |Δx(t)| ≤ Ce−a(t−t0 ) |Δx(t0 )|,

(20)

where the numbers C > 0 and a > 0 are independent of x(t), w(t) and u = U (Δx, t).

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Proof. Let us first prove the second part of the lemma. The Jacobian of the right-hand side of system (19) equals ∂f ∂h ∂G (x + Δx, u, w) = (x + Δx, u, w) + L (x + Δx, w). ∂Δx ∂x ∂x Due to Assumption A1 it holds that ∂G (x + Δx, u, w) ∈ co{(Ai + LCi ), i = 1, . . . , s} ∂Δx for all x, u, w and Δx. Since the LMI (17) is feasible, for the matrix L := Po−1 X it holds that Po (Ai + LCi ) + (Ai + LCi )T Po < 0,

i = 1, . . . , s.

Therefore, by Remark 2 system (19) with inputs x, u and w satisfies the Demidovich condition with the matrix P := Po > 0 and some matrix Q > 0. Consider the function V (Δx) := 1/2ΔxT P Δx. By Lemma 1 the derivative of this function along solutions of system (19) satisfies dV = ΔxT P (G(x + Δx, u, w) − G(x, u, w)) ≤ −2aV (Δx). dt

(21)

In inequality (21) the number a > 0 depends only on the matrices P and Q and does not depend on the particular values of x, u and w. This inequality, in turn, implies that there exists C > 0 depending only on the matrix P such that if the inputs x(t) and w(t) are defined for all t ≥ t0 then the solution Δx(t) is also defined for all t ≥ t0 and satisfies (20). It remains to show that system (18) is an observer for system (14). Denote Δx := x ˆ − x(t). Since x(t) is a solution of system (14), Δx(t) satisfies equation (19). By the previous analysis, we obtain that Δx(t) satisfies (20). Therefore, the observation error Δx exponentially tends to zero.  Lemmas 3 and 4 show how to design a state feedback controller that makes the closed-loop system input-to-state convergent and an observer for this system with an exponentially stable error dynamics. In fact, for such controllers and observers one can use the separation principle in order to design an output feedback controller that makes the closed-loop system inputto-state convergent. This statement follows from the next theorem. Theorem 2. Consider system (14). Suppose LMIs (15) and (17) are feasible. Denote K := YPc−1 and L := Po−1 X . Then system (14) in closed loop with the controller x ˆ˙ = f (ˆ x, u, w) + L(h(ˆ x, w) − y), u = Kx ˆ with w as an input is input-to-state convergent.

(22) (23)

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Proof. Denote Δx := x ˆ −x. Then in the new coordinates (x, Δx) the equations of the closed-loop system are x˙ = f (x, K(x + Δx), w), Δx˙ = G(x + Δx, u, w) − G(x, u, w), u = K(x + Δx),

(24) (25) (26)

where G(x, u, w) = f (x, u, w) + Lh(x, w). By the choice of K, system (24) with (Δx, w) as inputs is input-to-state convergent (see Lemma 3). By the choice of the observer gain L, for any inputs x(t), w(t) and for the feedback u = K(x(t) + Δx), any solution of system (25), (26) satisfies |Δx(t)| ≤ Ce−a(t−t0 ) |Δx(t0 )|,

(27)

where the numbers C > 0 and a > 0 are independent of x(t) and w(t) (see Lemma 4). Applying Property 7 (see Section 3), we obtain that the closed-loop system (24)-(26) is input-to-state convergent.  Although the proposed controller and observer structures do not significantly differ from the ones proposed in literature, they achieve the new goal of rendering the closed-loop system convergent (as opposed to asymptotically stable). The output-feedback controller design presented in Theorem 2 relies on the separation principle which, in turn, is based on the input-to-state convergence of the system/state-feedback controller combination. This inputto-state convergence property serves as a counterpart of the input-to-state stability property often used to achieve separation of controller and observer designs in rendering the closed-loop system asymptotically stable (as opposed to convergent).

6 Conclusions In this paper we have extended the notion of convergent systems defined by B.P. Demidovich and introduced the notions of (uniformly, exponentially) convergent systems as well as input-to-state convergent systems. These notions are coordinate independent, which makes them more convenient to use than the notions of incremental stability and incremental input-to-state stability. We have presented basic properties of convergent systems and studied parallel, series and feedback interconnections of input-to-state convergent systems. These properties resemble the properties of asymptotically stable LTI systems. Due to this fact (input-to-state) convergent systems are convenient to deal with in many control and system analysis problems. We have presented a simple sufficient condition for the input-to-state convergence property. In certain cases this condition can be reduced to checking the feasibility of certain LMIs. Finally, for a class of nonlinear systems we have presented an (output) feedback controller that makes the closed-loop system input-to-state

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convergent. The presented controller consists of a state-feedback controller that makes the closed-loop system input-to-state convergent and an observer with an exponentially stable error dynamics. For such controllers and observers the separation principle holds. This allows us to unite the obtained controller and observer. The conditions for such controller and observer designs are formulated in terms of LMIs. The results presented in this paper are mostly for systems with smooth right-hand sides. Convergent systems with non-smooth and discontinuous right-hand sides are considered in [16, 12, 13]. Extensions of convergent systems to non-global settings, further convergence properties and controller design techniques as well as applications to the output regulation problem, controlled synchronization problem and nonlinear observer design problem can be found in [16]. Acknowledgement. This research has been supported by the Netherlands Organization for Scientific Research NWO.

Appendix Proof of Property 4. Consider two inputs w1 and w2 ∈ PCm satisfying w1 (t) − w2 (t) → 0 as t → +∞ and the corresponding limit solutions z¯w1 (t) and z¯w2 (t). By the definition of convergence, both z¯w1 (t) and z¯w2 (t) are defined and bounded for all t ∈ R. Consider the system Δz˙ = F (¯ zw2 (t) + Δz, w2 (t) + Δw) − F (¯ zw2 (t), w2 (t)).

(28)

This system describes the dynamics of Δz = z(t) − z¯w2 (t), where z(t) is some solution of system (3) with the input w2 (t) + Δw(t). Since F (z, w) ∈ C 1 , and z¯w2 (t) and w2 (t) are bounded, the partial derivatives ∂F (¯ zw2 (t) + Δz, w2 (t) + Δw), ∂z

∂F (¯ zw2 (t) + Δz, w2 (t) + Δw) ∂w

are bounded in some neighborhood of the origin (Δz, Δw) = (0, 0), uniformly in t ∈ R. Also, for Δw ≡ 0 system (28) has a uniformly globally asymptotically stable equilibrium Δz = 0. This implies (Lemma 5.4, [8]) that system (28) is locally ISS with respect to the input Δw. Therefore, there exist numbers kz > 0 and kw > 0 such that for any input Δw(t) satisfying |Δw(t)| ≤ kw for all t ≥ t0 and Δw(t) → 0 as t → +∞, it holds that any solution Δz(t) starting in |Δz(t0 )| ≤ kz tends to zero, i.e. Δz(t) → 0 as t → +∞. Choose t0 ∈ R such that |w1 (t) − w2 (t)| ≤ kw for all t ≥ t0 . Consider a solution of the system (29) z˙ = F (z, w1 (t)) starting in a point z(t0 ) satisfying |z(t0 ) − z¯w2 (t0 )| ≤ kz . By the reasoning presented above, Δz(t) := z(t) − z¯w2 (t) → 0 as t → +∞. At the same time,

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z¯w1 (t) is a uniformly globally asymptotically stable solution of system(29). Hence, z(t) − z¯w1 (t) → 0 as t → +∞. Therefore, z¯w2 (t) − z¯w1 (t) → 0 as t → +∞.  Proof of Property 6. Consider some input w ∈ PCm . Since the y-subsystem is input-to-state convergent, there exists a solution y¯w (t) which is defined and bounded for all t ∈ R. Since the z-subsystem with (y, w) as inputs is inputto-state convergent, there exists a limit solution z¯w (t) corresponding to the input (¯ yw (t), w(t)). This z¯w (t) is defined and bounded for all t ∈ R. Let (z(t), y(t)) be some solution of system (6) with some input w(t). ˜ Denote Δz := z − z¯w (t), Δy := y − y¯w (t) and Δw = w ˜ − w(t). Then Δz and Δy satisfy the equations Δz˙ = F (¯ zw (t) + Δz, y¯w (t) + Δy, w(t) + Δw) − F (¯ zw (t), y¯w (t), w(t)) Δy˙ = G(¯ yw (t) + Δy, w(t) + Δw) − G(¯ yw (t), w(t)).

(30) (31)

Since both the z-subsystem and the y-subsystem of system (6) are input-tostate convergent, system (30) with (Δy, Δw) as input is ISS and system (31) with Δw as input is ISS. Therefore, the cascade interconnection of ISS systems (30), (31) is an ISS system (see e.g. [19]). In the original coordinates (z, y) this means that system (6) is ISS with respect to the solution (¯ zw (t), y¯w (t)). This implies that system (6) is input-to-state convergent.  Proof of Theorem 1. The proof of exponential convergence can be found in [2, 11]. We only need to show that system (3) is input-to-state convergent. Consider some input w(t) and the corresponding limit solution z¯w (t). Let z(t) be a solution of system (3) corresponding to some input w(t). ˆ Denote Δz := z − z¯w (t) and Δw := w ˆ − w(t). Then Δz satisfies the equation Δz˙ = F (¯ zw (t) + Δz, w(t) + Δw) − F (¯ zw (t), w(t)).

(32)

We will show that system (32) with Δw as input is ISS. Due to the arbitrary choice of w(t), this fact implies that system (3) is input-to-state convergent. Consider the function V (Δz) = 12 (Δz)T P Δz. Its derivative along solutions of system (32) satisfies dV = Δz T P {F (¯ zw (t) + Δz, w(t) + Δw(t)) − F (¯ zw (t), w(t))} (33) dt zw (t) + Δz, w(t) + Δw(t)) − F (¯ zw (t), w(t) + Δw(t))} = Δz T P {F (¯ +Δz T P {F (¯ zw (t), w(t) + Δw(t)) − F (¯ zw (t), w(t))}.

Applying Lemma 1 to the first component in (33), we obtain Δz T P {F (¯ zw (t) + Δz, w(t) + Δw(t)) − F (¯ zw (t), w(t) + Δw(t))} ≤ −a|Δz|2P ,

(34)

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where |Δz|2P := (Δz)T P Δz. Applying the Cauchy inequality to the second component in formula (33), we obtain the following estimate: |Δz T P {F (¯ zw (t), w(t) + Δw(t)) − F (¯ zw (t), w(t))}| ≤ |Δz|P |δ(t, Δw)|P , (35) where

zw (t), w(t)). δ(t, Δw) := F (¯ zw (t), w(t) + Δw(t)) − F (¯

Since F (z, w) is continuous and z¯w (t) and w(t) are bounded for all t ∈ R, the function δ(t, Δw) is continuous in Δw uniformly in t ∈ R. This, in turn, implies that ρ˜(r) := supt∈R sup|Δw|≤r |δ(t, Δw)|P is a continuous nondecreasing function. Define the function ρ(r) := ρ˜(r) + r. This function is continuous, strictly increasing and ρ(0) = 0. Thus, it is a class K function. Also, due to the definition of ρ(r), we obtain the following estimate |δ(t, Δw)|P ≤ ρ(|Δw|). After substituting this estimate, together with estimates (35) and (34), in formula (33), we obtain dV ≤ −a|Δz|2P + |Δz|P ρ(|Δw|). dt

(36)

From this formula we obtain a dV ≤ − |Δz|2P , dt 2

∀ |Δz|P ≥

2 ρ(|Δw|). a

(37)

By the Lyapunov characterization of the ISS property (see e.g. [8], Theorem 5.2), we obtain that system (32) is input-to-state stable. This completes the proof of the theorem. 

References 1. D. Angeli. A Lyapunov approach to incremental stability properties. IEEE Trans. Automatic Control, 47:410–421, 2002. 2. B.P. Demidovich. Dissipativity of a nonlinear system of differential equations, part I. Vestnik Moscow State Univiersity, ser. matem. mekh., (in Russian), 6:19–27, 1961. 3. B.P. Demidovich. Lectures on stability theory (in Russian). Nauka, Moscow, 1967. 4. V. Fromion, S. Monaco, and D. Normand-Cyrot. Asymptotic properties of incrementally stable systems. IEEE Trans. Automatic Control, 41:721–723, 1996. 5. V. Fromion, S. Monaco, and D. Normand-Cyrot. A link between input-output stability and Lyapunov stability. Systems and Control Letters, 27:243–248, 1996. 6. V. Fromion, G. Scorletti, and G. Ferreres. Nonlinear performance of a PI controlled missile: an explanation. Int. J. Robust Nonlinear Control, 9:485–518, 1999.

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7. Z.-P. Jiang, A. Teel, and L. Praly. Small-gain theorem for ISS systems and applications. Math of Control, Signals, and Systems, 7:104–130, 1994. 8. H.K. Khalil. Nonlinear systems, 2nd ed. Prentice Hall, Upper Saddle River, 1996. 9. J.P. LaSalle and S. Lefschetz. Stability by Liapunov’s direct method with applications. Academic press, New York, 1961. 10. W. Lohmiller and J.-J.E. Slotine. On contraction analysis for nonlinear systems. Automatica, 34:683–696, 1998. 11. A. Pavlov, A. Pogromsky, N. van de Wouw, and H. Nijmeijer. Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Systems and Control Letters, 52:257–261, 2004. 12. A. Pavlov, A. Pogromsky, N. van de Wouw, and H. Nijmeijer. On convergence properties of piecewise affine systems. In Proc. of 5th EUROMECH Nonlinear Dynamics Conference, 2005. 13. A. Pavlov, A. Pogromsky, N. van de Wouw, and H. Nijmeijer. The uniform global output regulation problem for discontinuous systems. In Proc. of 16th IFAC World Congress, 2005. 14. A. Pavlov, N. van de Wouw, and H. Nijmeijer. The global output regulation problem: an incremental stability approach. In Proc. of 6th IFAC Symposium on Nonlinear Control Systems, 2004. 15. A. Pavlov, N. van de Wouw, and H. Nijmeijer. The uniform global output regulation problem. In Proc. of IEEE Conf. Decision and Control, 2004. 16. A.V. Pavlov. The output regulation problem: a convergent dynamics approach. PhD thesis, Eindhoven University of Technology, Eindhoven, 2004. 17. V.A. Pliss. Nonlocal problems of the theory of oscillations. Academic Press, London, 1966. 18. A. Pogromsky. Passivity based design of synchronizing systems. International J. Bifurcation Chaos, 8(2):295–319, 1998. 19. E.D. Sontag. On the input-to-state stability property. European J. Control, 1:24–36, 1995. 20. V.A. Yakubovich. Matrix inequalities method in stability theory for nonlinear control systems: I. absolute stability of forced vibrations. Automation and Remote Control, 7:905–917, 1964. 21. T. Yoshizawa. Stability theory by Liapunov’s second method. The Mathematical Society of Japan, Tokio, 1966. 22. T. Yoshizawa. Stability theory and the existence of periodic solutions and almost periodic solutions. Springer-Verlag, New York, 1975.

Smooth฀and฀Analytic฀Normal฀Forms: A฀Special฀Class฀of฀Strict฀Feedforward฀Forms Issa฀A.฀Tall1฀ and฀Witold฀Respondek2 1฀

Department฀of฀Mathematics,฀Tougaloo฀College,฀500฀W.฀County฀Line฀Road, Jackson,฀MS,฀39174,฀USA. [email protected],฀[email protected] 2฀ Laboratoire฀de฀Math´ematiques,฀INSA฀de฀Rouen,฀Pl.฀Emile฀Blondel,฀76131฀Mont Saint฀Aignan,฀France. [email protected] Summary.฀ In฀ this฀ paper฀ we฀ prove฀ that฀ any฀ smooth฀ (resp.฀ analytic)฀ strict฀ feedforward฀system฀can฀be฀brought฀into฀its฀normal฀form฀via฀a฀smooth฀(resp.฀analytic) feedback฀transformation.฀This฀allows฀us฀to฀identify฀a฀subclass฀of฀strict฀feedforward systems,฀called฀systems฀in฀special฀strict฀feedforward฀form,฀shortly฀(SSFF),฀possessing฀ a฀ normal฀ form฀ which฀ is฀ a฀ smooth฀ (resp.฀ analytic)฀ counterpart฀ of฀ the฀ formal Kang฀normal฀form.฀For฀(SSFF)-systems,฀the฀step-by-step฀normalization฀procedure leads฀to฀smooth฀(resp.฀convergent฀analytic)฀normalizing฀feedback฀transformations. We฀illustrate฀the฀class฀of฀(SSFF)-systems฀by฀a฀model฀of฀an฀inverted฀pendulum฀on฀a cart.

Keywords:฀Normal฀form,฀strict฀feedforward฀system,฀feedback฀transformations.

1 Introduction The problem of transforming the nonlinear control single-input system Π : x˙ = f (x, u),

x ∈ Rn , u ∈ R m

by an invertible feedback transformation of the form Γ :

z = φ(x) u = γ(x, v)

to a simpler form has been extensively studied during the last twenty years. The transformation Γ brings Π to the system ˜ : z˙ = f˜(z, v), Π

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 147–164, 2005. © Springer-Verlag Berlin Heidelberg 2005

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whose dynamics are given by f˜(z, v) = dφ(φ−1 (z)) · f (φ−1 (z), γ(φ−1 (z), v)). If the control u is not present, that is, the system Π is actually a dynamical system of the form x˙ = f (x), x ∈ Rn ,

then the transformation Γ consists solely of a change of coordinates z = φ(x). A classical problem addressed by Poincar´e is whether it is possible to find local coordinates around an equilibrium point in which the dynamical system x˙ = f (x) becomes linear. If this is the case the system is said to be linearizable around that equilibrium point. Poincar´e’s method to attack the problem was to simplify the linear part of the system by a linear change of coordinates and then to apply, step by step, homogeneous changes of coordinates in order to normalize the corresponding homogeneous parts of the same degree of the system. If all homogeneous parts can be annihilated, we formally linearize the system. If not, the result of this normalization procedure gives a formal normal form, which contains nonlinearizable terms only, called resonant terms, (see e.g. [1]). Similarly, for control systems, a natural question is whether we can take ˜ to be linear, i.e., whether we can linearize the system Π via the system Π feedback. Necessary and sufficient geometric conditions for this to be the case have been given in [7] and [10]. Those conditions turn out to be, except for the planar case, restrictive and a natural problem which arises is to find normal forms for non linearizable systems. Although being natural, this problem is very involved and has been extensively studied during the last twenty years (see, e.g., [4, 5, 11, 13, 14, 15, 18, 27, 44, 45] and the recent survey [31]). A very fruitful approach to obtain normal forms has been proposed by Kang and Krener [15] and then followed by Kang [13, 14]. Their idea is to analyze, step by step, the action of the Taylor series expansion of the feedback transformation Γ on the Taylor series expansion of the system Π, which is closely related with classical Poincar´e’s technique for linearization of dynamical systems (see e.g. [1]). Using that approach, results on normal forms of single-input control systems with controllable linearization have been obtained by Kang and Krener [15] for the quadratic terms, and then generalized by Kang [13] for higher order terms. The results of Kang and Krener [15, 13] have been completed by Tall and Respondek who obtained canonical forms and dual canonical forms for single-input nonlinear control systems with controllable [36, 37] and then with uncontrollable linearization [38] (see also [20]). Recently those results have been generalized by Tall [34, 35] to multi-input nonlinear control systems. Although these normal forms are formal, they are very useful in studying bifurcations of nonlinear systems [16], [17], in obtaining a complete description of symmetries around equilibria [28, 29], in characterizing systems equivalent to feedforward forms [40, 41, 39].

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Challenging questions are thus whether these normal forms have their counterparts in the C ∞ and analytic categories and what are conditions for the normalizing procedure to be convergent. In other words, what are obstructions for obtaining smooth and/or analytic normal forms for control systems? It is well known that the problem of convergence of the normalizing transformations is difficult already for dynamical systems. It was solved (in terms of locations of the eigenvalues of the linearization) by Sternberg and Chen in the C ∞ -category and by Poincar´e, Dulac, Siegel, and others in the C ω -category (see [1] for details and references). For control systems, the eigenvalues of the linearization are not invariant under feedback and the convergence problem seems to be even more involved. Kang [13] derived from [18], and [19] (see also [8]) that if an analytic control system is linearizable by a formal transformation, then it is linearizable by an analytic transformation. Kang [13] also gave a class of non linearizable 3-dimensional analytic control systems which are equivalent to their normal forms by analytic transformations. Those are the only results about the convergence of the step-by-step normalization transformations known to us to this date (see, however, the C ∞ -smooth and/or analytic normal forms of [4, 9, 11, 26, 32, 46] obtained via singularity theory methods). In this paper we study the problem of smooth (resp. analytic) normal forms for smooth (resp. analytic) strict feedforward systems. A single-input nonlinear control system of the form Π : x˙ = f (x, u), where x ∈ Rn and u ∈ R, is in strict feedforward form if we have (SF F )

x˙ 1 = f1 (x2 , . . . , xn , u) ... x˙ n−1 = fn−1 (xn , u) x˙ n = fn (u).

A basic structural property of systems in strict feedforward form is that their solutions can be found by quadratures. Indeed, knowing u(t) we integrate fn (u(t)) to get xn (t), then we integrate fn−1 (xn (t), u(t)) to get xn−1 (t), we keep doing that, and finally we integrate f1 (x2 (t), . . . , xn (t), u(t)) to get x1 (t). Notice that, in view of the above, systems in feedforward form can be considered as duals of flat systems. In the single-input case, flat systems are feedback linearizable and are defined as systems for which we can find a function of the state that, together with its derivatives, gives all the states and the control of the system [6]. In a dual way, for systems in feedforward form (SFF), we can find all states via a successive integration starting from a function of the control. Another property, crucial in applications, of systems in (strict) feedforward form is that we can construct for them a stabilizing feedback. This important result goes back to Teel [42] and has been followed by a growing literature

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on stabilization and tracking for systems in (strict) feedforward form (see e.g. [12, 22, 33, 43, 3, 23]). It is therefore natural to ask which systems are equivalent to (strict) feedforward form. In [21], the problem of transforming a system, affine with respect to controls, into (strict) feedforward form via a diffeomorphism, i.e., via a nonlinear change of coordinates, was studied. A geometric description of systems in feedforward form has been given in [2]. Another approach has been used by the authors who have proposed a step-by-step constructive method to bring a system into a strict feedforward form in [40, 39] and feedforward form in [41]. Recently the authors [30] have shown that feedback equivalence (resp. statespace equivalence) to the strict feedforward form can be characterized by the existence of a sequence of infinitesimal symmetries (resp. strong infinitesimal symmetries) of the system. We will show in this paper that any smooth (resp. analytic) strict feedforward system can be brought to its normal form via an smooth (resp. analytic) feedback transformation. This will allow us to identify a subclass of strict feedforward systems, called special strict feedforward systems, possessing a normal form which is a smooth (resp. analytic) counterpart of the Kang normal form. The paper is organized as follows. In Section 2 we will recall the Kang normal form for single-input systems. Our main results: normal forms for strict feedforward and special strict feedforward systems are given in Section 3 and their proofs in Section 5. We illustrate our results by transforming a cart-pole system into the special strict feedforward normal form in Section 4

2 Notation and Definitions All objects, i.e., functions, maps, vector fields, control systems, etc., are considered in a neighborhood of 0 ∈ Rn and assumed to be C ∞ -smooth (or real analytic, if explicitly stated). Let h be a smooth function. By h(x) = h[0] (x) + h[1] (x) + h[2] (x) + · · · =

∞

h[m] (x)

m=0

we denote its Taylor expansion around zero, where h[m] (x) stands for a homogeneous polynomial of degree m. Similarly, for a map φ of an open subset of Rn to Rn (resp. for a vector field f on an open subset of Rn ) we will denote by φ[m] (resp. by f [m] ) the [m] term of degree m of its Taylor expansion at zero, i.e., each component φj of [m]

φ[m] (resp. fj of f [m] ) is a homogeneous polynomial of degree m in x. Consider the Taylor series expansion of the single-input system Π, given by Π ∞ : x˙ = f (x, u) = F x + Gu +

∞

f [m] (x, u),

m=2

(1)

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151

∂f where F = ∂f ∂x (0) and G = ∂u (0). We will assume throughout the paper that f (0, 0) = 0 and that the linear approximation around the origin is controllable. Consider also the Taylor series expansion Γ ∞ of the feedback transformation Γ given by

Γ

∞

:

z = φ(x) = T x +

∞

φ[m] (x)

m=2

∞

u = γ(x, v) = Kx + Lv +

γ [m] (x, v),

(2)

m=2

where the matrix T is invertible and L = 0. The action of Γ ∞ on the system Π ∞ step by step leads to the following normal form obtained by Kang [13] (see also [15] and [36]) Theorem 1. The control system Π ∞ , defined by (1), is feedback equivalent, by a formal transformation Γ ∞ of the form (2), to the formal normal form ∞ ΠN F : z˙ = Az + Bv +

∞

f¯[m] (z, v),

m=2

where (A, B) is the Brunovsky canonical form and for any m ≥ 2, we have ⎧ n+1 [m−2] ⎪ ⎪ (z1 , . . . , zi ), 1 ≤ j ≤ n − 1, zi2 Pj,i ⎨ [m] i=j+2 ¯ fj (z, v) = (3) ⎪ ⎪ ⎩ 0, j = n, [m−2]

being homogeneous polynomials of degree m − 2 of the indicated with Pj,i variables, and zn+1 = v.

The problem whether an analogous result holds in the smooth (resp. analytic) category is actually a challenging question, which can be formulated as whether for a smooth (resp. analytic) system Π the normalizing feedback transformation Γ ∞ gives rise to a smooth (resp. convergent) Γ and thus leads to a smooth (resp. analytic) normal form ΠN F . One of difficulties resides in the fact that it is not clear at all how to express, in terms of the original system, homogeneous invariants transformed via an infinite composition of homogeneous feedback transformations. We will study in this paper a special class of smooth (resp. analytic) control systems, namely special strict feedforward systems, that can be brought to their normal form by smooth (resp. analytic) transformations.

3 Main Results Consider the class of smooth (resp. analytic) single-input control systems Π : x˙ = f (x, u),

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either locally in a neighborhood X × U of (0, 0) ∈ Rn × R or globally on Rn × R, in strict feedforward form (SFF), that is, such that (SF F )

fj (x, u) = fj (xj+1 , . . . , xn , u),

1 ≤ j ≤ n.

Notice that for any 1 ≤ i ≤ n, the subsystem Πi , defined as the projection of Π onto Rn−i via π(x1 , . . . , xn ) = (xi+1 , . . . , xn ), is a well defined system whose dynamics are given by x˙ j = fj (xj+1 , . . . , xn , u) for i + 1 ≤ j ≤ n. Define the linearizability index of the (SFF)-system to be the largest integer p such that the subsystem Πr , where p + r = n, is feedback linearizable. Clearly, the linearizability index is feedback invariant and hence the linearizability indices of two feedback equivalent systems coincide. Notice that each component of a strict feedforward system (SFF) decomposes uniquely, locally or globally, as: fj (x, u) = hj (xj+1 ) + Fj (xj+1 , . . . , xn , u), for 1 ≤ j ≤ n (we put Fn = 0), where

Fj (xj+1 , 0, . . . , 0) = 0.

(4)

(5)

A strict feedforward form for which hj (xj+1 ) = kj xj+1 ,

1 ≤ j ≤ r − 1,

(6)

for some non zero real numbers k1 , . . . , kr−1 , will be called a special strict feedforward form (SSFF). The first main result of this paper is as follows. Theorem 2. Consider a smooth (resp. analytic) special strict feedforward form (SSFF) given by (4)-(5)-(6) in a neighborhood of (0, 0) ∈ Rn × R. There exists a smooth (resp. analytic) local feedback transformation that brings the system (4)-(5)-(6) into the normal form

ΠSSF N F :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

z˙1 = z2 +

n+1 i=3

zi2 P1,i (z2 , . . . , zi )

... z˙j = zj+1 +

⎪ ⎪ ⎪ z˙r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z˙r+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z˙n−1 z˙n

n+1 i=j+2

zi2 Pj,i (zj+1 , . . . , zi )

... = zr+1 + = zr+2 ... = zn = v,

n+1

zi2 Pr,i (zr+1 , . . . , zi ) i=r+2

(7)

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153

where Pj,i (zj+1 , . . . , zi ) are smooth (resp. analytic) functions of the indicated variables and zn+1 = v. Moreover, if the system is defined globally on Rn ×R, then so are the feedback transformation and the normal form. A proof of this result follows from Theorem 4 completed by feedback linearization of the subsystem Πr . A main observation is that the above normal form ΠSSF N F given by (7) is itself a (SSFF)-system and, on the other hand, it constitutes a smooth ∞ (resp. analytic) counterpart ΠN F of the formal normal form ΠN F given by Theorem 1 (see a detailed comparison of various normal forms at the end of this section). It is therefore a natural question to ask whether it is always possible to transform a strict feedforward form, given by (4)-(5), into a special strict feedforward form (4)-(5)-(6). Consider another analytic system ˜ : z˙ = f˜(z, v), Π in strict feedforward form (SFF), that is, such that

where

˜ j (zj+1 ) + F˜j (zj+1 , . . . , zn , v), 1 ≤ j ≤ n f˜j (z, u) = h

(8)

F˜j (zj+1 , 0, . . . , 0) = 0.

(9)

It is in the special strict feedforward form (SSFF) if ˜ j (zj+1 ) = k˜j zj+1 , h

1 ≤ j ≤ r˜ − 1

(10)

for some non zero real numbers k˜1 , . . . , k˜r˜−1 . Theorem 3. If two analytic (SFF)-systems given by, respectively, (4)-(5) and (8)-(9) are feedback equivalent, then r = r˜ and ˜ j (lj+1 zj+1 ) = lj hj (zj+1 ), h

1 ≤ j ≤ r − 1,

for some non zero real numbers l1 , . . . , lr−1 . Corollary 1. A strict feedforward system (SFF), given by (4)-(5), is feedback equivalent to the special strict feedforward form (SSFF), given by (8)-(9)-(10), if and only if hj (xj+1 ) = kj xj+1 , for 1 ≤ j ≤ r − 1, that is, the (x1 , . . . , xr−1 )-part of the system is already in (SSFF) in its original coordinates. Notice that the terms hj (xj+1 ) for 1 ≤ j ≤ n are feedback linearizable (ac¯ j form exactly the feedback linearizable terms of the form ΠSF N F tually h given by Theorem 4). Basically, Theorem 3 and Corollary 1 imply that the linearizable terms hj (xj+1 ), for 1 ≤ j ≤ r − 1, of a strict feedforward form

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(SFF) cannot be linearized (unless they are already linear) via any feedback transformation that preserves the strict feedforward structure of the system. This means that special strict feedforward forms (SSFF) define the only subclass of strict feedforward systems that can be brought to the Kang normal form ΠN F still being in the strict feedforward form. Whether it is possible to bring a (SFF)-system into its normal form ΠN F by an analytic transformation is unclear but if true, then the normal form ΠN F will loose the structure of (SFF) (unless the system is (SSFF)). On the other hand, any strict feedforward form (SFF) can be brought to a form ΠSF N F , called strict feedforward normal form (introduced by the authors in [40] in the formal category), which is close as much as possible to the normal form ΠN F . Indeed, we have the following second main result of the paper (which implies Theorem 2): Theorem 4. Any smooth (resp. analytic) strict feedforward form (SFF), given by (4)-(5) in a neighborhood of (0, 0) ∈ Rn × R, is equivalent via a smooth (resp. analytic) local feedback transformation to the following strict feedforward normal form (SFNF) ⎧ n+1 ⎪ ¯ 1 (z2 ) + ⎪ z ˙ zi2 P1,i (¯ zi ) = h ⎪ 1 ⎪ ⎪ i=3 ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ n+1 ⎪ ⎨ ¯ r (zr+1 ) + zi2 Pr,i (¯ zi ) z˙r = h ΠSF N F : (11) i=r+2 ⎪ ⎪ ¯ ⎪ z˙r+1 = hr+1 (zr+2 ) ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¯ n (v) , z˙n = h

¯ j (zj+1 ), for 1 ≤ j ≤ r − 1 with lj ∈ R, where zn+1 = v, hj (lj+1 zj+1 ) = lj h and Pj,i (¯ zi ) = Pj,i (zj+1 , . . . , zi ) are smooth (resp. analytic) functions of the indicated variables. Moreover, if the system is defined globally on Rn × R, then so are the feedback transformation and the normal form.

In order to explain relations between various results proved in this section, let us recall that according to Theorem 1, we can bring the infinite Taylor ∞ expansion Π ∞ of a control system into its formal normal form ΠN F: f ormal

∞ Π ∞ −−−−−−−→ ΠN F.

Theorem 4 says that any smooth (resp. analytic) strict feedforward system can be transformed into its strict feedforward normal forms via a smooth (resp. analytic) feedback transformation, locally or globally: C∞

ΠSF F −−−−ω−−→ ΠSF N F . C

In particular, the last transformation can be applied to a smooth (resp. analytic) special strict feedforward system ΠSSF F (such systems form a subclass

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155

of strict feedforward systems) yielding its special strict feedforward normal form ΠSSF N F which (after having linearizaed the subsystem Πr , see Theorem 2) actually is the smooth (resp. analytic) normal form ΠN F of the system (in other words, smooth (resp. analytic) counterpart of the formal normal form ∞ ΠN F ): C∞

ΠSSF F −−−−ω−−→ ΠSSF N F = ΠN F . C

Provided that the linear approximation is controllable, the linearizability index of a general (SFF)-system on R2 is at least one while the linearizability index of a general control-affine system on R3 is at least two. It follows that in those two cases the functions hj are not invariant (compare Theorem 3), which implies the following: Corollary 2. Any smooth (resp. analytic) strict feedforward form (SFF) on R2 , given by (4)-(5), is feedback equivalent to the normal form ΠSSF N F :

z˙1 = z2 + v 2 P1,3 (z2 , v) z˙2 = v,

where P1,3 is a smooth (resp. analytic) function of the indicated variables. Any smooth (resp. analytic) control-affine strict feedforward (SFF) on R3 is feedback equivalent to the normal form ΠSSF N F

z˙1 = z2 + z32 P1,3 (z2 , z3 ) : z˙2 = z3 z˙3 = v.

where P1,3 is a smooth (resp. analytic) function of the indicated variables. Normal forms for strict feedforward systems on R2 with noncontrollable linearization are given in [30].

4 Example: Cart-Pole System In this example we consider a cart-pole system that is represented by a cart with an inverted pendulum on it [24, 25]. The Lagrangian equations of motion for the cart-pole system are q1 + m2 l cos(q2 )¨ q2 = m2 l sin(q2 )q˙22 + F (m1 + m2 )¨ cos(q2 )¨ q1 + l¨ q2 = g sin(q2 ), where m1 and q1 are the mass and position of the cart, m2 , l, q2 ∈ (−π/2, π/2) are the mass, length of the link, and angle of the pole, respectively. Taking q¨2 = u and applying the feedback law (see [24]) F = − ul(m1 + m2 sin2 (q2 ))/ cos(q2 )

+ (m1 + m2 )g tan(q2 ) − m2 l sin(q2 )q˙22 ,

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the dynamics of the cart-pole system are transformed into x˙ 1 = x2 , x˙ 3 = x4 ,

x˙ 2 = g tan(x3 ) − lu/cos(x3 ) x˙ 4 = u,

where, for convenience, we take x1 = q1 , x2 = q˙1 , x3 = q2 , and x4 = q˙2 . This system is in special strict feedforward form with the linearizability index p = 2. The feedback transformation defined by (compare [25]) x3

z1 = x1 + l 0

ds cos s

x4 cos x3 z3 = g tan x3 x4 z4 = g 2 cos x3 v = gu/ cos2 (x3 ) + 2gx3 x4 sin(x3 )/ cos3 (x3 )

z2 = x2 + l

takes the system into the special strict feedforward normal form z˙1 = z2 , z˙3 = z4 ,

z˙2 = z3 + z˙4 = v.

lz3 z2 (g2 +z32 )3/2 4

5 Proofs In this section we will prove our results: Theorem 4 and then Theorem 3. The following lemmas play a key role in the proof of Theorems 3 and 4. Lemma 1. (i) A smooth system in (SFF) remains in (SSF) after applying any smooth change of coordinates z = Φ(x) and feedback v = γ(x, u), defined by z1 = Φ1 (x) = x1 + φ1 (x2 , . . . , xn ) z2 = Φ2 (x) = x2 + φ2 (x3 , . . . , xn ) ... zn−1 = Φn−1 (x) = xn−1 + φn−1 (xn )

(12)

zn = Φn (x) = xn + φn (xn ) v = γ(x, u) = (1 + φn (xn )) u, where for any 1 ≤ j ≤ n,

φj (0) = 0

and

dφj (0) = 0.

(13)

(ii) Moreover, if the transformation (12)-(13) takes the system (4)-(5) into (8)-(9), then ˜ j (zj+1 ), for all 1 ≤ j ≤ n − 2. hj (zj+1 ) = h The proof of this lemma is straightforward. Indeed,

Smooth and Analytic Normal Forms n

z˙j = x˙ j + k=j+1 n

+ k=j+1

157

∂φj x˙ k = hj (xj+1 ) + Fj (xj+1 , . . . , xn , u) ∂xk

∂φj hk (xk+1 ) + Fk (xk+1 , . . . , xn , u) . ∂xk

Since the inverse x = Φ−1 (z) = z+ψ(z) is such that xj = zj +ψj (zj+1 , . . . , zn ), the claim follows from the fact that Fk (xk+1 , 0, . . . , 0) = 0, j ≤ k ≤ n. Lemma 2. Given any integer 2 ≤ i ≤ n, suppose that Θ1i (x2 , . . . , xi ), i (xi ) are smooth (resp. analytic). Then there exist Θ2i (x3 , . . . , xi ), . . . , Θi−1 smooth (resp. analytic) functions φ1 (x2 , . . . , xi ), φ2 (x3 , . . . , xi ), . . . , φi−1 (xi ) such that ∂φj ∂φj ∂φj i i + · · · + Θi−1 + = −Θji (14) Θj+1 ∂xj+1 ∂xi−1 ∂xi for any 1 ≤ j ≤ i−1. Moreover, φj (0) = 0 and, if Θji (0) = 0, then dφj (0) = 0. If the functions Θji , for 1 ≤ j ≤ i − 1, are defined globally, then so are the solutions φj . Local solvability is obvious. Global solvability follows easily from the fact that ∂ ∂ ∂ i i the vector field f = Θj+1 ∂xj+1 + · · · + Θi−1 ∂xi−1 + ∂xi is globally rectifiable. 5.1 Proof of Theorem 4 Let us consider a system in the strict feedforward form (SFF), given by (4)-(5), with the linearizability index p, and let r = n − p. n By a series of linear transformations replacing xj by i=j aji xi , where aji ∈ R (and applied successively from j = n up to j = 1), we arrive at the system x˙ j = hj (xj+1 ) + Fj (xj+1 , . . . , xn , u), for 1 ≤ j ≤ n, where hj (xj+1 ), and Fj (xj+1 , . . . , xn , u) are smooth (resp. analytic) functions such that hj (xj+1 ) = xj+1 + x2j+1 bj (xj+1 ) Fj (xj+1 , 0, . . . , 0) = 0, dFj (0) = 0, for any 1 ≤ j ≤ n (recall that, by definition, xn+1 = u and that Fn = 0). We can decompose each function Fj , for 1 ≤ j ≤ n − 1, uniquely as Fj (xj+1 , . . . , xn , u) =Fj,n (xj+1 , . . . , xn ) + uΘjn (xj+1 , . . . , xn ) + u2 Qj,n+1 (xj+1 , . . . , xn , u) with Fj,n (xj+1 , 0, . . . , 0) = 0.

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Then we look for a change of variables z1 = x1 + φ1 (x2 , . . . , xn ), z2 = x2 + φ2 (x3 , . . . , xn ), ... zn−1 = xn−1 + φn−1 (xn ), zn = xn

(15)

that annihilates the linear terms in u, that is, transforms the system into z˙j = hj (zj+1 ) + Fˆj,n (zj+1 , . . . zn ) + u2 Pj,n+1 (zj+1 , . . . , zn , u), z˙n−1 = hn−1 (zn ) + u2 Pn−1,n+1 (zn , u), z˙n = hn (u), for 1 ≤ j ≤ n − 2, where Fˆj,n and Pj,n+1 are suitable functions with Fˆj,n (zj+1 , 0, . . . , 0) = 0 for any 1 ≤ j ≤ n − 1. Since hj (xj+1 ) = xj+1 + x2j+1 bj (xj+1 ) and n

z˙j = x˙ j + k=j+1

∂φj x˙ k = hj (xj+1 ) + Fj (xj+1 , . . . , xn , u) ∂xk n

+ k=j+1

∂φj hk (xk+1 ) + Fk (xk+1 , . . . , xn , u), , ∂xk

it follows that the components of the transformation (15) should satisfy n Θj+1

∂φj ∂φj ∂φj n + · · · + Θn−1 + = −Θjn ∂xj+1 ∂xn−1 ∂xn

for any 1 ≤ j ≤ n − 1. Applying Lemma 2 (with i = n) we can find smooth (resp. analytic) functions φj (xj+1 , . . . , xn ), φj (0) = 0, that solve the above partial differential equations. We then conclude that we can annihilate linear terms in the variable u. Now let us assume that, for some 2 ≤ i ≤ n, we have transformed the system into the form x˙ j = hj (xj+1 ) + Fj (xj+1 , . . . , xn , u),

(16)

with n+1

Fj (xj+1 , . . . , xn , u) = Fj,i+1 (xj+1 , . . . , xi+1 ) +

x2k Pj,k (xj+1 , . . . , xk ), k=i+2

where xn+1 = u, and Fj,i+1 (xj+1 , 0, . . . , 0) = 0, and the functions Pj,k are identically zero when k ≤ j. We decompose Fj,i+1 (xj+1 , . . . , xi+1 ) as

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159

Fj,i+1 (xj+1 , . . . , xi+1 ) = Fj,i (xj+1 , . . . , xi ) + xi+1 Θji (xj+1 , . . . , xi ) + x2i+1 Qj,i+1 (xj+1 , . . . , xi+1 ). Following Lemma 2, we can find smooth (resp. analytic) functions φ1 , . . . , φi−1 , φ(0) = 0, such that i Θj+1

∂φj ∂φj ∂φj i + · · · + Θi−1 + = −Θji , ∂xj+1 ∂xi−1 ∂xi

for any 1 ≤ j ≤ i − 1. It then follows that the change of variables zj = xj + φj (xj+1 , . . . , xi ),

1≤j ≤i−1

zj = xj ,

i≤j ≤n

applied to the system, annihilates the linear terms in xi+1 . Since the variables xi , . . . , xn , u are unchanged, the structure of the terms x2k Pj,k (xk+1 , . . . , xk ) remains unchanged for all k ≥ i. Thus, this change of coordinates takes the system into the form z˙j = hj (zj+1 ) + Fˆj (zj+1 , . . . , zn , u),

(17)

for 1 ≤ j ≤ n, with Fˆj (zj+1 , . . . , zn , u) = Fˆj,i (zj+1 , . . . , zi ) +

n+1

zk2 Pj,k (zj+1 , . . . , zk ), k=i+1

where zn+1 = u, and Fˆj,i (zj+1 , 0, . . . , 0) = 0, and the functions Pj,k being identically zero when k ≤ j. Applying recursively the above procedure (starting from i = n down to i = 2) we arrive at the desired form ¯ j (zj+1 ) + z˙j = h

n+1

zi2 Pj,i (zj+1 , . . . , zi )

(18)

i=j+2

for any 1 ≤ j ≤ n. We may remark that there is a finite number of coordinates changes (actually n − 1) and all changes are smooth (resp. analytic) thus implying that normalizing transformations are smooth (resp. analytic) in the whole domain where the system is defined. Now recall that the linearizability index is p and hence the subsystem Πr , given by (18), for r + 1 ≤ j ≤ n is feedback linearizable. Prolong it by adding u˙ = v to get a feedback linearizable control-affine system. Then the involutiveness of the linearizability distributions (see, e.g., [10]) implies that all Pj,i , for r + 1 ≤ j ≤ n, j + 2 ≤ i ≤ n + 1 vanish identically. Finally, notice that the only transformation changing hj is the preliminary ¯ j (zj+1 ), where lj = ajj .  linear transformation giving hj (lj+1 zj+1 ) = lj h

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5.2 Proof of Theorem 3 Consider two analytic systems in the form (4)-(5) and (8)-(9), and assume that they are feedback equivalent. Using Theorem 4 we may assume that (4)-(5) is in the form n+1

x2i Pj,i (¯ xi )

ΠSF N F : x˙ j = hj (xj+1 ) +

(19)

i=j+2

where hj (xj+1 ) and Pj,i (¯ xi ) = Pj,i (xj+1 , . . . , xi ) are analytic functions of the indicated variables, and xn+1 = u, and that (8)-(9) is in the form ˜ j (zj+1 ) + ˜ SF N F : z˙j = h Π

n+1

zi2 P˜j,i (¯ zi )

(20)

i=j+2

˜ j (zj+1 ) and P˜j,i (¯ zi ) = P˜j,i (zj+1 , . . . , zi ) are analytic functions of the where h indicated variables, and zn+1 = v. Moreover, by Theorem 4, the functions ˜j hj (xj+1 ) of (4) are replaced in (19) by lj hj (xj+1 /lj+1 ) and analogously for h of (8). Recall that (A, B) denote the Brunovsk´ y canonical form and m0 the degree of the first nonlinearizable term. Let us consider the Taylor series expansion of (19), ∞ ΠSF ˙ = Ax + Bu + NF : x

∞

h[m] (x) +

m=2

where for any m ≥ m0 [m] f¯j (x, u) =

and for any m ≥ 2,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

n+1

[m−2]

i=j+2

x2i Pj,i

∞

f¯[m] (x, u)

m=m0

(xj+1 , . . . , xi ), 1 ≤ j < n,

0,

j=n

[m]

hj (x) = hm,j xm j+1 , 1 ≤ j ≤ n.

We also consider Taylor series expansion of (20), ˜∞ Π SF N F : z˙ = Az + Bu +

∞

˜ [m] (z) + h

m=2

where for any m ≥ m0 [m] f˜j (z)

=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

n+1 i=j+2

0,

zi2 P˜j,i

[m−2]

∞

f˜[m] (z, v),

m=m0

(zj+1 , . . . , zi ), 1 ≤ j < n, j=n

Smooth and Analytic Normal Forms

161

and for any m ≥ 2, [m] m ˜ , 1 ≤ j ≤ n. hj (z) = ˜hm,j zj+1

By Lemma 1(ii), it is enough to prove that any change of coordinates z = Φ(x) = x + φ(x) and feedback v = γ(x, u) that take the system (19) into (20) is of the form (12)-(13). Suppose this is not true. Consider the Taylor series expansion of this feedback transformation

Γ

∞

∞

z = Φ(x) = x +

:

φ[m] (x)

m=2 ∞

v = γ(x, u) = u +

γ [m] (x, u),

(21)

m=2

and let l be the largest nonnegative integer such that for 2 ≤ m ≤ l, we have [m]

[m]

φj (x) = φj (xj+1 , . . . , xn ), 1 ≤ j ≤ n − 1

φn[m] (x) = φn[m] (xn ) [m]

γ [m] (x, u) =

dφn u. dxn

Because of Lemma 1 we can assume, without loss of generality, that φ[m] = 0 for 2 ≤ m ≤ l. This means that ∞

z = Φ(x) = x + Γ

∞

:

φ[m] (x)

m=l+1 ∞

γ [m] (x, u).

v = γ(x, u) = u +

m=l+1

Since (21) takes the (SFF) into a (SFF), it structurally preserves (see [39] or [40] for the definition) the terms of degree smaller than m0 + l − 1. This means that Γ ∞ = Γ ≥m0 +l ◦ Γ ≥l+1 with [≥l+2]

z1 = x1 + σ1 xl+1 + φ1 (¯ x2 ) 1 [≥l+2] l+1 z2 = x2 + σ1 LAx xl+1 + σ x + φ2 (¯ x3 )) 2 2 1 ... j

zj = xj +

k=1

Γ ≥l+1 :

... zn = xn + v = − −

n k=1 n k=1

[≥l+2]

l+1 + φj σk Lj−k Ax xk

n k=1

(¯ xj ) (22)

n−k l+1 σk LAx xk

+

[≥l+2] φn (¯ xn )

n−k+1 l+1 σk LAx xk n−k+1 l+1 σk LB LAx xk u + γ [≥l+2] (x, u)

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that preserves structurally terms of degree smaller than m0 + l − 1 of (19). Moreover, Γ ≥l+1 brings the homogeneous terms of degree m0 + l of (19) into h[m0 +l] (z) + f¯[m0 +l] (z) +

n

σk [f¯[m0 ] , Yk

[l+1]

],

k=1

where [l+1]

Yk

= zkl+1

∂ ∂ n−k l+1 ∂ + LAz zkl+1 + · · · + LAz zk . ∂zk ∂zk+1 ∂zn

The transformation Γ ≥m0 +l should preserve structurally terms of degree m0 + n [l+1] σk [f¯[m0 ] , Y ] into the strict feedforward l, which means transforming k

k=1

form. This is possible if and only if σ1 = · · · = σn−1 = 0. The latter implies that [l+1]

φj γ

(x) = 0, 1 ≤ j ≤ n − 1

φ[l+1] (x) n [l+1]

= σn xl+1 n

(x, u) = (l + 1)σn xln u.

This contradicts the definition of l and hence prove that the feedback transformation that takes the system (19) into (20) is of the form (12)-(13). The proof then follows from Lemma 1 (ii). 

References 1. V. I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations, Second Edition, Springer-Verlag, 1988. 2. A. Astolfi and F. Mazenc. A geometric characterization of feedforward forms, in Proc. MTNS, Perpignan, France, 2000. 3. S. Battilotti. Semiglobal stabilization of uncertain block-feedforward forms via measurement feedback, in Proc. NOLCOS, Enschede, the Netherlands, (1998), pp. 342-347. 4. B. Bonnard. Quadratic control systems, Mathematics of Control, Signals, and Systems, 4 (1991), pp. 139–160. 5. R. W. Brockett. Feedback invariants for nonlinear systems, Proc. IFAC Congress, Helsinski, 1978. 6. M. Fliess, J. L´evine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples, Int. Journal of Control, 6 (1995), pp. 1327-1361. 7. L. R. Hunt and R. Su. Linear equivalents of nonlinear time varying systems, in: Proc. MTNS, Santa Monica, CA, (1981), pp. 119-123. 8. A. Isidori. Nonlinear Control Systems, Second Edition, New York: SpringerVerlag, 1989.

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9. B. Jakubczyk. Equivalence and Invariants of Nonlinear Control Systems, in Nonlinear Controllability and Optimal Control, H.J. Sussmann (ed.), Marcel Dekker, New York-Basel, (1990), pp. 177-218. 10. B. Jakubczyk and W. Respondek. On linearization of control systems, Bull. Acad. Polon. Sci. Ser. Math., 28 (1980), pp. 517-522. 11. B. Jakubczyk and W. Respondek. Feedback classification of analytic control systems in the plane, in: Analysis of Controlled Dynamical Systems, B. Bonnard et al. (eds.), Birkh¨ auser, Boston, (1991), pp. 262–273. 12. M. Jankovic, R. Sepulchre, and P. Kokotovic. Constructive Lyapunov stabilization of nonlinear cascade systems, IEEE TAC, 41 (1996), pp. 1723-1735. 13. W. Kang. Extended controller form and invariants of nonlinear control systems with single input, J. Math. Sys, Estim. and Control, 4, (1994), pp. 1-25. 14. W. Kang. Quadratic normal forms of nonlinear control systems with uncontrollable linearization, Proc. 34th IEEE CDC, New Orleans, (1995). 15. W. Kang and A.J. Krener. Extended quadratic controller normal form and dynamic feedback linearization of nonlinear systems, SIAM J. Control and Optim., 30 (1992), pp. 1319-1337. 16. W. Kang. Bifurcation and normal form of nonlinear control systems − part I and II, SIAM J. Control and Optim., 36, (1998), 193-212 and 213-232. 17. W. Kang. Bifurcation control via state feedback for systems with a single uncontrollable mode, SIAM J. Control and Optim., 38, (2000), 1428-1452. 18. A.J. Krener. Approximate linearization by state feedback and coordinate change, Systems and Control Letters, 5 (1984), pp. 181-185. 19. A. J. Krener, S. Karahan, M. Hubbard, and R. Frezza. Higher order linear approximations to nonlinear control systems, Proc. 26th IEEE CDC, Los Angeles, (1987) pp. 519-523. 20. A. J. Krener, W. Kang, and D. E. Chang. Control bifurcations, IEEE TAC, 49 (2004), pp. 1231-1246. 21. A. Marigo. Constructive necessary and sufficient conditions for strict triangularizability of driftless nonholonomic systems,in Proc. 34th IEEE CDC, Phoenix, Arizona, USA, (1999), pp. 2138-2143. 22. F. Mazenc and L. Praly. Adding integrations, saturated controls, and stabilization for feedforward forms, IEEE TAC, 41 (1996), pp. 1559-1578. 23. F. Mazenc and L. Praly. Asymptotic tracking of a reference state for systems with a feedforward structure, Automatica, 36 (2000), pp. 179-187. 24. R. Olfati-Saber. Fixed point controllers and stabilization of the cart-pole system and the rotating pendulum, Proc. 38th IEEE CDC, Phoenix, AZ, 1999, pp. 1174-1181. 25. R. Olfati-Saber. Normal forms for underactuated mechanical systems with symmetry, IEEE TAC, 47 (2002), pp. 305-308. 26. W. Respondek. Feedback classification of nonlinear control systems in R2 and R3 , in: Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek (eds.), Marcel Dekker, New York, (1998), pp. 347-382. 27. W. Respondek. Transforming a single input system to a p-normal form via feedback, Proc. 42nd IEEE CDC, Maui, Hawai, pp. 1574-1579. 28. W. Respondek and I.A. Tall. How Many Symmetries Does Admit a Nonlinear Single-Input Control System around Equilibrium, in Proc. 40th IEEE CDC, pp. 1795-1800, Florida, (2001). 29. W. Respondek and I.A. Tall. Nonlinearizable single-input control systems do not admit stationary symmetries, Systems and Control Lett., 46 (2002), pp 1-16.

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30. W. Respondek and I.A. Tall. Strict feedforward form and symmetries of nonlinear control systems, Proc. 43th IEEE CDC, Atlantis, Bahamas. 31. W. Respondek and I.A. Tall. Feedback equivalence of nonlinear control systems: a survey on formal approach, to apper in Normal Forms, Bifurcations and Chaos in Automatic: From Theory Towards Applications, W. Perruquetti and J.-P. Barbot (eds.), Marcel Dekker, 2005. 32. W. Respondek and M. Zhitomirskii. Feedback classification of nonlinear control systems on 3-manifolds, Mathematics of Control, Signals and Systems, 8 (1995), pp. 299-333. 33. R. Sepulchre, M. Jankovi´c, and P. Kokotovi´c. Constructive Nonlinear Control, Springer, Berlin-Heidelberg-New York, 1996. 34. I.A. Tall. Normal Forms of Multi-Inputs Nonlinear Control Systems with Controllable Linearization, in New Trends in Nonlinear Dynamics and Control, and their Applications, W. Kang, M. Xiao, C. Borges (eds.), LNCIS vol. 295, Springer, Berlin-Heidelberg, 2003, pp. 87-100. 35. I.A. Tall. Feedback Classification of Multi-Inputs Nonlinear Control Systems, submitted in SIAM J. Control and Optim.. 36. I.A. Tall and W. Respondek. Feedback classification of nonlinear single-input control systems with controllable linearization: normal forms, canonical forms, and invariants, in SIAM J. Control and Optim., 41 (5), pp 1498-1531, (2003). 37. I.A. Tall and W. Respondek. Normal forms, canonical forms, and invariants of single-input control systems under feedback, Proc. 39th IEEE CDC, Sydney, (2000), pp. 1625-1630. 38. I.A. Tall and W. Respondek. Normal forms and invariants of nonlinear singleinput systems with noncontrollable linearization, NOLCOS, Petersburg, Russia, (2001). 39. I.A. Tall and W. Respondek. Feedback Equivalence to a Strict Feedforward Form for Nonlinear Single-Input Systems, to appear in Int. Journal of Control. 40. I.A. Tall and W. Respondek. Transforming a Single-Input Nonlinear System to a Strict Feedforward Form via Feedback, Nonlinear Control in the Year 2000, A. Isidori, F. Lamnabhi, and W. Respondek, (eds.), Springer-Verlag, 2, pp. 527-542, London, England, (2001). 41. I.A. Tall and W. Respondek. Feedback Equivalence to Feedforward Form for Nonlinear Single-Input Systems, Dynamics, Bifurcations and Control, F. Colonius and L. Gr¨ une (eds.), LNCIS, 273, pp. 269-286, Springer-Verlag, Berlin Heidelberg, (2002). 42. A. Teel. Feedback stabilization: nonlinear solutions to inherently nonlinear problems, Memorandum UCB/ERL M92/65. 43. A. Teel. A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE TAC, 41 (1996), pp. 1256-1270. 44. M. Zeitz. Controllability canonical (phase-variable) form for non-linear timevariable systems, Int. J. Control, 37 ( 1983), pp. 1449-1457. 45. M. Zeitz. Canonical forms for nonlinear systems, Nonlinear Control Systems Design, A. Isidori (ed.), IFAC-Symp., Pergamon Press, Oxford, 1989, pp. 33-38. 46. M. Zhitomirskii and W. Respondek. Simple germs of corank one affine distributions, Banach Center Publications, 44, (1998), 269-276.

Constructive฀Nonlinear฀Dynamics฀– Foundations฀and฀Application฀to฀Robust Nonlinear฀Control Johannes฀Gerhard,฀Martin฀M¨ onnigmann,฀and฀Wolfgang฀Marquardt Lehrstuhl฀f¨ ur฀Prozesstechnik,฀RWTH฀Aachen฀University,฀Templergraben฀55, D-52056฀Aachen,฀Germany. {gerhard,moennigmann,marquardt}@lpt.rwth-aachen.de Summary.฀ This฀paper฀summarizes฀two฀recently฀presented฀strategies฀based฀on฀bifurcation฀theory฀and฀nonlinear฀dynamics฀that฀address฀robust฀control฀design฀of฀nonlinear฀systems.฀The฀first฀strategy฀applies฀numerical฀bifurcation฀analysis฀to฀the฀robust controller฀tuning฀of฀uncertain฀nonlinear฀systems.฀Continuation฀in฀the฀controller฀and system฀parameters฀is฀used฀to฀compute฀stability฀boundaries฀for฀the฀controller฀tuning parameters.฀Restrictions฀on฀the฀controller฀tuning฀are฀determined฀that฀must฀hold฀to guarantee฀stability฀for฀the฀complete฀operating฀range฀of฀the฀closed฀loop฀system.฀The second฀strategy,฀the฀recently฀presented฀optimization-based฀Constructive฀Nonlinear Dynamics฀(CNLD)฀extends฀ideas฀from฀nonlinear฀dynamics฀and฀bifurcation฀theory฀to address฀the฀synthesis฀rather฀than฀analysis฀problems.฀CNLD฀is฀based฀on฀constraints that฀enforce฀a฀minimal฀back฀off฀from฀critical฀manifolds.฀Critical฀manifolds฀are฀boundaries฀in฀the฀space฀of฀system฀and฀controller฀ parameters฀that฀separate฀regions฀with qualitatively฀different฀system฀behavior,฀e. g.฀a฀region฀ with฀stable฀operating฀points from฀a฀region฀with฀unstable฀system฀behavior.฀The฀generality฀of฀the฀concept฀of฀critical฀manifolds฀permits฀to฀address฀feasibility,฀stability฀and฀performance฀of฀a฀nonlinear (closed-loop)฀ system.฀The฀back฀off฀from฀ the฀critical฀ manifolds฀ admits฀to฀robustly account฀ for฀ parametric฀ uncertainty฀ during฀ the฀ design฀ step฀ and฀ to฀ guarantee฀ the operability฀of฀the฀optimal฀solution.฀For฀closed-loop฀systems฀the฀approach฀admits฀simultaneous฀plant฀and฀control฀system฀design.฀This฀work฀summarizes฀the฀background and฀the฀basic฀ideas฀of฀both฀approaches.฀The฀application฀to฀robust฀nonlinear฀control design฀is฀illustrated฀by฀several฀case฀studies. Keywords:฀Robust฀control,฀nonlinear฀dynamics,฀parametric฀uncertainty,฀stability, optimization.

1฀Introduction Mathematical฀models฀employed฀in฀systems฀and฀control฀design฀are฀subject฀to uncertainties฀ and฀ usually฀ do฀ not฀ match฀ the฀ real฀ systems฀ perfectly.฀ Due฀ to tighter฀ profit฀ margins฀ and฀ stricter฀ safety฀ or฀ environmental฀ regulations฀ advanced฀ control฀ strategies฀ have฀ to฀ account฀for฀ nonlinearity฀ and฀ uncertainty.

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 165–182, 2005. © Springer-Verlag Berlin Heidelberg 2005

166

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The main task in robust control design is to find a suitable controller for a nonlinear system that guarantees stability and satisfactory performance of the closed-loop system in the presence of uncertainties. A large number of robust control approaches for uncertain nonlinear systems can be found in the literature. Frequently, models are linearized around the desired operating point. After linearization mature methods of robust linear control theory (e. g. [23]) or simple tuning rules for PID type controllers [26] can be applied. Neglecting system nonlinearities, however, may lead to instability because of disturbances or set point variations. To overcome the limitations of robust controllers based on linearization, a large variety of nonlinear control design techniques have been developed. Robust nonlinear control methods include e. g. Lyapunov-based methods [1, 7], the construction of robust Lyapunov control functions [8] and nonlinear H∞ control [15]. A review of several robust control methods can be found, for example, in [18]. Certainly, there is not a single approach which is capable to solve all problems arising in robust nonlinear control. Most of the robust control methods assume a specific structure of the model uncertainties. For example, so called matching conditions have to be satisfied (see e. g. [16]). The methods presented in this paper address the task of robust nonlinear control from the perspective of bifurcation theory. Similar to control theory, bifurcation theory investigates the stability of nonlinear systems though both research areas have been developed independently from each other. Two bifurcation based strategies for the robust control of nonlinear systems have been suggested by the authors in recent years. The first strategy is an analysis based method [13, 14]. It applies bifurcation analysis to closed-loop nonlinear systems by considering controller parameters, set points, as well as system parameters as bifurcation parameters. This way, controller settings can be determined that result in stable operation of the closed-loop system for a specified degree of model uncertainty. The second approach called Constructive Nonlinear Dynamics (CNLD) is optimization based. It has originally been developed by the authors for the robust design of nonlinear process systems for which stability and feasibility in the presence of parametric uncertainty must be guaranteed [20, 21]. With some extensions, however, this method has also successfully been applied for robust design of closed-loop systems [9, 10, 22]. The method is based on imposing additional constraints on the system to guarantee a specified distance from critical boundaries in the space of uncertain parameters. As will be shown below the general concept of critical boundaries allows to consider robust stability, feasibility, as well as robust performance of closed loop systems. It should be mentioned here that other optimization based methods exist that also address robust system and control design under uncertainty by additional constraints. Lyapunov matrix equations of the linearized system, e.g., are used in [3] as additional constraints to ensure that all eigenvalues of the linearized system stay in the open left half of the complex plane.

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167

The paper is organized as follows. The next section specifies the class of nonlinear closed-loop systems that are addressed and states the assumptions that must hold in order to apply the approaches. Section 2.2 briefly reviews bifurcation theory. In Sect. 3 the analysis based controller tuning procedure is introduced and shortcomings inherent to all analysis based approaches are illustrated. Section 4 sketches the basic idea of CNLD and presents two illustrative case studies from chemical and automotive engineering. Section 4.3 shows an extension of the approach to include a broader class of uncertainties.

2 Preliminaries In this section the problem class considered in this paper is introduced and a short review on bifurcation analysis is given. 2.1 Nonlinear Systems with Parametric Uncertainty We treat nonlinear uncertain dynamical systems that can be represented by a system of ordinary differential equations (ODE) with uncertain parameters. Consider a nonlinear open-loop system of the form x˙ = f (x, u, d, ϑ) ,

y = h(x, u, d, ϑ) ,

(1)

where x, u, d, ϑ and y represent the nx state variables, nu manipulated input variables, nd time-varying bounded disturbances, nϑ potentially uncertain parameters and ny outputs. The functions f are smooth and map from a subset of Rnx ×Rnp ×Rnu ×Rnϑ into Rnx . In case of closed loop systems, nu controller equations ˆ u = k(d, y, ysp , ϑ, ϑ) are introduced in (1) to define the manipulated input variables. The control law is a function of the disturbances, the parameters and the outputs of the open-loop system. Additionally the set points ysp and control parameters ϑˆ are introduced. Incorporating this control law into (1) results in the closed-loop system ˆ , y = h(x, ˆ d, ysp , ϑ, ϑ) ˆ . x˙ = fˆ(x, d, y, ysp , ϑ, ϑ) (2) Next we introduce some restrictions that must hold at this stage of development of CNLD for control system design. Most importantly, the system is supposed to be driven by inputs or disturbances which are slow in comparison to the time scales of the system (2). Specifically, we assume that the dynamics of the disturbances d (and variations in the set point ysp ) can be partitioned into constant means and bounded time-dependent variations according to di (t) = d¯i + d˜i (t),

d˜li ≤ d˜i (t) ≤ d˜ui ,

∀ i,

(3)

and that the variations d˜i (t) are much slower than the time scales of the system (2), i. e.

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1 d(d˜i ) d˜i dt

|Re(λj )|, ∀ i, j ,

(4)

where Re(λj ) denote the real parts of the eigenvalues λj of the Jacobian of fˆ evaluated at steady state. In other words di (t) and yi,sp (t) are expected to vary only quasi-statically with respect to the system dynamics. In this case di (t) and yi,sp can be modeled by a constant uncertain parameter, rather than by a time-varying function. The parameters ϑ, ϑˆ are also potentially subject to uncertainty complying with equations of type (3) and (4). Thus, with these assumptions, set point variations, disturbances, as well as system and control T , dT , ϑT , ϑˆT ). parameters can be concatenated into one new vector η T = (ysp Elimination of the outputs in (2) finally results in x˙ = f (x, η) ,

(5)

to represent the closed loop behavior of the system. The choice of a candidate robust controller must ensure that a desired system behavior, e. g. stability or performance, can be guaranteed for all values of the uncertain parameters η. It should be noted that conditions (3) and (4) are quite restrictive since only systems with quasi-stationary dynamics can be considered. Current research work aims at relaxing these assumptions — see also Sect. 4.3. 2.2 Brief Review of Bifurcation Analysis In this section, basic concepts of bifurcation analysis are introduced. We consider models of type (5) with constant (or slowly varying) parameters η. Bifurcation analysis by means of parameter continuation is an established method which has been used in engineering for roughly three decades. The popularity of bifurcation analysis can be attributed to the development of reliable numerical parameter continuation methods [17] which enable the analysis of large-scale nonlinear systems. Today these mature numerical methods are implemented in various software packages, e. g. AUTO 2000 [6]. Early applications of numerical bifurcation analysis treated low order nonlinear models such as the frequently studied continuous stirred tank reactor (CSTR) [27]. Thanks to the continuous improvements of the available methods complex and industrially relevant systems can be considered today. Continuation analysis permits to find and analyze critical boundaries that divide the space of the parameters η into regions where stationary operating points of the system f (x, η) = 0 exhibit qualitatively different behavior. Most frequently, bifurcation analysis is used to detect stability boundaries due to saddle node and Hopf points. In numerical bifurcation analysis local stability of each stationary point is determined by evaluating the eigenvalues of the linearized model along the curve of steady states. Generally a stability boundary corresponds to those steady states for which at least one eigenvalue crosses the imaginary axis. A manifold composed of such critical steady states then separates the region

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with only stable steady states from the region with unstable steady states. Note that this stability boundary is valid for the nonlinear model itself even though its computation involves linearization of the model. Critical manifolds of stability boundaries are computed by applying numerical parameter continuation to augmented systems of the nonlinear model (5) with additional equations g˜. The augmented system for saddle node points, e. g., reads fxT v = 0. (6) f = 0 , g˜sn = T v v−1

Here fxT denotes the transpose of the Jacobian of system (5) with respect to x and v ∈ Rnx the left eigenvector corresponding to the eigenvalue λ = 0. For augmented systems of other critical points the reader is referred to [20]. As elaborated in [20] stability boundaries are a special instance of a critical manifold. Other types of boundaries which mark qualitative changes in system behavior, such as feasibility boundaries, can also be described by critical manifolds. All critical manifolds can be cast into the form M c = {(x, x˜, η) ∈ U : 0 = f (x, η), 0 = g˜(x, x˜, η)}

(7)

with auxiliary variables x˜ ∈ Rnx˜ required for the defining equations g˜(x, x˜, η) of the particular type of critical manifold. In the augmented system of the saddle node type bifurcation (6), e. g., the nx components of the eigenvector v are the auxiliary variables x ˜. For stability and feasibility boundaries g˜ consists of nx˜ + 1 equations. The augmented system (7) thus determines a steady state with nx states, one parameter η1 and nx˜ auxiliary variables that exhibits the dynamic properties defined by the augmented system. An extension to stability boundaries that is interesting for controller design is a boundary which confines both the real part σ and the imaginary part ω of the leading eigenvalues to a specified sector in the open left half of the complex plane. This type of boundary is related to pole placement regions which are frequently used in control theory, see e. g. [4].

3 Analysis-based Robust Control Tuning Numerical bifurcation analysis can be used to assess robust stability for nonlinear systems. We assume that a control system structure has been decided upon and the main sources of model uncertainty have been identified. Parameter continuation is performed to compute stability boundaries of the closedloop systems which separate the regions with stable steady states from those with unstable behavior in the parameter space. Bifurcation parameters include controller parameters, the set point of the controller, as well as the parameters describing the mismatch between the model and the real system. By considering the set point to be a bifurcation parameter, the analysis can be carried out over an entire operating region of the system rather than for a particular value of the set point. The resulting stability diagrams can then

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be used to determine by visual inspection the region for the controller tuning parameters for which the closed-loop system remains stable over the entire operating range in the presence of model uncertainty. Hahn et al. [13, 14] apply the analysis based approach to the robust controller tuning for a temperature controlled CSTR considering both parametric and model uncertainties. Bifurcation analysis reveals a lower limit for the parameter tuning in the presence of unmodeled dynamics and an upper limit for the controller tuning in case parametric uncertainty is considered. The critical manifolds provide information what parameter tuning is required to guarantee stability over the entire operating region. 3.1 Limitations of the Analysis-based Approach In general models of real physical systems are complex and comprise a larger number of uncertain parameters. In such a case the main drawback of the analysis becomes apparent. As an analysis relies on the visualization and interpretation of the results, the approach is in principle restricted to two or three dimensions, i. e. two or three parameters. If variations of more than three parameters are to be analyzed, a collection of diagrams has to be created. This procedure becomes time consuming and tedious when a larger number of parameters nη is involved. Assume for simplicity that for each parameter involved a fixed number of points along its axis nap is required for the computation and visualization of the diagrams. The total number of grid points ngp required for the analysis is then ngp = nnapη .

(8)

Obviously the required computational effort grows exponentially with the number of considered parameters, making a thorough analysis too expensive for problems with a larger number of variable parameters.

4 CNLD for Control System Design As pointed out all methods based on analysis, including the robust controller tuning procedure, become infeasible for a larger number of parameters. M¨ onnigmann and Marquardt [20, 21] have developed a new approach that is not based on the explicit evaluation of the critical boundaries in the parameter space but merely on the distance between a candidate operating point and the closest critical point on the determining critical boundaries. Initially the method was developed to address robust design of process systems in the presence of parametric uncertainty. Due to its generality, however, the method can also be applied to robust control problems for nonlinear systems with given control structure [9, 10, 22]. This section first gives a short introduction into the theoretical background of CNLD. The concept of parametric distance based on the normal space of the critical boundaries and its application to the formulation of constraints in

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process optimization are briefly presented. If all determining critical manifolds in the vicinity of the candidate operating point are included, the constraints guarantee robustness of the optimum in the sense that parametric uncertainty cannot lead to undesired process behavior. In Sect. 4.2 the application of the method to robust control of nonlinear systems is elaborated and two illustrative case studies are presented. 4.1 Constraints Based on Parametric Distance The distance between a candidate operating point and the closest point on a critical manifold is measured along the normal direction in the space of parameters. Fig. 1 presents a sketch of a stability boundary in a two-dimensional parameter space (η1 , η2 ). Note that the normal space will be one-dimensional for any dimension of the parameter space. The effort required to compute the normal direction of a critical boundary increases linearly in the number of parameters considered. Therefore an approach based on the distance to critical boundaries is not limited by the number of parameters. stable

 unstable




unstable

stable r













Fig. 1. Left: Surface of steady states and stability boundary in (η1 , η2 , x)–space; right: projection of the stability boundary on the (η1 , η2 )-plane. The closest connection from a candidate operating point to the critical manifold is along direction r which is normal to the critical manifold

Figure 1 illustrates the idea of robustness of systems with uncertain parameters. Assume first that η1 and η2 are known exactly. In this sketch any point to the right of the stability boundary corresponds to stable process behavior even if this point is arbitrarily close to the boundary. If η1 , η2 are uncertain, the operating point has to remain in the stable region for all combinations of values the uncertain parameters (η1 , η2 ) can attain. In Fig. 1 (right) a typical uncertainty region is indicated by the circle around a nominal operating point. In order to distinguish the uncertain parameters from the others, η is split into parameters p ∈ Rnp which are known exactly and parameters α ∈ Rnα which are subject to uncertainty. System (5) then reads x˙ = f (x, α, p) .

(9)

It is common practice to describe parametric uncertainty by specifying upper and lower bounds

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J. Gerhard, M. M¨ onnigmann, and W. Marquardt (0)

αi ∈ αi

(0)

− Δαi , αi

+ Δαi ,

i = 1 . . . nα .

(10)

Scaling of the parameters with the intervals Δαi results in the dimensionless parameters and uncertainty regions (0)

αi ∈ αi

(0)

− 1, αi

+1 ,

i = 1 . . . nα .

(11)

The hypercube of length 2 defined by (11) can be overestimated by the nα √ dimensional hyperball of radius nα . For a 2-dimensional parameter space, as displayed in Fig. 1 (right), the minimal distance can be visualized by a (0) (0) circle around a nominal operating point (α1 , α2 ). This circle may touch the critical boundary or stay off it but must not cross it. The shortest distance √ l = nα between the critical manifold and the operating point occurs along the direction of the normal vector r to the critical manifold, cf. Fig. 1. Thus the robustness condition of the operating point can be stated as α(0) = α(i) + l(i)

r(i) , r(i)

l(i) ≥

√ nα .

(12)

In general, more than one critical manifold exists. An upper index (i) is introduced to enumerate the critical manifolds M (c,i) and all related quantities. If (12) holds for all closest critical points on M (c,i) , i = 1 . . . I, the system will not cross any of the critical boundary despite parametric uncertainty. As detailed in [21] the approach is not restricted to simple uncertainty descriptions by upper and lower bounds. It is shown that a general description of uncertainty can be realized if the boundary of the uncertainty region is described by a set of smooth functions M r . Such a robustness manifold M r can be defined analogously to the critical manifolds (7). For general M r , the minimal distance constraints (12) must ensure that the distance between the locally closest points on the robustness manifold and the critical manifold is larger than or equal to zero. For simplicity, however, only parametric uncertainty as described by (11) is considered here. The use of normal vectors to find the parametrical closest critical point to a fixed candidate point was first suggested by Dobson [5]. M¨onnigmann and Marquardt [20] presented a scheme for the derivation of normal vector systems of minimal order for more general types of boundaries. The augmented system required for evaluating the normal direction to a critical manifold consists of the defining equations of the specific critical manifold (7) and some additional equations. As an illustrative example we state here the equations defining the normal direction r ∈ Rnα for a critical manifold of saddle node points G(sn) =

eq. (6) fαT v − r

= 0,

(13)

where fα denote the partial derivatives of the system equations with respect to the uncertain parameters α. According to the general derivation scheme the

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normal direction r to the general critical manifold M (c,i) can be calculated by equations of the form G(c,i) x(i) , x ˜(i) , α(i) , p(0) , r(i)

= 0,

(14)

where G(c,i) is defined on some appropriate subset of, and maps into, Rnx × Rnx˜ × Rnr × Rnp . For further details on the structure of G(c,i) for different types of critical manifolds the reader is referred to [20]. The minimal distance conditions (12) and the defining equations of the normal vector (14) can now be used as parametric robustness constraints in the following nonlinear program: min

x(0) ,α(0) ,p(0)

s. t.

φ(x(0) , α(0) , p(0) )

(15a)

0 = f (x(0) , α(0) , p(0) ) ,

(15b)

0 = G(c,i) (x(i) , x ˜(i) , α(i) , p(0) , r(i) ) , 0 = α(i) − α(0) + l 0 ≤ l(i) −

√ nα ,

r(i) , r(i)

(15c)

i = 1, . . . , I .

The objective φ(x, α, p) in (15a) measures profitability of system operation at a nominal operating point x(0) which is fixed by the parameters α(0) and p(0) of the closed-loop system. Equations (15b) ensure that the operating point is a steady state of the system. The constraints (15c) enforce the minimal back off between the operating point and the closest critical point on M (c,i) . Hence the optimization problem may simultaneously determine the operating point x(0) and the parameters α(0) , p(0) of the closed-loop system. Parameters of the plant and the controller are not distinguished between, thus facilitating the integration of plant and control design in a seamless way. In the case of robust control tuning, the nominal design parameters are fixed and the control parameters are the only degrees of freedom in the optimization problem (15). In this case the objective φ could refer to a quadratic tracking functional. Obviously, normal vector constraints can only be formulated for critical manifolds whose location is already known, e. g. through previous analysis studies. For most of the models, however, data on critical manifolds are not available from the literature and a compulsory analysis step prior to the application of the constructive methods certainly would foil the fundamental idea of the method that was developed to avoid the tedious and expensive analysis procedure. M¨onnigmann and Marquardt [22] propose an algorithm where critical manifolds are detected as the optimization proceeds. Suitable test functions for the detection of critical manifolds are known from numerical bifurcation theory [17]. Assuming that the location of the determining critical manifolds is not known a priori, optimization has to start without any

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J. Gerhard, M. M¨ onnigmann, and W. Marquardt

robustness constraints (15c). The test functions are used to detect previously unknown critical manifolds along the optimization path. Once detected, minimal distance constraints for the known manifolds (15c) are added to (15). By repeatedly solving the optimization problem and monitoring for new critical points the set of known critical manifolds can be built up iteratively. If no further critical manifolds are detected along the optimization path the robustness region has to be examined rigorously for critical manifolds which were not crossed by the nominal operating point during optimization, but which nevertheless might exist inside the robustness region. If both tests reveal no further critical manifolds an optimal operating point that is robust with respect to the specified parametric uncertainty is found and the algorithm terminates. Several technical details need to be resolved for an implementation of the method and algorithm sketched here. Most importantly higher order derivatives of the nonlinear system (9) are required for the setup of the defining normal vector equations (14) and corresponding minimal distance constraints. Furthermore partial derivatives of these constraints with respect to the optimization variables have to be calculated for any gradient based NLP solver. The derivatives required for stating the normal vector constraints are currently evaluated by symbolic and automatic differentiation in MAPLE. MAPLE is also used for assembling the constrained NLP (15), which is exported with MAPLE’s codegen device to FORTRAN code. The derivatives required by the NLP solver – currently NPSOL [12] is used – are calculated by automatic differentiation of the FORTRAN code with ADIFOR [2]. The search for critical points inside the robustness region is implemented by applying the test functions to a set of sample points inside the robustness region. 4.2 Robust Controller Design In this application of CNLD to robust controller design we consider the closedloop system (9) with system parameters partitioned into known and uncertain parameters p and α, respectively. According to the discussion in Sect. 2, the uncertain parameters α typically correspond to disturbances d, set point variations ysp and uncertain system parameters ϑ, whereas the controller parameters ϑˆ are considered to be parameters p not affected by uncertainty. Depending on the application, different types of critical manifolds and corresponding minimal distance constraints have to be taken into account. Robust stability of a nonlinear closed-loop system in the presence of parametric uncertainty can be guaranteed by including minimal distance constraints to manifolds of Hopf and saddle node points. Robust performance can be addressed by minimal back off from critical manifolds bounding the leading eigenvalues. Robust feasibility is achieved by considering minimal distance constraints to manifolds defined by feasibility constraints. Assuming that all determining critical manifolds are taken into account, solving (15) results in a robust system and controller design which is optimal with respect to the objective φ.

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Temperature Controlled CSTR As an illustrative example a case study considering a cooled CSTR with an exothermic first order reaction A → B with temperature control and perfect level control [13] is presented. The CSTR model consists of nonlinear state equations stemming from material and energy balances including reaction kinetics and heat transfer. The temperature of the cooling fluid Tc is the manipulated variable and the reactor temperature T is the controlled variable. Temperature control is realized by means of linearizing state feedback. The control law comprises one tuning parameter ε which corresponds to the time constant of the closed-loop dynamics, i. e. the smaller ε, the faster the dynamics of the closed-loop system. If the model used for the controller design and the real plant exactly match, the controller stabilizes the process for all possible set point temperatures Tsp . In such a case ε can be tuned to arbitrary small values resulting in very fast system dynamics. In real applications, however, there will always exist mismatch between the plant and the model. In this case study, we consider parametric uncertainty in the feed rate. In addition unmodeled dynamics as a second source of model mismatch is introduced into the model by an overdamped second order process between the controller output u and the manipulated variable Tc with parameter εv to represent the time constant of the fast unmodeled dynamics. The analysis presented in [13] shows that a critical manifold of a particular type of bifurcation points, so called nontransversal (NT) Hopf points, separates the parameter space into two regions with qualitatively different process behavior. In one region the process is unstable for some values of the set point temperature Tsp while in the other region the process is stable for all variations of Tsp . CNLD is used to find an optimal operating point and a robust controller design to guarantee stability of the CSTR over the complete range of operating conditions in the presence of parametric uncertainty and unmodeled dynamics. As the determining critical manifold is already known from the previous analysis the search for new critical points along the optimization path and inside the robustness region is not required in this case study. By backing off from the manifold of NT Hopf points stability of the CSTR can be guaranteed for the complete operating region despite parametric uncertainty. In this application, the profit function φ of eq. (15a) is the yield of the reactor and the constraints (15b) – (15c) consist of the steady state closed-loop equations of the CSTR and minimal distance constraints to the manifold of NT Hopf points. For details on the definition of the normal vector equations for the manifold of NT Hopf points the reader is referred to [9]. The feed rate q and the time constant of the unmodeled dynamics εv are considered to be uncertain parameters α. The robustness ball in Fig. 2 (right) overestimates the uncertainties Δα1 = Δq = 10 L min−1 and Δα2 = Δεv = 0.01 min. For reference the optimization without the robustness constraints (15c) has been carried out. As can be inferred from Fig. 2 (left) the optimal operating point ( ) lies in the unstable region and the process becomes un-

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J. Gerhard, M. M¨ onnigmann, and W. Marquardt

15

15

A

q /Δq

0.5

unstable

unstable

0.5

stable 10

0

C

CA

q /Δq

stable (dotted line) due to a Hopf bifurcation () in a range of values of Tsp . Optimization with robustness constraints yields a controller setting (×) that guarantees stability over the entire region of Tsp as displayed in Fig. 2 (right) despite the parametric uncertainty in q and εv .

stable

5

εv/ Δεv

0 300 350 400 Tsp

10

0

5

ε / Δε v

v

0 300 350 400 Tsp

Fig. 2. Left: optimization of the CSTR without robustness constraints (optimal operating point marked by ); right: optimization with robustness constraints (optimal operating point marked by ×)

Automotive Body Control We will now apply CNLD to the robust controller design problem for a vehicle with active yaw control [10]. From previous investigations [24] it is known that vehicle models exhibit a saddle node bifurcation for deteriorating road conditions, beyond which trajectories become divergent causing the vehicle to fall into spin. The vehicle model considered here is based on the nonlinear single track model [19] including a tyre model [25]. A system of three ODEs is used to describe the dynamics of the vehicle side slip v, the vehicle yaw rate r and the roll angle ϕ. Obviously, road conditions play a crucial role for the dynamics of a vehicle. Here, the important case of different road conditions for the right and left side of the vehicle is modeled by considering two road adherence coefficients μr and μl . A yaw controller and a roll controller are employed. The aim of the body controller is to track the body yaw rate reference signal rref . This signal is a function of the longitudinal velocity u and the road steering angle δn which in turn is proportional to the hand wheel angle δhwa . An upper bound δˆhwa is assumed to ensure the lateral acceleration resulting from the yaw rate reference rref not to exceed the physical limit of gravity acceleration g. Yaw rate tracking is realized by two simple saturated P-controllers. The saturation nonlinearity has been modified to render the function f in system (1) sufficiently smooth necessary for an application of CNLD. The first body controller determines the yaw torque, Mr = Kr (r − rref ), Mr ∈ [−Mmax, Mmax ]. It directly influences the dynamics of r. The second controller determines the axis distribution λ = (1 − ay /g)Kλ (rref − r) + 0.5, λ ∈ [0.15, 0.85] of the roll torque Mϕ with ay denoting the lateral acceleration. It affects the yaw rate dynamics indirectly by influencing the corner loads of the vehicle.

Constructive Nonlinear Dynamics

177

μ

l

1

0.5

0.5

μr

1

0.5

μr

1

0.5

μr

1

Fig. 3. Gain scheduling in μ: left: icy road μ = (0.3, 0.6); middle: wet road μ = (0.6, 0.9); right: dry road μ = (0.9, 1.2)

We are interested in finding a controller tuning that guarantees stable vehicle behavior within specified ranges of the road adherence coefficients μl , μr , the vehicle velocity u, and the hand wheel angle δhwa . Specifically μl and μr can vary between 0.1 (slippery road) and 1.2 (road with good grip), the vehicle velocity u is assumed to vary between 15 and 50 m s−1 , and the hand wheel angle is bounded from above by δˆhwa . Performance of the controller is addressed by the objective φ = (r − rref )2 which penalizes deviations of the yaw rate from the reference signal. An analysis of the vehicle model confirms that its dynamics is dominated by a saddle node bifurcation. Robustness can be achieved by backing off the nominal operating point from this stability manifold. When Δα1 = Δα2 = Δμl = Δμr = 0.65, and Δα3 = Δu = 17.5 m s−1 (0) (0) with corresponding nominal values μl = μr = 0.65 and u(0) = 32.5 m s−1 are considered, no controller setting can be found that satisfies the requested robustness properties. It can be inferred, though, that the road adherence coefficients have a greater impact on stability than the velocity u. Therefore, a gain scheduling strategy for three ranges of μ is investigated. The nominal (0) (0) (0) values μi,1 = 0.45, μi,2 = 0.75, μi,3 = 1.05, i = (l, r) can be envisioned as three typical types of road conditions, the first representing an icy road, the second a wet road and the third a dry road. The road adherence uncertainties are Δμi,j = 0.15, i = (l, r), j = 1, . . . , 3. The nominal value and uncertainty of u are the same as above. For each of the three parameter sets an optimization problem of type (15) has to be solved. The results are illustrated in Fig. 3. For simplicity the third uncertain parameter u is omitted in the diagrams. Three sets of controller parameters are found that guarantee the stability of the vehicle within the specified ranges of uncertainty. For the icy road only the uncontrolled vehicle enables stable driving performance for the specified uncertainties. The tuning becomes tighter for the wet road and is most aggressive for the dry road. This is reflected by the region of instability that reaches further into the (μr , μl )-plane with tighter control tuning. 4.3 Extension to Fast Disturbances The presented case studies demonstrates the feasibility and the power of CNLD for robust controller design of uncertain nonlinear systems. One major

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drawback of the approach, however, is the assumption that disturbances, inputs and time varying parameters may only vary quasi-statically (see Sect. 2). While this assumption is necessary to apply CNLD in its present state of development, it is inadequate when disturbances and inputs vary on the same time scales as the system or even faster. In this section an extension is presented to overcome current limitations of CNLD. This extension aims at considering bounds on trajectories of a dynamic system rather than bounds on its steady-states. Corresponding critical manifolds do not rely on the steady state assumption and allow to incorporate fast variations of uncertain quantities. CNLD based controller design for this new type of critical boundary guarantees that the bounds on the trajectories are met even in the presence of fast disturbances. Instead of the simplified nonlinear system (9) with only quasi-statically varying uncertain parameters α we consider the original uncertain closedloop system (2). For simplicity, we summarize the disturbance and set point T T ) . Assume that there are (possibly time variables by replacing d ← (dT , ysp varying) bounds 0 < g(x, t, ϑ) on the state trajectories of the system that have to hold at any time despite disturbances and variations d and uncertainties in the parameters ϑ. The system class reads ˆ x˙ = fˆ(x, d, y, ϑ, ϑ), ˆ d, ϑ, ϑ) ˆ , y = h(x,

x(t0 ) = x0 , (16)

0 < g(x, t, ϑ) . Figure 4 illustrates how bounds g can be used to define critical manifolds for the response of a system to time-varying disturbances. Assume that a state x with initial condition x0 is not allowed to exceed a certain upper bound xmax , i. e., g = xmax − x. Further assume that the system is subject to a step disturbance at t = t0 whose magnitude is parameterized with d1 . Responses of x to these disturbances are shown in Fig. 4 (left). A parabolalike curve (thick line) connects the points where x attains the critical value xmax . The extremum of this curve belongs to the transient of x that touches but does not cross the upper bound. Consider now a second step disturbance at t = t0 parameterized by d2 . Then the parabola like curve of points crossing the boundary unfolds into a surface and the extremal point from Fig. 4 (left) becomes a curve in the three-dimensional space (t, d1 , d2 ) as shown in Fig. 4 (middle). This curve can be interpreted as a new type of critical manifold that splits the (d1 , d2 )-space into regions with qualitatively different trajectories of x. This is further illustrated in Fig. 4 (right) by projecting this curve on the (d1 , d2 )-plane. To the left of the curve only trajectories exist that never exceed the bound xmax whereas trajectories on the right side of the curve always violate the bound for some t. Augmented systems and normal vector constraints can be derived for these new type of critical manifolds following the general scheme developed by M¨ onnigmann and Marquardt [20]. Due to space limitations, we refrain here

179

2

Constructive Nonlinear Dynamics

t

x

max

d

x=x

d

1

d t

1

d

d

2

1

Fig. 4. Critical manifold for bounds on trajectories of state x of a dynamic system

from giving the technical details but refer the interested reader to an upcoming publication [11]. Robust Control of a CSTR The CSTR including feedback linearizing temperature control presented in Sect. 4.2 is reconsidered to demonstrate the applicability of normal vector constraints based on the new type of critical manifolds for the optimization problem (15). In this scenario we consider a nominal steady state operating point with a reactor temperature of Tsp = 370 K. We assume that the process is subject to stepwise disturbances of the feed concentration CAf (t) at t = t0 and sinusoidal variations of the feed rate q(t) around the nominal value q (0) = 100 L min−1 for t > t0 , (0) CAf (t) = CAf + C˜Af (t),

q = q (0) + q˜(t),

C˜Af (t) = q˜(t) =

0, t ≤ t0 , ΔCAf , t > t0 0, t ≤ t0 , Δq sin ω(t − t0 ), t > t0 .

The step disturbance of the feed concentration is parameterized by the uncertain height of the step disturbance ΔCAf (mol L−1 ) ∈ [−0.1, 0.1] and the sinusoidal disturbance of the feed rate by the uncertain amplitude Δq (L min−1 ) ∈ [−10, 10]. The frequency is set to ω = 0.5 min−1 . We now want to find a controller tuning and a process design which guarantees that the reactor temperature T stays within specified bounds despite the presence of disturbances. To be more specific we define a corridor for the reactor temperature around Tsp 1 2 1 0 < g2 = T − Tsp + 2

0 < g1 = Tsp − T +

Te + (Ti − Te ) exp Te + (Ti − Te ) exp

t − t0 τ t − t0 τ

,

t > t0 ,

,

t > t0 .

At t = t0 the gap between the upper and the lower limit is Ti and for increasing t the difference approximates Te . Here we choose Ti = 3 K, Te = 0.5 K and τ = 3 min.

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J. Gerhard, M. M¨ onnigmann, and W. Marquardt

For the nominal operating point (0) steady state constraints 0 = f (0) are employed. The initial values for the critical points (i) are set to the steady (i) state values of the nominal system x0 = x(0) . Minimal distance constraints on critical manifolds defined by the upper and lower limit as sketched in Fig. 4 are employed to ensure that the reactor temperature T stays within the bounds for all time despite the disturbances. The objective φ is the maximization of the reactor yield. Degrees of freedom for the optimization are the tuning pa(0) rameter of the controller , the nominal feed concentration CAf , the nominal (i) steady states x(0) and the uncertain variables ΔCAf , Δq (i) , t(i) . In this case study the location of the critical manifolds is not known beforehand. Therefore application of CNLD has to follow the steps of the algorithm sketched in Sect. 4, detecting the determining critical manifolds as the optimization proceeds. After termination of the algorithm there are two active normal vector constraints, corresponding to critical manifolds of the bounds g1 and g2 . In Fig. 5 (left) the two critical manifolds, the nominal operating point, and the robustness ellipse that touches both critical manifolds are displayed in the plane (Δq, ΔCAf ) of the uncertain parameters. As shown in Fig. 5 (middle) the trajectory corresponding to the point (1) on the critical manifold of g1 touches the upper bound at t(1) = 27.4 min whereas the trajectory of the critical point (2) of g2 shown in Fig. 5 (right) touches the lower bound at t(2) = 4.3 min. The results show that minimal distance constraints permit to find a controller setting that guarantees that the specified bounds on the reactor temperature robustly hold in the presence of dynamic disturbances.

ΔC

0.1 0 −0.1 −0.2

g 1 g

372

371

371

370 369

2

−0.3 −20 −10 0 10 Δ q [L/min]

372

T [K]

0.2 T [K]

Af

[mol/L]

0.3

20

368 0

370 369

10 20 t(1) 30 t [min]

368 (2) 0 t 10 20 t [min]

30

Fig. 5. Left: Robust operating point and critical manifolds in the plane of the disturbance parameters (ΔCAf , Δq); middle: trajectory of the critical point (1) touching the upper bound g1 at t(1) = 27.4 min; right: trajectory of the critical point (2) touching the lower bound g2 at t(2) = 4.3 min

5 Summary The work presented in this article shows successful applications of methods originating from applied bifurcation theory to problems arising in robust nonlinear control. Analysis based approaches can be used to determine bounds for controller tuning in the presence of parametric uncertainty. Analysis based

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methods, however, are limited, as they rely on visualization and interpretation of bifurcation diagrams. Any approach based on visualization of diagrams becomes cumbersome if a large number of varying parameters has to be considered. These limitations are tackled by a recently presented approach denoted as Constructive Nonlinear Dynamics. The method does not require interpretation of visualized data, it systematically incorporates dynamical properties such as stability and parametric uncertainty into the robust design of nonlinear systems, and it enables seamless integration of economic optimization and control design. The method is based on critical manifolds which separate the parameter space of nonlinear systems into regions with qualitatively different system behavior. Due to the universality of the concept of critical manifolds, the approach can be used to address feasibility constraints, stability boundaries as well as performance constraints based on boundaries of the leading eigenvalues. This turns the approach into a versatile method applicable also for the robust control of nonlinear systems as shown by the presented case studies. In the last section of the paper an extension of the approach was presented that allows to relax the restrictive assumption that disturbances, inputs and reference signals may vary only quasi-statically. A case study showed that critical manifolds based on dynamic boundaries are promising as they permit to consider fast disturbances. Current investigations therefore further explore the application of these new kind of critical manifolds to guarantee robustness with respect to fast disturbances. Acknowledgement. This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under grant no. MA1188/22-1.

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Optimal Control of Piecewise Affine Systems: A Dynamic Programming Approach Frank J. Christophersen, Mato Baoti´c , and Manfred Morari Automatic Control Laboratory, ETH Zentrum, Physikstrasse 3, ETL K13.1, CH – 8092 Z¨ urich,฀Switzerland. {fjc,baotic,morari}@control.ee.ethz.ch Summary.฀ We฀ consider฀ the฀ constrained฀ finite฀ and฀ infinite฀ time฀ optimal฀ control problem฀for฀the฀class฀of฀discrete-time฀linear฀piecewise฀affine฀systems.฀When฀a฀linear performance฀index฀is฀used฀the฀finite฀and฀infinite฀time฀optimal฀solution฀is฀a฀piecewise affine฀ state฀ feedback฀ control฀ law.฀ In฀ this฀ paper฀ we฀ present฀ an฀ algorithm฀ to฀ compute฀the฀optimal฀solution฀for฀the฀finite฀time฀case฀where฀the฀algorithm฀combines฀a dynamic฀programming฀exploration฀strategy฀with฀multi-parametric฀linear฀programming฀and฀basic฀polyhedral฀manipulation.฀We฀extend฀the฀ideas฀to฀the฀infinite฀time case฀and฀show฀the฀equivalence฀of฀the฀dynamic฀programming฀generated฀solution฀with the฀solution฀to฀the฀infinite฀time฀optimal฀control฀problem.

Keywords:฀Constrained฀systems,฀finite฀time,฀infinite฀time,฀optimal฀control,฀ discrete-time,฀ hybrid฀ systems,฀ piecewise฀ affine฀ systems,฀ dynamic฀ programming,฀multi-parametric฀linear฀program.

1 Introduction In the last few years several different techniques have been developed for the analysis and controller synthesis for hybrid systems [29, 19, 26, 23, 4, 11, 20, 10]. A significant amount of the research in this field has focused on solving constrained optimal control problems, both for continuous-time and discretetime hybrid systems. We consider the class of discrete-time linear hybrid systems. In particular the class of constrained piecewise affine (PWA) systems with polyhedral constraints on states and inputs that are obtained by partitioning the extended state-input space into polyhedral regions and associating with each region a different affine state update equation, cf. [26, 18]. As shown in [18], the class of piecewise affine systems is of rather general nature and equivalent to many Current Affiliation: Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, HR – 10000 Zagreb, Croatia. Email: mato.baotic @fer.hr

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 183–198, 2005. © Springer-Verlag Berlin Heidelberg 2005

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other hybrid system formalisms, such as for example mixed logical dynamical systems or linear complementary systems. For PWA systems the constrained finite time optimal control (CFTOC) problem can be solved by means of multi-parametric programming [11]. The solution is a piecewise affine state feedback control law and can be computed by using multi-parametric mixed-integer quadratic programming (mp-MIQP) for a quadratic performance index and multi-parametric mixed-integer linear programming (mp-MILP) for a linear performance index, cf. [11, 13]. As recently shown by Borrelli et al. [12] for a quadratic performance index and by [1, 21] for a linear performance index, it is possible to obtain the optimal solution to the CFTOC problem without the use of integer programming. In [12, 1] the authors propose efficient algorithms based on a dynamic programming strategy combined with multi-parametric quadratic or linear program (mp-QP or mp-LP) solvers. However, stability and feasibility (constraint satisfaction) of the closedloop system are not guaranteed if the solution to the CFTOC problem is used in a receding horizon control strategy. To remedy this deficiency various schemes have been proposed in the literature. For constrained linear systems stability can be (artificially) enforced by introducing ‘proper’ terminal set constraints and/or a terminal cost to the formulation of the CFTOC problem [25]. For the class of constrained PWA systems very few and restrictive stability criteria are known, e.g. [5, 25]. Only recently ideas used for enforcing closed-loop stability of the CFTOC problem for constrained linear systems have been extended to PWA systems [17]. However, the technique presented in [17] is generating suboptimal solutions since the objective function is altered compared to the original problem. The main advantages of the infinite time solution, compared to the corresponding finite time solution of the optimal control problem, are the inherent guaranteed stability and feasibility as well as optimality of the closed-loop system [28, 25, 9, 16]. In this paper we present algorithms to solve the constrained finite time optimal control problem and the constrained infinite time optimal control (CITOC) problem with a linear performance index for PWA systems. The algorithms combine a dynamic programming exploration strategy with a multiparametric linear programming solver and basic polyhedral manipulation. In the case of the CITOC problem we show the equivalence of the dynamic programming generated solution with the solution to the infinite time optimal control problem. Therefore the presented algorithm is non-conservative and generates the solution to the Bellman equation (if a bounded solution exists) which corresponds to the solution to the CITOC problem and thus avoids pitfalls of other conservative approaches.

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2 Linear Hybrid Systems Piecewise affine (PWA) systems [26] are equivalent to many other hybrid system classes [18], such as, for example, mixed logical dynamical systems [5], linear complementary systems [19], or min-max-plus-scaling systems and thus form a rather general class of linear hybrid systems. Discrete-time PWA systems can describe a large number of processes, such as discrete-time linear systems with static piecewise linearities, discrete-time linear systems with logic states and inputs or switching systems where the dynamic behavior is described by a finite number of discrete-time linear systems, together with a set of logic rules for switching among these systems. Furthermore, piecewise affine systems present themselves to be a powerful class for identifying or approximating generic nonlinear systems via multiple linearizations at different operating points [26, 14]. Even though hybrid systems (and in particular PWA systems) are a special class of nonlinear systems, most of the nonlinear system and control theory does not apply because it requires certain smoothness assumptions. For the same reason we also cannot simply use linear control theory in some approximate manner to design controllers for PWA systems. We focus on the class of discrete-time, stabilizable, linear hybrid systems that can be described as constrained continuous2 piecewise affine systems of the following form x(t + 1) = fPWA (x(t), u(t)) := Ai x(t) + Bi u(t) + ai ,

if

x(t) u(t)

∈ Di

(1)

where t ≥ 0, x ∈ Rn is the state, u ∈ Rm is the control input, the domain D of fPWA is a non-empty compact set in Rm+n d Di , D := ∪ni=1

n

d with nd < ∞ the number of system dynamics, and {Di }i=1 denotes the polyhedral partition of the domain D, i.e.,

Di := [ xu ] ∈ Rn+m | Dix x + Diu u ≤ Di0 , int(Di ) ∩ int(Dj ) = ∅,

∀i = j.

Note that linear state and input constraints of the general form C x x + C u u ≤ C 0 can be incorporated in the description of Di . Without loss of generality the standing assumption throughout this paper is: Assumption 3 (Equilibrium at the origin) The origin in the extended state-input space is an equilibrium point of the PWA system (1), i.e.

!n+m ∈ D and !n = fPWA (!n , !m )

(2)

where !n := [0 0 . . . 0] ∈ Rn . 2

Here a PWA system defined over a disjoint domain D is called continuous if fPWA is continuous over connected subsets of the domain.

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3 Constrained Finite Time Optimal Control We consider the piecewise affine system (1) and define the constrained finite time optimal control (CFTOC) problem as JT∗ (x(0)) := min JT (x(0), UT )

(3)

UT

subj. to

x(t + 1) = fPWA (x(t), u(t)) , x(T ) ∈ X f

where JT (x(0), UT ) := P x(T )

p+

T −1

Qx(t)

p

+ Ru(t) p ,

(4)

(5)

t=0

is the cost function (also called performance index ), UT is the optimization variable defined as input sequence T −1

UT := {u(t)}t=0 ,

(6)

T < ∞ is the prediction horizon, X f is a compact terminal set in Rn , and p with p ∈ {1, ∞} in (5) denotes the corresponding standard vector 1- or ∞-norm. The optimal value of the cost function, denoted with JT∗ , is called the value function. The optimization variable that achieves JT∗ is called the T −1 optimizer and we denote it with UT∗ := {u∗ (t)}t=0 . With a slight abuse of notation, when the CFTOC problem (3)–(4) has multiple solutions, i.e. when the optimizer is not unique, UT∗ denotes one (arbitrarily chosen) realization from the set of possible optimizers. Note that it is common practice to use the term linear performance index when referring to (5) even though, strictly speaking, the cost function JT (x(0), UT ) in (5) is a piecewise affine function of its arguments. The CFTOC problem (3)–(4) implicitly defines the set of feasible initial states XT ⊂ Rn (x(0) ∈ XT ) and the set of feasible inputs UT −t ⊂ Rm (u(t) ∈ UT −t , t = 0, . . . , T − 1). Our goal is to find an explicit (closed form) expression for the set XT , and for the functions JT∗ : XT → R and u∗ (t) : XT → UT −t , t = 0, . . . , T − 1.

·

Remark 1 (Choice of P , Q, R). The problem (3)–(4) can be posed and solved for any choice of the matrices P , Q, and R. However, from a practical point of view if we want to avoid unnecessary controller action while steering the state to the origin, the choice of a full column rank R is a necessity. Moreover, for stability reasons (as it will be shown in Section 4) a full column rank Q is assumed. Remark 2 (Time-varying system and/or cost). The CFTOC problem (3)–(4) naturally extends to PWA system and/or cost functions with time-varying parameters, i.e. Ai (t), Bi (t), ai (t), Di (t), as well as Q(t) and R(t) for t = 0, . . . , T − 1. For simplicity we focus on the time-invariant case but the

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CFTOC problem with time-varying parameters is of the same form and complexity as the CFTOC problem with time-invariant parameters and therefore it can be solved in an analog manner. We summarize the main result concerning the solution to the CFTOC problem (3)–(4) which is proved in [24, 11]. Theorem 1 (Solution to CFTOC). The solution to the optimal control problem (3)–(4) with p ∈ {1, ∞} is a piecewise affine value function JT∗ (x(0)) = ΦT,i x(0) + ΓT,i ,

if

x(0) ∈ PT,i

(7)

of the initial state x(0) and the optimal input u∗ (t) is a time-varying piecewise affine function of the initial state x(0) u∗ (t) = KT −t,i x(0) + LT −t,i ,

if

x(0) ∈ PT,i

(8) N

T where t = 0, . . . , T − 1, the sets PT,i , i = 1, . . . , NT , are polytopic, {PT,i }i=1 is a polyhedral partition of the set of feasible states x(0)

NT XT = ∪i=1 PT,i ,

(9)

x 0 with the closure of PT,i given by P¯T,i = {x ∈ Rn | PT,i x ≤ PT,i }.

3.1 The CFTOC Solution via Dynamic Programming Making use of Bellman’s optimality principle [3, 6], the constrained finite time optimal control problem (3)–(4) can be solved in a computationally efficient way by solving an equivalent dynamic program (DP) backwards in time [1, 7, 8]. The corresponding DP has the following form Jk∗ (x(t)) := min Qx(t) u(t)

subj. to

p

+ Ru(t)

p

∗ + Jk−1 (fPWA (x(t), u(t))),

fPWA (x(t), u(t)) ∈ Xk−1

(10) (11)

for k = 1, . . . , T , with t = T − k, cf. Figure 1, and with boundary conditions X0 = X f , and J0∗ (x(T )) = P x(T )

p

(12)

where Xk := {x ∈ Rn | ∃ u ∈ Rm , fPWA (x, u) ∈ Xk−1 }

(13)

is the set of all states at time t = T − k for which the problem (10)–(12) is feasible. Since p ∈ {1, ∞} the dynamic programming problem (10)–(12) can be solved by multi-parametric linear programs, cf. [11, 1], where the state x(t) is treated as a parameter and the control input u(t) as an optimization variable.

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... x(0)

...

x(T − k)

... 0

1

T

T −1

t = T −k

...

Iteration

x(T )

...

Time

T −1

T

1

0

...

k =T −t

Fig. 1. Relation of the time axis and iteration step of the dynamic program.

By solving such programs at each iteration step k, going backwards in time starting from the target set X f , we obtain the set Xk ⊂ Rn , the optimal control law u∗ (t) : Xk → Uk , with t = T − k, and the value function Jk∗ : Xk → R that represents the so called ‘cost-to-go’. Properties of the solution are given in the following theorem, cf. [24, 11]. Theorem 2 (Solution to CFTOC via DP). The solution to the optimal control problem (10)–(12) with p ∈ {1, ∞} is a piecewise affine value function Jk∗ (x(t)) = Φk,i x(t) + Γk,i ,

if

x(t) ∈ Pk,i

(14)

and the optimal input u∗ (t) is a time-varying piecewise affine function of the state x(t), i.e. it is given as a state feedback control law u∗ (t) = μ∗k (x(t)) := Fk,i x(t) + Gk,i ,

if

x(t) ∈ Pk,i

(15)

where k = 1, . . . , T , t = T − k, the sets Pk,i , i = 1, . . . , Nk , are polytopic, Nk is a polyhedral partition of the set of feasible states x(t) at time t {Pk,i }i=1 Nk Xk = ∪i=1 Pk,i ,

(16)

x 0 with the closure of Pk,i given by P¯k,i = {x ∈ Rn | Pk,i x ≤ Pk,i }.

Theorem 1 states that the solution to the CFTOC problem (3)–(4), i.e. the optimal input sequence UT∗ given by (8), is a function of the initial state x(0) only. On the other hand, Theorem 2 describes the solution to the dynamic program (10)–(12) as the optimal state feedback control law μ∗k (x(t)). Since we know that both solutions must be identical (assuming that the optimizer is unique), this implies that there is a connection between the matrices Kk,i and Lk,i in (8) and the matrices Fk,i and Gk,i in (15). It is easy to see that KT,i = FT,i and LT,i = GT,i . To establish the connection for the other coefficients one would have to carry out the tedious sequence of substitutions x(t) = fPWA (x(t − 1), μ∗T −t+1 (x(t − 1))), which would eventually express x(t) in (15) as a function of x(0) only. However, in this paper we focus on the DP approach in solving the CFTOC problem and since both approaches give the same

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solution, we will not go beyond this note in establishing an explicit connection between those coefficients. Having this in mind, from this point onwards, when we speak of the solution to the CFTOC problem we consider the solution in the form given in Theorem 2. In the rest of the paper with μ we denote a generic state feedback control law that maps a set of states X to a set of control actions U. Thus μ specifies the control action (or input action) u(t) = μ(x(t)) that will be chosen at time t when the state is x(t). Furthermore, with π we denote a control policy that maps a set of states to a sequence of control actions. For instance, in the case of the CFTOC problem (3)–(4) with prediction horizon T the optimal control policy is defined by πT∗ := {μ∗T , . . . , μ∗1 }. 3.2 Receding Horizon Control In the case that the receding horizon (RH) control policy [25] is used in closedloop the control is given as a time-invariant state feedback control law of the form μRH (x(t)) := FT,i x(t) + GT,i ,

if

x(t) ∈ PT,i

(17)

with u(t) = μRH (x(t)) and the time-invariant value function is JRH (x(t)) := ΦT,i x(t) + ΓT,i ,

if

x(t) ∈ PT,i

(18)

for t ≥ 0. Thus only NRH := NT (in the worst case different) control laws have to be stored. Note that in general JRH in (18) does not represent the value function of the closed-loop system when the receding horizon control law μRH is applied because JRH (x(0)) denotes the cost-to-go from x(0) to x(T ) when the openloop input sequence is applied. In the special case when the finite time solution ∗ for some T < ∞, JRH in is equivalent to the infinite time solution, i.e. JT∗ ≡ J∞ fact does represent the value function of the closed-loop system when applying μRH , see also Remark 3. 3.3 Example: Constrained PWA System Consider the piecewise affine system [5] ⎧ cos α(x(t)) − sin α(x(t)) 0 ⎪ ⎪ ⎪ x(t + 1) = 0.8 sin α(x(t)) cos α(x(t)) x(t) + [ 1 ] u(t), ⎪ ⎪ ⎪ ⎪ ⎪ π ⎨ 3 if [1 0]x(t) ≥ 0, α(x(t)) = π − 3 if [1 0]x(t) < 0, ⎪ ⎪ ⎪ ⎪ ⎪ x(t) ∈ [−10, 10] × [−10, 10], ⎪ ⎪ ⎪ ⎩ u(t) ∈ [−1, 1].

(19)

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F.J. Christophersen, M. Baoti´c, and M. Morari

1.5

45 40

1

JT∗ (x)

μ∗T (x)

35 0.5

0

−0.5

−10

30 25 20 15

10

10 −1 0

−1.5 10

5

−5

8

6

4

2

5 0

x1

−2

−4

−6

−8

−10

5

x2

0

0 −10

−8

−6

−4

−2

−5 0

x1

10

2

4

6

8

10

x2

−10

(b) Value function JT∗ (x). Same color corresponds to the same cost value.

(a) State feedback control law μ∗T (x).

Fig. 2. (In)finite time solution with T = T∞ = 11 for system (19) derived with the dynamic programming algorithm.

The CFTOC problem (3)–(4) is solved with Q = [ 10 01 ], R = 1, P = [ 00 00 ], and X f = [−10, 10] × [−10, 10] for p = ∞. Figure 2 depicts the finite time solution, i.e. optimal control law and value function, computed with the dynamic programming procedure of Section 3.1. For an efficient algorithmic implementation confer [1]. A posteriori it can be shown with the dynamic programming procedure that the finite time solution for a horizon T ≥ 11 = T∞ is in fact identical to the infinite time solution of the constrained optimal control problem, cf. Section 4.1. The (in)finite time solution with T = 11 for this example was solved in 1515 seconds on a Pentium 4, 2.2 GHz machine running Matlab 6.1. There exist 23 different affine expressions defining the control law μ∗T∞ (x) in 252 polyhedral regions, cf. Figure 2(a). Figure 2(b) reveals the corresponding value function. Same color corresponds to the same cost; moreover, the minimum cost is naturally achieved at the origin.

!

4 Constrained Infinite Time Optimal Control As in the previous section we consider the piecewise affine system (1) subject to state and input constraints and by letting T → ∞ the cost function (5) takes the following form (assuming that the limit exists) T

J∞ (x(0), U∞ ) := lim

T →∞

g(x(t), u(t))

(20)

t=0

where the function g : Rn × Rm → R≥0 also called the stage-cost is defined by

Optimal Control of PWA Systems: A Dynamic Programming Approach

g(x, u) := Qx

p

+ Ru p ,

191

(21)

with p ∈ {1, ∞}. Moreover, (3)–(4) becomes the constrained infinite time optimal control (CITOC) problem ∗ J∞ (x(0)) := min J∞ (x(0), U∞ ), U∞

subj. to

x(t + 1) = fPWA (x(t), u(t))

(22) (23)

where by U∞ := {u(t)}∞ t=0 we denote the optimization input sequence and by ∗ := {u∗ (t)}∞ U∞ t=0 the optimizer of (22)–(23). In order to guarantee closed-loop stability we assume that Q is of full column rank as it will be shown in the following, cf. Lemma 1. Additionally, also in the infinite time case it can be assumed that R is of full column rank even though these assumptions are not strictly needed, cf. Remark 1. ∗ Assumption 4 (Boundedness of J∞ ) The CITOC problem (22)–(23) is well defined, i.e. the minimum is achieved for some feasible input sequence ∗ ∗ , and J∞ (x) < ∞ for any feasible state x on a compact set X∞ . U∞

This assumption is hardly a limitation to the applicability of the presented method since in most practical applications we want to steer the state from some given state x(0) or set to some equilibrium point (here the origin, cf. Assumption 3) by spending a finite amount of ‘energy’. In the following example we illustrate the reasoning behind Assumption 4. Example 1 (Constrained LTI system). Consider the simple CITOC problem T ∗ J∞ (x(0)) = min lim U∞

T →∞

subj. to

t=0

|x(t)|,

x(t + 1) = 2x(t) + u(t), |x(t)| ≤ 1, and |u(t)| ≤ 1

(24) (25)

for the constrained one-dimensional LTI system (25). Problem (24)–(25) is feasible for all initial states in X¯∞ = [−1, 1] and one can observe that the closed-loop system for the optimal state feedback control law ⎧ if x ∈ [−1, −0.5], ⎨1 (26) μ∗∞ (x) = −2x if x ∈ [−0.5, 0.5], ⎩ −1 if x ∈ [0.5, 1]

has three equilibria at −1, 0, and 1. However, the closed-loop system is asymptotically stable only for the open set X∞ = (−1, 1). Figure 3 illustrates that ∗ (x(0)) → ∞ as x(0) → ±1 and therefore the the optimal value function J∞ problem is not well defined in the sense of Assumption 4. In practice, one can ∗ compute a δ-close approximation of J∞ on a closed subset X δ ⊂ X∞ as was done for obtaining Figure 3. Note that choosing any R = 0 does only influ∗ and μ∗∞ but does not influence the above mentioned ence the ‘shape’ of J∞ characteristic behavior of the solution.

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∗ Value Function J∞ (x)

20

10

0 −1

0

1

State x ∗ Fig. 3. δ-close approximation of the optimal value function J∞ (x) for Example 1. The colored x-axis denotes the different regions over which the piecewise affine value ∗ (x) is defined. function J∞

Please note that most of the following results hold also (or are straight forwardly extended) for general continuous nonlinear systems and are not restricted to the considered class of PWA systems. Lemma 1 (Stability of the CITOC solution). Consider the CITOC problem (22)–(23) and let its solution fulfill Assumption 4. Then the following holds: (a) The origin [x u ] = !n+m is part of the infinite time solution, i.e. !n ∈ X∞ ∗ (!n ) = 0 and u∗ (t) = !m for t ≥ 0. with J∞ ∗ to the system, any system state x(0) ∈ X∞ is (b) By applying the optimizer U∞ driven to the origin (attractiveness), i.e. if x(0) ∈ X∞ then limt→∞ x(t) = !n . ∗ (x) is continuous at x = !n (it can be discontinuous elsewhere), then (c) If J∞ ∗ by applying the optimizer U∞ to the system, the equilibrium point !n ∈ X∞ of the closed-loop system is asymptotically stable in the Lyapunov sense. Proof. (a) Because [x u ] = !n+m is an equilibrium point of the system (1) and g(x, u) ≥ 0 for all [x u ] ∈ D, the minimum of J∞ (x(0), U∞ ) ≥ 0 for x(0) = !n is achieved with e.g. u∗ (t) = !m for all t ≥ 0. That means ∗ (!n ) = 0 which is the smallest possible value of J∞ ( , ). J∞ ∗ ( ) < ∞, i.e. (b) By Assumption 4 and g( , ) ≥ 0 we have 0 ≤ J∞ ∗ J∞ ( ) is bounded from above and below. Additionally, the sequence {JT } with JT := Tt=0 g(x(t), u(t)) for any sequence {(x(t), u(t))}Tt=0 , as T increases, is ∗ ∗ . non-decreasing. Since we are using U∞ , the sequence {JT } converges to J∞ ∗ Consequently, for every ε1 > 0 there exists a T0 < ∞ with |JT − J∞ | < ε1 for all T ≥ T0 . Therefore necessarily limT →∞ g(x(T ), u(T )) = 0 and because Q is of full column rank it follows limT →∞ x(T ) = !n . (c) The origin !n ∈ X∞ is an equilibrium point of the closed-loop system. Because Q is of full column rank, we have for any t > 0 and x(0) = !n

·

··

··

·

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∗ ∗ ∗ (x(0) ∈ X∞ ) that J∞ (x(t)) ≤ J∞ (x(0)) − Qx(0) p < J∞ (x(0)). In general, for full column rank Q there exists a finite α1 > 0 such that α1 x 2 ≤ Qx p . ∗ Therefore, we have α1 x(t) 2 ≤ Qx(t) p ≤ J∞ (x(t)). On the other side, ∗ with Assumption 4 and the continuity of J∞ at x = !n (it can be discontinuous elsewhere), there always exists a K-class function [30] J¯ with ∗ ¯ x 2 ) for all x ∈ X∞ . Thus for all t ≥ 0 and x(0) ∈ X∞ it J∞ (x) ≤ J( ¯ x(0) 2 ). Clearly, for each ε¯ > 0 the choice of follows that x(t) 2 ≤ α11 J( −1 ¯ ε) := J¯ (α1 · ε¯) satisfies the Lyapunov stability definition in [30, 15], i.e. δ(¯ ¯ ε) > 0 such that from x(0) 2 < δ(¯ ¯ ε) follows for all ε¯ > 0 there exists δ(¯ x(t) 2 < ε¯ for all t ≥ 0. Hence, the equilibrium !n is a Lyapunov stable point and together with the attractiveness of !n (Lemma 1(b)) we have that  !n is asymptotically stable.

4.1 The CITOC Solution via Dynamic Programming Similar to the recasting of the CFTOC problem into a recursive dynamic program as presented in Section 3.1, it is possible to formulate for the CITOC problem (22)–(23) the corresponding dynamic program (DP) as follows Jk (x(t)) := min g(x(t), u(t)) + Jk−1 (fPWA (x(t), u(t)), u(t)

subj. to

fPWA (x(t), u(t)) ∈ Xk−1

(27) (28)

for k = 1, . . . , ∞, with initial conditions X0 = {x ∈ Rn | ∃ u, [x u ] ∈ D} , and J0 (x) = 0 ∀x ∈ X0 .

(29)

The set of all initial states for which the problem (27)–(29) is feasible at iteration step k is given by Xk := {x ∈ Rn | ∃ u, fPWA (x, u) ∈ Xk−1 } =

(30)

Nk ∪i=1 Pk,i .

Furthermore, we define the feasible set of states as k → ∞ by X∞ := lim Xk , k→∞

(31)

and the limit value function of the dynamic program (27)–(29) by ∗ J∞,DP := lim Jk . k→∞

(32)

From !n ∈ X0 , J0 (!n ) = 0, and Assumption 3 it is easy to see that the following properties

!n ∈ Xk and Jk (!n ) = 0 for all k ≥ 0 of the dynamic program (27)–(29) hold.

(33)

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Lemma 2 (Stability of the DP solution). Consider the DP problem ∗ (27)–(29). In addition, let J∞,DP (x) be bounded on X∞ and continuous at x = !n . Then, when applying the corresponding optimal control law u∗ (t) = μ∗∞,DP (x(t)) for all x(t) ∈ X∞ to the system, the following holds:

∗ ∗ (a) the limit value function J∞,DP (x) for all x ∈ X∞ , with J∞,DP (!n ) = 0, is a global Lyapunov function for the closed-loop system, (b) any system state x(0) ∈ X∞ is driven to the origin (attractiveness), i.e. if x(0) ∈ X∞ then limt→∞ x(t) = !n , and (c) the origin x = !n is an asymptotically stable equilibrium point.

∗ ∗ Proof. (a) From property (33) we have that J∞,DP (!n ) = 0. Because J∞,DP ∗ ∗ is a limit function of the DP (27)–(29) we have J∞,DP (x) = g(x, μ∞,DP (x)) + ∗ ∗ (fPWA (x, μ∗∞,DP (x))) for all x ∈ X∞ . Thus, with J∞,DP J∞,DP ( ) ≥ 0, ∗ ∗ ∗ it follows that −ΔJ∞,DP (x) := J∞,DP (x) − J∞,DP (fPWA (x, μ∗∞,DP (x))) = g(x, μ∗∞,DP (x)) ≥ Qx p for all x ∈ X∞ . Because Q is of full column rank, ∗ there exists a finite α1 > 0 with Qx p ≥ α1 x 2 and thus −ΔJ∞,DP (x) ≥ ∗ α1 x 2 for all x ∈ X∞ . This means that −ΔJ∞,DP ( ) is always bounded below ∗ (x) ≥ g(x, μ∗∞,DP (x)) by some K-class function. Similarly, we have that J∞,DP ≥ Qx p ≥ α2 x 2 for all x ∈ X∞ and some finite α2 > 0. By similar ar∗ gument as in Lemma 1(c) there exists a K-class function J¯ bounding J∞,DP ∗ from above. From these statements it follows directly that J∞,DP is a global Lyapunov function [30, 15, 31] for the closed-loop system. (b)+(c) Because a global Lyapunov function exists (Lemma 2(a)) for the closed-loop system on the set X∞ it follows immediately that the origin !n is a global asymptotically stable [30, 15, 31], and thus if x(0) ∈ X∞ then  limt→∞ x(t) = !n .

·

·

Note that in the infinite time case, in contrast to the finite time case discussed in Section 3, the equivalence of the solution of the dynamic program ∗ ∗ and the optimal solution J∞ of the CITOC problem is not immediJ∞,DP ate. Before we prove this equivalence it is useful to introduce the following operators. Definition 1 (Operator T and Tμ ). For any function J : X → R≥0 we define the following mapping (TJ)(x) := min g(x, u) + J (fPWA (x, u)) u∈U

(34)

where the set of feasible control actions U ⊂ Rm is defined implicitly through the domains of J and fPWA , i.e. U := {u ∈ Rm | ∃x, [x u ] ∈ D, fPWA (x, u) ∈ X }. T transforms the function J on X into the function TJ : X → R≥0 . Tk denotes the k-times operator of T with itself, i.e. Tk J = T(Tk−1 J) and T0 J = J with k ∈ N≥0 . Accordingly, we use

Optimal Control of PWA Systems: A Dynamic Programming Approach

(Tμ J)(x) := g(x, μ(x)) + J (fPWA (x, μ(x)))

195

(35)

for any function J : X → R≥0 and any control function μ : X → U defined on the state space X .

The DP procedure (27)–(29) can now be simply stated as Jk := TJk−1 , ∗ with J0 ( ) = 0, J∞,DP = limk→∞ Tk J0 . The solution to the DP procedure, ∗ J∞,DP , satisfies the Bellman equation [7]

·

J = TJ

(36)

which is effectively being used as a stopping criterion to decide when the DP procedure has terminated. ∗ To prove that the solution to the CITOC problem (22)–(23), J∞ , is identi∗ cal to the solution of the dynamic program (27)–(29), J∞,DP , we actually have to answer two questions: first, under which conditions does the DP procedure ∗ (27)–(29) converge, and second, when is J∞,DP a unique solution to the DP procedure (27)–(29). Theorem 3 (Equivalence: CITOC solution–DP solution). Let the so∗ lution to the CITOC problem (22)–(23), J∞ , satisfy Assumption 4 and let it be continuous at x = !n . Moreover, let the solution to the dynamic program ∗ , be bounded over the feasible set X∞ and be continuous at (27)–(29), J∞,DP x = !n . ∗ ∗ ∗ (x) = J∞,DP (x) for all x ∈ X∞ . Moreover, the solution J∞,DP is a Then J∞ unique solution of the dynamic program (27)–(29). ∗ Proof. According to [7, Sec. 3], the solution J∞ to the CITOC problem (22)– ∗ ∗ (23) satisfies the dynamic program (27)–(28), that is J∞ = TJ∞ . On the other hand, in general, the Bellman equation (36) may have no solution or it may have multiple solutions, but at most one solution has the property

lim J(x(t)) = 0,

(37)

t→∞

·

∗ cf. [27, Sec. 4.]. Now, if the DP (27)–(29) has a solution J∞,DP ( ) < ∞ fulfilling property (37) then it is unique and according to [27, Thm. 4.3] this solution satisfies the CITOC problem. From Lemma 2 it follows immediately that the DP solution does in fact satisfy property (37). 

Having established this result, in the following we will denote the solution to both problems, the CITOC problem (22)–(23) and the DP problem (27)– ∗ . (29), with J∞ Lemma 3 (Optimal control law μ∗∞ , [7, Prop. 3.1.3]). A stationary ∗ ∗ ) (x) = Tμ∗∞ J∞ control law μ∗∞ is optimal if and only if (TJ∞ (x) for all ∗ x ∈ X∞ . In other words, μ∞ (x) is optimal if and only if the minimum of (27) is obtained with μ∗∞ (x) for all x ∈ X∞ .

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As in the CFTOC case, the infinite time problem might have multiple optimizers μ∗∞ . With a slight abuse of notation, in the case when the optimizer is not unique, we denote with μ∗∞ one (arbitrarily chosen) realization from the set of possible optimizers. Now we are ready to state the theorem that characterizes the optimal ∗ and the optimal state feedback control law μ∗∞ . solution J∞ Theorem 4 (Solution to the CITOC and DP problem). Under Assumption 4 the solution to the optimal control problem (22)–(23) with p ∈ {1, ∞} is a piecewise affine value function ∗ J∞ (x(t)) = Φ∞,i x(t) + Γ∞,i ,

if

x(t) ∈ P∞,i

(38)

and the optimal state feedback control law is of the time-invariant piecewise affine form μ∗∞ (x(t)) = F∞,i x(t) + G∞,i ,

if

x(t) ∈ P∞,i

(39)

N∞ , is a polyhedral partition of the set X∞ of feasible states x(t) where {P∞,i }i=1 at time t with t ≥ 0.

Proof. Theorem 4 follows from the construction of the DP iterations (27)– (29), Theorem 2, and Assumption 4.  Remark 3 (Receding Horizon Control). Note that the infinite time optimal control law (39) has the same form as the corresponding receding horizon control policy (17). This means that with the optimal control law μ∗∞ the closed-loop and open-loop response of the system (1) are identical. Moreover, ∗ the value J∞ (x(t)) is the total ‘cost-to-go’ from x(t) to the origin, applying the optimal control policy. 4.2 Example: Constrained PWA System Consider again the constrained piecewise affine system (19) reported in [5]. The CITOC problem (22)–(23) is solved with Q = [ 10 01 ], R = 1, and X0 = [−10, 10] × [−10 10] for p = ∞. Also here, Figure 2 depicts the infinite time solution (T → ∞) computed with the dynamic programming procedure of Section 4.1. For an efficient algorithmic implementation, which does not rely on the exploration of the whole feasible state space at every iteration step of the dynamic program, is reported in [2]. The existing 23 different affine expressions defining the optimal control law μ∗∞ (x(0)) over 252 polyhedral regions is depicted in Figure 2(a). Figure 2(b) reveals the corresponding infinite time value function. The same color corresponds to the same cost value and the minimum cost is naturally achieved at the origin. The infinite time solution to (22)–(23) for this example was solved in 184 seconds on a Pentium 4, 2.2 GHz machine running Matlab 6.1. This shows

!

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the superiority of the algorithm proposed in [2] compared to the approach given in Section 3 or [1] where the computation of the infinite time solution took 1515 seconds on the same machine.

5 Software Implementation The presented algorithms to solve the CFTOC problem (3)–(4) and the CITOC problem (22)–(23) are implemented in the Multi-Parametric Toolbox (MPT) [22] for Matlab . The toolbox can be downloaded at:

!

http://control.ee.ethz.ch/~mpt/

References 1. M. Baoti´c, F. J. Christophersen, and M. Morari. A new Algorithm for Constrained Finite Time Optimal Control of Hybrid Systems with a Linear Performance Index. In Proc. of the European Control Conference, Cambridge, UK, September 2003. Available from http://control.ee.ethz.ch/index.cgi? page=publications. 2. M. Baoti´c, F. J. Christophersen, and M. Morari. Infinite Time Optimal Control of Hybrid Systems with a Linear Performance Index. In Proc. of the Conf. on Decision & Control, pages 3191–3196, Maui, Hawaii, USA, December 2003. Available from http://control.ee.ethz.ch/index.cgi?page= publications&action=details&i%d=311. 3. R. Bellman. Dynamic Programming. Princeton University Press, Princeton, N.J., 1957. 4. A. Bemporad, F. Borrelli, and M. Morari. Optimal Controllers for Hybrid Systems: Stability and Piecewise Linear Explicit Form. In Proc. of the Conf. on Decision & Control, Sydney, Australia, December 2000. 5. A. Bemporad and M. Morari. Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3):407–427, March 1999. 6. D. P. Bertsekas. Dynamic Programming and Optimal Control, volume I. Athena Scientific, Belmont, Massachusetts, 2nd edition, 2000. 7. D. P. Bertsekas. Dynamic Programming and Optimal Control, volume II. Athena Scientific, Belmont, Massachusetts, 2nd edition, 2001. 8. D. P. Bertsekas and S. Shreve. Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific, 1996. 9. R. R. Bitmead, M. Gevers, and V. Wertz. Adaptive Optimal Control: The Thinking Man’s GPC. International Series in Systems and Control Engineering. Prentice Hall, 1990. 10. F. Blanchini. Set invarinace in control — A survey. Automatica, 35:1747–1767, 1999. 11. F. Borrelli. Constrained Optimal Control of Linear and Hybrid Systems, volume 290 of Lecture Notes in Control and Information Sciences. Springer, 2003. 12. F. Borrelli, M. Baoti´c, A. Bemporad, and M. Morari. An Efficient Algorithm for Computing the State Feedback Solution to Optimal Control of Discrete Time Hybrid Systems. In Proc. on the American Control Conference, pages 4717– 4722, Denver, Colorado, USA, June 2003.

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13. V. Dua and E. N. Pistikopoulos. An algorithm for the solution of multiparametric mixed integer linear programming problems. Annals of Operations Research, 99:123–139, 2000. 14. G. Ferrari-Trecate, M. Muselli, D. Liberati, and M. Morari. A clustering technique for the identification of piecewise affine systems. Automatica, 39(2):205– 217, February 2003. 15. G. C. Goodwin, M. M. Seron, and J. A. De Don´ a. Constrained Control and Estimation: An Optimisation Approach. Communications and Control Engineering. Springer-Verlag, London, 2005. 16. P. Grieder, F. Borrelli, F. Torrisi, and M. Morari. Computation of the constrained infnite time linear quadratic regulator. Automatica, 40:701–708, 2004. 17. P. Grieder, M. Kvasnica, M. Baoti´c, and M. Morari. Stabilizing low complexity feedback control of constrained piecewise affine systems. accepted to Automatica, 2005. 18. W. P. M. Heemels, B. De Schutter, and A. Bemporad. Equivalence of hybrid dynamical models. Automatica, 37(7):1085–1091, 2001. 19. W. P. M. H. Heemels. Linear Complementarity Systems: A Study in Hybrid Dynamics. PhD thesis, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, November 1999. 20. M. Johansson. Piecewise Linear Control Systems: A Computational Approach, volume 284 of Lecture Notes in Control and Information Sciences. SpringerVerlag, 2003. 21. E. C. Kerrigan and D. Q. Mayne. Optimal control of constrained, piecewise affine systems with bounded disturbances. In Proc. of the Conf. on Decision & Control, pages 1552–1557, Las Vegas, Nevada, USA, December 2002. 22. M. Kvasnica, P. Grieder, M. Baoti´c, and M. Morari. Multi-Parametric Toolbox (MPT), 2003. Available from http://control.ee.ethz.ch/~mpt/. 23. J. Lygeros, C. Tomlin, and S. Sastry. Controllers for reachability specifications for hybrid systems. Automatica, 35(3):349–370, 1999. 24. D. Q. Mayne. Constrained Optimal Control. European Control Conference, Plenary Lecture, September 2001. 25. D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, 36(6):789–814, June 2000. 26. E. D. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE Trans. on Automatic Control, 26(2):346–358, April 1981. 27. N. L. Stokey and R. E. Lucas. Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge, Massachusetts, 2001 edition, 1989. 28. M. Sznaier and M. J. Damborg. Suboptimal control of linear systems with state and control inequality constraints. In Proc. of the Conf. on Decision & Control, volume 1, pages 761–762, December 1987. 29. A. van der Schaft and H. Schumacher. An Introduction to Hybrid Dynamical Systems, volume 251 of Lecture Notes in Control and Information Sciences. Springer-Verlag, 2000. 30. M. Vidyasagar. Nonlinear Systems Analysis. Prentice Hall, 2nd edition, 1993. 31. L. Weiss. Controllability, realization and stability of discrete-time systems. SIAM Journal on Control, 10(2):230–251, May 1972.

Hierarchical฀Hybrid฀Control฀Synthesis฀and฀its Application฀to฀a฀Multiproduct฀Batch฀Plant J¨ org฀Raisch12 and฀Thomas฀Moor3 1฀

Lehrstuhl฀f¨ ur฀Systemtheorie฀technischer฀Prozesse,฀Otto-von-Guericke Universit¨ at,฀Postfach฀4120,฀D-39016฀Magdeburg,฀Germany. [email protected] 2฀ Systems฀and฀Control฀Theory฀Group,฀Max฀Planck฀Institute฀for฀Dynamics฀of Complex฀Technical฀Systems 3฀ Lehrstuhl฀f¨ ur฀Regelungstechnik,฀Friedrich-Alexander฀Universit¨ at,฀D-91058 Erlangen,฀Germany. [email protected] Summary.฀ This฀paper฀presents฀a฀hierarchical฀approach฀to฀hybrid฀control฀systems synthesis.฀It฀is฀set฀in฀a฀general฀behavioural฀framework฀and฀therefore฀allows฀a฀combination฀of฀continuous฀and฀discrete฀layers฀in฀the฀resulting฀overall฀controller.฀Compared฀to฀unstructured฀approaches,฀it฀drastically฀reduces฀computational฀complexity and฀hence฀enlarges฀the฀class฀of฀continuous-discrete฀control฀problems฀that฀can฀be฀addressed฀using฀methods฀from฀hybrid฀systems฀theory.฀The฀potential฀of฀our฀approach is฀demonstrated฀by฀applying฀it฀to฀a฀multiproduct฀batch฀control฀problem.

Keywords:฀Hybrid฀systems,฀hierarchical฀control,฀discrete฀abstraction,฀multiproduct฀batch฀plant.

1฀Introduction In฀ hybrid฀ dynamical฀ systems,฀ discrete-event฀ components฀ (realised,฀ e.g.,฀ by finite฀automata)฀and฀continuous฀components฀(realised,฀e.g.,฀by฀ordinary฀differential฀equations)฀interact฀in฀a฀nontrivial฀way.฀The฀fundamental฀problems฀in analysing฀and฀synthesising฀such฀systems฀stem฀from฀the฀nature฀of฀their฀state sets.฀ While฀ the฀ state฀ space฀ of฀ a฀ continuous฀ system฀ usually฀ exhibits฀ vector space฀structure,฀and฀the฀state฀set฀of฀a฀discrete฀event฀system฀(DES)฀is฀often finite,฀the฀state฀set฀of฀the฀overall฀system฀inherits฀none฀of฀theses฀amenities:฀as a฀product฀of฀the฀constituting฀state฀sets,฀it฀is฀neither฀finite฀nor฀does฀it฀exhibit vector฀space฀structure.฀Hence,฀neither฀methods฀from฀discrete฀event฀systems theory,฀which฀rely฀on฀exploiting฀finiteness,฀nor฀concepts฀from฀continuous฀control฀theory฀carry฀over฀readily฀to฀hybrid฀problems.฀Nevertheless,฀as฀inherently hybrid฀application฀problems฀are฀very฀common,฀hybrid฀control฀systems฀have become฀an฀increasingly฀popular฀subject฀during฀the฀last฀decade฀(e.g.฀[2,฀1,฀6]).

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 199–216, 2005. © Springer-Verlag Berlin Heidelberg 2005

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A considerable part of hybrid systems research has gone into investigating approximation-based approaches (e.g. [7, 5, 16, 9]). There, the core idea is to approximate continuous dynamics by discrete event systems, and hence to transform the hybrid control problem into a purely discrete one. Of course, care has to be taken to guarantee that the resulting (discrete event) control system enforces the specifications not only for the discrete approximation but also for the underlying hybrid system. In [10, 14], the authors of this contribution developed an approximation-based synthesis approach which is based on the notion of l-complete abstractions. Like other approximationbased methods, our approach suffers from the “curse of complexity”: state sets of approximating DESs may become very large, and, as the subsequent control synthesis step involves forming the product of approximating DESs and a realisation of the specifications, computational effort can become excessive even for seemingly “small” applications. Obviously, complexity also represents a major problem in other control contexts, and it is common engineering knowledge that suitable decomposition techniques form a necessary ingredient for any systematic treatment of complex control problems. Hierarchical approaches are a particularly attractive way of problem decomposition as they provide an extremely intuitive control architecture. This contribution presents a hierarchical synthesis framework which is general enough to encompass both continuous and discrete levels and is therefore especially suited for hybrid control problems. It is based on two previous (rather technical) conference papers [13, 12]. To keep exposition reasonably straightforward, we focus on the case of two control layers. Unlike heuristic approaches, our synthesis framework guarantees that the control layers interact “properly” and do indeed enforce the overall specifications for the considered plant model. Its elegance stems from the fact that the specifications for the lower control level can be considered a suitable abstraction which may be used as a basis for the synthesis of the high-level controller. Formulating specifications for the lower control level may rely on engineering intuition. In fact, our approach allows to encapsulate engineering intuition within a formal framework, hence exploiting positive aspects of intuition while preventing misguided aspects from causing havoc within the synthesis step. In the context of discrete event and hybrid systems, where the “curse of dimensionality” seems to be particularly prohibitive, a number of hierarchical concepts have been discussed in the literature. Our approach has been inspired by the hierarchical DES theory in [19], but is technically quite different because we employ an input/output structure to adequately represent both time and event driven dynamics for hybrid systems. There is also a strong conceptual link to [8], where, as in [15, 4] and in our work, the preservation of fundamental properties across levels of abstraction is of prime concern. We demonstrate the potential of our hierarchical synthesis framework by applying it to a multiproduct batch control problem which is simple enough to serve as illustration for our main ideas, but of enough complexity to make it hard to handle for unstructured synthesis methods.

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discrete event controller (e.g. finite automaton) u(k) ∈ U

y(k) ∈ Y quantisation

1 0 0 1

ξ(t) ∈ Rp continuous plant model (e.g. ODEs) cont. dynamics with discrete external behaviour

Fig. 1. Continuous plant under discrete control.

This contribution is organised as follows: in Section 2, we briefly summarise our abstraction-based approach to hybrid control systems synthesis. In Section 3, it is shown how this approach can be extended to a two-level control structure. Section 4 describes the multiproduct batch control problem. Finally, in Section 5, we apply our results to this test case.

2 Abstraction Based Supervisory Control The purpose of this section is to briefly summarise key results from our earlier work [10, 14]. They apply to the scenario depicted in Fig. 1. There, the plant model is continuous, realised, e.g., by a set of ODEs, but communicates with its environment exclusively via discrete events. Input events from the set U may switch the continuous dynamics, and output events from a set Y are typically generated by some sort of quantisation mechanism. Hence both the input and the output signal are sequences of discrete events, denoted by u and y, respectively. Note that we do not need to specify at this point whether events occur at equidistant instants of time (“time-driven sampling”, “clock time”) or at instants of time that are defined by the plant dynamics, e.g., by continuous signals crossing certain thresholds (“event-driven sampling”, “logic time”). In J.C. Willems’ terminology (e.g. [18]), the (external) behaviour of a dynamic system is the set of external signals that the system can evolve on. Hence, with w := (u, y) and W := U × Y , the external plant behaviour Bp is a set of maps w : N0 → W ; i.e. Bp ⊆ W N0 , where N0 is the set of nonnegative integers and W N0 := {w : N0 → W } represents the set of all sequences in W . To clarify the input/output structure, we use a slightly weakened version of Willems’ I/O-behaviours: Definition 1. B ⊆ W N0 is a (strict) I/- behaviour with respect to (U, Y ), if (i) the input is free, i.e. PU B = U N0 and

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(ii) the output does (strictly) not anticipate the input4 , i.e. PU w| ˜ [0,k] = PU w| ˆ [0,k] ⇒ (∃ w ∈ B)PY w|[0,k] = PY w| ˜ [0,k] and PU w = PU w ˆ for all k ∈ N0 , w, ˜ w ˆ ∈ B; for the strict case the premiss on the l.h.s. is weakened to PU w| ˜ [0,k) = PU w| ˆ [0,k) .

Item (ii) in Def. 1 says that we can change the future (and, in the strict case, the present) of the input without affecting present and past of the output. We now focus on the role of a controller, or supervisor, evolving on the same signal space as the plant model. Adopting the concepts of supervisory control theory for DESs [17] to the behavioural framework, the task of a supervisor Bsup ⊆ W N0 is to restrict the plant behaviour Bp ⊆ W N0 such that the closed loop behaviour contains only acceptable signals. The closed loop behaviour Bcl is Bp ∩ Bsup , because a signal w ∈ Bp “survives” closing the loop if and only if it is also in Bsup . We collect all acceptable signals in the specification behaviour Bspec and say that the supervisor Bsup enforces the specification if Bcl ⊆ Bspec . It is immediately clear that any supervisor must exhibit two additional properties: (i) it must respect the I/O structure of the plant, i.e., it may restrict the plant input but then has to accept whatever output event the plant generates; (ii) it must ensure that, at any instant of time, there is a possible future evolution for the closed loop. This is formalised by the following definition: Definition 2. A supervisor Bsup ⊆ W N0 is admissible to Bp ⊆ W N0 if

(i) Bsup is generically implementable, i.e. k ∈ N0 , w|[0,k] ∈ Bsup |[0,k] , w| ˜ [0,k] ∈ W k+1 , w| ˜ [0,k] ≈y w|[0,k] implies w| ˜ [0,k] ∈ Bsup |[0,k] ; and (ii) Bp and Bsup are non-conflicting, i.e. Bp |[0,k] ∩ Bsup |[0,k] = (Bp ∩ Bsup )|[0,k] for all k ∈ N0 . This leads to the following formulation of supervisory control problems.

Definition 3. Given a plant Bp ⊆ W N0 , W = U × Y , and a specification Bspec ⊆ W N0 , the pair (Bp , Bspec )cp is a supervisory control problem. A supervisor Bsup ⊆ W N0 that is admissible to Bp and that enforces Bspec is said to be a solution of (Bp , Bspec )cp . In [10, 11], we adapt the set-theoretic argument of [17] to show the unique existence of the least restrictive solution for our class of supervisory control problems. Note that the least restrictive solution may be trivial, i.e. Bsup = ∅, 4

The restriction operator ( · )|[k1 ,k2 ) maps sequences w ∈ W N0 to finite strings w|[k1 ,k2 ) := w(k1 )w(k1 + 1) · · · w(k2 − 1) ∈ W k2 −k1 , where W 0 := { } and denotes the empty string. For closed intervals, ( · )|[k1 ,k2 ] is defined accordingly. For W = U × Y , the symbols PU and PY denote the natural projection to the resp. component, i.e. PU w = u and PY w = y for w = (u, y), u ∈ U N0 , y ∈ Y N0 . We use w| ˜ [0,k] ≈y w|[0,k] as an abbreviation for the two strings to be identical up ˜ [0,k] = PU w|[0,k] and PY w| ˜ [0,k) = PY w|[0,k) . to the last output event, i.e. PU w|

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as this is admissible to the plant and, because of Bp ∩ ∅ ⊆ Bspec , enforces the specifications. Obviously, only nontrivial solutions are of interest. Therefore, if ∅ turns out to be the least restrictive solution of (Bp , Bspec )cp , one would conclude that the specifications are “too strict” for the given plant model. If both Bp and Bspec could be realised by finite automata, we could easily compute (a realisation of) the least restrictive solution of (Bp , Bspec )cp by appropriately modifying standard DES tools. While a finite automaton realisation of Bspec ∈ W N0 is quite common for finite W , the hybrid plant is in general not realisable on a finite state space. In [16, 10], we suggest to approach this problem by replacing Bp with an abstraction 5 Bca that is realised by a finite automaton. We can then readily establish a solution Bsup of the (purely discrete) control problem (Bca , Bspec )cp . Clearly, because of Bp ∩ Bsup ⊆ Bca ∩ Bsup ⊆ Bspec , the resulting supervisor also enforces the specifications for the original plant Bp . To show that Bsup is also admissible to Bp and hence solves the original (hybrid) control problem (Bp , Bspec )cp , we employ the notion of completeness: Definition 4. [18] A behaviour B ⊆ W N0 is complete if w∈B

⇔

∀ k ∈ N0 : w|[0,k] ∈ B|[0,k] .

Hence, to decide whether a signal w belongs to a complete behaviour, it is sufficient to look at “finite length portions” of w. The external behaviour induced by a finite state machine is an example for completeness. B = {w ∈ RN0 | limk→∞ w(k) = 0}, on the other hand, is not complete. As a consequence of the following proposition, admissibility of a supervisor is independent of the particular plant dynamics provided that all involved behaviours are complete: Proposition 1. [11] Let Bp ⊆ W N0 be a complete I/- behaviour and Bsup ⊆ W N0 be complete and generically implementable. Then Bp and Bsup are nonconflicting. For the remainder of this paper, we restrict consideration to complete behaviours. Theorem 1 then follows immediately from Proposition 1. Theorem 1. Let Bca ⊆ W N0 , W = U × Y , be an abstraction of an I/behaviour Bp ⊆ W N0 , let Bspec ⊆ W N0 , and let Bsup ⊆ W N0 be a nontrivial solution of the supervisory control problem (Bca , Bspec )cp . If Bp and Bsup are complete then Bsup is a nontrivial solution of (Bp , Bspec )cp . In practice, to make our approach work, a sequence of increasingly refined abstractions Bl , l = 1, 2, . . ., of Bp , i.e. Bp ⊆ . . . Bl+1 ⊆ Bl . . . ⊆ B1 , is employed. In [16, 10], we suggest l-complete approximations as candidate abstractions Bl . One then begins with the “least accurate” abstraction B1 , checks whether a non-trivial solution of (B1 , Bspec )cp exists and, if this is 5

Bca is said to be an abstraction of Bp , if Bp ⊆ Bca .

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not the case, turns to the refined abstraction B2 . In this way, refinement and discrete control synthesis steps alternate until either a nontrivial solution to (Bl , Bspec )cp and hence to (Bp , Bspec )cp is found or computational resources are exhausted. Unfortunately, the latter case often turns out be true. This is the motivation for a hierarchical extension of our approach.

3 Hierarchical Control 3.1 Control Architecture To simplify exposition, we concentrate on the two-level control architecture shown in Fig. 2. Low-level control is implemented by an intermediate layer BLım [BHsup] BHsup: high-level supervisor uH

yH Bım: aggregation & low-level control

uL

yL BLp : low-level plant model

BHım [BLp]

Fig. 2. Plant perspective (dashed) and supervisor perspective (dotted).

Bım communicating with the plant6 BLp via low-level signals uL and y L and with the high-level supervisor BHsup via high-level signals uH and y H . Apart from implementing low-level control mechanisms corresponding to high-level commands uH , the intermediate layer Bım aggregates low-level measurement information y L to provide high-level information y H to BHsup . Aggregation may be both in signal space and in time, i.e. the time axis for high-level signals may be “coarser” than for low-level signals. Note that in this scenario Bım is a behaviour on WH × WL , where WH := UH × YH and WL := UL × YL represent the high and low-level signal sets. From the perspective of the (low-level) plant BLp , interconnecting Bım and BHsup provides the overall controller. Its external behaviour is denoted by BLım [BHsup ] and, as indicated by the dashed box in Figure 2, evolves on the low-level signal space WL . The behaviour BLım [BHsup ] is given by the projection of Bım into WLN0 with the internal high-level signal restricted to BHsup : BLım [BHsup ] := {wL | (∃ wH ∈ BHsup )[ (wH , wL ) ∈ Bım ]}. 6

(1)

To make notation easier to read, all high-level signals, signal sets and behaviours will be indicated by a sub- or superscript “H”, while low-level entities will be characterised by a sub- or superscript “L”. As the plant evolves on a physical, i.e. low-level signal set, its behaviour will be denoted by BLp from now on.

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Clearly, we require the overall controller BLım [BHsup ] to be a (nontrivial) solution of the original control problem (BLp , BLspec )cp . In particular, the overall controller is required to enforce the specification, i.e. we need BLp ∩ BLım [BHsup ] ⊆ BLspec .

(2)

We now re-examine Fig. 2: from the perspective of the high-level supervisor BHsup , interconnecting the intermediate layer Bım with the (low-level) plant model BLp provides a compound high-level plant model. Its external behaviour is denoted by BHım [BLp ] and, as indicated by the dotted box in Fig. 2, evolves on the high-level signal set WH . The behaviour BHım [BLp ] is the projection of Bım into WHN0 with the internal low-level signal restricted to BLp : BHım [BLp ] := {wH | (∃ wL ∈ BLp )[ (wH , wL ) ∈ Bım ]} .

(3)

By the same argument as before, the high-level supervisor BHsup is required to be admissible to the compound high-level plant model BHım [BLp ], i.e. BHsup must be generically implementable, and BHım [BLp ] and BHsup must be non-conflicting. We summarise our discussion of Figure 2 in the following definition. Definition 5. The pair (Bım , BHsup )tl is a two-level hierarchical solution of the supervisory control problem (BLp , BLspec )cp if (i) BLp ∩ BLım [BHsup ] ⊆ BLspec , and (iia) BLım [BHsup ] is admissible to BLp , and (iib) BHsup is admissible to BHım [BLp ]. We will now investigate how to make sure that the admissibility conditions (iia) and (iib) in Def. 5 hold. More precisely, we will discuss which property of Bım will help to enforce these conditions. To structure the discussion, we first address the case of uniform time scales on both signal levels. A layer suitably mediating between different time scales will be investigated subsequently. Uniform Time Scales – Type I Intermediate Layer From Fig. 2 it is obvious that uH and y L can be interpreted as inputs of the intermediate layer Bım , while uL and y H are outputs. It is therefore natural to require that Bım is a (strict) I/- behaviour w.r.t. (UH × YL , YH × UL ). If it is also complete, I/- and completeness properties are passed from BLp to to BHım [BLp ]. Formally, this can be stated as Lemma 1. If Bım is a complete strict I/- behaviour w.r.t. (UH ×YL , YH ×UL ), and if BLp is a complete I/- behaviour w.r.t. (UL , YL ), then BHım [BLp ] is a complete I/- behaviour w.r.t. (UH , YH ). The same property of Bım ensures that completeness and generic implementability carry over from BHsup to BLım [BHsup ]. Formally:

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Lemma 2. If Bım is a complete strict I/- behaviour w.r.t. (UH ×YL , YH ×UL ), and if BHsup is complete and generically implementable, then BLım [BHsup ] is complete and generically implementable. From Lemma 1 and 2, we can immediately deduce the following important statement: if the plant model BLp is a complete I/- behaviour, if the intermediate layer Bım is a complete strict I/-behaviour, and if the high-level supervisor is both complete and generically implementable, then the admissibility Conditions (iia) and (iib) are satisfied. Multiple Time Scales – Type II Intermediate Layer High-level and low-level signals are often defined on different time scales. Typically, in technical realisations, low-level signals “live” on a discrete time axis that is obtained by (fast) equidistant sampling. High-level signals mostly live on a time axis that is generated by low-level signals. A common scenario is that y L produces events, e.g. when certain thresholds are crossed. y H is then a sequence of events, and its time axis is constituted by event times. We assume that high-level commands are immediately issued after the occurrence of a high-level measurement event, hence uH lives on the same time axis as y H . This scenario is illustrated in Fig. 3. We call the resulting high-level time a dynamic time scale and formally define this notion as follows: Definition 6. Let T : YLN0 → N0 N0 . The operator T is said to be a dynamic time scale if T is strictly causal7 and if the time transformation T (y L ) : N0 → N0 is surjective and monotonically increasing for all y L ∈ YLN0 .

For a fixed low-level signal y L , the time transformation T (y L ) maps low-level time j ∈ N0 to high-level time k ∈ N0 . By requiring that T itself is a strictly causal operator, we ensure that at any instant of time the transformation T (y L ) only depends on the strict past of y L . We focus on measurement aggregation operators that are causal with respect to a dynamic time scale: Definition 7. The operator F : YLN0 → YHN0 is said to be causal w.r.t. T if T is a dynamic time scale and if y˜L |[0,j] = yˆL |[0,j]

⇒

F (˜ y L )|[0,k] = F (ˆ y L )|[0,k]

(4)

for k = T (˜ y L )(j) and all j ∈ N0 , y˜L , yˆL ∈ YLN0 .

We still have to link high-level control signals uH to low-level control signals u . This is done via a sample-and-hold device that is triggered by the time transformation T (y L ), i.e. successive high-level control actions are passed on to the lower level whenever a high-level measurement is generated. Formally, this is expressed by uL = uH ◦ T (y L ). In summary, an intermediate layer Bım L

7

Recall that an operator H : U N0 → Y N0 , i.e. an operator mapping signals u to ˆ|[0,k) ⇒ H(˜ u)|[0,k] = H(ˆ u)|[0,k] for signals y, is called strictly causal if u ˜|[0,k) = u ˜, u ˆ ∈ U N0 . all k ∈ N0 , u

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u H(k) y H(k) high-level time scale

k ∈ N0

low-level time scale

j ∈ N0

u L( j ) y L( j )

Fig. 3. Two time scales.

mediating between low-level and high-level time is completely defined by a dynamic time scale T and a measurement aggregation operator F that is causal w.r.t. T : Bım := {(uH , y H , uL , y L )| y H = F (y L ) and uL = uH ◦ T (y L )} .

(5)

It can be shown that (5) represents a complete behaviour and, like the intermediate layer discussed in the previous section, preserves the input/output structure of the plant and generic implementability of the supervisor. Lemma 3. For Bım given by (5) and BLp a complete I/- behaviour w.r.t. (UL , YL ), it follows that BHım [BLp ] is a complete I/- behaviour w.r.t. (UH , YH ). Lemma 4. If Bım is given by (5), and if BHsup is complete and generically implementable, it follows that BLım [BHsup ] is complete and generically implementable. In most practical situations, we will have to combine the two types of intermediate layers discussed on the previous pages. It is intuitively clear that combinations of type I and type II layers will also preserve the input/output structure of BLp and generic implementability of BHsup across the resulting intermediate layer. We will therefore omit a formal treatment (for this, the interested reader is referred to [13]) and conclude the discussion on structural properties of Bım by collecting the relevant facts in the following proposition. Proposition 2. If the plant model BLp is a complete I/- behaviour, if the intermediate layer Bım is an arbitrary combination of type I layers (i.e. complete strict I/-behaviours) and of type II layers (i.e. given by (5)), and if the high-level supervisor is both complete and generically implementable, then the admissibility Conditions (iia) and (iib) are satisfied. We are now in a position to discuss how to design Bım and BHsup (subject to the above constraints) such that BLp ∩ BLım [BHsup ] ⊆ BLspec , the last remaining condition from Definition 5, is also satisfied and (Bım , BHsup )tl therefore forms a two-level hierarchical solution of the control problem (BLp , BLspec )cp .

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3.2 A Bottom-Up Design Procedure We suggest an intuitive bottom-up procedure where we first design appropriate low-level control Bım and then find a suitable high-level supervisor BHsup . In a first step, we formalise the intended relation between high-level and HL N0 low-level signals by the specification BHL spec ⊆ (WH × WL ) . Hence Bspec H L denotes the set of all signal pairs (w , w ) that represent the desired effect of high-level control on the low-level plant BLp and, by implication, on the highlevel measurement. To ensure that wH and wL are related in the intended way, we require the intermediate layer Bım to enforce the specification BHL spec when connected to the low-level plant BLp . This condition is expressed by: {(wH , wL ) ∈ Bım | wL ∈ BLp } ⊆ BHL spec .

(6)

Suppose we have designed Bım to enforce (6), perhaps using classical continuous control methods. In principle, we could then base the design of BHsup on the compound high-level plant BHım [BLp ]. However, from a computational point of view — particularly for hybrid systems — it is preferable to use an ˜ Hp of BHım [BLp ] that does not explicitly depend on the low-level abstraction B dynamics or the precise nature of the implemented low-level control scheme. A ˜ Hp and a high-level specification B ˜ Hspec expressing BLspec suitable abstraction B in terms of high-level signals can be derived from (6): ˜ Hp := {wH | (∃ wL )[ (wH , wL ) ∈ BHL B spec ] } ;

(7)

˜ spec := {w | (∀ w ) [ (w , w ) ∈ Bspec ⇒ w ∈ Bspec ] } . B H

H

L

H

L

HL

L

L

(8)

˜ Hp of the compound high-level plant model is In other words, the abstraction B just the projection of the specification BHL spec onto its high-level signal compo˜ Hspec )cp ˜ Hp , B nents. Then, as desired, the resulting high-level control problem (B does not depend on the actual low-level plant under low-level control, BHım [BLp ], but only on the specification BHL spec of the preceding low-level design step. It follows immediately that any (nontrivial) solution BHsup of the high-level ˜ Hspec )cp will enforce the original low-level specification ˜ Hp , B control problem (B L Bspec when connected to BHım [BLp ], the plant model under low-level control: ˜ Hspec ˜ Hp ∩ BHsup ⊆ B B

=⇒

BLp ∩ BLım [BHsup ] ⊆ BLspec .

(9)

Hence Condition (i) from Def. 5 also holds, and the pair (Bım , BHsup )tl is a two-level hierarchical solution of the overall control problem (BLp , BLspec )cp . ˜ Hspec can be realised by finite automata, a slight modification ˜ Hp and B If B of standard DES methods, e.g. [17], may be used to synthesise BHsup . Such a situation is to be expected when all continuous signals are “handled” by the lower-level control scheme within Bım . Otherwise, another abstraction step is required; see e.g. [10]. The “degree of freedom” in the proposed bottom-up approach is the specification BHL spec . In general, its choice can be guided by the same engineering

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intuition that we would use in a hierarchical ad hoc design. However, unlike heuristic approaches, our method encapsulates intuition in a formal framework where we can prove that the composition of high-level controller and intermediate layer forms a valid solution of the original problem. 3.3 Minimising the Cost of Closed-Loop Operation In addition to a “hard” specification (i.e., a specification that must hold), many applications come with the control objective of minimising a certain cost function. To address this issue, we describe a straightforward extension of the hierarchical framework outlined in Section 3.1 and 3.2. From the low-level perspective, we want to solve the control problem: min max γ L (wL ) s.t. BLsup solves (BLp , BLspec )cp , wL ∈ BLp ∩ BLsup ,

BL sup

wL

(10)

where γ L : BLp ∩ BLspec → R is a (typically additive over time and positive) function to associate the cost γ L (wL ) with each low-level plant signal wL that satisfies the “hard” specification BLspec . Note that when the initial state of the plant is given and the supervisor is sufficiently restrictive, the closed-loop trajectory is unique and the maximum in (10) becomes obsolete. This will be the case in our multiproduct batch application; see also Sections 4 and 5. Suppose that the intermediate layer Bım has been designed as outlined in Sections 3.1 and 3.2. We then define a pessimistic high-level cost function by γ H (wH ) := max{γ L (wL )| (wH , wL ) ∈ Bım , wL ∈ BLp } , wL

(11)

and seek an optimal solution BHsup for the high-level min-max problem ˜ Hp ∩ BHsup . (12) ˜ Hp , B ˜ Hspec )cp and wH ∈ B min max γ H (wH ) s.t. BHsup solves (B

BH sup

wH

Note that the overall controller, i.e. the interconnection BLım [BHsup ] of Bım and the optimal BHsup does not necessarily form an optimal solution to the original problem (10). This is for two reasons: (i) the introduction of Bım reduces the available degrees of freedom; (ii) in the high-level control problem (12), ˜ Hp , resulting the behaviour BHım [BLp ] has been replaced by its abstraction B in over-approximation of actual costs. On the positive side, we expect the problem (12) to be computationally tractable in situations where (10) is not. We want to emphasise that, despite the tradeoff between computational effort and closed-loop performance, our bottom-up design method guarantees the “hard” specification BLspec to hold.

4 Discontinuously Operated Multiproduct Batch Plant In the chemical industries, discontinuously operated multiproduct plants are widely used for the production of fine, or specialty, chemicals. In the sequel,

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A1

A3

Reactors

R2

R1

Feed tanks

Vacuum

F1

Products

B1

F2

Products

B2

F3

B3

Filtersystems

Products

Fig. 4. Example plant.

we describe a specific example for a multiproduct batch control problem. It is idealised to a certain extent but general enough to capture most of the problems that characterise multiproduct batch plants. The plant is used to produce three kinds of colour pigments (Fig. 4): from one of the storage tanks B1, B2, or B3, solvent is pumped into either a large reactor R1 or a small reactor R2. Reactant Ai , i = 1, 2, 3, is added to start reaction i delivering the desired kW i kP i Pi . It is accompanied by a parallel reaction Ai −→ Wi resultproduct: Ai −→ ing in the waste product Wi . If concentration of Wi crosses a given threshold Wi,max , product quality becomes unacceptable and the batch is spoilt. For the duration of the reaction, there are two control inputs: the feed rate of the reactant and the heating/cooling rate for the reactor. After the reaction is finished, the contents of the reactors is filtered through either F1, F2, or F3, and the solvent is collected in the corresponding tank B1, B2, or B3. The solvent can subsequently be fed back into the reactors. If, in any of the filters, darker colours are filtered before lighter ones (say P3 before P1 or P2, and P2 before P1), a cleaning process between the two filtration tasks is needed, taking time tc . The feed rates into the reactors are discrete-valued control inputs as are the decision variables (realised by discrete valve positions) that determine whether a particular reactor is emptied through a particular filter. Heating/cooling rates are continuous-valued control inputs. The overall aim is to produce the demanded product volumes with minimal operating costs, while satisfying quality constraints (upper bounds for waste concentrations) and safety constraints (upper bounds for reactor temperatures). For simplicity, the following assumptions are made for the reactions: 1. all reactions are first order. 2. the volume of reactant Ai , product Pi and waste product Wi , i = 1, 2, 3, is negligible compared to overall reactor volume. The latter can therefore be considered constant during dosing and reaction.

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3. the time constants for heating/cooling of the reactors are small compared to the reaction time constants. The (scaled) reactor temperatures can therefore be considered to be the manipulated variables. With these assumptions, the model equations can be easily derived from component balances: q(t) d cA (t) = − (kP i (t) + kW i (t)) cAi (t), dt i V d cP (t) = kP i (t)cAi (t), dt i d cW (t) = kW i (t)cAi (t), dt i

(13) (14) (15)

where V is the volume of the considered reactor, q the corresponding dosing rate (in kmol/h), and cAi , cPi , cWi are reactant, product and waste concentration in the ith reaction (in kmol/m3 ), i = 1, 2, 3. The reaction kinetics EP i

kP i (t) = kP i0 e− Rθ(t) ,

EW i

kW i (t) = kW i0 e− Rθ(t)

(16)

are determined by temperature θ. Defining u(t) := kW 1 (t), βi := EP i /EW 1 , βi δi δi := EW i /EW 1 , αi := kP i0 /kW 10 , γi := kW i0 /kW 10 , we rewrite (13)–(15) as d q(t) cAi (t) = − αi u(t)βi + γi u(t)δi cAi (t), dt V d cP (t) = αi u(t)βi cAi (t), dt i d cW (t) = γi u(t)δi cAi (t). dt i

(17) (18) (19)

Note that δ1 = γ1 = 1, and that u is strictly monotonically increasing in θ and can therefore be considered as scaled temperature with unit [1/h] .

5 Hierarchical Control of Multiproduct Batch Plant 5.1 Low-level Plant Model The low-level plant model represents the continuous dynamics of filter and reaction processes in the various modes of operation. We consider low-level signals to evolve w.r.t. clock time, using a suitably small sampling period. Note that after a reaction is finished, the respective reactor has to be emptied, i.e., its contents has to be filtered before the reactor can be reused in another production step. Neglecting the time required to fill a reactor, there are at most two concurrent operations performed by the plant. Thus, our lowlevel plant model consists of two subsystems, each of which is being used for one out of three chemical reaction schemes or a subsequent filtering process.

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As a low-level signal space we choose WL = WL1 ×WL2 , where each component corresponds to one subsystem. The possible modes of operation for the subsystem j ∈ {1, 2} consist of the three chemical reactions and the filtering processes. The latter can use any nontrivial combination of the three filters. Including a cleaning and an “idle” mode, this gives a total of 3 + 2 + 7 = 12 modes for each subsystem; they can be conveniently encoded as a discrete input uDj , j = 1, 2, with range UDj = {P1, P2, P3, Clean, Idle, F001, F010, F011, . . . , F111} .

(20)

While in one of the reaction modes Pi, low-level dynamics is modelled by a sampled version of the ODEs (17), (18), (19). The parameters are as follows: β1 = 0.5, α1 = 2.0h−0.5 , β2 = 0.4, α2 = 2.0h−0.6 , β3 = 0.5, α3 = 3.0h−0.5 , δi = γi = 1, i = 1, 2, 3; the initial concentrations at the beginning of each reaction are all zero: cAi 0 = cPi 0 = cWi 0 = 0, i = 1, 2, 3. The product concentrations required at the end of each reaction are cP1 e = 10kmol/m3 , cP2 e = 8kmol/m3 , cP3 e = 12kmol/m3 , and the bounds for the waste concentrations cW1 , cW2 , cW3 are 2kmol/m3 , 1.5kmol/m3 , and 3kmol/m3 , respectively. The volumes of reactor R1 and R2 are 5m3 and 2.5m3 , respectively. The (on/off) dosing signal qj can take values in the set {0, 12kmol/h}, and the control signal uj is required to “live” within the interval [0.01h−1 , 3.0h−1 ], where the upper bound results from safety requirements. The signal (qj , uj ) is seen as an additional low-level input uCj with range UCj ⊆ R2 . We assume the continuous state is measured as a plant output y Cj with range YCj ⊆ R3 . For filtering, an integrator models the progress of time, where the integration constant depends on the number of filters used and the volume of the respective reactor. The time to empty the smaller of the two reactors through one filter is ctf = 6h. If two or three filters are being used simultaneously, this reduces to 3h and 2h. For the larger reactor, filtering takes twice as long. The continuous input uCj is ignored in filtering mode. The completion of either operation corresponds to reaching a target region within the continuous state space. This is indicated by a discrete low-level output y Dj which can take values in {Busy, Done}. The signal space of subsystem j ∈ {1, 2} is then given by the product WLj = ULj × YLj , ULj = UDj × UCj , YLj = YDj × YCj .

This is illustrated in Fig. 5, which summarises the overall control architecture. Note that in Fig. 5, the two subsystems are shown merged. With the above parameters, the typical time to finish a reaction step is between 5h and 10h, with filtering taking at least another two hours. 5.2 Low-level Specification and Cost Function The low-level specification BLspec for the multiproduct batch example includes the following requirements: (i) the mode of operation may only change immediately after the previous operation has been completed; (ii) chemical reactions

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BHsup: high-level supervisor switching strategy (DES) uD

yD

τ

Bım

convert to logic time controller 1 controller 2 controller m

uD

uC

yC

yD completion detection

operation 1 operation 2 operation m BLp : low-level plant

Fig. 5. Control architecture (subsystems merged).

and filtering alternate in each subsystem; (iii) each filter can only be used by one subsystem at a time; (iv) the filters have to be cleaned when need arises (see Section 4); (v) the demanded product volumes are produced. Note that BLspec contains only discrete requirements and therefore represents a typical discrete event specification, albeit in clock time. Formally, we therefore have N0 for some behaviour BD BLspec = BD spec × (UC × YC ) spec over UD × YD , implying that the continuous low-level signals are not restricted8 by BLspec . In a finite realisation of BD spec , the state needs to track the following: current major mode of operation (52 possibilities: three reactions, filtering, or cleaning in both subsystems); current allocation of filters (23 possibilities: three filters which can be allocated to either subsystem); recent usage of filters (33 possibilities: three filters, each of which could have been used for either of three products); product volumes produced so far (63 possibilities: this number results from the additional assumption that only integer multiples of a minimum batch size are allowed and that the maximum demand for each of the three products is known. In our example, 2.5m3 is the minimum batch size and 12.5m3 the maximum demand.) This amounts to 1.16 × 106 states. The integral cost function γ L only refers to the UCj components of the low-level signal. It includes energy cost (heating), material cost (feed rates) and an overhead cost depending on time spent. For a low-level signal wL , let Tf

γ L (wL ) := 0

Tf

(u1 (t) + 0.05q1 (t))dt + 0

Tf

(u2 (t) + 0.05q2 (t))dt +

0.15dt ,

0

(21) where Tf denotes the time when all demanded products have been delivered. 8

Note that safety and quality restrictions are imposed indirectly – the former via the restricted range for uCj = (qj , uj ), the latter via the completion signals y Dj , which are linked to the y Cj reaching their target regions.

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Given low-level plant model, specification and cost function, one could try to solve the optimisation problem (10) as it stands. This amounts to a nonlinear mixed discrete-continuous dynamic program with 2 continuous input signals (u1 , u2 ), discrete input signals (uDj , qj , j = 1, 2) that altogether can take (12 × 2)2 = 576 values, 6 continuous state variables, and a discrete state set with 1.16 × 106 elements. Over an adequate time horizon (about 50h), we found this computationally intractable for off-the-shelf optimisation software. Instead we apply the hierarchical procedure outlined in Section 3.2. 5.3 Hierarchical Design – Low-level Control Recall that low-level control is based on the specification BHL spec representing the intended relation between high-level and low-level signals. For example, we introduce high-level control symbols signifying the commands “run reaction i in subsystem j such that the batch is finished at minimal cost within time τj ∈ T = {1h, 2h, . . . 10h}”. This is implemented by 3 × 2 × 10 = 60 low-level controllers that can be selected by high-level control (see Fig. 5). Obviously, low-level controller design is local in the sense that it only refers to one reaction process and one subsystem at a time. The corresponding dynamic program has 1 binary input (qj ), 1 continuous input (uj ) and 3 continuous state variables (concentrations). It can be solved numerically by standard optimisation software. As an illustration, the minimal cost γ1,i (τ1 ) for reactor R1 to produce one batch of product Pi is given in Table 1. As low-level optimisation does not depend on the demanded overall amount of products, this design step only needs to be performed once over the life-cycle of the plant. Table 1. Minimal cost for reactor R1 to produce one batch. τ1 γ1,1 γ1,2 γ1,3

< 5h ∞ ∞ ∞

5h ∞ 2.81 ∞

6h 3.70 2.58 4.16

7h 3.42 2.46 3.71

8h 3.28 2.40 3.58

9h 3.21 2.36 3.51

10h 3.17 2.32 3.49

As indicated above, high-level control actions consist of modes from UDi and timing parameters from T . Because the timing for the filter process and the idle operation are determined by the mode, there are 3 × 10 + 9 = 39 relevant high-level control actions per subsystem to be encoded in UH . As high-level measurement, we choose the completion component from the lowlevel subsystems, i.e. YH = {Busy, Done1, Done2}. While low-level signals “live” on clock time, high-level signals evolve on logic (event-driven) time, where events are triggered by changes in the YDj -components. 5.4 Hierarchical Design – High-level Control Connecting the intermediate layer (i.e., the low-level control and measurement aggregation mechanism described above) to the plant model, results in a hybrid system with external behaviour BHım [BLp ]. To design a suitable high-level

Hierarchical Hybrid Control of a Multiproduct Batch Plant

215

supervisor, we need a discrete abstraction of BHım [BLp ], a high-level “image” of the original specification BLspec , and a high-level cost function γ H . ˜ Hp diAs pointed out in Section 3.2, we can derive a suitable abstraction B . The specified external rectly from the intermediate layer specification BHL spec behaviour of each subsystem under low-level control (w.r.t. the discrete variables uDj and y Dj ) can be modelled as a timed discrete event system (TDES, see e.g. [3]), where time is still clock time. To obtain a DES realisation of an ˜ Hp of BHım [BLp ], we form the synchronous product of the individabstraction B ual TDESs and remove tick events (which “count” the progress of clock time) by language projection. Note that in our design the first instance where a composition of subsystems needs to be computed occurs after the individual subsystems have undergone considerable simplification. ˜ Hspec can be directly obAccording to (8), the high-level-specification B D tained from Bspec by transforming clock time to logic time. Together with ˜ Hp , we obtain a transition system with 17 × 106 the high-level abstraction B states and an average of 13.1 relevant input events per state. Since every highlevel input event corresponds to a low-level mode (either chemical reaction or filtering) that will be completed at a known cost, the high-level cost function γ H is additive over high-level logic time (i.e. cost per transitions). Thus, the high-level optimisation problem (12) can be solved using standard methods from dynamic programming. On a decent desktop computer, synthesis of the high-level supervisor takes 61 minutes. Fig. 6 shows the obtained closed-loop operation to produce 12.5m3 , 12.5m3 and 7.5m3 of the products P1, P2 and P3, respectively. The overall cost amounts to 27.5. SubSys1:

P1

SubSys2: P1

time 0

P2 P1

P2 P1

20h

P3 P2

P3

40h

Fig. 6. Optimal schedule (filter processes grey, cleaning black).

6 Conclusions We discussed a hierarchical extension of our behavioural approach to hybrid control synthesis. In particular, we provided conditions for intermediate layers ensuring that all control layers interact in the desired sense. These conditions imply that if each layer enforces its specification for an appropriate plant model or abstraction, the resulting overall controller is guaranteed to enforce the overall specification for the underlying plant model. We also discussed how to add optimal performance objectives to hard specifications. We demonstrated the potential of our approach through a multiproduct batch example, which we found intractable using off-the-shelf optimisation software.

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Acknowledgement. We dedicate this paper to Michael Zeitz on the occasion of his 65th birthday. He has always been a role model for us in both research and teaching. We thank A. Kienle for helpful discussions on the multiproduct batch example, and B.V. Mishra and D. Gromov for their help in solving the low-level optimisation problems in Section 5.3. The first author gratefully acknowledges funding by Deutsche Forschungsgemeinschaft (DFG) in the context of Research Unit (Forschergruppe) 468. Part of this work has also been done in the framework of the HYCON Network of Excellence, contract number FP6-IST-511368.

References 1. P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors. Hybrid Systems V, LNCS 1567. Springer-Verlag, 1999. 2. P. J. Antsaklis and A. Nerode, editors. IEEE TAC, Special Issue on Hybrid Systems, volume 43. 1998. 3. B.A. Brandin and W.M. Wonham. Supervisory control of timed discrete-event systems. IEEE TAC, 39:329–342, 1994. 4. P.E. Caines and Y.J. Wei. Hierarchical hybrid control systems: a lattice theoretic formulation. IEEE TAC, 43:4:501–508, 1998. 5. J.E. Cury, B. Krogh, and T. Niinomi. Synthesis of supervisory controllers for hybrid systems based on approximating automata. IEEE TAC, 43:564–568, 1998. 6. S. Engell, G. Frehse, and E. Schnieder, editors. Modelling, Analysis, and Design of Hybrid Systems. LNCIS 279. Springer-Verlag, 2002. 7. X. Koutsoukos, P.J. Antsaklis, J.A. Stiver, and M.D. Lemmon. Supervisory control of hybrid systems. Proceedings of the IEEE, 88:1026–1049, July 2000. 8. R.J. Leduc, B.A. Brandin, W.M. Wonham, and M. Lawford. Hierarchical interface-based supervisory control. In Proc. 40th CDC, pp. 4116–4121, 2001. 9. J. Lunze, B. Nixdorf, and H. Richter. Hybrid modelling of continuous-variable systems with application to supervisory control. In Proc. ECC, 1997. 10. T. Moor and J. Raisch. Supervisory control of hybrid systems within a behavioural framework. Systems and Control Letters, 38:157–166, 1999. 11. T. Moor and J. Raisch. Think continuous, act discrete: DES techniques for continuous systems. Proc. 10th Mediterranean Conf. Contr. and Autom., 2002. 12. T. Moor and J. Raisch. Hierarchical hybrid control of a multiproduct batch plant. In Proc. 16th IFAC World Congress, Prague, Czech Republic, 2005. 13. T. Moor, J. Raisch, and J.M. Davoren. Admissibility criteria for a hierarchical design of hybrid control systems. In Proc. IFAC Conference on the Analysis and Design of Hybrid Systems (ADHS’03), pages 389–394, 2003. 14. T. Moor, J. Raisch, and S.D. O’Young. Discrete supervisory control of hybrid systems based on l-complete approximations. Discrete Event Dynamic Systems, 12:83–107, 2002. 15. G.J. Pappas, G. Lafferriere, and S. Sastry. Hierarchically consistent control systems. IEEE TAC, 45:6:1144–1160, 2000. 16. J. Raisch and S.D. O’Young. Discrete approximation and supervisory control of continuous systems. IEEE TAC, 43:569–573, 1998. 17. P.J. Ramadge and W.M. Wonham. The control of discrete event systems. Proceedings of the IEEE, 77:81–98, 1989. 18. J.C. Willems. Models for dynamics. Dynamics Reported, 2:172–269, 1989. 19. K.C. Wong and W.M. Wonham. Hierarchical control of discrete-event systems. Discrete Event Dynamic Systems, 6:241–306, 1996.

Closed-Loop Fault-Tolerant Control for Uncertain Nonlinear Systems Michel Fliess1 , C´edric Join2 , and Hebertt Sira-Ram´ırez3 1

2

3

´ ´ ´ Equipe ALIEN, INRIA Futurs & Equipe MAX, LIX (CNRS, UMR 7161), Ecole polytechnique, 91128 Palaiseau, France. [email protected] ´ Equipe ALIEN, INRIA Futurs & CRAN (CNRS, UMR 7039), Universit´e Henri Poincar´e (Nancy I), BP 239, 54506 Vandoeuvre-l´es-Nancy, France. [email protected] CINVESTAV-IPN, Secci´ on de Mecatr´ onica, Departamento de Ingenier´ıa El´ectrica, Avenida IPN, No. 2508, Col. San Pedro Zacatenco, AP 14740, 07300 M´exico D.F., M´exico. [email protected]

Summary. We are designing, perhaps for the first time, closed-loop fault-tolerant control for uncertain nonlinear systems. Our solution is based on a new algebraic estimation technique of the derivatives of a time signal, which

! ! !

yields good estimates of the unknown parameters and of the residuals, i.e., of the fault indicators, is easily implementable in real time, is฀ robust฀ with฀ respect฀ to฀ a฀ large฀ variety฀ of฀ noises,฀ without฀ any฀ necessity฀ of knowing฀their฀statistical฀properties.

Convincing฀numerical฀simulations฀are฀provided฀via฀a฀popular฀case-study฀in฀the฀diagnosis฀community,฀namely฀the฀three-tank฀system,฀which฀may฀be฀characterized฀as a฀flat฀hybrid฀system.

Keywords:฀Fault฀diagnosis,฀fault-tolerant฀control,฀uncertain฀nonlinear฀systems,฀ differential฀ algebra,฀ algebraic฀ estimation฀ techniques,฀ derivatives฀ of฀ a noisy฀time฀signal,฀three-tank฀system.

1 Introduction We are further developing recent works on closed-loop fault detection and isolation for linear [11] and nonlinear [10, 25] systems, which may contain uncertain parameters. This important subject which is attracting more and more attention (see, e.g., [2, 3, 5, 19] and the references therein) is treated

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 217–233, 2005. © Springer-Verlag Berlin Heidelberg 2005

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in the nonlinear case like in [10, 25], i.e., via differential algebra and the estimation techniques of [16]. Introducing on-line accommodation, or fault-tolerant control , i.e., the possibility of still controlling a nonlinear system if a fault does occur, is the main novelty of this article. We are therefore achieving in the context of diagnosis one of the fundamental aims of nonlinear control (see, e.g., [27, 30, 37] and the references therein), i.e., we are able to combine on-line parameter estimation, and closed-loop fault-tolerant control. The two main ingredients of our solution are: " "

an algebraic estimation technique [15] which permits to obtain the derivatives of various orders of a noisy time signal1 , and thus excellent estimates of the unknown parameters and of the residuals, i.e., of the fault indicators. Differential฀flatness (see฀[12,฀13]฀and฀[31,฀32,฀33,฀35]):฀we฀all฀know฀that฀this standpoint฀is฀already฀playing฀a฀crucial฀rˆ ole฀in฀many฀concrete฀and฀industrial control฀applications.

Our฀ solution฀ moreover฀ is฀ robust฀ with฀ respect฀ to฀ a฀ large฀ variety฀ of฀ noises, without฀any฀necessity฀of฀knowing฀their฀statistical฀properties. Our฀paper฀is฀organized฀as฀follows.฀Section฀2฀is฀introducing฀the฀basics฀of the฀ differential฀algebraic฀setting.฀ Its฀content฀with฀ respect฀to฀fault฀variables completes฀and฀supersedes฀[10].฀Section฀3฀recalls฀the฀techniques฀for฀estimating฀the฀derivatives฀of฀a฀noisy฀signal.฀Section฀4฀is฀devoted฀to฀the฀three-tank system,฀which฀is฀perhaps฀the฀most฀popular฀case-study฀in฀the฀fault-diagnosis community฀(see, e.g.,฀[29]฀and฀the฀references฀therein).฀Several฀simulations฀are illustrating฀our฀results฀which฀may฀be฀favorably฀compared฀to฀some฀recent฀studies฀on฀this฀subject฀(see, e.g.,฀[22]),฀where฀only฀off-line฀diagnosis฀was฀obtained. A฀short฀conclusion฀indicates฀some฀prolongations.

2฀Differential฀Algebra฀and฀Nonlinear฀Systems We฀will฀not฀recall฀here฀the฀basics฀of฀the฀approach฀to฀nonlinear฀systems฀via differential฀fields฀2 ,฀which฀is฀already฀well฀covered฀in฀the฀control฀literature฀(see, e.g.,฀[6,฀12,฀33,฀35]฀and฀the฀references฀therein). 1

2

This method was introduced in [16] where it gave a quite straightforward solution for obtaining nonlinear state reconstructors, i.e., nonlinear state estimation, which are replacing asymptotic observers and (sub)optimal statistical filters, like the extended Kalman filters. See [9, 14] for applications in signal processing. All differential rings and fields (see, e.g., [26] and [4]) are of characteristics zero d . A differential and are ordinary, i.e., they are equipped with a single derivation dt ring R is a commutative ring such that, ∀ a, b ∈ R, d (a + b) = a˙ + b˙ dt d (ab) = ab ˙ + ab˙ dt

A differential field is a differential ring which is a field. A constant is an element c ∈ R such that c˙ = 0.

Nonlinear Fault-Tolerant Control

219

Notation. Write k X (resp. k{X}) the differential field (resp. ring) generated by the differential field k and the set X. 2.1 Perturbed Uncertain Nonlinear Systems and Fault Variables Let k0 be a given differential ground field. Let k = k0 (Θ) be the the differential field extension which is generated by a finite set Θ = (θ1 , . . . , θα ) of uncertain parameters , which are assumed to be constant3 , i.e., θ˙ι = 0, ι = 1, . . . , α. A nonlinear system is a differential field extension K/k, which is generated by the sets S, π, W , i.e., K = k S, π, W , where 1. S is a finite set of system variables, 2. π = (π1 , . . . , πr ) denotes the perturbation, or disturbance, variables, 3. W = (w1 , . . . , wq ) denotes the fault variables. They satisfy the following properties: " " "

"

The perturbation and fault variables do not “interact”, i.e., the differential extensions k π /k and k W /k are linearly disjoint (see, e.g., [28]). The fault variables are assumed to be independent, i.e., W is a differential transcendence basis of k W /k. Set k{S nom , W nom } = k{S, π, W }/(π), where – (π) ⊂ k{S, π, W } is the differential ideal generated by π, – the nominal fault variables S nom , Wnom are the canonical images of S, W. Assume that the ideal (π) is prime4 . The nominal system is K nom /k, where K nom = k S nom , Wnom is the quotient field of k{S nom , Wnom }. Set k{S pure } = k{S nom , W nom }/(Wnom ) where – the differential ideal (W nom ) ⊂ k{S nom , W nom } is generated by W nom , – the pure system variables S pure are the canonical images of S nom . Assume that the ideal (W nom ) is prime. The pure system is K pure /k, where K pure = k S pure is the quotient field of k{S pure }.

A dynamics is a system where a finite subset u = (u1 , . . . , um ) ⊂ S of control variables has been distinguished, such that the extension K pure /k upure is differentially algebraic. The control variables verify the next two properties: 3

This assumption may be easily removed in our general presentation. An ideal I of a ring R is said to be prime [28] if, and only if, one of the two following equivalent conditions is verified: – the quotient ring R/I is entire, i.e., without non-trivial zero divisors, – ∀ x, y ∈ R such that xy ∈ I, then x ∈ I or y ∈ I. The assumptions for (π) and below for (Wnom ) being prime are thus natural. 4

220

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M. Fliess, C. Join, and H. Sira-Ram´ırez

they do not interact with the fault variables, i.e., the fields k u and k W are linearly disjoint over k. they are independent, i.e., the components of u are differentially algebraically independent over k.

An input-output system is a dynamics where a finite subset y = (y1 , . . . , yp ) ⊂ S of output variables has been distinguished. Only input-output systems will be considered in the sequel. 2.2 Differential Flatness A system K/k is said to be (differentially) flat if, and only if, there exists a finite set z = (z1 , . . . , zm ) of elements in the algebraic closure of K such that " "

its components are differentially algebraically independent over k, the algebraic closures of K pure and k zpure are the same.

The set z is called a flat output. It means that " " "

any pure system variable is a function of the components of the pure flat output and of their derivatives up to some finite order, any component of the pure flat output is a function of the pure system variables and of their derivatives up to some finite order, the components of the flat output are not related by any nontrivial differential relation.

The next property is well known [12, 13]: Proposition 1. Take a flat dynamics with independent control variables, then the cardinalities of z and u are equal. 2.3 Detectability, Isolability and Parity Equations for Fault Variables The fault variable wι , ι = 1, . . . , q, is said to be detectable if, and only if, , where W nom = W nom \{wnom }, the field extension K nom /k unom , W nom ι ι ι is differentially transcendental. It means that wι is indeed “influencing” the output. A subset W = (wι1 , . . . , wιq ) of the set W of fault variables is said to be "

Differentially algebraically isolable if, and only if, the extension k unom , y nom , W

nom

/k unom , y nom

(1)

is differentially algebraic. It means that any component of W nom satisfies a parity differential equation, i.e., an algebraic differential equations where the coefficients belong to k unom , y nom .

Nonlinear Fault-Tolerant Control

" "

221

Algebraically isolable if, and only if, the extension (1) is algebraic. It means that the parity differential equation is of order 0, i.e., it is an algebraic equation with coefficients k unom , y nom . Rationally isolable if, and only if, W nom belongs to k unom , y nom . It means that the parity equation is a linear algebraic equation, i.e., any component of W nom may be expressed as a rational function over k in the variables unom , y nom and their derivatives up to some finite order.

The next property is obvious: Proposition 2. Rational isolability ⇒ algebraic isolability ⇒ differentially algebraic isolability. When we will say for short that fault variables are isolable, it will mean that they are differentially algebraically isolable. Proposition 3. Assume that all fault variables belonging to W are isolable, then card(W ) ≤ card(y)

Proof. The differential transcendence degree5 of the extension k unom , y nom , W nom /k

(resp. k unom , y nom /k) is equal to card(u) + card(W ) (resp. is less than or equal to card(u) + card(y)). The equality of those two transcendence degrees implies our result. 2.4 Observability and Identifiability A system variable x, a component of the state for instance, is said to be observable [7, 8] if, and only if, xpure is algebraic over k upure , y pure . It means in other words that xpure satisfies an algebraic equation with coefficients in k upure , y pure . It is known [7, 8] that under some natural and mild conditions this definition is equivalent to the classic nonlinear extension of the Kalman rank condition for observability (see, e.g., [24]). A parameter θ is said to be algebraically (resp. rationally) identifiable [7, 8] if, and only if, it is algebraic over (resp. belongs to) k upure , y pure .

3 Estimation of the Time Derivatives6 Consider a real-valued time function x(t) which is assumed to be analytic on some interval t1 ≤ t ≤ t2 . Assume for simplicity’s sake that x(t) is analytic around t = 0 and introduce its truncated Taylor expansion 5

6

See, e.g., [28] for the definition of the transcendence degree of a field extension. See [26] for its obvious generalization to differential fields. See [16] and [9] for more details and related references.

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M. Fliess, C. Join, and H. Sira-Ram´ırez N

x(t) =

x(ν) (0) ν=0

tν + o(tN ) ν!

Approximate x(t) in the interval (0, ε), ε > 0, by a polynomial xN (t) = ν N (ν) (0) tν! of degree N . The usual rules of symbolic calculus in Schwartz’s ν=0 x distributions theory [34] yield (N +1)

xN

(N −1) (t) = x(0)δ (N ) + x(0)δ ˙ + · · · + x(N ) (0)δ

where δ is the Dirac measure at 0. From tδ = 0, tδ (α) = −αδ (α−1) , α ≥ 1, we obtain the following triangular system of linear equations for determining estimated values [x(ν) (0)]e of the derivatives7 x(ν) (0): (N −1) ˙ + · · · + [x(N ) (0)]e δ tα x(N +1) (t) = tα [x(0)]e δ (N ) + [x(0)] eδ α = 0, . . . , N

(2)

The time derivatives of x(t) and the Dirac measures and its derivatives are removed by integrating with respect to time both sides of equation (2) at least N times: (ν)

(ν)

(N −1) ˙ + · · · + [x(N ) (0)]e δ τ1α [x(0)]e δ (N ) + [x(0)] eδ ν ≥ N, α = 0, . . . , N (3) τ1 (ν) t τν−1 . . . 0 is an iterated integral. A quite accurate value of = 0 0 where the estimates may be obtained with a small time window [0, t].

τ1α x(N +1) (τ1 ) =

Remark 1. Those iterated integrals are moreover low pass filters8 . They are attenuating highly fluctuating noises, which are usually dealt with in a statistical setting. We therefore do not need any knowledge of the statistical properties of the noises (see [14]).

4 Application to the Three-Tank System 4.1 Process Description The three-tank system can be conveniently represented as in [1] by:

7 8

Those quantities are linearly identifiable [15]. Those iterated integrals may be replaced by more general low pass filters, which are defined by strictly proper rational transfer functions.

Nonlinear Fault-Tolerant Control 8 p > x˙ 1 = −Dμ1 sign(x1 − x3 ) |x1 − x3 | > > > > +u1 /S + w1 /S p > > > > = Dμ3 sign(x3 − p x2 ) |x3 − x2 | x ˙ 2 > > > > sign(x ) |x2 | −Dμ > 2 2 > < +u2 (t)/S + w2 /S p > x˙ 3 = Dμ1 sign(x1 − x3 ) |x1 − x3 | > p > > > −Dμ3 sign(x3 − x2 ) |x3 − x2 | + w3 /S > > > > > y1 = x1 + w4 > > > y2 = x2 + w5 > > : y3 = x3 + w6

223

(4)

where xi , i = 1, 2, 3, is the liquid level in tank i. The control variables u1 , u2 are the input flows. The actuator and/or system faults w1 , w2 , w3 represent power losses and/or leaks; w4 , w5 , w6 are sensor faults. The constant parameters D, S are well known physical quantities. The viscosity coefficients μi , i = 1, 2, 3, are constant but uncertain. The next result is an immediate consequence of proposition 3: Proposition 4. The fault variables w1 , . . . , w6 are not simultaneously isolable. The pure system corresponding to system (4) may be called a flat hybrid system: it is flat in each one of the four regions defined by x1 > x3 or x1 < x3 , and x2 > x3 or x2 < x3 . In all possible cases, x1 , x3 are the components of a flat output. 4.2 Control From the single outflow rate in tank 2 we may assume that system (4) is staying in the region defined by x1 < x3 and/or x3 < x2 . We obtain the following pure open loop control, where x∗1 = F1 and x∗3 = F3 ,

and where

” “ √ u∗1 = S F˙1 + Dμ1 F1 − F3

“ p p ” u∗2 = S F˙3 − Dμ3 F3 − x∗2 + Dμ2 x∗2

x∗2 = F3 −

√ −F˙3 +Dμ1 F1 −F3 Dμ3

2

The loop is closed via a nonlinear extension (see, also, [20, 21]9 ) of the classic proportional-integral (PI) controller: √ u1 = u∗1 + SDμ1 y1 − y3 − SDμ1 9

x∗1 − x∗3 − P1 Se1 − P2 S

e1

Those references also contain most useful material on the control of uncertain nonlinear systems.

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√ √ u2 = u∗2 − SDμ3 y3 − y2 + SC2 y2 + SDμ3 −SDμ2 x∗2 − P3 Se3 − P4 S e3

x∗3 − x∗2

(5)

where ei = yi − Fi∗ is the tracking error. Set for the gain coefficients P1 = P3 = 2.10−2 ,

P2 = P4 = 2.10−4

4.3 Simulation Results General Principles The estimations of the uncertain parameters and of the residuals10 are achieved via the estimations of the first order derivatives of the output variables. Remark 2. In order to test the robustness of our approach, we have added a zero-mean Gaussian noise of variance 0.005. Estimations of the Viscosity Coefficients The values of the known system parameters are D = 0.0144, S = 0.0154. The nominal flatness-based reference trajectories are computed via the following nominal numerical values of the viscosity coefficients μ1 = μ3 = 0.5,

μ2 = 0.675

whereas their true values are = μ1 (1 + 0.33), μreal 1

= μ2 (1 − 0.33), μreal 2

= μ3 μreal 3

The system behavior in the fault free case is presented figure 1. Those viscosity coefficients are algebraically identifiable: μ1 = μ3 = μ2 =

−(S y˙ 1 −u1 ) √ SD y1 −y3 −(S y˙ 1 +S y˙ 3 −u1 ) √ SD y3 −y2 −(S y˙ 1 +S y˙ 2 +S y˙ 3 −u1 −u2 ) √ SD y2

Their estimations, which yield [μ1 ]c = 0.6836,

[μ2 ]c = 0.4339,

[μ3 ]c = 0.4819

are presented in figure 2. After a short period of time has elapsed, those estimates become available for the implementation of our diagnosis and accommodation schemes. 10

A residual is a fault indicator which is usually deduced from some parity equation. Here it is obtained via the estimates of the unknown coefficients and of the derivatives of the control and output variables.

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225

−5

4.5

x 10

0.35

4

0.3

u1

3.5 0.25

3

u2

2.5

X1

X2

X3

0.2

2

0.15

1.5

0.1

1 0.05

0.5 0 0

1000

2000 Time (Te)

3000

4000

(a) Control inputs

0 0

1000

2000 Time (Te)

3000

4000

(b) Measured outputs Fig. 1. Fault-free case

Actuator and System Faults Fault diagnosis Assuming only the existence of the fault variables w1 , w2 yields their algebraic isolability: √ w1 = S [ˆy˙ 1 + μ1 D y1 − y3 ] − u1 √ √ ˜ w2 = S y˙ 2 − μ3 D y3 − y2 + μ2 D y2 − u2

Convenient residuals r1 , r2 are obtained by replacing in the above equations the viscosity coefficients μ1 , μ2 , μ3 by their estimated values [μ1 ]c , [μ2 ]c , [μ3 ]c : √ r1 = S [ˆy˙ 1 + [μ1 ]c D y1 − y3 ] − u1 √ √ ˜ r2 = S y˙ 2 − [μ3 ]c D y3 − y2 + [μ2 ]c D y2 − u2

Fault-tolerant control

Using the closed loop control u1 and u2 and the residual estimation, define a fault-tolerant control by = u1 + ua1 u[FTC] 1 = u2 + ua2 u[FTC] 2 where " "

u1 , u2 are given by formula (5), the additive control variables ua1 , ua2 are defined by ua1 = −r1 ,

ua2 = −r2

The simulations are realized by assuming a detection delay Tdi of the fault variable wi .

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2

2

1.5

1.5

1

1

0.5

0.5

0 0

1000

2000 Time (Te)

3000

4000

0 0

1000

(a) μ1

2000 Time (Te)

3000

4000

(b) μ2 2

1.5

1

0.5

0 0

1000

2000 Time (Te)

3000

4000

(c) μ3 Fig. 2. Estimations of the viscosity coefficients

Simulation comments Figures 3-(c)-(d), 4-(c)-(d) and 5-(c)-(d) indicate an excellent fault diagnosis for the following three classic cases (see, e.g., [17]): 1. w1 = −0.5u1, w2 = −0.5u2, for t > 1000Te, where Te is the sampling period (figures 3), 2. w1 = −0.5u1, w2 = −0.5u2 for t > 2000Te (figures 4), t t 3. w1 = (−0.5− 16000T )u1 , w2 = (−0.5− 16000T )u2 for t > 1000Te (figures 5). e e The behavior for the residuals changes at time t = 500Te. This is due to the fact that the nominal value of μi is being used for t < 500Te. The interest of the fault-tolerant control is demonstrated in figures 3, 4 and 5. Note that the

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227

simulations were realized with a delay of Tdi = 100T e for the fault-tolerant control. In figure 6 the system is corrupted by two major faults variables, where w1 = −0.9u1 , w2 = −0.9u2 for t > 1000Te. The fault-tolerant control is then saturating the actuator. The output references cannot be reached.

0.3

0.3

0.25

0.25

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fault occurrence

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2000 Time (Te)

(d) r2

Fig. 3. Fault occurrence in steady mode

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2000 Time (Te)

fault occurrence

−1

3000

4000

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1000

(c) r1

2000 Time (Te)

3000

4000

(d) r2

Fig. 4. Fault occurrence in dynamical mode

Combination of System and Sensor Faults Fault diagnosis We associate here the leak w2 and the sensor fault w4 , which is algebraically isolable: ”2 “ √ w4 = y1 − y3 −

y˙ 3 +μ3 D(y3 −y2 ) μ1 D

y3 −y2

It yields in the same way as before the residual r 4 = y1 − y3 −

“

y˙ 3 +[μ3 ]c D(y3 −y2 ) [μ1 ]c D

√

y3 −y2

”2

Nonlinear Fault-Tolerant Control

0.3

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2000 Time (Te)

3000

4000

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−4

1

1000

229

−4

x 10

1.5

x 10

1 0.5 0

0 −0.5 fault occurrence

−1 0

1000

2000 Time (Te)

−1

3000

4000

fault occurrence

−1.5 0

(c) r1

1000

2000 Time (Te)

3000

4000

(d) r2 Fig. 5. Fault of type 3

Fault-tolerant control The leak w2 is accommodated as in section 4.3. For the sensor fault w4 , accommodation is most simply achieved by subtracting r4 from the measurement y1 when closing the loop (5). Simulation comments Figures 7-(c)-(d) (resp. 7-(a)-(b)) show an excellent fault diagnosis (resp. ac√ commodation) for w2 (t) = −0.3[μ2 ]c D x2 , for t > 1000Te, and w4 (t) = 0.02, for t > 2500Te.

230

M. Fliess, C. Join, and H. Sira-Ram´ırez −4

1.6

x 10

1.4

0.3

1.2

0.25

1

0.2

0.8 0.15

0.6

u2

0.2 0 0

0.1

u1

0.4

1000

0.05

2000 Time (Te)

3000

(a) Control inputs

4000

0 0

1000

2000 Time (Te)

3000

4000

(b) Measured outputs

Fig. 6. Actuator saturations

Remark 3. Figures 8-(a) and 8-(b), when compared to figures 1-(b) and 7-(b), show quite better performances of the feedback loop (5) when the nominal values of the viscosity coefficients are replaced by the estimated ones.

5 Conclusion This communication should be viewed as a first draft of a full paper which will comprise also state estimation [16, 36] and many more examples. Those simple solutions of long-standing problems in nonlinear control are robust with respect to a large variety of perturbations and may be quite easily implemented in real time. They were made possible by a complete change of viewpoint on estimation techniques, where the classic asymptotic and/or probabilistic methods are abandoned11 . Further studies will demonstrate the possibility of controlling nonlinear systems with poorly known models, i.e., not only with uncertain parameters. Acknowledgement. It is an honor and a pleasure for the authors to dedicate this work to Prof. M. Zeitz for his 65th birthday as a tribute to his wonderful scientific achievements. At least a few words in German are in order: gewidmet Herrn Prof. Dr.-Ing. M. Zeitz zum 65. Geburtstag.

11

See [16] for more details.

Nonlinear Fault-Tolerant Control

0.3

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y

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−5

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x 10

0.1 fault occurrence 0.05

0

0

−0.05 fault occurrence −5 0

1000

2000 Time (Te)

3000

4000

−0.1 0

1000

(c) r2

2000 Time (Te)

3000

4000

(d) r4 Fig. 7. Leak and sensor faults

References 1. AMIRA-DTS2000, Laboratory setup three tank system. Amira Gmbh, Duisburg, 1996. 2. M. Basseville, V. Nikiforov, Detection of Abrupt Changes: Theory and Application. Prentice Hall, Englewood Cliffs, NJ, 1993. 3. M. Blanke, M. Kinnaert, J. Lunze, M. Staroswiecki, Diagnosis and FaultTolerant Control, Springer, Berlin, 2003. 4. A. Chambert-Loir, A Field Guide to Algebra, Springer, Berlin, 2005. 5. J. Chen, R. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems, Kluwer, Boston, 1999. 6. E. Delaleau, Alg`ebre diff´erentielle, in Math´ematiques pour les Syst`emes Dynamiques, J.P. Richard (Ed.), vol. 2, chap. 6, pp. 245-268, Herm`es, Paris, 2002.

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0.3

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2000 Time (Te)

3000

(a) Fault-free case

4000

0 0

y1 x1

1000

2000 Time (Te)

3000

4000

(b) Leak and sensor faults with faulttolerant control

Fig. 8. Control feedback with the estimated viscosity coefficients 7. S. Diop, M. Fliess, On nonlinear observability, Proc. 1st Europ. Control Conf., Herm`es, Paris, 1991, 152-157. 8. S. Diop, M. Fliess, Nonlinear observability, identifiability and persistent trajectories, Proc. 36th IEEE Conf. Decision Control, Brighton, 1991, 714-719. 9. M. Fliess, C. Join, M. Mboup, H. Sira-Ram´ırez, Compression diff´erentielle de transitoires bruit´es, C.R. Acad. Sci. Paris, s´ erie I, 339, 2004, 821-826. 10. M. Fliess, C. Join, H. Mounier, An introduction to nonlinear fault diagnosis with an application to a congested internet router, in Advances in Communication Control Networks, S. Tabouriech, C.T. Abdallah, J. Chiasson (Eds), Lect. Notes Control Inform. Sci., vol. 308, Springer, Berlin, 2005, pp. 327-343. 11. M. Fliess, C. Join, H. Sira-Ram´ırez, Robust residual generation for linear fault diagnosis: an algebraic setting with examples, Internat. J. Control, 77, 2004, 1223-1242. 12. M. Fliess, J. L´evine, P. Martin, P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples, Internat. J. Control, 61, 1995, 1327-1361. 13. M. Fliess, J. L´evine, P. Martin, P. Rouchon, A Lie-B¨ acklund approach to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control, 44, 1999, 922-937. 14. M. Fliess, M. Mboup, H. Mounier, H. Sira-Ram´ırez, Questioning some paradigms of signal processing via concrete examples, in Algebraic Methods in Flatness, Signal Processing and State Estimation, H. Sira-Ram´ırez, G. SilvaNavarro (Eds.), Editiorial Lagares, M´exico, 2003, pp. 1-21. 15. M. Fliess, H. Sira-Ram´ırez, An algebraic framework for linear identification, ESAIM Control Optim. Calc. Variat., 9, 2003, 151-168. 16. M. Fliess, H. Sira-Ram´ırez, Control via state estimations of some nonlinear systems, Proc. 6th IFAC Symp. Nonlinear Control Syst. (NOLCOS), Stuttgart, 2004.

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17. P.M. Frank, Fault diagnosis dynamic systems using analytical and knowledgebased redundancy - A survey and some new results, Automatica, 26, 1990, 459-474. 18. J. Gertler, Survey of model-based failure detection and isolation in complex plants, IEEE Control Systems Magazine, 8, 1988, 3-11. 19. J. Gertler, Fault Detection and Diagnosis in Engineering Systems, Marcel Dekker, New York, 1998. 20. V. Hagenmeyer, E. Delaleau, Exact feedforward linearization based on differential flatness, Internat. J. Control, 76, 2003, 537-556. 21. V. Hagenmeyer, E. Delaleau, Robustness analysis of exact feedforward linearization based on differential flatness, Automatica, 39, 2003, 1941-1946. 22. D. Henry, A. Zolghadri, M. Monsion, S. Ygorra, Off-line robust fault diagnosis using the generalized structured singular value, Automatica, 38, 2002, 13471358. 23. R. Isermann, P. Ball´e, Trends in the application of model-based fault detection and diagnosis of technical processes, Control Eng. Practice, 5, 1997, 709-719. 24. A. Isidori, Nonlinear Control Systems, 3rd ed., Springer, London, 1995. 25. C. Join, H. Sira-Ram´ırez, M. Fliess, Control of an uncertain three-tank-system via on-line parameter identification and fault detection, Proc. 16th IFAC World Congress on Automatic Control, Prague, 2005. 26. E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. 27. M. Krsti´c, I. Kanellakopoulos, P. Kokotovi´c, Nonlinear and Adaptative Control Design, Wiley, New York, 1995. 28. S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, MA, 1993. 29. J. Lunze, J. Askari-Marnani, A. Cela, P.M. Frank, A.L. Gehin, T. Marku, L. Rato, M. Staroswiecki, Three-tank control reconfiguration, in Control of Complex Systems, K.J. Astr¨ om, D. Blanke, A. Isidori, W. Schaufelberger, R. Sanz (Eds), Springer, Berlin, 2001, pp. 241-283. 30. R. Marino, P. Tomei, Nonlinear Control Design: Geometric, Adaptative and Robust, Prentice-Hall, London, 1995. 31. P. Martin, P. Rouchon, Syst`emes plats de dimension finie, in Commandes non lin´eaires, F. Lamnabhi-Lagarrigue, P. Rouchon (Eds), Herm`es, Paris, 2003. 32. P. Martin, P. Rouchon, Catalogue de syst`emes plats, in Commandes non lin´eaires, F. Lamnabhi-Lagarrigue, P. Rouchon (Eds), Herm`es, Paris, 2003. 33. J. Rudolph, Beitr¨ age zur flacheitsbasierten Folgeregelung linearer und nichtlinearer Syteme endlicher und undendlicher Dimension, Shaker Verlag, Aachen, 2003. 34. L. Schwartz, Th´eorie des distributions, 2e ´ed., Hermann, Paris, 1966. 35. H. Sira-Ram´ırez, S. Agrawal, Differentially Flat Systems, Marcel Dekker, New York, 2004. 36. H. Sira-Ram´ırez, M. Fliess, On the output feedback control of a synchronous generator, Proc. 43rd IEEE Conf. Decision Control, Atlantis, Bahamas, 2004. 37. J.T. Spooner, M. Maggiore, R. Ordonez, K.M. Passino, Stable Adaptative Control and Estimation for Nonlinear Systems, Wiley, New York, 2002.

Feedforward Control Design for Nonlinear Systems under Input Constraints Knut Graichen and Michael Zeitz Institut f¨ ur฀Systemdynamik฀und฀Regelungstechnik,฀Universit¨at฀Stuttgart, Pfaffenwaldring฀9,฀70569฀Stuttgart,฀Germany. {graichen,zeitz}@isr.uni-stuttgart.de Summary.฀ A฀new฀approach฀is฀presented฀for฀the฀feedforward฀control฀design฀of฀nonlinear฀SISO฀systems฀with฀the฀application฀to฀finite–time฀setpoint฀transitions฀under input฀constraints.฀The฀inversion–based฀design฀treats฀the฀transition฀task฀as฀a฀two– point฀boundary฀value฀problem฀(BVP)฀defined฀in฀the฀input–/output฀coordinates฀of the฀considered฀system.฀For฀its฀solvability,฀free฀parameters฀are฀provided฀in฀a฀set–up function฀for฀the฀right–hand฀side฀of฀the฀input–output฀dynamics฀ODE.฀This฀concept allows฀to฀directly฀incorporate฀input฀constraints฀within฀the฀BVP,฀which฀can฀be฀solved in฀a฀straightforward฀manner,฀e.g.฀with฀a฀standard Matlab function.฀Furthermore, an฀approach฀is฀proposed฀ for฀ the฀design฀ of฀the฀transition฀time฀in฀dependence฀of฀a desired฀ aggressiveness฀ of฀ the฀feedforward฀ control.฀ The฀ side–stepping฀ maneuver฀ of the฀inverted฀pendulum฀on฀a฀cart฀serves฀as฀an฀example฀to฀illustrate฀the฀concept.

Keywords:฀Nonlinear฀feedforward฀control,฀setpoint฀transition,฀system฀inversion,฀internal฀dynamics,฀input฀constraints,฀boundary฀value฀problem,฀trajectory planning.

1 Introduction In many practical applications where high performance trajectory tracking is required, the two–degree–of–freedom control scheme in Figure 1 is commonly used to enable the independent design of the feedforward and the feedback control [11]. The feedforward control ΣFF is applied to achieve the desired tracking performance of the output y, whereas the feedback control ΣFB is designed such that the system Σ is appropriately stabilized and robustified against model uncertainties and disturbances. The signal generator Σ∗ provides the reference trajectory y ∗ (t) for both the feedback and the feedforward control. In comparison to the broad spectrum of available design methods for feedback control, only few methods are known for a systematical feedforward control design, which forms a contrast to the respective demand in industry. The reason for this methodological gap is related to the system inversion

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 235–252, 2005. © Springer-Verlag Berlin Heidelberg 2005

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required in course of the feedforward control design and to the respective difficulties arising with nonlinear systems. A further difficulty arises if input constraints have to be satisfied, e.g. due to physical limitations of actuators or safety reasons in industrial applications.

ΣFF Σ∗

y∗ -

ΣFB

u∗ u

Σ

y

Fig. 1. Structure of the two–degree–of–freedom (2dof) control scheme [11] with system Σ, feedback control ΣFB , feedforward control ΣFF , and signal generator Σ∗ .

The most frequently studied feedforward design problem concerns the transition between two stationary setpoints within a finite time interval. Practical applications of this are e.g. position changes of mechatronical systems or load changes in process control. From a mathematical point of view, a finite–time transition between stationary setpoints forms a two–point boundary value problem (BVP). Its solution comprises the input trajectory u∗ (t) in dependence of a predefined output trajectory y ∗ (t), which requires the inversion of the considered system. If input constraints are given, they have to be considered in course of the feedforward control design. The feedforward control design is purely algebraic if the considered system is differentially flat [5]. This is related to the flatness property stating that all system variables can be parameterized in dependence of the flat output. Hence, the transition task can be defined in the coordinates of the flat output and the control trajectory is calculated without any time integration [6, 10]. If the considered system is not flat, the design of a feedforward control is not as trivial. Devasia, Chen, and Paden [4, 3] thoroughly investigated the inversion–based feedforward design with respect to a desired output trajectory. The proposed technique is based on a stable system inversion in the coordinates of the input–output normal form. Thereby, the internal dynamics of the system is split into stable and unstable parts and is numerically solved by applying an iterative forward and backward integration scheme. However, the integration leads to a pre– and/or postactuation time interval. This means that the feedforward control has to start in advance of the output trajectory (preactuation), or reaches the final stationary input value only asymptotically (postactuation), although the output transition is performed in a finite time. These effects are well–known and must be taken into account if stable system inversion is used, see e.g. its application to flexible structures [2, 20] and aircraft systems [14]. The pre– and postactuation intervals of the feedforward control are necessary if the focus is on tracking a predefined output trajectory, e.g. exact path following for autonomous vehicles. A new approach presented in [7, 8] treats the setpoint transition problem as a two–point BVP throughout all design steps of the inversion–based feed-

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forward control. The main idea is that the construction of the desired output trajectory requires free parameters for the solvability of the BVP of the internal dynamics and for the causality of the feedforward control trajectory. The BVP with free parameters can be solved with standard BVP solvers, e.g. with the Matlab function bvp4c [18]. Thereby, the determined free parameters distort the desired output trajectory compared to a predefined trajectory, which must be accepted for the causality of the feedforward control and in order to accomplish the transition problem in a finite time. So far, all these approaches do not directly consider input constraints within the feedforward control design, as it is possible in case of optimization– based techniques. For instance, in [17] constrained nonlinear optimal control problems are solved using flatness or inversion–based approaches. In this contribution, the approach [7, 8] developed by the authors is further elaborated by considering input constraints for the resulting feedforward control trajectory without the necessity of introducing inequality constraints or objective functions [17]. Thereby, the finite–time transition problem is treated as a BVP of both the input–output dynamics and the internal dynamics. The free parameters for the solvability of the BVP are provided in a set–up function for the right–hand side of the input–output dynamics. In order to incorporate the input constraints, this set–up is case–dependently re–planned if the constraints are violated. Due to this concept, the input constraints are directly incorporated into the BVP of the transition problem. Similar to the approach [7, 8], the constrained feedforward control is again calculated with the Matlab BVP solver bvp4c [18]. An important parameter of the feedforward control design is the transition time. If it is chosen too small with respect to the system dynamics and input constraints, the transition is not realizable and the BVP has no solution. Therefore, a concept is proposed to calculate the transition time within the BVP in dependence of a design parameter for the aggressiveness of the feedforward control. The paper is outlined as follows: in Section 2, the feedforward control task for the setpoint transition problem is defined as a two–point BVP for the state and input of the considered nonlinear system. The coordinates of the input–output normal form are used to reformulate the BVP in an appropriate manner, and the inversion–based feedforward design approach [7, 8] proposed by the authors is shortly recapitulated. In Section 3, the input constraints are considered by introducing a case–dependent set–up function for the right– hand side of the input–output dynamics ODE. The numerical solution of the resulting BVPs is discussed in Section 4. Furthermore, Section 5 describes the calculation of the transition time within the BVP in dependence of a desired aggressiveness of the feedforward control. Finally, the side–stepping of the inverted pendulum on a cart serves as an example to illustrate the developed feedforward design concept.

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2 Finite–time Transition Problem for Nonlinear Systems Considered are nonlinear SISO systems Σ:

x˙ = f (x, u),

x(0) = x0 ,

y = h(x)

(1)

with time t ∈ R, state x ∈ Rn , input u ∈ R, and output y ∈ R. The vector field f : Rn × R → Rn and the output function h : Rn → R are sufficiently smooth. Furthermore, the input u(t) is assumed to be constrained by u(t) ∈ [umin, umax ].

(2)

This condition accounts for the fact that many applications require to satisfy certain input constraints due to physical limitations, e.g. maximum available torque of mechatronic devices or limited cooling/heating power for chemical reactors. A widespread control problem concerns the transition between two stationary setpoints (u∗0 , x∗0 , y0∗ ) and (u∗T , x∗T , yT∗ ) of system (1) within a finite time interval t ∈ [0, T ]. It is assumed that the stationary solutions (x∗0 , u∗0 ) : (x∗T , u∗T ) :

f (x∗0 , u∗0 ) = 0, f (x∗T , u∗T ) = 0,

y0∗ = h(x∗0 ), yT∗ = h(x∗T ).

(3)

are uniquely determined, whereby the stationary inputs u∗0 , u∗T must also satisfy the constraints (2). The transition between the setpoints (3) in a finite time T imposes the following boundary conditions (BCs) on the system (1): x(0) = x∗0 ,

x(T ) = x∗T .

(4)

Note that the initial state x0 = x∗0 has to be consistent with the stationary setpoint x∗0 . From a mathematical point of view, the n first–order differential equations in (1) and the 2 n BCs (4) form a two–point boundary value problem (BVP) for the states xi (t), i = 1, . . . , n in dependence of the constrained input trajectory u(t), t ∈ [0, T ]. Thereby, the transition time T must be chosen large enough with respect to both the system dynamics and the constraints. 2.1 Transition Problem in Input–Output Coordinates The inversion–based feedforward control design is based on the input–output representation of the considered system [4, 3, 7]. The prerequisite for the derivation of the input–output coordinates of the SISO system (1) is the definition of the relative degree r: Definition 1 ([12, 16]). The nonlinear SISO system (1) with the output y = h(x) has the relative degree r at point x0 if

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∂ i L h(x) = 0, ∂u f ∂ r L h(x) = 0 ∂u f

(5)

i ∈ {0, . . . , r − 1},

for all x in a neighborhood of x0 . The operator Lf represents the Lie derivative along the vector field f . Literally, the relative degree r denotes the number of times the output y has to be differentiated until the input u appears explicitly. The input–output coordinates of the system (1) are determined via the diffeomorphism [12, 16] [y, y, ˙ . . . , y (r−1) , η]T = φ(x) y

(i)

=

Lif h(x)

with

= φi+1 (x),

η = φη (x) ∈ R

i = 0, . . . , r − 1

n−r

(6)

,

where η is a supplementary state vector to complete the diffeomorphism φ(x) ∈ Rn to a coordinate system. With (6), the system (1) can be transformed into the nonlinear input–output normal form y (r) = α(y, y, ˙ . . . , y (r−1) , η, u) η˙ = β(η, y, y, ˙ ...,y

(r−1)

, u)

(7) (8)

with α(·) = Lrf h ◦ φ−1 and βi (·) = Lf φη,i ◦ φ−1 , i = 1, . . . , n − r. The chain of r integrators (7) with input u represents the input-output dynamics. The internal dynamics is defined by the differential equation (8) for the state η ∈ R(n−r) . The BCs (4) of the considered finite–time transition problem can be transformed via the diffeomorphism (6) into the input–output coordinates: y(0) = y0∗ , y(T ) = yT∗ , y (i) (0) = y (i) (T ) = 0, i = 1, . . . , r − 1

η(0) =

η ∗0

=

φη (x∗0 ),

η(T ) =

η ∗T

=

φη (x∗T ).

(9) (10)

Thereby, the BVP (1)–(4) is split into two coupled BVPs (7)–(10) for y(t) and η(t) in dependence of the constrained input u(t). 2.2 Inversion–based Feedforward Control Design The determination of a feedforward control trajectory u∗ (t) requires the solution of the two coupled BVPs (7)–(10). The inversion–based feedforward control design [4, 3, 7] is based on the inverse input-output dynamics (7) u∗ = α−1 y ∗ , . . . , y ∗(r) , η∗

(11)

in dependence of a desired output trajectory y ∗ (t) ∈ C r and the state η ∗ (t) of the internal dynamics (8).1 The asterisk (∗ ) is used to characterize the 1

In view of the definition (5) of the relative degree r, it is assumed that the inverse function α−1 exists in the transition region between x∗0 and x∗T .

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feedforward variables. In order to determine η ∗ (t), the internal dynamics (8) can be rewritten in a more appropriate manner by substituting the feedforward control (11) into (8), i.e. ¯ ∗ , y ∗ , y˙ ∗ , . . . , y ∗(r) ), η˙ ∗ = β(η

η ∗ (0) = η∗0 ,

η∗ (T ) = η ∗T .

(12)

subject to the 2(n − r) BCs (10). The output y ∗ (t) and its time derivatives serve as inputs to (12). If the corresponding zero dynamics (by setting y ∗(i) = 0, i = 0, 1, . . . , r in (8)) is stable, the system (1) is minimum–phase; it is nonminimum–phase, if the zero dynamics is unstable. Note that the BVP (12) of the internal dynamics is overdetermined with (n − r) ODEs and 2(n − r) BCs. Its solution plays a key role in the feedforward control design. The main idea of the new approach presented in [7, 8] is that the BVP of the internal dynamics generally requires (n − r) free parameters p = (p1 , . . . , pn−r ) for its solvability. The free parameters are provided in a set–up Υ (t, p) for the output trajectory y ∗ (t) = Υ (t, p), which satisfies the 2r BCs (9). The set–up Υ (t, p) can be constructed e.g. with polynomials or trigonometric basis functions [7, 8]. Substituting y ∗ (t) = Υ (t, p) and its time derivatives into the internal dynamics (12) results in a BVP with free parameters, which can be solved e.g. with the standard Matlab solver bvp4c [18], see Section 4. Its solution comprises the trajectory η ∗ (t), t ∈ [0, T ] as well as the parameter set p, which yields the desired output trajectory y ∗ (t) = Υ (t, p). The shape of y ∗ (t) is determined by the parameter set p in Υ (t, p), which may result e.g. in an initial under– or overshoot occurring in the trajectory y ∗ (t). This can be seen as a distortion of a predefined monotonic transition trajectory, which must be accepted to realize the finite–time transition [7, 8]. Finally, the feedforward control u∗ (t), ∈ [0, T ] follows from (11) with y ∗ (t) and η ∗ (t). This approach performs the setpoint change in the finite time interval t ∈ [0, T ], since all the BCs (9)–(10) for y ∗ (t) and η ∗ (t) are satisfied and the feedforward control u∗ (t) is constant outside the transition interval. However, input constraints for the feedforward control u∗ (t) ∈ [umin , umax ] can only be considered “manually”, i.e. by increasing the transition time T or choosing different polynomials or trigonometric basis functions for the set–up Υ (t, p), and subsequently recalculating the feedforward control.

3 Boundary Value Problems for Constrained Input In order to directly incorporate the input constraints (2) within the feedforward control design, a closer look at the feedforward control (11) reveals how u∗ (t) may be influenced. The trajectory η ∗ (t) is determined by the BVP (12) of the internal dynamics. The only way to directly affect u∗ (t) is through the highest time derivative y ∗(r) (t) of the output, since the remaining derivatives y ∗(i) (t), i = 1, . . . , r − 1 and y ∗ (t) can be solved by integration. Therefore, a

Feedforward Control Design under Input Constraints

241

new function α ˆ = y ∗(r) is introduced to parametrize the highest time derivative of the output y ∗ (t), such that the setpoint transition is defined by the two BVPs ˆ y ∗(r) = α,

y ∗ (0) = y0∗ , y

∗(i)

(0) = y

y ∗ (T ) = yT∗ , ∗(i)

¯ ∗ , y ∗ , y˙ ∗ , . . . , y ∗(r−1) , α), η˙ ∗ = β(η ˆ η ∗ (0) = η ∗0 ,

(T ) = 0 ,

(13)

i = 1, . . . , r − 1

η ∗ (T ) = η ∗T .

(14)

The solutions y ∗ (t) and η ∗ (t) of the two BVPs (13)–(14) as well as the feedforward trajectory u∗ (t) in (11) mainly depend on the set–up of the function α ˆ = y ∗(r) with respect to the following objectives: (i) In order to guarantee that the feedforward trajectory u∗ (t) in (11) is con-

tinuous at the bounds t = 0 and t = T , the output trajectory y ∗ (t) must meet two additional BCs y ∗(r) (0) = 0,

y ∗(r) (T ) = 0 .

(15)

These BCs have to be satisfied by the function α ˆ = y ∗(r) (t). (ii) The solvability of the BVPs (13)–(14) defined by n first–order ODEs and

2 n BCs generally requires n free parameters. Similar to [8], the parameters p = (p1 , . . . , pn ) are provided in a set–up function Φ(t, p), which can be used to parametrize α ˆ = Φ(t, p). Thereby, the function Φ(t, p) has to satisfy the BCs (15), i.e. Φ(0, p) = 0 and Φ(T, p) = 0. Two suitable alternatives to construct Φ(t, p) are ⎧ n n k+1 t t ⎪ ⎪ ⎪ + p polynomial (16a) − p k k ⎪ ⎨ T T k=1 k=1 Φ(t, p) = n ⎪ kπt ⎪ ⎪ ⎪ sine series (16b) pk sin ⎩ T k=1

In the polynomial set–up (16a), the free parameters p = (p1 , . . . , pn ) are the coefficients of the highest order terms. The polynomial (16a) directly satisfies the BCs (15), which is easily seen for t = 0 and t = T . The sine series (16b) has a particularly simple structure because the BCs (15) are already satisfied by the sine functions. For many higher order systems, e.g. n > 6, this sine set–up has proven to be numerically more robust than the polynomial set–up. This is mainly due to the fact that the polynomial coefficients in (16a) increase significantly in magnitude for higher order terms. (iii) The consideration of the input constraints (2) requires to check if the

resulting feedforward control

u∗Φ = α−1 y ∗ , . . . , y ∗(r−1) , Φ(t, p), η ∗

(17)

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– following from (11) with the set–up y ∗(r) = Φ(t, p) – lies within the constraints (2). (iv) If u∗Φ is outside the bounds umin and umax , the right–hand side α ˆ of ODE

(13) must be “re–planned” such that the constraints are met. This is accomplished by the following case–dependent definition: ⎧ Φ(t, p) if u∗Φ ∈ [umin , umax ] ⎪ ⎨ α ˆ=

⎪ ⎩

α(y ∗ , . . . , y ∗(r−1) , η ∗ , umin ) if u∗Φ < umin ∗

α(y , . . . , y

∗(r−1)

, η , umax ) if ∗

u∗Φ

(18)

> umax

Note that the set–up Φ(t, p) always determines the function α ˆ = Φ(t, p) at the interval bounds t = 0, T , since u∗Φ (0, p) = u∗0 and u∗Φ (T, p) = u∗T must lie within the interval [umin, umax ]. Hence, the two additional BCs in (15) are always satisfied by Φ(t, p). The calculation of the feedforward control u∗ (t), t ∈ [0, T ] in (11) requires the solution of the two BVPs (13)–(14) with (16)–(18) in dependence of the free parameters p. Due to the case–dependent choice of α ˆ in (18), the input constraints are directly incorporated in the formulation of the BVPs. A difficult question is the existence and uniqueness of a solution to the nonlinear BVPs (13)–(14). Some results and theorems for nonlinear two–point BVPs are discussed e.g. in [1, 13] from both analytical and numerical viewpoints. However, explicit results are only available for linear systems and for certain nonlinear second– and third–order systems. The case–dependent set– up of the function α ˆ in (18) further complicates this problem. On the other hand, the question of solvability of the BVPs (13)–(14) implies the existence of a feedforward control and therefore the solution of the BVP can be seen as a constructive controllability analysis of the considered system. However, the analytic investigation of controllability is still an unsolved problem for general nonlinear systems. A “heuristical” controllability criterion can be stated if the considered setpoints of the transition problem are linked by a quasi–stationary curve via a connected set of equilibria. Then, for a sufficiently large transition time T , the state trajectories will approach the stationary curve which implies existence of a solution to the BVPs.

4 Numerical Solution of the BVPs The solution of BVPs with free parameters as given in (13)–(14) with (16)– (18) for the output y ∗ (t) and the internal dynamics state η ∗ (t) is a standard task in numerics. For instance, Matlab provides the function bvp4c [18], which is designed to numerically solve two–point BVPs of the form ξ˙ = f (ξ, t, p) ,

t ∈ (a, b)

g (ξ(a), ξ(b), p) = 0

(19) (20)

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243

for the state ξ(t) on a finite interval t ∈ [a, b]. The function bvp4c involves also the determination of unknown parameters p, which directly corresponds to the set of free parameters p in the BVPs (13)–(14) with (16)–(18). Thereby, it is assumed that the residual form BCs (20) must suffice to determine the value of the parameter set p. The function bvp4c requires the condition dim p = dim g − dim ξ

(21)

concerning the number of free parameters and the number of BCs and to be satisfied. The Matlab function bvp4c is a finite difference code that implements a collocation formula. The collocation technique uses a mesh of points in order to divide the interval of integration into subintervals. The bvp4c–function determines a numerical solution by solving a system of algebraic equations resulting from the collocation conditions imposed on the subintervals and from the BCs (20). Moreover, bvp4c estimates the error of the numerical solution on each subinterval. If the solution does not satisfy a tolerance criterion, the solver adapts the mesh and repeats the procedure. The user must provide the initial points of the mesh as well as a guess of the solution at the mesh points and of the unknown parameters. The Matlab function bvp4c can be used in a straightforward manner to solve the BVPs (13)–(14) with (16)–(18), if the ODEs, the time interval, the case–dependent definition of α, ˆ and the BCs are appropriately adapted to the forms of (19) and (20). This requires to rewrite the rth–order ODE (13) of the output y ∗ (t) in state–space notation with r first–order ODEs. Furthermore, the linear BCs in (13)–(14) must be formulated in the residual representation (20) y ∗ (0) − y0∗ = 0,

y ∗ (T ) − yT∗ = 0,

η∗ (0) − η ∗0 = 0, η ∗ (T ) − η ∗T = 0.

y ∗(i)

0,T

= 0,

i = 1, . . . , r − 1

(22)

The use of the function bvp4c requires the specification of suitable initial mesh points tk ∈ [0, T ] as well as guesses of the solutions y ∗(i) (t), i = 0, . . . , r − 1 and η ∗ (t) at the points t = tk . In addition, an initial value of the unknown parameter set p is needed. Reasonable guesses for the trajectories are linear interpolations between the respective BCs on a uniform mesh grid over t ∈ [0, T ]. An initial guess for the unknown parameter set can be p = 0 corresponding to φ(t, p) = 0, t ∈ [0, T ]. Due to the algebraic solution technique of the BVP solver bvp4c, there is no distinction whether the considered internal dynamics (14) is stable or unstable, since its solution is obtained without numerical time integration, in contrast to the stable system inversion [4]. Therefore, this approach for feedforward control design under input constraints is applicable in the same manner for linear and nonlinear systems with stable and unstable internal dynamics, i.e. for both minimum–phase and nonminimum–phase systems.

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5 Calculation of Transition Time The transition time T is an important parameter in course of the feedforward control design and must be chosen appropriately with respect to the system dynamics and input constraints. To simplify this problem, this section describes an approach to determine T as part of the BVP solution. Thereby, a normalized design parameter κ ∈ (0, 1) serves as a measure for the aggressiveness of the feedforward control u∗ (t) with respect to the input constraints (2). A necessary condition is the existence of a quasi–stationary connection between the two stationary setpoints (3), i.e. a connected set of equilibria between x∗0 and x∗T . By choosing κ → 0, the trajectories y ∗ (t) and η ∗ (t) will approach the quasi–stationary connection with y ∗(i) (t) → 0, i = 1, . . . , r, which corresponds to a large transition time T .2 For κ → 1, the feedforward control u∗ (t) hits the input constraints [umin, umax ] and approaches a bang– bang behavior performing a nearly time–optimal transition. The input constraints [umin , umax ] of the feedforward control u∗ (t) can be projected to time–dependent constraints [α(t), α(t)] for the set–up function y ∗(r) = α ˆ with α(t) = min{α1 (t), α2 (t)},

α(t) = max{α1 (t), α2 (t)},

t ∈ [0, T ]

and α1 (t) = α(y ∗ , . . . , y ∗(r−1) , η ∗ , umin ),

α2 (t) = α(y ∗ , . . . , y ∗(r−1) , η ∗ , umax ).

Figure 2a shows a typical trajectory of y ∗(r) (t) = α(t) ˆ and the corresponding limits α(t), α(t). Due to the BCs (15), the conditions α(0) ˆ =α ˆ (T ) = 0 hold. This also ensures α(t) < 0 < α(t) for t = 0 and t = T , since the stationary input values u∗0 and u∗T have to lie inside the constraints [umin , umax ]. The aggressiveness of the feedforward control u∗ (t) can be rated by comparing the area between the curve α ˆ (t) and the time axis with the maximum available area determined by α(t) or α(t) and the time axis (depending on the sign of α ˆ (t)). Thereby, the time axis, i.e. α ˆ = 0, is used as reference since ˆ → 0 holds for it represents the quasi–stationary connection, and y ∗(r) (t) = α a sufficiently large transition time T . On the other hand, the trajectory α ˆ (t) may touch the limits α(t) or α(t), corresponding to a maximum feedforward control u∗ (t) = umin or u∗ (t) = umax , see Figure 2a. During this time interval the areas between the time axis and the curve α(t) ˆ as well as between the time axis and the respective limit α(t) or α(t) are the same. However, a problem also evident in Figure 2a is that the limits α(t), α(t) may cross the time axis such that 0 ∈ / [α(t), α(t)] holds for a certain time interval. In this case, the time axis cannot be used as reference to evaluate the 2

For instance, the pendulum on cart or the rotary double pendulum possess no quasi–stationary connection between the downward and upward equilibria. Hence, the pendulum swing–up cannot be performed arbitrarily slowly [9].

Feedforward Control Design under Input Constraints

α ˆ ∈ [α, α]

4

(a)

y∗(r) = α ˆ α, α

2

245

0

−2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time α ˆ n ∈ [−αn , αn ]

4

(b)

α ˆn ±αn

2 0

−2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

Fig. 2. Typical trajectories of (a) α ˆ (t) ∈ [α(t), α(t)] and (b) corresponding normalized ones α ˆ n (t) ∈ [−αn (t), αn (t)].

areas under the curve α ˆ (t) and the respective limit α(t) or α(t). Therefore, the curves α ˆ (t), α(t), and α(t) have to be transformed accordingly in this case. In a first step, the limits α(t), α(t) are symmetrically mapped to the normalized values αn (t) and −αn (t) with αn (t) = min{|α(t)|, |α(t)|}.

(23)

This ensures that the time axis lies within the interval [−αn (t), αn (t)], see Figure 2b. Furthermore, the trajectory α(t) ˆ ∈ [α(t), α(t)] has to be mapped to α ˆn (t) ∈ [−αn (t), αn (t)]. If the time axis is outside the interval [α(t), α(t)], see Figure 2a, α ˆ n (t) is determined via a linear interpolation function α ˆn = ˆ ) between the points αn = (α, −αn ) and αn = (α, αn ).3 Otherwise, g(αn , αn , α if 0 ∈ [α, α] holds, α ˆ n (t) is determined in dependence of the origin 0 = (0, 0) and the limits αn or αn according to ⎧ ˆ ) if 0 ∈ / [α, α] ⎨ g(αn , αn , α ˆ≤0 ˆ) if 0 ∈ [α, α] and α α ˆ n = g(0, αn , α (24) ⎩ g(0, αn , α ˆ) if 0 ∈ [α, α] and α ˆ > 0. 3

For a linear interpolation between two given points Q = (q1 , q2 ) and R = (r1 , r2 ), the ordinate coordinate of a point S = (s1 , s2 ) is determined according to s2 = g(Q, R, s1 ) =

q2 r1 − q1 r2 r2 − q2 + s1 . r1 − q1 r1 − q1

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The latter two cases ensure that the time axis α ˆ = 0 stays at α ˆ n = 0 if 0 ∈ ˆ (t) → 0 for a [α, α] holds. This is of importance to ensure that if y ∗(r) (t) = α large transition time T , the normalized value α ˆn (t) also approaches α ˆ n (t) → 0. Figure 2b shows the normalized profiles α ˆn (t) and ±αn (t) corresponding to Figure 2a. The definition (24) provides the opportunity to determine the areas under the curves |ˆ αn (t)| and αn (t) by introducing the two ODEs θ˙1 = |ˆ αn (t)|,

θ˙2 = αn (t),

θ1 (0) = θ2 (0) = 0,

(25)

which extend the previous BVPs (13)–(14). Thereby, θ1 (T ) and θ2 (T ) denote the areas under the respective curves. Since |ˆ αn (t)| ≤ αn (t) is ensured, the ratio θ1 (T )/θ2 (T ) between the two areas lies within the interval (0, 1). This ratio can be set to a desired value by introducing a design parameter κ ∈ (0, 1) and demanding the BC (26) θ1 (T ) − κ θ2 (T ) = 0 .

to be satisfied. For κ → 1, both curves |ˆ αn (t)| and αn (t) approach the same area values θ1 (T ) and θ2 (T ), which signifies an aggressive transition leading to a bang–bang behavior of the feedforward control u∗ (t) in a nearly time– optimal transition time T .4 On the other hand, κ → 0 means that the surface αn (t)| tends to zero compared to the maximum availθ1 (T ) under the curve |ˆ able area under |αn (t)|, i.e. y ∗(r) (t) = α ˆ → 0 holds. The feedforward control u∗ (t) and the trajectories y ∗ (t), η ∗ (t) approach the quasi–stationary connection for a corresponding large transition time T . The discussion shows that the design parameter κ ∈ (0, 1) can be used to influence the aggressiveness of the feedforward control u∗ (t) – and hence the transition time T . Since the previous BVPs (13)–(14) are extended by the two ODEs for θ1 (t), θ2 (t) and three BCs in (25) and (26), another free parameter is required beside p = (p1 , . . . , pn ). The transition time T can be incorporated as free parameter by using the time transformation t = τ,

T = ,

di 1 di = i i, i dt dτ

i≥1

(27)

with the normalized time coordinate τ ∈ [0, 1] and the scaling factor . Applying this time transformation to the formulation of the BVPs (13)–(18) and (25)–(26) results in a fixed time interval τ ∈ [0, 1] with the scaling factor = T as the new parameter, which can be influenced by the aggressiveness measure κ ∈ (0, 1). 4

An alternative to the right–hand side of the ODEs (25) is θ˙1 = α ˆ 2n (t) and ˙θ2 = α2n (t). Then, θ1 (T )/θ2 (T ) weights the volume of both curves α ˆ n (t) and αn (t) rotating around the time axis. If a jacobian is provided analytically for the numerical solution of the BVP (see Section 4), this has the advantage that no case differentiation is necessary as in (25) to account for |α ˆ n (t)|, and the jacobian has no discontinuities at α ˆ n (t) = 0.

Feedforward Control Design under Input Constraints

247

6 Example – the Inverted Pendulum on a Cart The presented feedforward control design under input constraints is illustrated for the m inverted pendulum on a cart, which is a frequently used benchmark problem in nonlinl ear control, see e.g. [19]. As depicted in Figθ ure 3, a mass m is mounted on top of a pendulum with length l and the angle θ(t) to the vertical. The pendulum is pivoted to the cart with the mass M and the displacement u M x(t). The cart is manipulated by the input x force u(t) which is assumed to be limited by 0 u(t) ∈ [umin, umax ]. The equations of motion of the pendulum Fig. 3. The inverted pendulum are given by two second–order ODEs for the on a cart. cart position x(t) and the angle θ(t):5 mlθ˙ 2 sin θ − mg sin θ cos θ + u M + m sin2 θ (M + m)g sin θ − mlθ˙ 2 sin θ cos θ − cosθ u θ¨ = . M l + ml sin2 θ

x¨ =

(28) (29)

The inversion–based feedforward control design is based on the input–output representation (7)–(8) of the considered system. With the coordinates y = x,

η = θ,

(30)

the pendulum model (28)–(29) has relative degree r = 2 and is already in input–output normal form (7)–(8) y¨ = α(η, η, ˙ u),

(31)

η¨ = β(η, η, ˙ u)

(32)

subject to the input constraints (2). The input–output dynamics (31) and the internal dynamics (32) are both affected by the input u(t), but not by the car position y = x. An interesting transition problem is the side–stepping of the inverted pendulum, i.e. to traverse a distance yT∗ between two upward unstable equilibria within a finite time interval t ∈ [0, T ]. This means that the solutions of the ODEs (31)–(32) have to satisfy the 2n = 8 BCs y(0) = 0, y(T ) = yT∗ , y(0) ˙ = y(T ˙ )=0 η(0) = 0, η(T ) = 0, η(0) ˙ = η(T ˙ ) = 0. 5

(33) (34)

The following parameter values are taken from [19]: g = 9.81 m/s2 , l = 0.15 m, M = 1 kg, m = 0.25 kg.

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The inverse of the input–output dynamics (31) enables the calculation of the feedforward control (11) u∗ = mg sin η ∗ cos η ∗ − mlη˙ ∗2 sin η ∗ + (M + m sin2 η ∗ )¨ y∗ = α−1 (¨ y ∗ , η ∗ , η˙ ∗ )

(35)

in dependence of the desired acceleration y¨∗ (t) of the output and the trajectories η ∗ (t), η˙ ∗ (t) of the internal dynamics. The new representation (12) for the internal dynamics is obtained by substituting (35) into (32): ¯ ∗ , y¨∗ ) η¨∗ = (g sin η ∗ − y¨∗ cos η ∗ )/l = β(η

(36)

with the highest time derivative y¨∗ (t) serving as input. Note that the zero dynamics corresponding to (36) (with y¨∗ (t) = 0) is unstable in the upward position, i.e. the inverted pendulum is nonminimum–phase. In a next step, the function α ˆ is introduced to parametrize the right–hand side of (31). This yields the two BVPs for the output y ∗ (t) and η ∗ (t): y¨∗ = α ˆ, y ∗ (0) = 0, y ∗ (T ) = yT∗ , y˙ ∗ (0) = y˙ ∗ (T ) = 0 (37) ∗ ∗ ∗ η¨ = (g sin η − α ˆ cos η )/l, η ∗ (0) = 0, η ∗ (T ) = 0, η˙ ∗ (0) = η˙ ∗ (T ) = 0. (38) These BVPs depend on the function α ˆ and are overdetermined due to 2n = 8 BCs for n = 4 ODEs. Therefore, a set of four free parameters p = (p1 , . . . , p4 ) is necessary for the solution of the BVPs. The polynomial (16a) is used to construct the set–up function 4

Φ(t, p) = −

pk k=1

t + T

4

pk k=1

t T

k+1

(39)

in order to parametrize α ˆ = Φ(t, p) if the respective feedforward control (17) u∗Φ = mg sin η ∗ cos η ∗ − mlη˙ ∗2 sin η ∗ + (M + m sin2 η ∗ )Φ(t, p)

(40)

satisfies the constraints [umin , umax ]. Otherwise, y¨∗ = α ˆ is re–planned according to ⎧ Φ(t, p) if u∗Φ ∈ [umin , umax ] ⎪ ⎨ α ˆ = α(η ∗ , η˙ ∗ , umin ) if u∗Φ < umin (41) ⎪ ⎩ ∗ ∗ ∗ α(η , η˙ , umax ) if uΦ > umax

such that the constraints are exactly fulfilled. The re–planning function α(·) in the latter two cases of (41) is given by the right–hand side of the input–output dynamics (31). The transition time T is designed according to Section 5 in dependence of the design parameter κ ∈ (0, 1) as a measure for the aggressiveness of the side–stepping maneuver. Note that a quasi–stationary connection exists between the setpoints in (33)–(34), since the upward equilibrium y = 0 with y˙ = y¨ = 0 and u = 0 exists for any constant cart position η.

Feedforward Control Design under Input Constraints

249

The BVPs (37)–(41) are extended by the two ODEs and the three BCs (25)–(26) for the states θ1 (t), θ2 (t), and are solved with the Matlab function bvp4c.6 Thereby, the time transformation (27) is applied with the normalized time coordinate τ ∈ [0, 1] and the scaling factor as the fifth free parameter beside p = (p1 , . . . , p4 ) in (39). The starting profiles for the states y ∗ (τ ), y˙ ∗ (τ ), and η ∗ (τ ), η˙ ∗ (τ ) are linear interpolations between the respective BCs on a uniform mesh with 50 grid points over the time interval τ ∈ [0, 1]. The guesses of the unknown parameters are p = 0 and = 1. The bvp4c–function delivers the output trajectory y ∗ (τ ), the internal dynamics state η ∗ (τ ), and their respective time derivatives, as well as the parameter set p = (p1 , . . . , p4 ) and the scaling factor . The transition time follows to T = . Figure 4 shows the resulting transition time T for a side–stepping of yT∗ = 1 m by varying the design parameter κ ∈ (0, 1) for three different input constraints [umin , umax ]. For smaller input constraints, the side–stepping generally requires a larger transition time T . Furthermore, by increasing κ → 1 corresponding to a more aggressive feedforward control u∗ (t), T approaches a time–optimal minimal value. 10

*

u ∈[−1,1] N * u ∈[−5,5] N * u ∈[−10,10] N

8

T [s]

6

4

2

0

0

0.2

0.4

κ [−]

0.6

0.8

1

Fig. 4. Transition time T in dependence of the aggressiveness measure κ ∈ (0, 1) for three different input constraints.

The upper part of Figure 5 shows some time–discrete snapshots of the side– stepping to illustrate the maneuver for the constraints u∗ (t) ∈ [−10, 10] N and κ = 0.6.7 The cart of the pendulum counterswings at the beginning and the end which is necessary to accelerate and decelerate the pendulum. The lower part of Figure 5 displays the profiles of the output y ∗ (t), the state η ∗ (t) of 6

7

The implementation in Matlab requires to write the second–order ODEs in (37)– (38) in state–space representation, see Section 4. The trajectories y ∗ (τ ) and η ∗ (τ ) are transformed back to the original time coordinate t ∈ [0, T ]. Finally, the feedforward control u∗ (t) is determined by substituting y ∗ (t), η ∗ (t) and their time derivatives in (35).

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the internal dynamics and the feedforward control u∗ (t) for the constraints u∗ (t) ∈ [−10, 10] N and three different values of κ. The counterswing behavior of the pendulum occurs in the output trajectory y ∗ (t) as an initial undershoot and a final overshoot, which reveals the nonminimum–phase characteristic of the pendulum. Furthermore, Figure 5 clearly shows the aggressiveness of the feedforward control u∗ (t) for increasing κ–values and the respectively decreasing transition time T . Especially κ = 0.95 results in a bang–bang like behavior of u∗ (t) sharply hitting the constraints umin and umax . The exact values of T = and the parameters for the set–up function Φ(t, p) in (39) are determined to κ = 0.95 : κ = 0.6: κ = 0.25:

T = = 1.03 s, T = = 1.19 s, T = = 1.63 s,

p = ( 1.79, -6.13, 7.40, -2.96 ) · 104 p = ( 1.83, -5.86, 6.97, -2.79 ) · 103 p = ( 0.60, -1.95, 2.33, -0.93 ) · 103 .

Snapshots at discrete time points (κ = 0.6) t=0.14 s t=0.32 s

0

0.1

t=0.5 s

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

0

0

−0.5 0

0.5

1

1.5

0

0.5

time [s]

1

κ = 0.95 κ = 0.6 κ = 0.25

5 0

*

2

1.5

time [s]

10 u [m/s ]

1

0.5 η* = θ* [rad]

1 y* = x* [m]

t=0.86 s t=1 s

t=0.68 s

−5 −10 0

0.2

0.4

0.6

0.8 time [s]

1

1.2

1.4

1.6

Fig. 5. Time–discrete snapshots of the inverted pendulum to illustrate the side– stepping as well as trajectories y ∗ (t), η ∗ (t), and feedforward control u∗ (t) for the input constraints u∗ (t) ∈ [−10, 10] N and three different values of κ ∈ (0, 1).

Feedforward Control Design under Input Constraints

251

The re–planning of the function α ˆ is illustrated in Figure 6, where the second time derivative of the output y¨∗ (t) = α ˆ is plotted together with the set–up Φ(t, p) for u∗ ∈ [−10, 10] N and κ = 0.6, also see Figure 5. Obviously, α ˆ in (41) is re–planned twice such that the feedforward control u∗ (t) meets the constraints. In view of the 2–degree–of–freedom control scheme in Figure 1, the pendulum still requires a feedback control ΣFB to stabilize and robustify the pendulum in the unstable upward setpoints and during the side–stepping maneuver. For instance, a simple linear LQR controller could be used for the feedback ΣFB [9]. ^ α(t) Φ(t,p*)

^ Φ [m/s2] α,

10 5 0 −5 −10 0

0.2

0.4

0.6 0.8 time [s]

1

Fig. 6. Second derivative y¨∗ (t) = α ˆ of the output and set–up function Φ(t, p) in (39) for the pendulum side–stepping (u∗ ∈ [−10, 10] N, κ = 0.6).

7 Conclusions The transition between stationary setpoints of nonlinear SISO systems is considered for the presentation of a new design approach for inversion–based feedforward control subject to input constraints. This approach treats the finite–time transition task as a two–point BVP in the input–output coordinates of the system, which comprises the BVPs for the input–output dynamics and the internal dynamics. A key element for the solution of the BVPs is to provide a sufficient number of free parameters in a set–up function for the right–hand side of the input–output dynamics ODE. The input constraints are exactly met by case–dependently re–planning the set–up function. This has the advantage that the constraints are directly incorporated into the BVPs, which can be solved in straightforward manner, e.g. with the Matlab BVP solver bvp4c. This concept has also been applied to the swing–up problem of the pendubot [9], an underactuated polar double pendulum with constrained torque, as well as to distributed reaction-diffusion–convection systems with constrained boundary control [15]. Furthermore, the determination of the transition time can also be incorporated in the specification and solution of the BVP for the finite–time setpoint transition. In the unconstrained case, this approach is consistent with the feedforward control design approach proposed by the authors [7, 8].

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References 1. U.M. Ascher, R.M.M. Mattheij, and R.D. Russell. Numerical solution of boundary value problems of ordinary differential equations. Prentice Hall, 1988. 2. E. Bayo. A finite–element approach to control the end–point motion of a single– link flexible robot. Journal of Robotic Systems, 4:63–75, 1987. 3. D. Chen and B. Paden. Stable inversion of nonlinear non–minimum phase systems. International Journal of Control, 64:81–97, 1996. 4. S. Devasia, D. Chen, and B. Paden. Nonlinear inversion–based output tracking. IEEE Transactions on Automatic Control, 41:930–942, 1996. 5. M. Fliess, J. L´evine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. International Journal of Control, 61:1327–1361, 1995. 6. M. Fliess and R. Marquez. Continuous–time linear predictive control and flatness: a module-theoretic setting with examples. International Journal of Control, 73:606–623, 2000. 7. K. Graichen, V. Hagenmeyer, and M. Zeitz. A new approach to inversion–based feedforward control design for nonlinear systems. Automatica (accepted). 8. K. Graichen, V. Hagenmeyer, and M. Zeitz. Van de Vusse CSTR as a benchmark problem for nonlinear feedforward control design techniques. In Preprints NOLCOS, pages 1415–1420, Stuttgart (Germany), 2004. 9. K. Graichen and M. Zeitz. Nonlinear feedforward and feedback tracking control with input constraints solving the pendubot swing–up problem. IFAC World Congress, Prague/CZ, 2005. 10. V. Hagenmeyer and M. Zeitz. Flachheitsbasierter Entwurf von linearen und nichtlinearen Vorsteuerungen. Automatisierungstechnik, 52:3–12, 2004. 11. I.M. Horowitz. Synthesis of Feedback Systems. Academic Press, New York, 1963. 12. A. Isidori. Nonlinear Control Systems. Springer, 3rd edition, 1995. 13. H.B. Keller. Numerical methods for two–point boundary value problems. Blaisdell, Massachusetts, 1968. 14. P. Martin, S. Devasia, and B. Paden. A different look at output tracking: Control of a VTOL aircraft. Automatica, 32:101–107, 1996. 15. T. Meurer, K. Graichen, and M. Zeitz. Motion planning and feedforward control for distributed parameter systems under input constraints. IFAC World Congress, Prague/CZ, 2005. 16. H. Nijmeijer and A. van der Schaft. Nonlinear Dynamical Control Systems. Springer, 1990. 17. J. Oldenburg and W. Marquardt. Flatness and higher order differential model representations in dynamic optimization. Computers & Chemical Engineering, 26:385–400, 2002. 18. L.F. Shampine, J. Kierzenka, and M.W. Reichelt. Solving boundary value problems for ordinary differential equations in Matlab with bvp4c. http://www.mathworks.com/access/pub/bvp.zip, 2000. 19. D.G. Taylor and S. Li. Stable inversion of continuous–time nonlinear systems by finite–difference methods. IEEE Transactions on Automatic Control, 47:537– 542, 2002. 20. W. Yim and S.N. Singh. Nonlinear inverse and predictive end point trajectory control of flexible macro–micro manipulators. Journal of Dynamic Systems, Measurement, and Control, 119:412–420, 1997.

System Inversion and Feedforward Control via Formal Power Series and Summation Methods Marc Oliver Wagner, Thomas Meurer, and Michael Zeitz Institut f¨ ur฀Systemdynamik฀und฀Regelungstechnik,฀Universit¨at฀Stuttgart, Pfaffenwaldring฀9,฀70569฀Stuttgart,฀Germany. {wagner,meurer,zeitz}@isr.uni-stuttgart.de Summary.฀ This฀paper฀provides฀initial฀results฀on฀the฀use฀of฀formal฀power฀series฀parameterizations฀and฀summation฀methods฀for฀linear฀and฀nonlinear฀finite฀dimensional systems.฀The฀parameterizations฀prove฀to฀be฀suitable฀for฀system฀inversion,฀trajectory planning,฀and฀feedforward฀control.฀In฀addition,฀the฀parameterization฀offers฀a฀possibility฀of฀incorporating฀conditions฀imposed฀on฀the฀desired฀output฀and฀the฀necessary input฀trajectory,฀which฀are฀formulated฀directly฀in฀the฀time฀domain.฀The฀potential฀of the฀method฀is฀demonstrated฀by฀a฀nonlinear฀example฀system฀with฀unstable฀internal dynamics.

Keywords:฀System฀inversion,฀feedforward฀control,฀trajectory฀planning,฀internal฀dynamics,฀formal฀power฀series,฀summation฀methods.

1 Introduction In [11] and [13], formal power series and summation methods have recently been shown to be a powerful tool for feedforward and feedback control of infinite dimensional systems. In consequence, the question has been posed whether these concepts could also be beneficial for the feedforward and feedback control of finite dimensional systems, especially when considering questions that still remain to be answered. One of the open questions concerns system inversion, trajectory planning, and feedforward control of finite dimensional systems exhibiting internal dynamics. A common task – which will be considered in this contribution – is the calculation of the input trajectory that produces a transition between steady states from a given initial state to a final state within a finite time interval [0, T ] along a prescribed trajectory. For this task, three major concepts have been suggested so far. In [5], the trajectory is planned for the system output. Then, the input trajectory is expressed in terms of the desired output and the state of the internal dynamics. Inserting this expression into the equations for the internal dynamics yields a

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 253–270, 2005. © Springer-Verlag Berlin Heidelberg 2005

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system of differential equations for the internal dynamics which depends on the desired output trajectory. Depending on the stability property of these differential equations, forward and/or reverse-time integration is applied. This yields a solution for the internal dynamics which is used to determine the necessary input. However, the resulting trajectory for the control input is generally non-constant outside of the transition interval [0, T ]. Therefore, either noncausal inputs or an infinite transition time is required. Furthermore, an extension of this approach to nonlinear problems is hardly possible. In [7, 9], flat systems with internal dynamics are considered whose system output and flat output differ. Trajectories on [0, T ] are planned via trajectories for the flat output to ensure that the internal dynamics of the system reaches its steady state at t ∈ {0, T }. Hence, a finite transition time is achieved. However, the system inversion is only achieved for the relation between input and flat output, not the input and the system output. Thus, although any feasible trajectory may be prescribed for the flat output, the possibilities of deliberately shaping trajectories for the system output are limited. A different approach is taken in [8], where the finite-time transition problem is formulated as a boundary value problem for the system output and the internal dynamics. The procedure differs from the previously mentioned methods, because it does not require the system to have any flatness properties and solves the problem of the stable numerical integration of the differential equations. However, it lacks a general formulation of the conditions under which the boundary value problem can be solved and again the possibility of deliberately shaping the output trajectory. Neither of the three above mentioned approaches provides the solution to the trajectory planning and feedforward control problem for systems with nonflat system output without the integration of a differential equation, which is either written as an initial value problem or as a boundary value problem. Using the proposed concept of formal power series parameterizability, this contribution shows exemplarily how to solve the feedforward control problem for an example system with polynomial nonlinearities without integrating any differential equation. Moreover, the proposed method allows a straightforward formulation of arbitrary conditions imposed on the desired output and the necessary input in the time domain. It thus establishes a powerful tool for the analysis and the design of feedforward controls. The paper is organized as follows: In Section 2, the problem is formulated for an example system of second order and the idea of formal power series parameterization for finite dimensional systems is introduced. In Section 3 the calculation of the feedforward control for a prescribed trajectory via formal power series is demonstrated. Additional conditions that account for the internal dynamics of the example system are formulated and incorporated in the feedforward design in Section 4. In Section 5, the ideas are generalized and conceptualized. Section 6 summarizes the results and gives an overview over future research.

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255

2 Problem Formulation Consider the second order differential equation a2 y¨ + a1 y˙ + a0 y = b1 u˙ + b0 [1 − y]u, y(0) = 0, y(0) ˙ =0

t>0

(1) (2)

representing the minimal representation of a system with input u ∈ C 0 , output y(t), and a parameter causing the system to be nonlinear for = 0. In the context of system inversion, (1) is seen as a differential equation in u with a determined function y(t). Hence an initial condition u(0) = 0

(3)

must be defined whenever b1 = 0. Remark 1. Note that if b1 = 0, Eqn. (1) represents a flat system with y(t) being the flat output. If b1 = 0, the system is still flat. However, y(t) is not a flat output and the system shows an internal dynamics which is difficult to account for. In this paper, the feedforward control design problem consists of calculating the input ud (t) that steers the output from its steady state output trajectory at yd (t) = 0 for t ≤ 0 to a new steady state yd (t) = 1 for t ≥ 1 along a prescribed path yd (t) within the finite time interval [0, 1]. In the case b1 = 0, a unique determination of the steady states (in the classical sense) requires the additional specification of u(0) = 0 or u(1) = us , respectively. This is due to the fact that a constant output y(t) = y¯ does not necessarily imply that u(t) = u ¯ = const., since there obviously exist functions u(t) whose superposition ˙ + b0 [1 − y¯]u(t) = y¯ for any t. satisfies b1 u(t) In order to provide detailed insight in the proposed approach, only the condition u(0) = 0 is imposed on the input trajectory while neglecting the final condition u(1) = us . This in addition allows to classify the ideas within the methods described in Section 1. In a second step, it is shown in Section 4, that the incorporation of u(1) = us is essential for the solution of the control problem. For the subsequent analysis, the following sets of parameters are used, representing a linear system Σl and a nonlinear system Σnl . The two parameters b0 and b1 having opposing signs assure that the internal dynamics is unstable. Therefore, the system shows inverse response behavior. Table 1. Parameters and steady state conditions of system (1), (2). System a2 a1 a0 b1 b0 Σl Σnl

1 3 2 1 −1 0 1 3 2 1 −1 0.2

yd (0) y˙ d (0) u(0) yd (1) y˙ d (1) u(1) 0 0

0 0

0 0

1 1

0 0

−2 −2.5

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2.1 Formal Power Series Parameterization Assume that yd (t) and ud (t) can be expanded into formal power series and thus be written as yd (t) → yˆ(t) =

∞

yn

n=0

tn , n!

ud (t) → uˆ(t) =

∞

un

n=0

tn . n!

(4)

Inserting the power series and their formal derivatives into (1) and taking into account the initial condition ud (0) = 0 yields the formal recurrence u0 = 0 =: f0

(5) n

un+1 = yn+2 + 3yn+1 + 2yn + un − =: fn+1 (y0 , y1 , . . . , yn+2 ),

j=0

n j

yj un−j

n≥0

(6)

for the coefficients un depending only on the coefficients yn . Thus, a formal parameterization of the required input ud (t) is found, whereby the important question arises if a solution ud (t) to (1)–(2) together with a prescribed trajectory yd (t) can be deduced from the parameterization (4)–(6).

3 Trajectory Planning Neglecting the Internal Dynamics Whether the recurrence (5), (6) produces a series uˆ(t) that can be mapped to a solution function ud (t) depends – among other factors – on the choice of the coefficients yn and thus on a thorough planning of the desired output trajectory yd (t). In a first step, a simple polynomial trajectory yd (t) = yˆ(t) =

3t2 − 2t3 1

0≤t≤1 , t>1

(7)

1 = 1, n!

(8)

is chosen, which satisfies the four conditions 3

yd (0) = y0 = 0,

yd (1) =

yn n=0 2

y˙ d (0) = y1 = 0,

y˙ d (1) =

yn+1 n=0

1 = 0, n!

(9)

in order to achieve a set–point transition from yd (0) = 0 to yd (t) = 1, t ≥ 1 within the time interval [0, 1]. Since the coefficients yn differ between the interval [0, 1], where y0 = 0, y1 = 0, y2 = 6, y3 = −12, and yn = 0 for n ≥ 4, and the interval [1, ∞), where y0 = 1, and yn = 0 for n ≥ 1, the evaluation of the recurrence (5), (6) to determine ud (t) must be carried out separately

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for t ∈ [0, 1] and t > 1. Thereby, the series coefficient u0 must be chosen consistently, i.e. u0 = ud (0) = 0 for the first interval and u0 = ud (1) = us = 1 for the second interval, in order to ensure continuity of the input ud (t). In the following only the first case is evaluated explicitly, whereby the results for the second interval follow similarly. 3.1 Linear Case Before studying the nonlinear case, consider first the linear problem with = 0 in order to illustrate the approach. Here, the recurrence (5), (6) reduces to u0 = 0

(10)

un+1 = yn+2 + 3yn+1 + 2yn + un ,

n ≥ 0.

(11)

This set of recursive equations can be explicitly solved under the assumption nmax t yn n! of a polynomial output trajectory yd (t) of degree nmax , i.e. yd (t) = n=0 such that yn = 0 for n > nmax . This yields ⎧ 0 n=0 ⎪ ⎪ ⎪ ⎨ fn (y0 , y1 , ..., yn+1 ) 1 ≤ n ≤ nmax − 1 un = (12) ⎪ ⎪ fn (y0 , y1 , ..., ynmax ) nmax ≤ n ≤ nmax + 1 ⎪ ⎩ n ≥ nmax + 2. un−1 Hence the formal power series for u ˆ(t) evaluates to nmax

u ˆ(t) = n=0

(un − unmax +1 )

∞

tn tn + unmax +1 n! n! n=0

(13)

=exp(t)

with the latter term representing the exponential function, which converges uniformly and absolutely on each compact interval. As a result, the series u ˆ(t) converges on each bounded interval, which also justifies the required time–differentiation of the series (4) in order to derive the recurrence relation. Hence, the formal solution u ˆ(t) and the solution function ud (t) coincide, i.e. nmax

ud (t) = n=0

(un − unmax +1 )

tn + unmax +1 exp(t). n!

(14)

For the chosen desired trajectory yd (t) from (7), evaluation of (14) with coefficients un from (12) on both intervals yields the closed form solution ud (t) =

36(1 − exp(t)) + 42t + 24t2 + 4t3 , (108 − 36e) exp(t − 1) − 2,

0≤t≤1 , 1≤t≤τ

(15)

with τ ≥ 1 being an arbitrary but finite constant. Inserting the solution and (7) in (1) shows that the solution is valid for t ≥ 0.

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As can be seen from the solution on [1, ∞), tracking the desired output (7) requires a non-constant input ud (t) for t > 1. The input even grows beyond any bound as t approaches infinity. This is the well known case of an unstable internal dynamics considered in system theory. Hence, the planned transition given in (7) does not provide the desired set–point transition, which would require ud (t) to be constant for t > 1. This corresponds to results from [5], where the tracking of a prescribed trajectory requires either a non-causal input or an input that is non-constant beyond the end of the transition interval. Remark 2. Obviously, (15) can be obtained by applying the Laplace transform to (1), transforming the desired output trajectory into the Laplace domain, solving the algebraic equation and transforming it back into the time domain. This approach, in contrast, would be limited to linear systems. As will be shown in the next paragraph, this is not the case for the formal power series parameterization. 3.2 Nonlinear Case If = 0, there is no obvious way of determining a closed form solution. Therefore, instead of evaluating the infinite series uˆ(t), the series is approximated by a finite sum with N coefficients ∞ n=0

N

un

tn tn → un . n! n! n=0

(16)

Such approximations require series convergence with N being chosen with respect to the respective speed of convergency. As can be seen from Figure 1 (a), the series seems to converge and converge sufficiently fast. The approximated input u(t), which is used for a simulation model, is shown in (b), whereas the results of the simulation are shown in (c) and (d) for the system output y(t) compared to the desired output yd (t), and the internal dynamics x2 (t) as defined in Appendix B, respectively. The results show that approximate tracking is accomplished. As it is the case for the linear system, the necessary input ud (t) grows considerably for large t. Since in addition the internal dynamics — compare to Appendix B — does not display a constant behavior for times t > 1, the desired steady state cannot be reached at t = 1.

4 Trajectory Planning in View of the Internal Dynamics So far, only the minimal requirements (8)–(9) and u(0) = 0 have been imposed on the desired output trajectory yd (t) and the input ud (t). It is of general interest to incorporate further conditions, either on the desired output trajectory yd (t) or on the necessary input ud (t).

Feedforward Control via Formal Power Series and Summation Methods Series coefficients

System input

6

....

4

259

20

(a)

un n!

(b)

15

2 10 0 5

−2

u(t) −4

0

10

20 index n

30

0

40

0

0.5

System output

1 time t

1.5

2

Internal dynamics 0

(c)

(d)

x (t) 2

1 −5

0.8 0.6

−10

0.4 0.2

−15

y(t) y (t)

0

d

0

0.5

1 time t

1.5

−20

2

0

0.5

1 time t

1.5

2

Fig. 1. Results for Σnl defined in (1)–(2) using partial summation (16) for the desired output trajectory (7). (a) Values of the series coefficients un /n!. (b) Calculated input u(t) from (16) using N = 40. (c) Simulation results of the output y(t) compared to the desired trajectory yd (t). (d) Simulation results for the internal dynamics x2 (t) defined in Appendix B.

In view of the results of the previous section where an unbounded growth of the input for times t outside the transition interval [0, 1] was observed, it seems natural to impose a further condition on the final value of the input ud (t), i.e. ud (t) = us = const. for t ≥ 1. In terms of the formal series uˆ(t) from (4), this reads as ud (1) = u ˆ(1) =

∞ n=0

un

∞

1 1 (6) = fn (y0 , y1 , ..., yn+2 ) = us . n! n! n=0

(17)

Incorporating this equation with the imposed conditions (8)–(9) on the desired output trajectory yd (t), results in five restrictions on the series coefficients yn of the series expansion yˆ(t) for the desired output yd (t). As a result, a polynomial of order 4 is required to realize the desired set–point transition in the finite time interval [0, 1], i.e. yd (t) = 4n=0 yn tn /n! for t ∈ [0, 1]. If t ≥ 1, a constant output yd (t) = 1 is postulated as before. In summary, for t ∈ [0, 1] the resulting complete set of conditions reads as

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yd (0) = y0 = 0,

yd (1) =

yn 0 3

y˙ d (0) = y1 = 0,

y˙ d (1) =

1 = 1, n!

yn+1 0

1 =0 n!

(18) (19)

for the output trajectory yd (t) and u(0) = u0 = 0 ud (1) =

∞

(20)

fn (y0 , ..., y4 )

0

1 = us n!

(21)

for the input ud (t). Due to the recurrence (5), (6), the conditions on the coefficients un can be rewritten in terms of the coefficients yn . This is in particular easy for the case = 0. 4.1 Linear Case For

= 0 and 0 ≤ t ≤ 1, it follows directly from (10)–(11) that u0 = 0, u 1 = y2 , u2 = y3 + 4y2 ,

u3 = y4 + 4y3 + 6y2 u4 = 4y4 + 6y3 + 6y2 un = 6(y4 + y3 + y2 ),

n ≥ 5.

(22)

Calculating ud (1) yields ∞

4

ud (1) =

1 1 un + 6(y4 + y3 + y2 ) n! n! n=5 n=0 =e− 65 24

= [6e − 12] y2 + 6e − A31

89 191 y3 + 6e − y4 . 6 12

A32

(23)

A33

Thus, summarizing all conditions (18), (19), (23) imposed on the coefficients yn , n = 2, 3, 4 of the desired trajectory yd (t) allows the computation of the coefficients yn , n = 2, 3, 4, which are necessary to solve the recursion, i.e. ⎛ ⎞ ⎛ −1 −1 −1 ⎞−1 ⎛ ⎞ ⎛ ⎞ y2 yd (1) 1 2! 3! 4! ⎝ y3 ⎠ = ⎝ 1!−1 2!−1 3!−1 ⎠ ⎝ y˙ d (1) ⎠ = A ⎝ 0 ⎠ . (24) y4 ud (1) −2 A31 A32 A33 A

Since matrix A is invertible, the coefficients yn , n = 2, 3, 4 and thus ud (t) can be calculated in a strictly algebraic manner for 0 ≤ t ≤ 1. It can be verified, that ud (1) = −2.

Feedforward Control via Formal Power Series and Summation Methods

261

System input

5

u(t)

0

−5

−10 0

0.2

0.4

0.6

0.8

1

time t

1.2

1.4

1.6

1.8

2

1.2

1.4

1.6

1.8

2

System output

1.5

y(t) y (t)

1

d

0.5 0 −0.5 −1 −1.5 0

0.2

0.4

0.6

0.8

1

time t

Fig. 2. Simulation results for finite time transition for Σl with an output trajectory yd (t) of polynomial order 4 as defined in Tables 1 and 2.

For t > 1, the steady state input ud (t) = −2 is used. This input can also be obtained by formulating a constant trajectory yd (t) = 1 for t > 1 and evaluating the recurrence (10)–(11) as done for the interval [0, 1]. Thus, ud (t) is completely determined and can be used for simulation or analytical verification of the solution. The respective input ud (t) is shown in Figure 2 (upper graph). The corresponding system output y(t) obtained by solving the system ODE (1)–(2) is depicted in the lower graph. Since Σl is a non-minimum phase system, a change in the set–point from 0 to 1 requires an undershoot in the planned trajectory yd (t), which can be seen in the lower graph of Figure 2. This corresponds to results shown in [8], where a certain deformation of the desired trajectory was shown to be necessary for the solution of the control problem. Note that it is straightforward and easy to impose additional conditions on the input and output trajectories directly in the time domain by enlarging the matrix A and using a polynomial of higher order for yd (t). In addition, conditions stemming from input or output constraints can also be formulated in terms of linear inequalities. Further research will be directed towards the possibility to fulfill certain sets of conditions by using the theory of linear matrix inequalities [3]. 4.2 Nonlinear Case In the nonlinear case = 0 – as opposed to the linear case – the conditions (18)–(21) include the nonlinear equation (21) relating the series coefficients of the output to the desired steady state input. This makes a solution in closed form harder, if not impossible. Therefore, in the following an approximate solution of (18)–(21) is proposed by replacing the infinite series uˆ(t) by a finite

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approximation. Hence, a system of equations needs to be solved in order to obtain the values of the series coefficients yn , n = 2, 3, 4 and thus both an approximation of the desired output trajectory yd (t) and the necessary input ud (t). As is shown in the sequel, the results depend on the chosen approximation method, such that sophisticated approaches are required. Thereby, two approaches are considered: habitual partial summation and k-summation (see e.g. [12]). Results for Partial Summation For the determination of an approximation for u ˆ(t), partial summation with N coefficients is applied. Substituting this in (18)–(21) results in the set of equations 1 1 1 + y3 + y4 = 1 2! 3! 4! 1 1 1 y2 + y3 + y4 = 0 1! 2! 3!

y2

(25)

N

1 f˜n (y2 , y3 , y4 ) = us n! n=0 where f˜n (y2 , y3 , y4 ) = fn (0, 0, y2 , y3 , y4 ) = un since y0 = y1 = 0 and fn (·) as defined in (6). Solving (25) for yn , n = 2, 3, 4, first provides the respective out4 put trajectory yd (t) = yˆ(t) = n=0 yn tn /n! and second allows to determine an approximation (in the sense of partial summation) to the corresponding N input ud (t) = upartial (t) = n=0 f˜n (y2 , y3 , y4 )/n!. In the case of multiple real solutions, the solution with the smallest absolute value is taken, since the other coeffcients then tend to be smaller as well and the convergence behavior of u ˆ(t) tends to be better. Remark 3. Considering the solution of the set of equations, some comments follow: "

"

The nonlinear equation may have multiple solutions or no solutions at all. However, for nonlinear equations in one variable and polynomials in particular, both easily verifiable conditions for solvability and efficient algorithms for the numerical computation of the solution exist (see [4]). If additional conditions are imposed on the input trajectory ud (t), more than one nonlinear equation will appear. Then, the analysis obviously becomes more difficult.

Numerical results are given in Figure 3. In graph (a), the value of the coefficients in the partial summation un /n! are shown to illustrate the convergence behavior of the series u ˆ(t). Although uˆ(t) seems to converge, more coefficients may be necessary to obtain a sufficient approximation. Graph (b) shows the calculated input upartial (t) whereas graph (c) displays the simulated output

Feedforward Control via Formal Power Series and Summation Methods Series coefficients 200

System input

....

(a)

un n!

(b)

2 0

100

−2

0

−4

−100

−6

upartial(t)

−200 0

10

20 index n

30

−8

40

0

0.5

System output

1 time t

1.5

2

Internal dynamics

1.5

(c)

1

7

(d)

x (t) 2

6 5

0.5

4

0

3

−0.5

2 y(t) yd(t)

−1 −1.5

263

0

0.5

1 time t

1.5

1 2

0

0

0.5

1 time t

1.5

2

Fig. 3. Results for finite time transition for Σnl defined in (1)–(2) using partial summation for a desired output trajectory as defined in Table 2 with yd (0) = 0, yd (1) = 1. (a) Values of the series coefficients un /n!. (b) Calculated input upartial (t) using partial summation. (c) Simulation results of the output y(t) compared to its desired trajectory yd (t). (d) Simulation results for the state x2 (t) of the internal dynamics as defined in Appendix B.

y(t) along with the desired output yd (t). The state x2 (t) of the internal dynamics — compare to Appendix B — is depicted in graph (d). As can be seen from the simulated output y(t), the desired trajectory yd (t) is not perfectly tracked and an offset to the desired set–point appears. This defect is due to an insufficient approximation of u ˆ(t) obtained by partial summation with N = 40 coefficients. Therefore, the computed desired trajectory yd (t) is not correctly determined, and the state x2 (t) of the internal dynamics does not reach its steady state value as can be seen from graph (d). Hence the determined input upartial (t) does not allow for adequate tracking. Motivation for k-Summation Although an expression can be calculated for u ˆ(t), there is no knowledge about the convergence behavior of uˆ(t). It might be a diverging series, or a series which does not converge within the number of coefficients that can be computed in practice. In these cases, partial summation is not suitable as an approximation and therefore, different methods need to be applied to determine an adequate approximation for u ˆ(t).

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The classical theory of formal power series and their summation methods (see e.g. [10]) provides an easily applicable method to retrieve the solution function from the respective formal power series. This is in particular not restricted to convergent series. Motivated by the successful application of formal series and k–summation methods to feedforward and feedback control of distributed parameter systems [13, 11], in the following these results are utilized for the considered finite– dimensional control problem. Thereby, the necessary definitions and results for summation methods in this context are summarized in Appendix A. Results for k-Summation Due to the possibly divergent character of u ˆ(t), the so–called (N, ξ)-approximate k-sum (38) as defined in Appendix A is applied to determine the series coefficients yn , n = 2, 3, 4 following 1 1 1 + y3 + y4 = 1 2! 3! 4! 1 1 1 y2 + y3 + y4 = 0 1! 2! 3! ⎞ ⎛ N n N ξn 1 ξn ⎝ = us . f˜j (y2 , y3 , y4 ) ⎠ j! Γ (1 + nk) Γ (1 + nk) n=0 n=0 j=0 y2

(26)

As a consequence, the coefficients yn , n = 2, 3, 4 directly depend on the summation parameters N , ξ and k, who have to be chosen appropriately as explained in Appendix A.2. In order to illustrate the convergence behaviour of the series uˆ(t) using the (N, ξ)-approximate k-summation, it suffices to consider the outer sum in the denominator ∞

n

n=0 j=0

ξn uj j! Γ (1 + nk)

(27)

of (37) as defined in Appendix A. The behaviour of (27) is depicted in Figure 4 (a), which indicates the improved convergence and verifies the validity of chosen approximation. The systems input uksum (t) and the state x2 (t) of the internal dynamics perfectly reach their respective value at t = 1, which can be seen from graphs (b) and (d), respectively. Hence the output can be tracked in an excellent manner as shown in graph (c). Remark 4. It should be noted that the desired trajectories yd (t) differ depending on the used summation method. This is due to the fact that the determination of the coefficients y2 , y3 , y4 for the series ansatz yˆ(t) for y(t) depends on the nonlinear equation (21). Thereby, either partial summation (25) or (N, ξ)–approximate k–summation (26) can be utilized for the computation of y2 , y3 , y4 , which results in the differing desired trajectories yd (t) illustrated in Figures 3 (c) and 4 (c).

Feedforward Control via Formal Power Series and Summation Methods Series coefficients

4

3

x 10

2

....

n P

j=0

1

265

System input

uj ξn j! Γ (1+nk)

(a)

(b)

2 0 −2

0 −4 −1 −6 −2

u

−8 −3

0

10

20 index n

30

40

(t)

ksum

0

0.5

System output

1 time t

1.5

2

Internal dynamics

1.5

(c)

1

10

(d)

x2(t) 8

0.5

6

0 4

−0.5

2

y(t) y (t)

−1

d

−1.5

0

0.5

1 time t

1.5

2

0

0

0.5

1 time t

1.5

2

Fig. 4. Results for Σnl defined in (1)–(2) using (N, ξ)-approximate k-summation with k = 1 for a desired polynomialP output trajectory of order 4 as defined in uj ξn appearing in the outer sum Table 2. (a) Values of the coefficients n j=0 j! Γ (1+nk) of (27). (b) Calculated input uksum (t) using k-summation. (c) Simulation results of the output y(t) compared to its desired trajectory yd (t). (d) Simulation results for the state x2 (t) of the internal dynamics.

5 Generalizations In order to provide a systematic approach to parameterization and output trajectory planning, some definitions are necessary. Definition 1 (Formal power series parameterizability). Let a dynamical system be given in minimal input–output realization D1 {y}(t) = D2 {u}(t), y

(n)

(0) = y0,n ,

t>0

(28)

n = 0, ..., d1 − 1

(29)

n = 0, ..., d2 − 1.

(30)

where D1 and D2 are differential operators of order d1 and d2 , respectively, and u ∈ C d2 −1 , together with a set of initial conditions for the inverse system u(n) (0) = u0,n ,

Then the system is called formal power series parameterizable, if expressing input and output as formal power series yˆ(t) =

∞ n=0

yn

tn , n!

uˆ(t) =

∞ n=0

un

tn n!

(31)

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in t with coefficients yn and un and taking into account formal derivatives, yields an explicitly solvable recurrence un = fn (u0 , u1 , ..., un−1 , y0 , y1 , ...)

(32)

for the coefficients un , where the term fn represents functions depending on the differential equation (28). Remark 5. Clearly, controllable linear SISO systems like Σl defined by (1)–(2) for = 0 are formal power series parameterizable. ˆ(t) in general deThe possibility of assigning a function ud (t) to the series u pends on the choice of the coefficients yn and thus on the desired output trajectory. Therefore, a suitable trajectory needs to be chosen for yd (t). Definition 2 (Exact output trackability (adapted from [6])). Let Σ be a formal power series parameterizable system defined by (28)–(30) and yd (t) a desired output trajectory which is defined on an interval [0, T ] and which can be expanded into a power series yˆ(t) converging uniformly on [0, T ] with (n) yd (0) = y0,n , n = 0, ..., d1 − 1. Σ is called exactly output trackable with respect to yd (t), if the formal series u ˆ(t) resulting from the recurrence (32) can be mapped to a function ud (t) solving the differential equation (28)–(30),i.e., D2 {ud }(t) = D1 {yd }(t),

t>0

(33)

and (n)

ud (0) = u0,n ,

n = 0, ..., d2 − 1.

(34)

Proposition 1. Σl defined in (1)–(2) with the parameters given in Table 1 is exactly output trackable with respect to every desired output trajectory yd (t) of finite polynomial order. Proof. Let a trajectory yd (t) be given as a polynomial of order nmax . From the recurrence (10)–(11) it follows, that all coefficients un up to the index n = nmax + 1 can be expressed in term of the coefficients yj , j = 0, ..., nmax . All coefficients un for n ≥ nmax + 2, depend only on unmax +1 . Following (14), ud (t) is the sum of a polynomial of order nmax +1 in t and an exponential function unmax +1 exp(t). Since the power series representation of the exponential function converges uniformly on the interval [0, τ ], where τ is an arbitrary but fixed number, all formal operations are valid and ud (t) solves the differential equation. Remark 6. For nonlinear systems where no closed form solution can be determined in general, exact output trackability is hard to prove. However, approximative tracking is often sufficient in real world applications. Future research will be directed towards formulating approximate output trackability in the context of formal power series and summation methods.

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6 Conclusions Using the concepts of formal power series parameterizability and output trackability, this contribution provides initial results on a new tool for the analysis and synthesis of feedforward controls. The proposed method neither demands for the determination or even the existence of a flat output, nor the integration of a differential equation. For the linear example system, the feedforward control problem is reduced to a system of linear algebraic equations, which can be exactly solved. On the other hand for the nonlinear example, an approximate solution is obtained by solving a system of linear and nonlinear algebraic equations. The results for the example system confirm previous results which suggest, that in the trajectory planning for a change of steady state within a finite interval [0, T ] for systems with internal dynamics one fundamental decision has to be drawn: Either the trajectory is prescribed, then the necessary input may vary beyond the transition interval [0, T ]. This is the approach followed in [5] and [6]. Or the input is forced to be constant beyond the transition interval [0, T ]. Then, the trajectory has to be deformed to meet the necessary conditions at the end of transition interval [0, T ]. This concept is used in [7, 8, 9]. The proposed concept of power series parameterization covers both approaches and clarifies that the difference lies in the formulation of one specific condition. In addition, the parameterization approach can be interpreted as a constructive proof of controllability. Future research will be directed towards the definition of approximate output trackability, the inclusion of constraints imposed on the system input or output by using the theory of linear matrix inequalities, the generalization to multiple input multiple output systems, and the application to real–world problems.

A Summation Methods Throughout this contribution, infinite series uˆ(t) must be mapped to functions ud (t) that produce the desired output yd (t) when applied to the corresponding system as the system input u(t). As shown in Section 4.2, habitual partial sums may not produce the correct results. Therefore, more powerful maps need to be found. The general setting of such maps is the theory of summation methods. Definition 3 (Summation methods and summable series [10]). Let a power series ∞ tn (35) un n! n=0 in t be given. A linear map M that assigns a function ud (t) to the formal power series (35) is called a summation method. The set of all power series which can be mapped to functions using M is called the set of M-summable series.

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The most commonly used summation method is the limit of the partial sums as the number of coefficients in the partial sums tends to infinity, i.e. Mpartial :

∞ n=0

N

un

tn tn → lim un . N →∞ n! n! n=0

(36)

However, since power series must converge in the conventional sense in order to be mapped to a function, the space of series that can be summed via the limit of partial sums is relatively small. In order to enlarge the space of summable series, other methods have been introduced (see [10]). Summation methods and the corresponding sets of summable series must have certain properties in order to be used for the solution of differential equations. Firstly, the summation method must be regular, i.e. it must assign to each convergent series the limit of its partial sums. Secondly, the set of summable series must have the structure of a differential algebra. A summation method that ensures both properties is the so–called k-summation as defined in [12] or [1]: Definition 4 (k-summation [2]). The map from the space of power series to the space of functions defined by ∞

Mk :

n

un

n

t n=0 j=0 → lim ∞ ξ→∞ n!

j

n

ξ uj tj! Γ (1+nk)

n=0

ξn Γ (1+nk)

(37)

is called k-summation. The great advantage of k-summation when compared to partial summation is twofold. First, the number of coefficients that are necessary to obtain an accurate approximation of the series is smaller than for partial summation if the series shows some alternating behavior. Second, even some divergent series can be summed via k-summation [1]. A.1 Finite Approximations All feasible summation methods include at least one limiting process. As such processes cannot in general be numerically evaluated, an approximation must be defined. For the summation process using partial sums, a finite partial sum can serve as an approximation. However, the quality of the approximation depends greatly on the series and the number of terms that enter the partial sum. For the k-sum, which includes two limiting processes, an adequate approximation must be defined. Definition 5 ((N, ξ)-approximate k-sum [13]). The (N, ξ)-approximate ∞ k-sum of a formal k-summable power series n=0 un tn /n! is defined as

Feedforward Control via Formal Power Series and Summation Methods N

Sk (N, ξ) :=

n

n=0 j=0

j

269

n

ξ uj tj! Γ (1+nk)

N n=0

.

(38)

ξn Γ (1+nk)

The quality of the approximation using the (N, ξ)-approximate k-sum depends on the suitable choice of k and the two summation parameters N and ξ. A.2 Choice of Summation Parameters In general, the quality of the approximation using the (N, ξ)-approximate ksum increases with N . Thus, N should be chosen as large as possible. However, due to complexity issues, there is a limit to the number of coefficients that can be calculated in practice. A feasible number of coefficient for the systems studied in this contribution is N = 40. Furthermore, an optimal parameter ξ must be found depending on the choice of N . Here, two facts should be taken into consideration. On the one hand, ξ should be as large as possible due to the fact that the limit ξ → ∞ is of interest. On the other hand, the coefficients of the outer sum in the numerator and the coefficients of the sum in the denominator of (37) should have reached a value close to 0 for n = N . This limits the maximum possible value for ξ. Therefore, ξ usually has an optimal value somewhere in between, and the optimal value can often be easily determined. For the denominator, the optimal value for ξ can easily be verified independantly of the series that should be summed. For k = 1 and N = 40, a value of ξ = 20 constitutes a good choice since the error made in the sum is smaller than 0.003%. In order to check the convergence behavior of the sum in the numerator, the corresponding expression must be evaluated and examined.

B State Space Representation of Σ Defined in (1)-(2) The input–output relation (1), (2) can be expressed in state space following x˙ 1 = x2 + u, x˙ 2 = −2x1 − 3x2 + ( x1 − 4)u, y = x1 ,

x1 (0) = 0, x2 (0) = 0,

(39)

with the initial condition x˙ 1 (0) = 0 or equivalently u(0) = 0. Obviously since y = x1 , the internal dynamics are governed by the second ODE (39) for the state x2 (t).

C Trajectories and Summation Parameters Table 2 shows the trajectories and parameters used in the this contribution.

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Table 2. Trajectories and summation parameters for finite time transitions yd (0) = 0 → yd (1) = 1 for Σl and Σnl defined in Table 1. System 0 ≤ t ≤ 1 Σl Σnl Σl Σnl Σnl

2

3

yd (t) = 3t − 2t yd (t) = 3t2 − 2t3 yd (t) ≈ −27.213t2 + 58.427t3 − 30.213t4 yd (t) ≈ −21.096t2 + 46.191t3 − 24.096t4 yd (t) ≈ −24.707t2 + 53.414t3 − 27.707t4

t>1

N

k

ξ

yd (t) = 1 yd (t) = 1 yd (t) = 1 yd (t) = 1 yd (t) = 1

∞ 40 40 40 40

– – – – – – – – 1 20

Fig. – 1 2 3 4

References 1. W. Balser. Formal power series and linear systems of meromorphic ordinary differential equations. Springer Verlag, 2000. 2. W. Balser and R.W. Braun. Power series methods and multisummability. Math. Nach,, p. 37–50, 2000. 3. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Volume 15 of Studies in Applied Mathematics, SIAM, 1994. 4. I.N. Bronstein, K.A. Semendjajew, G. Musiol, and H. M¨ uhlig. Taschenbuch der Mathematik. Verlag Harri Deutsch, Frankfurt am Main, 3. edition, 1997. 5. D. Chen and B. Paden. Stable inversion of nonlinear non-minimum phase systems. Int. J. Contr., 64(1):81–97, 1996. 6. S. Devasia and B. Paden. Stable inversion for nonlinear nonminimum-phase time-varying systems. IEEE TAC, 43(2):283–288, 1998. 7. M. Fliess, H. Sira-Ram´ırez, and R. Marquez. Regulation of non-minimum phase outputs: a flatness based approach. In D. Normand-Cyrot, editor, Perspectives in Control - Theory and Applications: A tribute to Ioan Dor´ e Landau, p. 143– 164. Springer, London, 1998. 8. K. Graichen, V. Hagenmeyer, and M. Zeitz. Van de Vusse CSTR as benchmark problem for nonlinear feedforward control design techniques. In Proc. 6th IFAC Symposium on Nonlinear Control Systems – NOLCOS 2004, volume 3, p. 1415– 1420, Stuttgart, Germany, 2004. 9. V. Hagenmeyer and M. Zeitz. Flachheitsbasierter Entwurf von linearen und nichtlinearen Vorsteuerungen. at - Automatisierungstechnik, 1:3–12, 2004. 10. G.H. Hardy. Divergent Series. Oxford at the Clarendon Press, 3. edition, 1964. 11. T. Meurer and M. Zeitz. Flatness–based feedback control of diffusion– convection–reaction systems via k–summable power series. In Proc. 6th IFAC Symposium on Nonlinear Control Systems – NOLCOS 2004, volume 1, p. 191– 196, Stuttgart, Germany, 2004. 12. J.-P. Ramis. Les series k–sommables et leurs applications. In Analysis, Micrological Calculus and Relativistic Quantum Theory, volume 126 of Lecture Notes in Physics, p. 178–199. Springer, 1980. 13. M.O. Wagner, T. Meurer, and M. Zeitz. K–summable power series as a design tool for feedforward control of diffusion–convection–reaction systems. In Proc. 6th IFAC Symposium on Nonlinear Control Systems – NOLCOS 2004, volume 1, p. 149–154, Stuttgart, Germany, 2004.

Flatness-Based Improved Relative Guidance Maneuvers for Commercial Aircraft Thierry Miquel1 , Jean L´evine2 , and F´elix Mora-Camino3 1

CENA, 7 av. E. Belin, BP4005, 31055 Toulouse, France. [email protected] 2 Centre Automatique et Syst`emes ´ Ecole฀des฀Mines฀de฀Paris,฀35฀rue฀Saint-Honor´ e,฀77300฀Fontainebleau,฀France. [email protected] 3฀ ENAC,฀7฀av.฀E.฀Belin,฀BP4005,฀31055฀Toulouse,฀France,฀and฀LAAS,฀CNRS,฀7 av.฀du฀Colonel฀Roche,฀31077฀Toulouse,฀France. [email protected] Summary.฀ With฀the฀sustained฀increase฀of฀air฀traffic฀leading฀to฀airspace฀saturation, new฀ flight฀ maneuvering฀ capabilities฀ are฀ expected฀ for฀ commercial฀ aircraft.฀ Among these,฀ relative฀ guidance฀ of฀ airliners,฀ a฀ set฀ of฀ maneuvers฀ that฀ doesn’t฀ require฀ air traffic฀control฀support,฀appears฀as฀a฀promising฀solution฀to฀ease฀air฀traffic฀controllers’ workload.฀Relative฀guidance฀means฀path฀convergence฀of฀following฀aircraft฀to฀delayed position฀of฀leading฀ones฀and฀station฀keeping฀The฀convergence฀and฀station฀keeping phases฀ are฀ realized฀ onboard฀ the฀ aircraft฀ using฀ their฀ sensing฀ and฀ communication facilities,฀in฀a฀decentralized฀way,฀under฀aircraft฀maneuverability฀constraints฀due฀to safety฀ and฀ passenger฀ comfort฀ issues.฀ In฀ this฀ paper,฀ after฀ introducing฀ the฀ relative positioning฀ dynamics฀ of฀ two฀ aircraft,฀ their฀ flatness฀ property฀ is฀ shown฀ and฀ a฀ flat control฀ law฀ is฀ proposed.฀ Reference฀ trajectories฀ are฀ designed฀ in฀ two฀ steps:฀ if฀ the separation฀ between฀ aircraft฀ is฀ too฀ small,฀ the฀ follower’s฀ trajectory฀ is฀ stretched฀ by imposing฀a฀sinusoidal฀movement฀at฀constant฀speed;฀in฀the฀opposite฀case,฀the฀follower copies฀ the฀leader’s฀ delayed฀trajectory.฀ The฀feedback฀ tracking฀is฀done฀by฀feedback linearization.฀Simulation฀results,฀using฀a฀real฀scenario฀involving฀wide฀body฀aircraft in฀a฀merging฀situation฀are฀displayed฀and฀discussed.

Keywords:฀Aircraft฀control,฀relative฀guidance,฀nonlinear฀control,฀differential flatness.

1฀Introduction In฀the฀management฀of฀terminal฀traffic฀(i.e.฀concerning฀flights฀approaching฀the airport),฀the฀main฀task฀of฀air฀traffic฀controllers฀consists฀in฀sequencing,฀merging฀ and฀ spacing฀ aircraft฀ before฀ landing.฀ Roughly฀ speaking,฀ the฀ sequencing Corresponding author

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 271–284, 2005. © Springer-Verlag Berlin Heidelberg 2005

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3

Meter fix 1

2

Meter fix 2

1

4

Meter fix 3

Airport

Fig. 1. Example of sequencing and merging operations for arriving aircraft at airport.

procedure consists in establishing an order between the approaching aircraft, compatible with their respective flight capabilities. Spacing the aircraft means putting the sequenced aircraft on a line, with a prescribed minimum distance or time delay between the aircraft. Indeed, the following aircraft has to be protected at least from wake turbulence generated by the leading aircraft. The minimum wake turbulence separation adopted by the civil aviation authorities depends upon the maximum takeoff weights of the aircraft involved [7]. The merging phase concerns the choice of a path joining the present aircraft position to the leading aircraft path at a fixed point called meter fix, the spacing requirement being met at this point. In addition, this path must satisfy various comfort and safety constraints. An example of typical merging flight path for arriving aircraft at an airport is depicted in Fig. 1. The station keeping task, i.e. keeping the required spacing all along up to landing, remains under each crew’s responsibility. On the traffic controller’s side, establishing properly spaced landing sequences is very demanding in heavy traffic conditions. As a consequence, an automation tool named Arrival Manager (AMAN) often helps air traffic controllers to build a sequence of aircraft in order to safely and expeditiously land them [8]. Unfortunately, the airborne counterpart of the Arrival Manager, which could help the flight crew to merge its aircraft towards a meter fix according

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273

to a sequencing constraint, and ensure the station keeping, is not yet available. This yet to be designed onboard function assumed to ensure the merging and station keeping tasks thanks to the information delivered by some surveillance and communication capabilities is generally called relative guidance and should be included in the Automatic Flight Pilot. Indeed, despite the fact that current aircraft’s Flight Management System (FMS) have the ability to navigate over predefined paths, they are not capable to generate a trajectory with a prescribed delay relatively to another aircraft over meter fixes. Indeed, such a new capability onboard aircraft needs on-line information on the aircraft environment, and more specifically the knowledge of the leading aircraft position and velocity. In this respect, the Automatic Dependent Surveillance-Broadcast (ADS-B) is a potential key enabler to support these surveillance requirements [6]. Aircraft equipped with ADS-B capabilities broadcast their position, velocity and identification periodically (e.g. every second). Any neighboring aircraft capable of receiving those data will therefore be able to track the surrounding traffic. Clearly, automatic merging and station keeping operations could relieve air traffic controllers of providing time consuming radar vectoring instructions to the trailing aircraft once the flight crew has accepted the relative guidance clearance. Thus, the expected benefit of such new capabilities onboard aircraft is an increase of air traffic controller availability, which could result in increased air traffic efficiency and / or capacity. Enhancement of flight crew airborne traffic situational awareness with associated safety benefits is also expected. Preliminary studies have mainly investigated the station keeping phase without taking into consideration the merging phase. This field is addressed for UAVs or military aircraft by means of linear and nonlinear techniques for example [9, 10]. However, research for civil aircraft where safety and passenger comfort are crucial issues is still in its initial stage. Indeed, in [1], the authors focus on station keeping performed manually, whereas in [11] a proportional, integral and derivative (PID) law to control longitudinal station keeping is developed. In this context, this paper investigates the nonlinear design of a relative guidance mode, i.e. dedicated to both the merging at a specified meter fix and the station keeping. The paper is organized as follows: after introducing the relative positioning dynamics of two aircraft, their flatness property is displayed. A flat control law with reference trajectories satisfying a given set of conditions and feedback linearization is proposed. The trajectory generation is then detailed. Finally, simulation results, involving wide body aircraft in a merging situation are displayed and discussed.

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ψL

Wx Wy

VL

W : Wind speed Leading aircraft

yL(t) μ

τ

ψ

θ

χ

υ

ρ

V : airspeed Gs : Ground speed

y(t) Trailing aircraft (Inertial Frame) x(t)

xL(t)

Fig. 2. Reference frame.

2 Relative Motion Kinematics 2.1 Inertial Position Dynamics The here considered reference frame is affixed to the trailing aircraft, whose coordinates in the inertial frame are denoted by (x, y), as shown in Fig. 2. The along track distance, denoted by τ , is aligned with the trailing aircraft ground speed vector, whereas the cross track distance, denoted by υ, is the distance from the trailing to the leading aircraft perpendicularly to the ground speed. The heading angle of the trailing aircraft is denoted by ψ, its airspeed by V . Subscript L is added to all variables related to the leading aircraft, e.g. (xL , yL ) for its coordinates. Since wind is considered in this paper, the track angle of the trailing aircraft is denoted by χ. The track angle χ is the direction followed by the aircraft with respect to this inertial frame, whereas the heading angle ψ is the direction followed by the aircraft with respect to the air. The track angle χ and the ground speed Gs are defined as follows: x˙ = Gs sin χ,

y˙ = Gs cos χ

(1)

Assuming that Earth is flat and non-rotating, it may be considered as an inertial frame. From Fig. 2, the inertial position dynamics of the trailing

Flatness-Based Improved Relative Guidance Maneuvers

275

aircraft are given by the following relation, where ψw denotes the direction from where the wind is blowing and W its velocity: x˙ = V sin ψ + W sin(ψw − π) y˙ = V cos ψ + W cos(ψw − π)

(2)

Those relations hold even if the motion of the aircraft in the vertical plane is considered as far as the flight path angle γ is small, which is a realistic assumption for commercial aircraft. Referring to (1) and (2), the track angle χ and the heading angle ψ are linked by the following relationship: ⎧ 1 ⎪ ⎪ ⎨ sin χ = G (V sin ψ − W sin ψw ) s (3) 1 ⎪ ⎪ ⎩ cos χ = (V cos ψ − W cos ψw ) Gs

Note that we also have Gs = V 2 + W 2 − 2V W cos(ψ − ψw ). This relation is not needed since ground speed direct measurements by Doppler or GPS are generally available. 2.2 Relative Position Dynamics From Fig. 2, the relative position of the leading aircraft in the reference frame affixed to the trailing aircraft can be expressed in terms of the inertial positions of the trailing aircraft and leading aircraft: τ = (xL (t) − x(t)) sin χ + (yL (t) − y(t)) cos χ υ = (xL (t) − x(t)) cos χ − (yL (t) − y(t)) sin χ

(4)

We assume the same wind for the leading and the trailing aircraft since the wind encountered by the leading aircraft is not part of the Automatic Dependent Surveillance-Broadcast (ADS-B) messages [6]. Taking into account the inertial position dynamics expressed in (2), the time derivative of (4) yields: τ˙ = χυ ˙ + VL cos(ψL − ψ) − V cos(χ − ψ) υ˙ = −χτ ˙ + VL sin(ψL − ψ) + V sin(χ − ψ)

(5)

˙ is obtained by differentiating (3). Note that it The expression χ(V, ˙ V˙ , ψ, ψ) ˙ , ψw , ψ˙ w ) that are generally also depends on the wind characteristics (W, W available on-board through the Air Data Computer (ADC). 2.3 State Space Representation ˙ V˙ )T the control vector and by X = Denoting by u = (u1 , u2 )T = (ψ, T (τ, υ, ψ, V ) the state vector, equations (5) reduce to the following state space representation of the relative guidance kinematics:

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⎛

⎞ ⎛ ⎞ τ˙ χυ ˙ + VL cos(ψL − ψ) − V cos(χ − ψ) ⎜ υ˙ ⎟ ⎜ −χτ ⎟ ⎟ ⎜ ˙ + VL sin(ψL − ψ) + V sin(χ − ψ) ⎟ = f (X, u). X˙ = ⎜ ⎝ ψ˙ ⎠ = ⎝ ⎠ u1 ˙ u2 V

(6)

In this state space representation, the dependence of f with respect to VL and ψL have been omitted since these data are independent of the trailing aircraft dynamics, and are communicated by the leader to the trailing aircraft. This model, which indeed represents very simplified relative dynamics, will be used for the controller design described in the next section.

3 Controller Design 3.1 Relative Guidance Maneuver Description It is assumed in the following that two autopilot functions dealing with airspeed control and bank angle control and allowing for coordination between throttle, aileron and rudder are available onboard the trailing aircraft, which is the case in many modern jets. Furthermore, and as explained in the introduction, the purpose of the relative guidance control system is first to guide the trailing aircraft towards a specified meter fix and then to maintain station keeping behind the leading aircraft. As a consequence, the relative guidance maneuver is divided into two phases in the proposed approach: the merging phase and the station keeping phase. During the merging phase, the focus is on the air traffic situation awareness. More specifically, the knowledge of the merging point and the shape of the merging trajectory of the trailing aircraft are of interest from the flight crew and air traffic control perspectives. In this respect, particular trajectories ensuring the convergence of the trailing aircraft to the delayed leading one at the meter fix will be planned in section 3.4 On the other hand, when the trailing aircraft maintains station keeping towards the leading aircraft, the focus is on safety. Indeed, in radar environment, the trailing aircraft will comply with radar separation standards provided by ICAO [7], in which case variable ρ describing the horizontal range between the leading and the trailing aircraft is of major interest. As a consequence, the problem of relative guidance maneuver requires explicit trajectory generation, which naturally suggests to investigate the model flatness. Note that, if the system is flat, trajectories in the flat output space can be easily planned and then mapped to appropriate inputs. Moreover, differential flatness not only allows to design a relative guidance reference trajectory but also a tracking feedback controller.

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3.2 Differential Flatness Flatness was originally introduced more than a decade ago by M. Fliess, J. L´evine, Ph. Martin and P. Rouchon (see e.g. [4, 5]). Recall that, roughly speaking, a flat system is a system for which there exists a generalized output vector such that all states and inputs can be expressed in terms of this output vector and a finite number of its derivatives. More precisely, a nonlinear system: x˙ = f (x, u),

x ∈ Rn ,

u ∈ Rm

(7)

is differentially flat if one can find an output z ∈ Rm of the form: z = ζ(x, u, . . . , u(s) )

(8)

where u(s) denotes the sth order derivative of u with respect to time, and such that ˙ . . . , z (r) ), u = ϕ1 (z, z, ˙ . . . , z (r+1) ). (9) x = ϕ0 (z, z, Output z is called flat output. In addition, system (7) is said Lie-B¨ acklund equivalent to the following system (called trivial system), where vector v is the new input: v = z (r+1) . (10) Imposing a given arbitrary trajectory to z yields a trajectory for all the system variables x and u, without integrating any differential equation. Remark that the time derivatives involved in the above formulas do not imply to take derivatives of noisy signals since it involves precomputed open-loop time functions. 3.3 Differential Flatness of the Relative Dynamics The purpose of this section is to show that pair (ρ, μ) is a flat output vector of system (6), where ρ is the (horizontal) range between the leading and the trailing aircraft and μ the relative bearing between those aircraft, as shown in Fig. 2. First of all, it is worth noticing that (ρ, μ) are related to the state variables (τ, υ) by: ρ = τ 2 + υ2 (11) υ μ = χ + arctan τ In addition, the following relation between the leading aircraft’s inertial position and the trailing aircraft’s one is immediate from Fig. 2 : x = xL − ρ sin μ,

y = yL − ρ cos μ

(12)

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So the goal is, according to (12), to find the relationship between the state vector X and the control vector u defined in (6) and the flat output components ρ and μ and their derivatives. For that purpose, it is assumed that the leading aircraft and wind characteristics, i.e. (xL , yL , VL , ψL , W, ψW ), are available on-line through Automatic Dependent Surveillance (ADS-B) communications and Air Data Computer (ADC). Moreover, we assume that the trailing and the leading aircraft are subject to the same wind. Taking into account (2) and differentiating (12) with respect to time leads to: V sin ψ = VL sin ψL − ρ˙ sin μ − ρμ˙ cos μ (13) V cos ψ = VL cos ψL − ρ˙ cos μ + ρμ˙ sin μ This leads to the expression of airspeed and heading:

V 2 = VL2 + ρ˙ 2 + ρ2 μ˙ 2 − 2VL (ρ˙ cos(ψL − μ) + ρμ˙ sin(ψL − μ)) and tan ψ =

(14)

VL sin ψL − ρ˙ sin μ − ρμ˙ cos μ VL cos ψL − ρ˙ cos μ + ρμ˙ sin μ

(15)

V sin ψ − W sin ψw V cos ψ − W cos ψw

(16)

Thus, airspeed and heading are clearly expressed in terms of (ρ, μ, ρ, ˙ μ). ˙ Furthermore, relation (3) leads to: tan χ =

Combined with (14) and (15), we conclude that χ is also function of (ρ, μ, ρ, ˙ μ) ˙ only. Finally, referring again to Fig. 2, the cross-track and the along-track distances read: τ = ρ cos(μ − χ), υ = ρ sin(μ − χ). (17)

This shows that all the system variables can be expressed as functions of (ρ, μ) and their first and second order derivatives. It achieves to prove that the pair (ρ, μ) constitutes a flat output vector for system (6). 3.4 Merging Trajectory Planning During the merging phase, the current leading aircraft position is delayed by a specified time and projected on the trailing aircraft flight plan, as shown in Fig. 3. This projection simply consists in finding the virtual position of the leading aircraft on the trailing aircraft trajectory such that the curvilinear distance between this point and the meter fix is the same as the corresponding curvilinear distance on the leading aircraft flight plan. At the beginning of the merging phase, once the targeted leading aircraft position has been projected on the flight plan of the trailing aircraft, two cases are possible:

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Leading aircraft targeted position (ΔT seconds before the beginning of the merging phase) Leading aircraft position at the beginning of the merging phase

τ*

Trailing aircraft position at the beginning of the merging phase τ0

Δτ0

Meter fix Reference trajectory

2A

υ*

Leading aircraft targeted position projected on the flight plan of the trailing aircraft Fig. 3. Leading aircraft targeted position mapped to the flight plan of the trailing aircraft.

"

"

when the projected leading aircraft position is in front of the current position of the trailing aircraft, the problem is the same as in the station keeping phase: the trailing aircraft must follow the delayed leader trajectory and the tracking error must be compensated by feedback. This will be developed in section 3.5. During this phase, ρref is simply set to the range between the current position of the leading aircraft and the projected position, whereas μref is set to the direct route towards the merging meter fix. when the projected leading aircraft position is behind the current position of the trailing aircraft, then a new reference trajectory is computed online.

The purpose of this reference trajectory is to stretch the covered distance by the trailing aircraft while maintaining a constant airspeed. From an operational point of view, this might keep jet engine regime variations small and avoid being confronted with flight domain limitations. In this paper, the adopted reference trajectory is shaped as a sinusoid in the cartesian coordinates. Indeed in this phase, since we aim at stretching the follower path at constant speed, the initial and final positions and veloc-

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ities being given, it is more natural to replace the flat output (ρ, μ) by the equivalent one (τ, υ), in virtue of (11). Let τ0 stand for the distance between the position of the trailing aircraft and the meter fix at the beginning of the merging phase. The reference lateral deviation is given by (see Fig. 3): υ ∗ (τ ) = A 1 − cos

2πτ τ0

(18)

.

Amplitude A is adjusted so that the length of the reference trajectory is equal to the distance τ0 + Δτ0 between the mapped leading aircraft position and the meter fix at the beginning of the merging phase: τ0 + Δτ0 =

τ0 2π

2π

1+

0

2

2πA sin x τ0

dx

(19)

The reference trajectory is then built assuming a constant speed V0 along it. This speed is equal to the speed of the leading aircraft at the beginning of the merging phase: V0 =

τ˙ ∗ (t)2 +

d ∗ (υ ◦ τ ∗ ) (t) dt

2

(20)

which yields the differential equation: V0

τ˙ ∗ = 1+

2πA τ0

sin

∗ 2π ττ0

2

(21)

Finally, the required reference trajectory (ρ∗ , μ∗ ) of the original flat output is obtained by using (11). This reference trajectory will be tracked using a feedback linearizing control described in the next section (see (23), with (ρ∗ , μ∗ ) in place of (ρref , μref )). 3.5 Station Keeping Control Recall that, in this phase, ρref is set to the range between the current position of the leading aircraft and the projected position, whereas μref is set to the leader route. Using again the fact that the pair (ρ, μ) is a flat output, the proposed tracking feedback is a feedback linearizing controller. A consequence of the flatness property of system (6) is that it is Lie-B¨ acklund equivalent (see[5]) to the following system: ρ¨ = v1 ,

μ ¨ = v2

(22)

Since the previous relations indicate that the second derivatives of ρ and μ can be interpreted as new controls, they are chosen as follows:

Flatness-Based Improved Relative Guidance Maneuvers

v1 = −2ξρ wρ (ρ˙ − ρ˙ ref ) − wρ2 (ρ − ρref ) v2 = −2ξμ wμ (μ˙ − μ˙ ref ) − wμ2 (μ − μref )

281

(23)

(ρref , μref ) refers to the reference trajectory for the pair (ρ, μ). The damping ratios (ξρ , ξμ ) and the natural frequencies (wρ , wμ ) will be chosen to comply with the time scale of the autopilot functions dealing with airspeed and bank angle control. Given the positions and velocities of the trailing and leading aircraft, the outputs are computed as follows: the values of (ρ, μ, ρ, ˙ μ) ˙ are firstly computed thanks to (11) and its derivative. Then, these values are used with (23) to set the values of the new controls. Finally, the controlled bank angle and airspeed are derived thanks to (14) and the time derivative of (15). Note that the involved transformations are non singular in the flight domain under consideration.

4 Case Study 4.1 Scenario In this section, a scenario built around a real French air navigation procedure is used in order to evaluate the properties of the controller previously designed. "

The leading aircraft, a Boeing B767-300, starts its trajectory vertical to meter fix DPE (Dieppe) at 6000m (flight level 200) with a conventional airspeed of 160m/s (310 knots). It firstly converges towards meter fix SOKMU

DPE Leading aircraft: B767-300 37 NM

DVL

Trailing aircraft: A320 4 NM

44 NM

17 NM

SOKMU : Meter fix 1

Fig. 4. Case study scenario.

MERUE : Meter fix 2

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T. Miquel, J. L´evine, and F. Mora-Camino

and stabilizes its descent at 3000m (flight level 100) with conventional airspeed of 130m/s (250 knots). Then, it moves towards meter fix MERUE. The trailing aircraft, an Airbus A320, starts its trajectory 7400m (4NM) after meter fix DVL (Deauville) at 7315m (flight level 240) with a conventional airspeed of 155m/s (300 knots). As the leading aircraft, it firstly converges towards meter fix SOKMU and stabilizes its descent at 3000m (flight level 100) with conventional airspeed of 130m/s (250 knots). Then, it moves towards meter fix MERUE.

The altitude profiles of the two aircraft are compliant with those observed on radar data. The leading aircraft performances are simulated through the European aircraft performance database [2]. Without any relative guidance maneuver, the trailing aircraft is positioned 36 sec behind the leading aircraft at meter fix SOKMU, with a separation of 5370 m (2.9 NM). Since this separation is not acceptable from ICAO safety standards [7], an objective delay of 90 sec behind the leading aircraft after meter fix SOKMU has been selected for the trailing aircraft. In order to comply with the time response of the airspeed and bank angle control channels for a wide body aircraft such as an Airbus A320, the damping ratios (ξρ , ξμ ) and the natural frequencies (wρ , wμ ) have been set as follows [3]: wρ = wμ = 2.10−2 rad/sec (24) ξρ = ξμ = 0.8 In addition, the conventional airspeed variations have been limited to ±15m/s (± 30 knots) to preserve as far as possible jet engines regime. 4.2 Results The movements of aircraft in the horizontal plane and the evolution of the conventional airspeed of both aircraft resulting from the flatness-based relative guidance controller are shown in Fig. 5 and Fig. 6: The curvature of the trajectory at the beginning of the merging maneuver comes from the reference trajectory tracking. The conventional airspeed firstly decreases from 155m/s (300 knots) to 144m/s (280 knots) during the merging phases, and then decreases towards 130m/s (250 kts) during the station keeping phase. As expected, the specified delay of 90 sec is achieved at meter fix SOKMU, and maintained after it.

5 Conclusion In this paper, the design of a combined feedforward and feedback relative guidance control law for wide body aircraft has been considered.

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Leading aircraft

Trailing aircraft

MERUE SOKMU

Fig. 5. Aircraft movement in the horizontal plane (axis in NM).

The proposed approach is based on the flatness property owned by the relative aircraft dynamics. One of the key-point of such a design is that the trailing aircraft is driven along a reference trajectory computed on line that stretches the follower path at constant speed to obtain the required separation with respect to the leader. The design of the reference trajectory relies on smooth functions and, at every moment of the merging or station keeping phase, safety and comfort aspects can be easily monitored. This approach appears quite promising and is currently being considered for practical validation. Acknowledgement. This paper is dedicated to Prof. Michael Zeitz for his 65th birthday.

References 1. M. Agelii, C. Olausson. Flight Deck Simulations of Station Keeping. Paper 17. Proceeding of the 4th USA/Europe Air Traffic Management Research and Development Seminar. Santa Fe (USA), 2001. 2. BADA. Aircraft Performance Summary Tables for the Base of Aircraft Data, Revision 3.4. Eurocontrol Experimental Centre. EEC Note 06/02, 2002.

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320

Leading aircraft

300

Trailing aircraft

280

260

240

0

200

400

600

Fig. 6. Conventional airspeed (in knots) versus time (in sec).

3. C. Favre. Fly-by-wire for commercial aircraft: the Airbus experience. In: Advances in Aircraft Flight Control. Chap. 8, pp211-229. Taylor & Francis, London, 1996. 4. M. Fliess, J. L´evine, P. Martin, P. Rouchon. Flatness and Defect of Non-linear Systems: Introductory Theory And Examples. International Journal of Control, 61:1327-1361, 1995. 5. M. Fliess, J. L´evine, P. Martin, P. Rouchon. A Lie-B¨ acklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. on Automatic Control, vol. 44, N. 5, 922-937, 1999. 6. E. Hoffman, D. Ivanescu, C. Shaw, K. Zeghal. Effect of Automatic Dependent Surveillance Broadcast (ADS-B) transmission quality on the ability of aircraft to maintain spacing in sequence. Air Traffic Control Quaterly, Special Issue: Aircraft Surveillance Applications of ADS-B, 11:181-201, 2003. 7. ICAO. Annex 11, Air Traffic Services. International Civil Aviation Organization, Montr´eal, 2001. 8. M. Kayton, W.R. Fried. In: Avionics navigation systems, second edition. Chap. 14, pp.642-689. John Wiley & Sons, New York, 1997. 9. M. Pachter, J.J. D’Azzo, A.W. Proud. Tight formation flight control. Journal of Guidance, Control, and Dynamics, 24:246-254, 2001. 10. S.N. Singh, R. Zhang, P. Chandler, S. Banda. Decentralized nonlinear robust control of UAVs in close formation. International Journal of Robust and Nonlinear Control, 13:1057-1078, 2003. 11. P. Vinken, E. Hoffman, K. Zeghal. Influence of speed and altitude profile on the dynamics of in-trail following aircraft. Paper No. 2000-4362. Proceeding of AIAA Guidance Navigation and Control Conference. Denver (USA), 2000.

Vehicle Path-Following with a GPS-aided Inertial Navigation System Steffen Kehl1 , Wolf-Dieter P¨ olsler1, and Michael Zeitz2 1 2

Dr. Ing. h.c. F. Porsche AG, Stuttgart Institut f¨ ur฀Systemdynamik฀und฀Regelungstechnik,฀Universit¨at฀Stuttgart, Pfaffenwaldring฀9,฀70569฀Stuttgart,฀Germany. [email protected]

Summary.฀ This฀ article฀ describes฀ a฀ methodology฀ to฀ guide฀ a฀ passenger฀ car฀ along a฀predefined฀path฀in฀order฀to฀perform฀drive฀tests.฀A฀feedback/feedforward฀control scheme฀for฀this฀path-following฀problem฀is฀described.฀The฀controller฀is฀based฀on฀measurements฀which฀are฀provided฀by฀a฀GPS฀aided฀navigation฀system.฀Thereby,฀problems arise฀in฀course฀of฀the฀implementation฀of฀such฀a฀high-precision฀navigation฀system฀due to฀the฀realtime฀demands.฀Simulation฀results฀for฀a฀test฀manoeuvre฀illustrate฀the฀performance฀of฀the฀developed฀control฀scheme.

Keywords:฀Path฀following,฀vehicle฀control,฀two-degrees-of-freedom-control, inertial฀navigation.

1 Introduction During the development of a passenger car, many drive tests have to be performed in order to ensure the desired vehicle behaviour. This comprises durability tests and repeated maneuvers on the same track with different velocities. The torque on the wheels will be marginal in curves. A lateral vehicle controller can be helpful to improve the repeatability of these tests. The aim is not to simulate a human driver. The focus is on minimal lateral deviation and not on minimal laptime. The conditions for such a vehicle controller are different from those for a vehicle controller for public roads. The vehicle path is completely predefined, the terrain and road conditions are known. Moreover, there will be no obstacles on the path. Differential information for GPS receivers are available, which provides an accuracy for the GPS position in the range of centimeters in most regions of the test area. The vehicle reaches the limits of the road-to-tire contact force for many test manoeuvres and therefore a linear approximation of the tire forces cannot be used. The main goal of the controller is to guide the vehicle along the preplanned path. It is not crucial that the vehicle passes through a specific point at a

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 285–300, 2005. © Springer-Verlag Berlin Heidelberg 2005

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given time but that it stays close to this geometric path. The velocity may vary around the desired value. The chosen control approach is based on the one-track vehicle model. This model and the corresponding tire model are described in Section 2. The tracking control design is based on a two-degree-of-freedom control structure with a feedforward and a feedback component (Section 3). The generation of the desired path with smoothing splines is discussed in Section 4. The resulting lateral vehicle controller is tested in simulation studies with a rigorous and validated vehicle model for a specific passenger car. The results of one test are shown in Section 5. For the projected test on a real passenger car, a highprecision navigation system is necessary for the measurement of the actual vehicle state. This measurement system is described in Section 6 with proposals to eliminate problems arising due to the realtime requirements of the navigation system.

2 Vehicle and Tire Model In the following, the usually weak coupling between longitudinal and lateral vehicle dynamics is neglected. Even though this coupling is used by race car drivers during cornering, in case of the automated manoeuvres the drive train will not perform heavy momentum changes. By this approach, it is also possible to let a human driver control engine and gearbox for security reasons, while the controller handles the steering wheel. A commonly used vehicle model as shown in Figure 1 is known as the one-track vehicle model [7]. This model is described in [5] and [9] by the following differential equations for the coordinates β(t), ψ(t), x(t), and y(t) partly defined in Figure 1: 1 {Sv (αv ) cos(δ − β) + Sh (αh ) cos(β)} , β˙ = −ψ˙ + mv 1 ψ¨ = {Sv (αv ) lv cos(δ) − Sh (αh ) lh } , J x˙ = v cos (ψ + β) , y˙ = v sin (ψ + β) .

(1) (2) (3) (4)

The vehicle sideslip angle β describes the difference between the actual moving direction θ of the vehicle’s center of gravity and the yaw angle ψ of the vehicle body. The position is expressed by the coordinates x and y on the plane. The vehicle mass m, the moment of inertia J, and the horizontal velocity vh = v are considered quasi-constant. The tire sideslip angles αv and αh describe the difference between the direction of heading and the direction of travel of the front and rear tire (Figure 2). They are calculated as follows

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287

αv

δ

Sv

vh

lv

ψ

θ β

FN

CoG R

lh

M

αh

Sh H

Fig. 1. Kinematics of the vehicle model [9, 5, 1]

αv = δ − arctan αh = arctan

lv ψ˙ + v sin β v cos β

(5)

,

lh ψ˙ − v sin β v cos β

(6)

in dependence on the distances lv and lh of the front and rear wheel axles to the center of gravity and the front wheel steering angle δ. The contact force Sv between front wheel and road is interpreted as input. In (1) and (2), Sh is the rear wheel sideforce. An appropriate wheelslip-angle αv is needed to generate the desired front wheel sideforce Sv as illustrated in Figure 2. The wheelslip-angle αv can be obtained from the inverse steady-state tire characteristics which are described in Section 2.1. Under the assumption of no steering system elasticity, the steering wheel angle δs , which is required to obtain the wheelslip angle αv , is calculated as follows by transformation of (5) and introduction of the gear ratio of the steering gear Ks : δs = Ks αv + arctan

lv ψ˙ + v sin β v cos β

.

(7)

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Fig. 2. Tire sideforce is caused by tire sideslip [6]. 15

front rear

lateral wheel force [kN]

10

5

0

−5

−10

−15 −30

−20

−10

0 wheelslip [°]

10

20

30

Fig. 3. Summed front and rear wheels characteristics. The dotted line is the rear wheel sideforce Sh (αh ) in dependence of αh , the solid line is the front wheel sideforce Sv (αv ) in dependence of αv .

2.1฀ Inverse฀Tire฀Characteristics The฀ tire฀ characteristics฀(Figure฀ 3)฀ are฀ crucial฀ for฀ the฀ design฀ of฀ the฀ vehicle controller.฀The฀sideslip-force฀relationships฀S฀v฀(α฀v )฀and฀S฀h฀(α฀h )฀are฀obtained฀by simulating฀a฀rigorous฀vehicle฀model฀on฀a฀dry,฀flat฀and฀homogenous฀circular track฀with฀ slowly฀increasing฀speed.฀ The฀tire฀ sideslip฀ angles฀increase฀due฀ to the฀raising฀lateral฀vehicle฀acceleration฀and฀wheel฀sideforces.฀The฀simulation model฀is฀validated฀by฀measurements฀and฀corresponds฀very฀well฀to฀an฀existing vehicle.฀It฀delivers฀the฀sideforces฀and฀slip฀angles฀of฀all฀four฀wheels.฀The฀two front฀wheel฀sideforces,฀as฀well฀as฀the฀two฀rear฀wheel฀forces฀are฀summed฀up (Figure฀3).฀Consequently,฀the฀typical฀vertical฀wheel฀load฀distribution฀is฀taken into฀account฀in฀the฀tire฀characteristics.

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10 8

front wheel slip angle [°]

6 4 2 0 −2 −4 −6 −8 −10 −10

−8

−6

−4

−2 0 2 front wheel side force [kN]

4

6

8

10

Fig. 4. Inverse front wheel characteristics. It depicts the front wheel slip angle αv (Sv ) in dependence of Sv .

For a given front wheel sideforce Sv , the corresponding wheel sideslip angle αv is needed. As the function Sv (αv ) is not strictly monotonically increasing, αv is held constant for impossible sideforces as shown in Figure 4.

3 Tracking Control Design The proposed controller design is based on the two-degree-of-freedom control scheme [4, 3] as depicted in Figure 5. The plant consists of the vehicle and ˆ . The desired path is the navigation system which delivers all vehicle states x generated by piecewise defined polynomials. The distance between this path and the vehicle’s center of gravity is the tracking error dq . The dynamic feedforward block computes the desired front wheel sideforce Sv,f f based on the ˆ . The feedback controller planned path and the actual measured vehicle state x stabilizes the car along the desired path. The feedforward and feedback signals are summed up to Sv . With the known vehicle kinematics and the measured tire characteristics, the steering wheel angle δs is calculated with Equation (7) and serves as input u = δs to the plant. The feedforward and feedback control design is described in the following subsections. 3.1 Feedforward Control The main idea of the realized feedforward controller is to declare the side force Sv at the front wheel as steering input. With knowledge of the wheel characteristics and vehicle kinematics, the required sideforce can be achived in the nominal case by applying the proper steering wheel angle.

290

S. Kehl, W.-D. P¨ olsler, and M. Zeitz ^ x Feed− Sv,ff forward

Generated Path

+

dq −

Plant

^x Feedback

Sv,fb

S + v

Kinematics, u Tire Characteristics

Navigation System

Vehicle

^x

x,y Position

^ x

Fig. 5. The two-degree-of-freedom control scheme for the tracking control of the vehicle.

The desired wheel sideforces bring the vehicle to equilibrium with the dominating centripetal force FN at the actual vehicle speed and the desired path curvature. This force is partly provided by the rear wheels. The remainder Sv,f f has to be realized by the front wheels and is calculated by inverting the ordinary differential equation (1): Sv,f f = =

˙ − Sh (αh ) cos β mv(β˙ + ψ) , cos(δ − β) mv(vκ) − Sh ( lh ψ˙ − β) cos β v

cos(δ − β)

(8) ,

(9)

˙ the absolute vehicle speed v with the actual path curvature κ, the yaw rate ψ, and the vehicle sideslip angle β. For the online implementation, these variables are provided by the integrated navigation system (see Section 6) and enable an easy calculation of the feedforward force Sv,f f . 3.2 Stabilizing Feedback Control The feedforward control is based on the nominal model with various simplifications of the reality. In order to compensate for external disturbances and model errors, a stabilizing feedback control is used. The stabilizing controller is a proportional-derivative controller and the feedforward component Sv,f f is completed by the feedback component Sv,f b = dq p1 + v sin(θd − θ)p2

(10)

which is depending on the distance to the preplanned path dq and its derivative d˙q = v sin(θd − θ). Here, θd is the desired or planned moving direction and θ is the actual moving direction. The feedback gains p1 and p2 are chosen to let the vehicle approach the planned trajectory asymptotically.

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The block ’Kinematics and Tire characteristics’ in Figure 5 calculates the steering wheel angle δs . The needed wheel slip angle αv (Sv ) is attained by the inverted tire characteristics (Figure 4). Finally the steering wheel angle δ is calculated with Equation (7): δs = Ks αv (Sv,f f + Sv,f b ) + arctan

lv ψ˙ + v sin β v cos β

.

(11)

4 Generation of the Desired Path It is necessary to provide a carefully preplanned path to obtain a smooth input trajectory. Curvature change must be continuously differentiable to obtain smooth wheel velocity. This requirement is fulfilled by eighth order splines as described in [8]. The most obvious and simple approach for trajectory planning is to let a human driver steer the vehicle and to record the position of the navigation solution. The amount of this collected data has to be reduced which can be done by smoothing splines. Smoothing splines allow the optimization of the path for minimum acceleration change and thus minimizing the steering wheel effort. The result is a piecewise defined pair of polynomials for the desired vehicle position at the time t : T

T

pd (t ) = (px , py ) = (Sx (t ) , Sy (t )) ,

(12)

where Sx (t ) and Sy (t ) are the polynomials for east and north coordinate. Differentiating these polynomials yields the desired vehicle speed vd vd (t ) =

S˙x (t )2 + S˙y (t )2 .

(13)

The advantage of the piecewise polynomial form of the desired path is the possibility to analytically derive the curvature and its rate of change, thus avoiding interpolation from data tables. The parameter t can be updated for each small time step by Δt =

v(t) cos(θ(t) − θd (t ))Δt vd (t )

(14)

based on the measurements of the actual vehicle speed v and direction θ. Note that small errors in the measurements will accumulate to the longitudinal deviation dl (Figure 6). To compensate this effect, t needs to be corrected so that dl ≈ 0. With equation (12) and the knowledge of this corrected parameter t , no searching on the path has to be performed to find the point which is corresponding to the actual vehicle position.

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296

d

295.5

l

295

p(t)

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5 Simulation Results In order to test the performance of the described lateral vehicle controller, the rigorous vehicle simulation model is used as reference. A reasonable benchmark scenario for the designed tracking control is the double lane-change manoeuvre. The velocity is chosen so that the resulting lateral accelerations are close to 1g. To demonstrate the quality of the feedforward part of the controller, the feedback is switched off (p1 , p2 = 0). Figure 7 shows the path travelled by the vehicle under feedforward control. The three diagrams show the front wheel angle δ relative to the vehicle, the lateral error dq and the lateral vehicle acceleration aq = −FN /m. In the upper diagram, an additional dotted curve is plotted showing the steering angle which would be necessary under kinematic conditions. The difference between the needed angle in reality and under kinematic conditions shows the strong nonlinear behaviour during this manoeuvre. The lateral error increases almost linearly, mostly due to a slight offset in vehicle motion direction at the beginning. The lateral acceleration is relatively smooth, although the tires reach saturation at several points of the manoeuvre. In Figure 8, nearly the same manoeuvre simulation as in Figure 7 is shown. The only difference is, that the stabilizing feedback component of the controller is now activated. The two dotted lines in the front wheel angle diagram are for kinematic steering and the feedforward share, shown for reference only.

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The solid line is the applied front wheel angle. It shows differences at the points where the vehicle yaw angle changes rapidly. The lateral error is significantly smaller than the tire dimensions and could not be reached by a human driver. The lateral vehicle acceleration is smoother than under exclusive feedforward control. This example shows that the major part of the steering input is originating from the feedforward control justifying the choice of a rather primitive feedback part. The velocity can be forced even slightly further resulting in a vehicle drifting with front wheels in saturation most of the time. As the front wheel angle is always set in order to obtain the desired slip angle for huge vehicle sideslip angles, the controller acts like a perfectly countersteering driver. Simulations were also performed successfully for the Porsche race track in Weissach with a path previously defined by a human driver under severe cornering conditions.

6฀GPS-aided฀Inertial฀Measurement฀System To฀supply฀the฀controller฀with฀a฀precise฀vehicle฀state฀at฀an฀acceptable฀update rate,฀a฀GPS-aided฀inertial฀navigation฀system฀(INS)฀is฀used.฀The฀principles฀of aided฀inertial฀navigation฀are฀described฀among฀many฀other฀publications฀in฀the very฀comprehensive฀book฀of฀J.A.฀Farrell฀and฀M.฀ Barth฀[2].฀A฀short฀system description฀is฀given฀in฀the฀following. 6.1฀ System฀Description฀of฀the฀GPS-aided฀Inertial฀Navigation System Standalone฀GPS3฀ has฀the฀advantage฀of฀absolute฀position฀measurement,฀with the฀disadvantage฀of฀a฀low฀update฀rate฀with฀uncertain฀availability฀and฀accuracy. A฀further฀disadvantage฀of฀single฀antenna฀GPS฀concerns฀that฀it฀can฀not฀supply angular฀information. A฀standalone฀INS4฀ has฀a฀high฀availability,฀update฀rate฀and฀immunity฀to external฀disturbances฀but฀suffers฀from฀unbounded฀error฀growth฀due฀to฀twofold integration฀of฀the฀measured฀inertial฀sensor฀data฀during฀strapdown฀calculation5. A฀GPS-aided฀inertial฀navigation฀system฀combines฀the฀advantages฀of฀both systems฀by฀simultaneously฀eliminating฀the฀disadvantages฀of฀both฀stand-alone systems. 3

4

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A real-time kinematic differential GPS Receiver reaches a nominal accuracy of 2 cm at an update rate of 20Hz. Typically used tactical grade inertial measurement units (IMUs) have angular accuracies of about 1◦ /h (1σ), and accelerometer accuracies of about 10−3 g (1σ) at an update rate higher than 100Hz. A strapdown INS is rigidly mounted to the vehicle body. Levelling of the accelerometers is realized not by a mechanically gimballed platform but mathematically by coordinate transformations.

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To join these two systems, an extended Kalman filter (EKF) is used in a feedback configuration as shown in Figure 9. The functional block h(x) provides the same position coordinate system and units as the GPS receiver out of the INS solution. With each measurement of the GPS receiver, the INS error state is observed and corrected as described in [2]. 6.2 Realtime Navigation Solution The need of closed loop systems for realtime measurements raises the problem of sensor latency. A seldomly adressed theme in descriptions of integrated INS/GPS solutions is that the aiding GPS measurements reach the filter with

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Fig. 8. Simulation results of a double lane-change manoeuvre with feedforward and feedback control depicted in Figure 5. The upper diagram depicts the path travelled by the vehicle. The three diagrams below show the corresponding steering angle δ, the lateral error dq and the lateral acceleration aq . The dotted curves depict the steering angle δk under kinematic conditions and the feedforward share δf f of the used front wheel angle δ.

a non-negligible delay6 . Thus the INS corrections must be applied in the past, when the GPS measurement is valid. The straightforward approach which provides a realtime navigation solution is shown in Figure 10. The IMU data is stored in a buffer until the corresponding GPS measurement is available. Thus the INS delivers a delayed solution from the past. In order to obtain the realtime solution, a strapdown extrapolation from the last correction time to 6

Usually GPS latency is below 0.05 seconds, but under difficult reception conditions latency values of up to 0.5 seconds were observed.

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present time has to be performed with the buffered IMU data for each new available IMU measurement. This extrapolation is consuming much processortime and possibly cannot be performed by the available hardware. Several methods to bypass this problem have been examined. A very simple but effective approach showed to be accurate enough and consists of an inertial navigation system which is updated with each IMU measurement. A sufficient number of INS states are buffered. The states contained in this buffer and the INS state are corrected from GPS sampling time to the present whenever a new GPS measurement with its timestamp of validity is available (Figure 11). This solution has been compared to the exact solution by offline

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calculation of exemplary measurements (different maneuvers and GPS reception conditions). It shows only minimal inaccuracies and stable operation. The necessary number of floating-point operations could be significantly reduced compared to the straightforward extrapolation method. 6.3 Discontinuity Avoidance A typical problem with GPS is the occurence of discontinuities in the position solution due to multipath or integer ambiguities. An exemplary measurement detail is depicted in Figure 12. In an integrated navigation system (INS) one can use the GPS raw data to feed the Kalman filter. This facilitates detection and elimination of multipath and integer ambiguities. The drawback of this method is a centralized need of processing power. A decentralized solution lets the GPS receiver calculate a position solution and detects jumps with the knowledge of the physically possible vehicle dynamics and the fact that an INS is short-time accurate. If a significant and sudden increase of the difference between GPS and INS solution is detected, and if the GPS-receiver also shows reception problems by raising its estimated solution standard deviation, this jump can be considered as a GPS error. This error can be corrected in the subsequent GPS measurements. However, this correction is only reasonable as long as the estimated INS error is below the estimated GPS measurement error. When the GPS reception problems disappear, the estimated GPS error suddenly changes to a much better value which is then smaller than the estimated INS error. At this moment the applied correction has to be terminated as can be seen in Figure 13.

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6.4 Navigation Solution The integrated inertial navigation system provides 3D position, velocity, angles, and angular velocity with a high update rate (here: 400Hz). The position error as it is estimated by the Extended Kalman filter is better than 1cm with best GPS reception. Driving tests showed absolute dynamical position accuracy better than 4.5 cm (with an estimated 3 cm error of the position reference included). Velocity error is smaller than 0.02 m/s, yaw-angle error converges to below 0.1 degrees. This navigation solution is a solid basis for lateral vehicle control. An important vehicle state information is the sideslip angle which can be calculated from the navigation solution. Furthermore, the good absolute position solution is also invaluable for positioning accuracy. In case of GPS reception failure, the navigation system provides a valid position solution for an interval which is sufficient to bring the vehicle to a safe stop. 6.5 Sideslip Angle Calculation The calculation of the vehicle sideslip angle β is straightforward, since the integrated navigation system provides velocity in body coordinates:

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β = arctan2(vy , vx ) for (vx 2 + vy 2 ) > 0

(15)

where arctan2(y, x) is a four quadrant inverse tangent function. The sideslip angle accuracy suffers from system noise at very low vehicle speeds. For velocities smaller than 10 km/h, the lateral acceleration is below 0.2 g and the linearized or even a kinematic vehicle model can be used to derive the sideslip angle: (16) β = arcsin(lh /R). Here lh is the distance between rear axle and center of gravity and R is the curve radius of the path travelled by the center of gravity.

7 Conclusions This article presents a methodology for the realization of a path-following vehicle controller scheme. Problems as GPS latency and sudden jumps in the GPS solutions are addressed and practical solutions to these problems are proposed. This provides a high-precision GPS aided inertial navigation system usable in realtime for vehicle controlling purposes. The realized tracking control scheme comprises a path planning based on smoothing splines. The realized feedforward methodology provides a steering input very close to the necessary input even for severe manoeuvres. Due to

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the good performance of the feedforward controller, a simple stabilizing controller can be realized with low gains to complete the two-degree-of-freedom controller structure. Simulations with a validated vehicle model show very good lateral guidance behaviour of the vehicle along the predefined paths. Based on these promising simulation results, an integrated navigation and lateral guidance system will be realized to evaluate the performance of the described methodology in reality on flat surfaces for closed loop test manoeuvres as well as on race tracks. A high-precision sideslip-angle measurement is necessary for the correct functioning of the controller. Other less cost intensive methods for measuring yaw- and sideslip angle like low-cost ineartial sensors aided by multiple antenna GPS should be taken into account for future research. Acknowledgement. The authors would like to thank the Dr. Ing. h.c. F. Porsche AG and the Institut fr Systemdynamik und Regelungstechnik at the Universitt Stuttgart for supporting this work. Special thanks to Mr. Lothar Witte for providing the rigorous vehicle model, to Mr. Jochen Held for performing the navigation system test measurements, to Mr. Volker Stahl for the hardware development and to Benjamin Herzer for his work on the realtime improvement of the navigation system. Sincere thanks to Knut Graichen and Thomas Meurer for their helpful comments and suggestions.

References 1. DIN70000. Fahrzeugdynamik und Fahrverhalten, German Standard, 1994. 2. J. A. Farrell and M. Barth. The Global Positioning System and Inertial Navigation. McGraw-Hill, 1998. 3. V. Hagenmeyer and M. Zeitz. Flachheitsbasierter Entwurf von linearen und nichtlinearen Vorsteuerungen. Automatisierungstechnik, 52:3–12, 2004. 4. I.M. Horowitz. Synthesis of Feedback Systems. Academic Press, New York, 1963. 5. R. Mayr. Regelungsstrategien f¨ ur die automatische Fahrzeugf¨ uhrung. SpringerVerlag, 2001. 6. W. F. Milliken and D. L. Milliken. Race Car Vehicle Dynamics. SAE International, 1995. 7. P. Riekert and T. E. Schunck. Zur Fahrmechanik des gummibereiften Kraftfahrzeugs. Ingenieur-Archiv, XI:210–224, 1940. 8. D. Simon. Data smoothing and interpolation using eighth order algebraic splines. Technical report, Cleveland State University, Department of Electrical and Computer Engineering, 2003. 9. I. S¨ ohnitz. Querregelung eines autonomen Straßenfahrzeugs. Dissertation, Technische Universit¨ at Braunschweig, 2001.

Control of Switched Reluctance Servo-Drives Achim A.R. Fehn and Ralf Rothfuß Robert Bosch GmbH, CS/ESM, P.O. Box 1355, 74003 Heilbronn, Germany. {achim.fehn,ralf.rothfuss}@de.bosch.com

Summary.฀ The฀presented฀model-based฀solution฀for฀the฀tracking฀control฀of฀switched reluctance฀(SR)฀servo-drives฀is฀an฀outcome฀of฀its฀differential฀flatness.฀The฀controller shows฀a฀cascade฀structure฀as฀commonly฀used฀in฀drive฀engineering.฀The฀main฀difficulty by฀exploiting฀the฀flatness฀property฀for฀the฀control฀of฀SR฀drives฀is฀the฀calculation฀of the฀desired฀trajectories฀for฀the฀phase฀currents.฀This฀problem฀is฀solved฀by฀means฀of smooth฀reference฀surfaces,฀which฀are฀determined฀on฀the฀basis฀of฀the฀machine’s฀magnetic฀characteristics.฀Experiments฀illustrate฀the฀good฀performance฀of฀the฀designed control฀scheme.

Keywords:฀Switched฀reluctance฀servo-drives,฀tracking฀control,฀flatness,฀trajectory฀planning.

1 Introduction The robustness and the simple construction of switched reluctance (SR) machines suggest their use as drive for automotive x-by-wire actuators. SR machines are brushless and thus electronically commutated machines consisting of a salient stator and rotor. The driving torque is caused by the position dependent variation of the airgap reluctance (magnetic resistance). The task for the considered type of SR drives is the accurate positioning of the rotor in order to actuate the mechanism. Exemplary applications are electro-mechanical brake or steering systems. In these cases, the requirements w.r.t. system dynamics, accuracy and robustness are very high. In addition, the desired position and torque cover the complete operating range of the machine and are subject to unpredictable changes caused by the driver. Thus, a closed loop control is needed. For electrical servo drives, cascade controllers have proven their suitability in many applications [1]. The block diagram of such a cascade control for a SR drive is depicted in figure 1. The vectors v and i comprise the terminal voltages and the phase currents. The rotor position is given by ϕ. The corresponding

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 301–313, 2005. © Springer-Verlag Berlin Heidelberg 2005

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SR machine v ✲ el. circuits

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Fig. 1. Block diagram of the cascade control for a SR drive

target values are described by id and ϕd , resp. The inner current control loop is realized by an analogue PI-controller for each phase. By adequate tuning of the PI-controllers, it can be ensured that the underlying system is considerably faster than the rotor motion. Thus, the dynamics of the electric subsystem can be neglected for the design of the rotor position control, which is the focus of this contribution. The controller design is illustrated for a regular (symmetric) three phase SR machine. However, the presented approach can be directly extended to machines with other phase numbers. Provided that the desired position is given as a smooth reference trajectory ϕd (t) the control task represents a nonlinear tracking problem. This problem can be favorably solved based on an appropriate dynamic model by considering differential flatness [2, 3, 4], which is an inverse system approach. The main difficulty by exploiting the flatness property for the control of SR machines is the calculation of the desired trajectories for the phase currents, because they can not be chosen fully independently. This problem is solved by means of smooth reference surfaces, which allow the online calculation of the desired trajectories for any desired rotor motion within the operating range of the machine. In doing so, the trajectory planning considers the full system dynamics, i.e., the dynamic properties of the electric circuits are not neglected in this case. The structure of the contribution corresponds to the solution procedure of the tracking problem. At first, a dynamic model for the rotor motion of SR drives is recalled in Section 2. The flatness of this model is studied in Section 3.1 by determining a flat output y and calculating the corresponding inverse system. The inverse system allows the design of a feedforward control, which steers the system along a suitable smooth reference trajectory y d (t) for the flat output. The trajectory planning and the computation of the feedforward for the motion control is treated in Section 3.2. Based on this, a flatnessbased closed loop control is designed in Section 3.3. Experimental results for the closed loop controlled SR drive are presented in Section 4. A summary of this contribution is given in Section 5.

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2 Dynamic Model for the Rotor Motion The dynamic model for the rotor motion of a SR machine is recalled here. For details and full SR machine models, the reader is referred to [5, 6, 7, 8, 9]. The rotor position ϕ is given by Euler’s law J ϕ¨ = M1 − M2 ,

(1)

where J is the inertia and M1,2 are the generated and the load torque, resp. The driving torque M1 is caused by the magnetic field of the machine and is in general a nonlinear function of the three phase currents i = [i1 , i2 , i3 ]T and the rotor position ϕ (2) M1 = f (ϕ, i). Due to the varying airgap and saturation effects, an analytic description of the torque characteristic (2) is not possible for SR machines [6]. Therefore, the nonlinear function (2) is approximated by a table, which is calculated by means of finite element simulations [9]. The load torque including frictional losses is assumed to be a nonlinear function of the rotor position ϕ and the rotor speed ϕ˙ M2 = g(ϕ, ϕ). ˙

(3)

The torque characteristic (2) and the load (3) represent the main nonlinearities of the considered subsystem. The input u and the state vectors x are the phase currents u = [i1 , i2 , i3 ]T (4) and the rotor position and speed x = [ϕ, ϕ] ˙ T.

(5)

3 Controller Design For the solution of the tracking problem for the rotor motion, the flatness property is examined for the model (1)-(3). At first, a brief definition of differential flatness [2, 3] is recalled. A system x˙ = f (x, u),

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(7)

is said to be differentially flat, if there exists a so-called flat output, ˙ u ¨ , . . . , u(γ) , y = h x, u, u, such that ˙ y ¨ , . . . , y (δ) x = α y, y,

˙ y ¨ , . . . , y (δ+1) and u = β y, y,

are fulfilled with the finite integers γ and δ.

(8)

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Equations (8) define the algebraic inverse of the nonlinear system w.r.t. the flat output y and a finite number of its time derivatives. Thus, they can be used to compute a feedforward law, which solves a tracking problem, when a sufficiently smooth reference trajectory y d (t) for y is given. The flatness-based design of feedforward controls is more generally treated in [10]. Furthermore, the inverse system (8) allows the calculation of a linearizing state feedback, which provides the base for the flatness-based controller design [4, 11]. 3.1 System Inversion According to equations (6)-(8) the flatness property of the system can be shown by computing the inverse system (8) from the model (1)-(3). The following system inversion considers the fact that for any rotor position only up to two of the three phases are capable to generate a torque of the desired sign [9].1 These phases are denoted by main and auxiliary (indices m, a ∈ {1, 2, 3}, m = a) in the sequel. Thus, an efficient operation of the machine requires that the current in the remaining phase (index o) is zero. Note that the tasks of the phases (main, auxiliary, off) depend on the direction of the desired torque and sequentially interchange with ϕ [9]. In the sequel, it is shown that the three dimensional vector T

y = [ϕ, im , io ] .

(9)

which comprises the rotor position ϕ and the currents im and io is a flat output (dim y = dim u) for the system (1)-(3). Differentiating the rotor position ϕ from (9) twice w.r.t. time, using (1) and (3) and solving for the generated torque M1 yields M1 = J ϕ¨ + M2 = J y¨1 + g(y1 , y˙ 1 ).

(10)

Substituting (10) and (9) into the torque characteristic (2), one obtains the auxiliary phase current2 ia = f −1 (ϕ, im , io ) = f˜−1 (y, y˙ 1 , y¨1 ).

(11)

With this, all inputs (4) and states (5) are determined by (9). Thus, (9) is a flat output of the system (1)-(3). 3.2 Trajectory Planning and Open-loop Position Control The starting point for a flatness-based controller design is the adequate planning of the reference trajectories y d (t) for all components of the flat output 1 2

This is also the case for regular four-phase machines. The solvability of the function (10) for the current ia is considered within the trajectory planning in section 3.2.

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y. Based on this, an open loop rotor position control is developed, which provides the target currents id for the underlying current controllers (see figure 1). In order to get feasible reference trajectories id (t) for the phase currents, the time functions id (t) have to be at least one time differentiable and the trajectory ϕd (t) at least three times. This results from the consideration of the dynamics of the full system including the electro-dynamic properties of the machine [9]. The trajectory planning for the rotor position has to be solved individually for the considered application. Since in this contribution no special actuator is addressed, the trajectory ϕd (t) is chosen to be suitable for a representative experimental test. The used trajectory is depicted in figure 2. It is assembled by two polynomial time functions of order seven. The structure of the flat output (9) suggests that the trajectories idm (t) and d io (t) can be planned freely as long as they are sufficiently smooth. However, according to (10), the generated torque M1 depends on all phase currents.

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Therefore, the trajectory planning has to consider that the computed auxiliary phase current from equation (11) has to be real. This problem is solved in the following way. At first, ido (t) is chosen to ido (t) = 0,

(12)

because the phase with the index o is defined by the phase, which counteracts the desired motion. On that condition, the solvability of the function (2) for the current ia is examined in order to specify the constraints for the planning of the reference trajectory idm (t). In the sequel, this is exemplified for the three-phase SR machine with 18 stator and 12 rotor poles, which is depicted in figure 3.3 However, the presented approach can be directly extended to machines with other phase and pole numbers. Since the desired current for the off phase is zero ido (t) = 0, only the magnetic coupling4 of the main and the auxiliary phase have to be considered by the torque characteristic (2). Due to this and the symmetry of the SR machine, the examination of the solvability of (2) can be restricted to the interval ϕd ∈ [0◦ , 10◦ [ and one direction of the desired torque M1d [9]. From (1)-(3), it follows that the reference trajectories id (t) for the phase currents have to fulfill ! (13) Γ (id , ϕd ) = M1d . According to (10), the desired torque M1d for (13) is given by M1d = J ϕ¨d + g(ϕd , ϕ˙ d ).

(14)

Since ϕd (t) and M1d (t) are predetermined, condition (13) allows the calculation of the constraints for idm (t), by solving (13) for idm (t) and using the maximum and minimum norm of this function w.r.t. ida (t). Due to the symmetric construction of the machine, this operation can be restricted to the intervals ϕd ∈ [0◦ , 10◦ [ and M1d ∈ [−M1max , 0[ with the maximum drive torque given by M1max = 14 Nm. The resulting constraints for idm (t) are depicted in figure 4. Note that the maximum considered phase current amounts to 90 A and SR machines are operated by currents of one sign [6]. Appropriate reference trajectories for the main phase current have to lie inside the determined constraints imax and imin m m . Furthermore, it is known from the physical properties of the machine that idm has to be zero for ϕd = 0 and d ◦ to reach imin m at least for ϕ = 10 [9]. Considering this, a smooth reference d d d surface im (M1 , ϕ ), as shown in figure 5, allows the calculation idm (t) for any smooth desired rotor motion given by ϕd (t) and M1d (t) resp. Thereby, ϕd (t) 3

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The considered machine is equipped with a hollow shaft in order to provide high torques and low inertia. The expression magnetic coupling describes the fact, that the motor phases are not able to produce torque independently torque. The main reason for this are saturation effects in the rotor yokes [9].

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Fig. 3. Considered three-phase 18/12 SR machine

and M1d (t) have to be converted to the used intervals ϕd ∈ [0◦ , 10◦ [ and M1d ∈ [−M1max , 0[ according to the symmetric construction of the machine. The corresponding reference surface ida (M1d , ϕd ) for the auxiliary phase can be calculated from idm (M1d , ϕd ) by inserting idm (M1d , ϕd ) into (13) and solving for ida within the intervals ϕd ∈ [0◦ , 10◦ [ and M1d ∈ [−M1max , 0[. The surface ida (M1d , ϕd ) can be used in the same way as idm (M1d , ϕd ) for the calculation of ida (t) for the given desired rotor motion. Finally, the determined trajectories idm (t), ida (t) and ido (t) have to be assigned to the three phases 1, 2, 3 depending on the desired rotor position ϕd and the direction of M1d . This assignment describes the electronic commutation for the SR machine [6]. With this, the open loop position control is completed. The block diagram of the reference surface based open loop rotor position control is depicted in figure 6.

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A.A.R. Fehn and R. Rothfuß

90

imax m❍

80

❥ ❍

70 60

i [A]

✻50 40 30

bottle neck ✟✟ ✙

20 10

✯ ✟✟ imin m

0 −2 −4

✛ M1d

[Nm]

−6 −8 −10 −12 −14

0

2

6

✲

4

8

10

ϕd [◦ ]

Fig. 4. Upper imax and lower limit imin for the main phase current trajectory idm m m d over the rotor position ϕ and the desired drive torque M1d

3.3 Rotor Position Tracking Control In order to stabilize the rotor motion against model errors and disturbances, a state feedback controller is designed by the use of the well-known flatness methodology, which is described in detail in [4, 11] for example. Starting point is the introduction of a new input v, which is defined by v := ϕ. ¨

(15)

With this, it follows from (1) for the desired torque M1d = Jv + M2 .

(16)

According to (3), the load torque M2 can be calculated from the measured rotor position ϕ and velocity ϕ. ˙ The stabilizing feedback law is given by t

v = ϕ¨d − p2 eϕ˙ − p1 eϕ − p0 with

eϕ (τ ) dτ

(17a)

0

eϕ (t) = ϕ(t) − ϕd (t) and eϕ˙ (t) = ϕ(t) ˙ − ϕ˙ d (t).

(17b)

The computation of the desired currents id from M1d follows the approach described in section 3.2 (see also figure 6). However, in the case of feedback

Control of Switched Reluctance Servo-Drives

309

60 50 40

i [A]

✻

idm (M1d , ϕd )

30 20 10 0

✛

−2 −4 −6 d −8 M1 [Nm] −10 −12 −14

0

Fig. 5. Reference surface

ϕd✲ Euler’s law eqn. (14) ϕ˙ d ,ϕ ¨d

2

8

6

✲

4

idm (M1d , ϕd )

10

ϕd [◦ ] for the main phase m

ϕd

❄ ✲ electronic




M1d

commut.

id1,2,3

✲

✻ idm,a,o

✲ ref. surf’s ϕ ✲ and (12) d

Fig. 6. Block diagram of the open loop rotor position control

control the actual rotor position ϕ is used instead of its target value ϕd as input for the evaluation of the reference surfaces, i.e., idm (M1d , ϕ) and ida (M1d , ϕ), and the electronic commutation. The block diagram of the rotor position control is depicted in figure 7. Equation (16) describes together with the reference surfaces and the electronic commutation the linearizing state feedback for the considered system (1)-(3).

4 Experimental Results The performance of the designed rotor position tracking control is now investigated experimentally for the reference trajectory ϕd (t) depicted in figure 2.

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A.A.R. Fehn and R. Rothfuß

linearizing feedback ϕ

ϕ ✲ tracking ϕd ✲ ctrl. (17)

❄

v ✲ Euler’s law eqn. (16)




ϕ

M1d

❄

✲ electronic commut.

id1,2,3

✲

✻ idm,a,o ✲ ref. surf’s ϕ ✲ and (12)

Fig. 7. Block diagram of the rotor position control

The considered load torque is given by the Coulomb friction M2 = sign(ϕ) ˙ 0.3 Nm.

(18)

The rotor position control is implemented by means of the rapid-prototyping environment ASCET-SD [12] with a sampling period of 200 μs. Therefore, this represents a quasi-continuous realization of the controller. The rotor position ϕ is measured by an incremental sensor with a resolution of 0.02◦ . In order to compute the rotor velocity ϕ, ˙ the measured position ϕ is numerically differentiated. The PI current controllers are realized by analogue circuits. The measured rotor position ϕ and velocity ϕ˙ and the corresponding tracking errors eϕ and eϕ˙ are plotted in figures 8 and 9. The cascade controlled machine shows a good tracking behaviour. The maximum tracking error for the rotor position eϕ amounts to 0.2◦ . This accuracy is by far sufficient for automotive applications. In addition, the rotor position is transmitted by reduction gears and, thus the resulting tracking error is a fraction of 0.2◦ . The oscillations of the tracking error, especially eϕ˙ , is caused by a remaining torque ripple, which is characteristic for SR machines. Further investigations showed that the reason for the observed torque ripple are inaccuracies of the current control, which can be overcome by an optimized tuning of the PI controllers. If this measure is not sufficient, a more accurate nonlinear current control has to be designed.

5 Conclusions In this contribution, a cascade trajectory tracking control for the rotor position of SR drives has been presented. The cascade consists of linear PI-controllers for the phase currents and a flatness-based rotor position control circuit. The good performance of the tracking control and, thus, its suitability for the realization of electromechanical actuators with SR drives has been proven by experimental data.

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1200

ϕ [◦ ]

1000

✻

ϕ(t) ❍ ❥ ❍

800 600 400 200 0 0

0.5

1

1.5

✲

2

2.5

3

✲

2

2.5

3

t [s]

eϕ [◦ ]

0.2

✻

eϕ (t) = ϕ(t) − ϕd (t)

0.1 0

−0.1 −0.2

0

0.5

1

1.5 t [s]

Fig. 8. Measured rotor position ϕ(t) and position tracking error eϕ (t)

The main difficulty within the design procedure is the computation of the reference trajectories for the phase currents, which involves the trajectory planning for two of the three phase currents and the solution of the nonlinear torque characteristics for the remaining phase current. This problem has been solved under the consideration of the physical properties of the system by means of reference surfaces. This approach allows a cost-effective series realization of SR-drives, because the reference currents can be tabulated in the control unit and the phase currents are controlled by standard PI-controllers. Although, the approach has been exemplified for a three phase machine, it can be directly extended to regular SR machines with other phase and pole numbers and might be useful for the controller design for various brushless machines. Note that the stability of the developed cascade control has not been theoretically examined. This remains for future works.

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300

ˆ ˜ ϕ˙ min−1

✻

200

ϕ(t) ˙

✲

100 0

−100 −200 −300

0

0.5

1

1.5

✲

2

2.5

3

✲

2

2.5

3

t [s]

20 ˆ ˜ eϕ˙ min−1

✻ 10 0 −10 −20

˙ − ϕ˙ d (t) eϕ˙ (t) = ϕ(t)

0

0.5

1

1.5 t [s]

Fig. 9. Measured rotor velocity ϕ(t) ˙ and tracking error eϕ˙ (t)

References 1. Leonhard, W. Regelung elektrischer Antriebe. Springer-Verlag, Berlin, 2000. 2. Auflage. 2. Fliess, M., L´evine, J., Martin, P., and Rouchon, P. On differentially flat nonlinear systems. In M. Fliess, editor, Nonlinear Control Systems Design, pages 408–412. Pergamon Press, 1992. 3. Fliess, M., L´evine, J., Martin, P., and Rouchon, P. Flatness and defect of nonlinear systems: Introductory theory and examples. International Journal of Control, 61:1327–1361, 1995. 4. Rothfuß, R., Rudolph, J., and Zeitz, M. Flachheit: Ein neuer Zugang zur Steuerung und Regelung nichtlinearer Systeme. Automatisierungstechnik – at, 45(11), 1997. 5. Taylor, D.G. Nonlinear control of electric machines: An overview. IEEE Control Systems Magazine, 14:41–51, December 1994. 6. Miller, T.J.E., editor. Electronic Control of Switched Reluctance Machines. Newnes Power Engineering Series. Newnes, Oxford, 2001.

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7. Krishnan, R. Switched reluctance motor drives: Modeling, Simulation, Analysis, Design, and Applications. CRC Press UK/Parthenon Publishing, 2002. 8. Fehn, A.A.R., Rothfuß, R., and Zeitz, M. Modeling and flatness-based feasibility studies for electro-mechanical brake-by-wire systems. In Proc. 4th Mathmod 2003, Vienna, 2003. 9. Fehn, A.A.R. Modellierung und Regelung von Stellantrieben mit geschalteten Reluktanzmotoren. Dissertation. Universit¨ at Stuttgart, 2004. 10. Hagenmeyer, V. and Zeitz, M. Flachheitsbasierter Entwurf von linearen und nichlinearen Vorsteuerungen. Automatisierungstechnik – at, 52(1), 2004. 11. Rudolph, J. Beitr¨ age zur flachheitsbasierten Folgeregelung linearer und nichtlinearer Systeme endlicher und unendlicher Dimension. Shaker Verlag, 2003. 12. ETAS Dokument EC010001 R4.01 EN. ASCET-SD Handbuch Version 4.0, 2000.

Flatness-Based฀Two-Degree-of-Freedom Control฀of฀Industrial฀Semi-Batch฀Reactors Veit฀Hagenmeyer฀and฀Marcus฀Nohr BASF฀Aktiengesellschaft,฀Engineering฀Service฀Center฀Automation฀Technology, WLE/ED฀-฀L440,฀67056฀Ludwigshafen,฀Germany. {veit.hagenmeyer,marcus.nohr}@basf-ag.de Summary.฀ A฀flatness-based฀two-degree-of-freedom฀control฀is฀applied฀to฀industrial semi-batch฀reactors.฀The฀advanced฀process฀control฀scheme฀makes฀use฀of฀a฀calorimetric฀ model฀of฀ the฀reactor฀ in฀ order฀ to฀ calculate฀ the฀nominal฀ nonlinear฀ feedforward; the฀ feedback฀ part฀ consists฀ of฀ a฀ simple฀ PID฀ control.฀ Results฀ from฀ production฀ are presented,฀which฀show฀the฀performance฀and฀effectiveness฀of฀the฀applied฀control.

Keywords:฀Semi-batch฀reactor,฀advanced฀process฀control,฀two-degree-offreedom฀control,฀PID฀control,฀differential฀flatness,฀nonlinear฀feedforward฀control,฀disturbance฀feedforward,฀trajectory฀planning.

1 Introduction The control of industrial semi-batch reactors is a challenging control task, which is aggravated — with respect to academic laboratory environments — by the fact that the size of industrial semi-batch reactors tends to get larger and larger. This implies that the ratio of the surface of the walls to the volume of the reactor gets more and more disadvantageous for control (the heat of reaction takes place in three spatial dimensions, the cooling over the reactor walls in only two spatial dimensions). Hence, the highly exothermic and temperature-sensitive reaction is the main obstacle for keeping the reaction temperature at its desired value in order to guarantee the quality of the product. For this reason, the classic approach in process control of a cascaded PID structure shows unsatisfactory behaviour in industrial application to semibatch reactors. Thus, a more advanced robust control structure, which has to be moreover easily implementable in the on-site digital control system (DCS), is a desideratum. Recently, easily implementable robust two-degree-of-freedom control structures have been rediscussed in the context of differential flatness (cf. Fliess, Sira-Ram´ırez and Marquez [11, 10], Hagenmeyer and Delaleau [15], and Hagenmeyer and Zeitz [17] for a survey). Differential flatness has been introduced

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 315–332, 2005. © Springer-Verlag Berlin Heidelberg 2005

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by Fliess, L´evine, Martin and Rouchon [9]. It is a structural property of a class of nonlinear systems, for which, roughly speaking, all system variables can be written in terms of a set of specific variables — the so-called flat outputs — and their derivatives (cf. furthermore the survey paper of Rothfuß, Rudolph and Zeitz [29]). Moreover, flatness has allowed to develop the control of many practical examples and led, in several cases, to industrial applications mainly in the area of electromechanic systems (cf. e.g. [23]). Since the introduction of differential flatness, its connection with chemical reactors and process control has been an active area of acadamic research. For example, the early and well-known article of Rothfuß, Rudolph and Zeitz [28] treats the flatness-based control of the nonlinear Van de Vusse reaction. Flatness-based control of nonlinear delay systems and chemical reactors is discussed by Mounier and Rudolph [25], and flatness-based nonlinear predictive control of fed-batch bioreactors is presented by Mahadevan et al. [24]. Recently, Vollmer and Raisch discussed the control of batch cooling crystallization processes based on orbital flatness [33]. Finally, a compact survey on chemical reactors and the property of differential flatness is given by Rouchon and Rudolph, cf. [30] and the references therein. However — to the knowledge of the authors — there is only one publication presenting an industrial application of flatness in the field of process control by Petit et al. [26], treating a large continuous polymerization reactor. Thus, the aim of this contribution is twofold: firstly to show how flatness-based twodegree-of-freedom control structures can be easily applied to industrial semibatch reactors, and secondly to present another successfully running industrial application of flatness in the field of process control. The contribution is organized as follows: first, flatness-based two-degreeof-freedom control is presented in Sec. 2. Then, industrial semi-batch reactors are discussed and modelled in Sec. 3. Thereafter, the results of the flatnessbased control of industrial semi-batch reactors are shown in Sec. 4. Some final remarks conclude the contribution in Sec. 5.

2 Flatness-based Two-Degree-of-Freedom Control In order to present the flatness-based two-degree-of-freedom control scheme, the property of differential flatness is briefly introduced in the following. Thereafter, the two-degree-of-freedom control structure is recalled, and the way how differential flatness supports a direct design of the dynamic feedforward control part of the two-degree-of-freedom control structure is shown. 2.1 Differential Flatness Differential flatness is a structural property of a class of multivariable nonlinear systems, for which, roughly speaking, all system variables can be written in terms of a set of specific variables (the so-called flat outputs) and their

Flatness-Based Control of Semi-Batch Reactors

317

derivatives. In this article only SISO flat systems are briefly presented for the sake of simplicity. Given the SISO nonlinear system Σ:

˙ x(t) = f (x(t), u(t)), x(0) = x0 y(t) = h(x(t))

(1) (2)

where the time t ∈ R, the state x(t) ∈ Rn , the input u(t) ∈ R and the real output y ∈ R. The vector field f : Rn × R → Rn and the function h : Rn → R is smooth. The system (1) is said to be (differentially) flat [9] if and only if there exists a (virtual) flat output z ∈ R such that z = F (x)

(3)

x = φ(z, z, ˙ ...,z ) u = ψ(z, z, ˙ . . . , z (n) ) (n−1)

(4) (5)

are smooth at least in an open dense subset of Rn , Rn and Rn+1 , respectively. These equations yield that for every given trajectory of the flat output t → z(t), the evolution of all other variables of the system t → x(t) and t → u(t) is also given without integration of any system of differential equations. Thus the flat output z(t) and its derivatives parameterize the state x(t) and the input u(t) via Eqns. (4) and (5). Thereby it is important to remark that a trajectory z(t) is such that its n-th derivative z (n) (t) admits a left and right limit everywhere. Furthermore z(t) has to be consistent with the initial condition of the system (1), which is given by x0 = φ z(0), z(0), ˙ . . . , z (n−1) (0)

(6)

This relation can also be expressed by z(0), z(0), ˙ . . . , z (n−1) (0)

T

= φ−1 (x0 )

(7)

since the function φ : Rn → Rn is (at least locally) bijective. Moreover, if the real output y is a flat output, i.e. z = y, then Eq. (5) evidently represents the left and right inversion of the system as defined by Respondek [27]. If the real output is not a flat output, i.e. z = y, then the evolution t → y(t) of the real output is also parameterized by the flat output z(t) and its derivatives, since considering Eq. (2), Eq. (4) and the results of Hagenmeyer and Zeitz [18] leads to y = h φ(z, z, ˙ . . . , z (n−1) ) = Γ z, z, ˙ . . . , z (n−r)

(8)

where r is the relative degree of the n–th order flat SISO system (1) with respect to the output (2). Thus Eq. (8) represents the parameterization of the real output by the flat output and its derivatives up to the order n − r. Furthermore, if Eq. (8) is considered as a differential equation for z with right hand side y, then it is a diffeomorphic representation of the (n−r)–dimensional internal dynamics of the n–th order flat SISO system (1) with respect to the non-flat output (2) of relative degree r (cf. [18] for further details).

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2.2 Flatness-based Two-Degree-of-Freedom Control In many practical applications a model-based feedforward is used in order to act on the behaviour of the reference reaction of a control loop (for a discussion of the classic static feedforwards in the context of advanced process control, cf. for instance the book of Schuler [31]). Thereby a simple closed–loop control structure consisting of a system Σ and a feedback control ΣF B is extended by an open loop feedforward control ΣF F as depicted in Fig. 1. The extended structure combining the feedforward control and the feedback control has two degrees of freedom for the independent design of both the reference reaction and the disturbance reaction. When using both degrees of freedom for a control with feedforward it becomes evident that for the design of the feedback control part ΣF B there are many different methods, whereas there are few systematic methods to design dynamic feedforwards ΣF F , which take the desired motion of the controlled variable into account.

✲ ΣF F Σ∗

∗ y✲ ♣ ✲ ❡✲ ΣF B −

✻

u∗

✲❄ ❡u✲

Σ

✲ ♣y

Fig. 1. Two-degree-of-freedom control structure with system Σ, feedback control control ΣF B , feedforward control ΣF F and reference generator Σ ∗ for a tracking control y(t) → y ∗ (t).

If the real output of a system is a flat one1 , i.e. z = y = F (x), then differential flatness is a simple and direct way to design a two-degree-of-freedom control scheme by system inversion: given a sufficiently smooth desired trajectory for the flat output t → z ∗ (t), Eq. (5) can be used to design the corresponding feedforward u∗ (t) directly: ΣF F = Σ −1 :

u∗ (t) = ψ(z ∗ , z˙∗ , . . . , z ∗ (n) (t))

(9)

Thereby consistency of the desired trajectory of the flat output z ∗ (t) with the initial condition has to be considered in view of (7) 1

If the real output of a flat system is not flat, i.e. z = y, then cf. for further details Fliess, Sira-Ram´ırez and Marquez [11, 10] and the survey paper by Hagenmeyer and Zeitz [17]. If the system is not flat, then cf., for instance, the seminal paper of Devasia, Chen and Paden [8], or most recently for set point changes the contributions of Graichen, Hagenmeyer and Zeitz [12, 13].

Flatness-Based Control of Semi-Batch Reactors

z ∗ (0), z˙ ∗ (0), . . . , z ∗ (n−1) (0)

T

= φ−1 (x0 )

319

(10)

Then, the flatness-based two-degree-of-freedom control structure can be represented by the block diagram shown in Fig. 1, in which z = y, z ∗ = y ∗ , and the flatness-based inversion feedforward ψ = Σ −1 = ΣF F as in Eq. (9) hold for a tracking control z(t) → z ∗ (t). Eq. (9) clarifies the necessity of the sufficient differentiability of the desired trajectory. For instance, a desired set-point change of the flat and real output z ∗ (t) = y ∗ (t) Σ∗ :

z ∗ (t), t ∈ [0, T ] : z ∗ (0) = z0∗ = F (x∗0 ) → z ∗ (T ) = zT∗ = F (x∗T ) with

z ∗ (i)

0,T

= 0, i > 0

(11)

has to combine both boundary points z0∗ and zT∗ in a sufficiently smooth way. For example, a polynomial solution is depicted in Fig. 2. The basic idea of flatness-based two-degree-of-freedom control is that the flatness-based feedforward control ΣF F = ψ as in Eq. (9) steers the system by inversion of its dynamic model Σ, such that the feedback part ΣF B has only to deal with small deviations stemming from parameter uncertainties, exogenous disturbances or modelling errors. This enables the use of simple linear PID-like structures for the feedback part ΣF B in the block diagram shown in Fig. 1 (for stability and robustness proofs of flatness-based two-degree-of-freedom control structures, cf. [14, 15, 16]). In an industrial context, the discussed structure is very useful for tracking control. If a differentially flat system is considered, for which there already exists a linear PID feedback control stabilizing the system in the respective vicinities of different operation points, a flatness-based feedforward combined with the existing disturbance rejection optimized PID feedback controller can lead to very good tracking of, for instance, guided set point changes.

zT∗ z ∗ (t) z0∗ 0

T

t

Fig. 2. Desired trajectory z ∗ (t) of the flat output for a set point change (11) in the time interval t ∈ [0, T ] .

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3 Industrial Semi-Batch Reactors In industry, semi-batch reactors are widely used for fast and highly exothermic reactions, e.g. polymerizations. Thereby, the monomer to be polymerized cannot be completely filled in the reactor before the reaction is started (as in batch-type processes), since the danger of getting into an uncontrollable run-away mode of the reaction is far too high, due to its highly exothermic and temperature-sensitive nature. Therefore, the monomer is fed to the reactor while the reaction has already started and the process is classified as semi-batch type. After the reaction is finished, the reactor is emptied and a new semi-batch process can be started. Thus, semi-batch reactors allow a multi-product use, i.e., different products can be produced sequentially. Due to fouling, which increases from batch to batch, the reactor has to be cleaned after a certain number of batches. In Fig. 3 the structure of a semi-batch polymerization reactor with a heating/cooling jacket is depicted. The size of the interior of the considered semibatch reactor is about 60 m3 and approximately 50 t of product can be produced per batch.

Fig. 3. Schematic of a semi-batch reactor with heating/cooling jacket

The main variables of the dynamic behaviour of a semi-batch reactor are the mass of the reaction mixture mR and the temperature of the reaction mixture TR . The mass flow of the educt feed is denoted by m ˙ E and the temperature of the educt feed by TE . The temperature of the water in the jacket is Tj . The mass flow of the heating/cooling water in the jacket is represented by m ˙ Wj and the temperature of the inlet of the jacket by Tjin .

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321

3.1 Nonlinear Model of Semi-Batch Reactors A nonlinear model of semi-batch reactors as presented in Fig. 3 is developed in the sequel. It is mainly based on the enthalpy balances of the reaction mixture and of the heating/cooling jacket. The following assumptions are made: The inlet temperature of the jacket Tjin is regulated via a very fast and stable split-range temperature control (cf. again for instance the book of Schuler [31]), which is not depicted in Fig. 3 ∗ can be regarded as the for the sake of simplicity. Thus, its desired value Tjin input to the dynamics of the jacket by using a classic singular perturbation argument [20] applied to cascaded control structures. It is furthermore assumed that the composition of the educt feed is constant. The reaction rate is modelled in dependence of the temperature of the reaction mixture. Available ˙ E, m ˙ Wj and Tjin . measurements are TR , TE , Tj , m For the calculation of the enthalpy balances, necessarily the mass content of the reactor has to be known at every instant of time. The mass content mR of the reactor is determined by t

mR (t) = mR (0) +

m ˙ E dτ,

mR (0) = mR0

(12)

0

By the algebraic equation A=

mR − V0 ρP

2 + A0 r

(13)

the level dependent area for heat exchange between the content of the reactor and the jacket can be calculated. Thereby, the coefficient ρP is the specific density of the product and r represents the radius of the reactor. Since the reactor has a torospherical end, the initial volume V0 and the respective initial area A0 have to be considered. To determine the heat of reaction, a temperature dependent reaction rate using Arrhenius’ law is assumed −EA d nP = k0 nM e RTR , dt

nP (0) = nP0

(14)

where nP is the molar amount of polymerized monomer, and nM denotes the molar amount of free monomer. The universal gas constant is represented by R and the monomer dependent constants by EA and k0 . Using (14), the molar amount of free monomer can be determined by −EA m ˙E d nM − k0 nM e RTR , = kM dt MM

nM (0) = nM0

(15)

Thereby the molar mass of the monomer is MM , and the monomer mass fraction of the feed mass flow is denoted by kM . The heat of reaction Q˙ R can then be calculated by −EA Q˙ R = −ΔhR k0 nM e RTR (16) where฀ΔhR฀ is฀the฀reaction฀enthalpy.

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After฀having฀determined฀these฀auxiliary฀variables,฀the฀enthalpy฀balances for฀the฀interior฀space฀of฀the฀reactor฀and฀of฀the฀heating/cooling฀jacket฀can฀be established.฀Solved฀for฀the฀free฀temperatures,฀one฀gets 1 d TR = Q˙ R + cpE m ˙ E (TE − TR ) + U A (Tj − TR ) , dt cpP mR

TR (0) = TR0 (17)

for the reaction mixture of the reactor, and ∗ ˙ Wj Tjin − Tj U A (TR − Tj ) m d Tj = + , dt cpW mWj mWj

Tj (0) = Tj0

(18)

for the heating/cooling jacket, which is modelled as an ideal continuous stirred tank reactor. The constants cpP , cpE and cpW represent the specific heat capacities of the product, the feed and the heating/cooling water in the jacket. Additionally the heat transfer coefficient U and the mass of water in the jacket mWj have to be considered. In summary, the following equations describe a four-dimensional dynamical system modelling a semi-batch reactor −EA m ˙E d nM = kM − k0 nM e RTR , nM (0) = nM0 dt MM −EA d nP = k0 nM e RTR , nP (0) = nP0 dt 1 d TR = Q˙ R + cpE m ˙ E (TE − TR ) + U A (Tj − TR ) , dt cpP mR

(19) (20) TR (0) = TR0 (21)

− Tj ˙ Wj U A (TR − Tj ) m d Tj = + , dt cpW mWj mWj ∗ Tjin

Tj (0) = Tj0

(22)

−E

A ∗ with the heat of reaction Q˙ R = −ΔhR k0 nM e RTR , the input Tjin and mea2 ˙ E, m ˙ Wj . surements TR , TE , Tj , m This model is sufficiently rigorous in view of control design. It is similar to the model of the famous Chylla-Haase benchmark [4, 5], which was further clarified by Helbig, Abel, M’hamdi and Marquardt [19] and by Clarke-Pringle and MacGregor [6], respectively.

4 Flatness-based Control of Industrial Semi-Batch Reactors A typical batch in an industrial context can be divided into two phases: a heat up phase and a production phase, cf. Fig. 4. In the heat up phase, a 2

The values of the model parameters cannot be given here for confidentiality reasons.

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323

TR∗ TR ∗ Tjin m ˙E

temperature [norm.]

1.2

1.4

1

1.2 1 0.8

0.8 0.6

mass flow [norm.]

high constant jacket inlet temperature is applied using the fast split range ∗ control by setting a constant desired value Tjin such that the temperature of the reaction mixture TR reaches the desired constant production temperature TR∗ = TRprod of the following phase as fast as possible. Due to the dynamics of the system and the fouling from batch to batch, the desired production temperature TRprod cannot be met exactly. This leads to a relatively broad distribution of initial conditions for the production phase in spite of a lot of optimization efforts. At the beginning of the production phase, the controller is switched on and the monomer is fed to the reactor (m ˙ E > 0). The task of the controller is to reach and maintain the desired temperature TR∗ = TRprod as good as possible by manipulating the desired value of the jacket inlet tem∗ perature Tjin , since the quality of the product is directly dependent on the temperature at which the reaction takes place. Thus the controller has two main challenges: to decay the initial errors due to the heat up of the reactor as fast as possible and to counteract the disturbance of the fast changing heat of reaction Q˙ R , which results from the feeding of monomer and the highly exothermic and temperature-sensitive reaction, cf. Eq. (16). Thereby, the control law to be developed faces the following obstacles: during a batch, the heat transfer coefficient U typically decreases to about one third of its initial value due to the increasing viscosity of the reaction mixture. Then, the initial heat transfer coefficient changes from batch to batch as a result of fouling. The reaction rates are poorly known for the different products. The monomer feed is switched on and off. Different products are

0.4 0.6 0

0.2 1

2

3

4

time [norm.]

5

6

0

Fig. 4. Experimental result of a typical production batch. Normalized reaction mixture temperature TR , its desired value TR∗ = TRprod, the normalized desired tem∗ and the normalized feed mass flow of monomer m ˙E perature of the jacket inlet Tjin are shown. Production phase starts in normalized time at t = 1.

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produced in an unpredictable batch to batch sequence. Moreover, in order to reduce complexity, one controller for all products has to be implemented instead of one controller for every product. 4.1 Conventional PID Control and Disturbance Feedforward The classic control law is a cascaded PID control [4], of which only the master ∗ controller regulating TR via the desired value of the slave controller Tjin is prod ∗ presented here as a simple PID controller initialized on TR = TR ∗,P ID Tjin = TR∗ + KP

(TR∗ − TR ) +

1 τN

t 0

d −τD TR dt

(TR∗ − TR ) dτ (23)

where KP and τN , τD are the respective controller coefficients3 . As already pointed out in [4], this PID control is not able to meet the desired quality of control in an industrial context even under optimized controller coefficients4 . This turned out to be true for the control of many industrial semi-batch reactors within BASF, cf. Fig. 5. Even though the initial error is very small, the presented oscillations in the evolution of TR are too large to be acceptable for production. Since the main disturbance is the fast changing heat of reaction Q˙ R due to the highly exothermic and temperature-sensitive reaction, a ˆ˙ (cf. for disturbance feedforward was implemented based on an estimated Q R instance [7]) ˆ˙ + Q˙ Q R E ∗,P ID+DF F ∗,P ID Tjin (24) = Tjin − UA with the cooling effect of the feed Q˙ E = cPE m ˙ E (TE − TR )

(25)

ˆ˙ is based on where m ˙ E and TE are measured variables. The estimation of Q R the calorimetric model (19) – (22) and makes use of an observer close to an extended Luenberger one (for extended Luenberger observers, cf. the articles of Zeitz [35] and Birk and Zeitz [3]5 ). In Fig. 6, the results of the application 3

4

5

In the following, the values of the coefficients cannot be given for confidentiality reasons. For the following application, the optimization of the coefficients was performed taking Kuhn’s T-sum-rule [22] into account. This rule uses certain areas of the step response and is thus far more robust than the classic tuning rules of Ziegler and Nichols or Chien, Hrones and Reswick using the tangent to the inflexion point of the step response. For an early work on nonlinear observer design for chemical reactors, cf. the monography of Zeitz [34]. For computer supported observability analysis and observer design of nonlinear systems in general, cf. the monography of Birk [2].

Flatness-Based Control of Semi-Batch Reactors TR∗ TR

1.08

temperature [norm.]

325

1.06 1.04 1.02 1 0.98 0.96 0.5

1

1.5

2

2.5

time [norm.]

3

3.5

Fig. 5. Experimental result of a typical production batch using cascaded PID control (23). Normalized reaction mixture temperature TR and its desired value TR∗ = TRprod are shown in an appropriate scale. Production phase starts in normalized time at t = 1.

of the disturbance feedforward (24) is shown. Clearly the oscillations were reduced in amplitude with respect to the results shown in Fig. 5, but nevertheless they were not satisfactory for production. The reason is that the controller has both the task of decaying the non-negligible initial error in Fig. 6, i.e. the behaviour of the reference reaction, and of counteracting the disturbance of the fast changing heat of reaction Q˙ R , i.e. the behaviour of the disturbance reaction. To design both behaviours independently, a two-degree-of-freedom control structure is appropriate. 4.2 Flatness-based Two-Degree-of-Freedom Control Under the above considerations of the very fast split range control and the ˆ˙ , the system represented by Eqns. (21)–(22) estimation of Q R 1 d TR ˆ˙ + c m Q = R pE ˙ E (TE − TR ) + U A (Tj − TR ) , dt cpP mR

TR (0) = TR0 (26)

˙ Wj − Tj d Tj U A (TR − Tj ) cpW m = + , dt cpW mWj cpW mWj ∗ Tjin

Tj (0) = Tj0

(27)

is a simple calorimetric model of the process (for a survey on calorimetric modelling, cf. the article of Schuler and Schmidt [32]). The system (26)–(27) ∗ and with known time-varying is flat with flat output TR , with input Tjin ˆ ˙ ˙ E , TE , m ˙ Wj , A and mR (cf. Eqns. (12) and (13)). Thus the parameters QR , m

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temperature [norm.]

1.08 1.06 1.04 1.02 1 0.98 0.96 0.5

1

1.5

2

2.5

time [norm.]

3

3.5

Fig. 6. Experimental result of a typical production batch using cascaded PID control plus disturbance feedforward (24). Normalized reaction mixture temperature TR and its desired value TR∗ = TRprod are shown. Production phase starts in normalized time at t = 1.

flatness-based two-degree-of-freedom control structure as discussed in Sec. 2 is directly applicable. However, a further simplification is advisable in view of implementation in the digital control system: since the value of the water mass flow in the jacket m ˙ Wj is very high, one has ˙ Wj cpW m cpW mWj

ˆ˙ Q ˙E UA cpE m UA R , , , cpW mWj cpP mR cpP mR cpP mR

(28)

Therefore a singular perturbation argument [20] applied to Eq. (27) leads to 0=

∗ − Tj ˙ Wj Tjin U A (TR − Tj ) cpW m + cpW mWj cpW mWj

(29)

∗ . Hence, and, taking again (28) into account, furthermore finally to Tj ≈ Tjin Eq. (26) can be written as

1 d TR ˆ˙ + c m ∗ Q = R pE ˙ E (TE − TR ) + U A Tjin − TR dt cpP mR

,

TR (0) = TR0

(30) This system is a one dimensional flat system with flat output TR , with input ∗ and the above mentioned time-varying parameters. Tjin The flatness-based two-degree-of-freedom control structure as discussed in Sec. 2 can directly be applied to system (30). To this end, a desired trajectory TR∗ = TR∗ (t) has to be planned as in Fig. 2. For the application, the beginning of the production phase of the batch is taken as the initial instant t = 0, and the actual reaction mixture temperature is therefore considered as TR (0) = TR0 .

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327

Even though this temperature TR0 is not a stationary set point in the general case, it is assumed to be one for the desired trajectory planning: A third order polynomial is used to guide the desired temperature trajectory as in Fig. 2 from TR∗ (0) = TR∗ 0 = TR0 to the a priori given desired production temperature TR∗ (T ) = TRprod , where it is kept constant for the rest of the batch onwards. With the initial error ΔTR∗ = TR∗ 0 − TRprod , one gets ⎧ T T ⎪ ⎨ TR∗ 0 − sign(ΔTR∗ )ΔTR∗ 3( ΔTR∗ )2 − 2( ΔTR∗ )3 , t ∈ [0, T ] ∗ (31) TR (t) = ⎪ ⎩ prod t>T TR , Thereby the transition time T is carefully chosen in dependence of the initial error ΔTR∗ . The nominal nonlinear feedforward control can then be derived from Eq. (30) according to Sec. 2 as

1 ˆ˙ − c m ∗ cpP mR T˙R∗ − Q (32) R pE ˙ E (TE − TR ) UA In combination with the PID control (23), in which the constant reference TR∗ = TRprod is replaced by TR∗ of (31) and the initialization is set to 0 ∗ Tjin = TR∗ +

∗

∗,P ID = KP Tjin

(TR∗ − TR ) +

1 τN

t 0

d (TR∗ − TR ) dτ + τD (T˙R∗ − TR ) dt

(33)

the flatness-based two-degree-of-freedom controller reads as 1 ˆ˙ − c m ∗,P ID,f lat ∗,P ID ∗ ∗ Tjin cpP mR T˙R∗ − Q = Tjin + TR∗ + R pE ˙ E (TE − TR ) UA (34) Thereby the proximity of this control to the disturbance feedforward control (24) is evident: the flatness-based two-degree-of-freedom controller encompasses elegantly the observer-based disturbance feedforward control for TR ≈ TR∗ . The result of the application of (34) is presented in Fig. 7. Even though the initial error of the production phase caused by the uncertainty of the dynamics during the heat up is very large with respect to the ones in Fig. 5 and Fig. 6, the reaction mixture temperature TR is well led to its desired production value TRprod , where it stays close to for the remaining time onwards. The deviation of TR from TR∗ at t ≈ 1.15 results from the disturbance of the started reaction. The respective fast evolution of Q˙ R can only be estimated approximately by the applied observer in the beginning. However, the oscillations, which occur with the PID plus disturbance feedforward (24) in Fig. 6, do not occur with the flatness-based two-degree-of-freedom controller (34) in Fig. 7, since the initial error is smoothly guided to 0 and therefore does not excite the system. 4.3 Relation to Other Work The advanced process control of batch reactors has a long scientific tradition and has been treated in the literature by a large number of groups, cf. the

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V. Hagenmeyer and M. Nohr TR∗ TR

temperature [norm.]

1.08 1.06 1.04 1.02 1 0.98 0.96 0.5

1

1.5

2

2.5

time [norm.]

3

3.5

Fig. 7. Experimental result of a production batch using flatness-based two-degreeof-freedom control (34). Normalized reaction mixture temperature TR and its desired value TR∗ are shown. Production phase starts in normalized time at t = 1.

survey of Berber [1]. Hence, a comparison of the developed flatness-based two-degree-of-freedom control (34) with existing control laws is appropriate. Thereby the problem arises that it was not possible to implement different control laws on the BASF production plant, since the project was limited in budget and time. Therefore, a direct comparison of different control laws by their respective production results cannot be given here. Nevertheless, a detailed comparison with the model-based methods discussed in [1] can be undertaken on the level of the respective control law equations. The control law of Kravaris, Wright and Carrier [21], which can be regarded as a two-degree-of-freedom control interpretation of globally linearizing control (GLC) introduced by Kravaris and Chung, reads as ˆ˙ ∗,P ID,GLC ∗,P ID ∗ + cpP mR T˙R∗ − Q = Tjin Tjin R

(35)

Thereby the cooling effect of the feed (25) is not modeled since in [21] only batch reactors are studied. The difference of control law (35) to (34) is that (35) lacks the additive term TR∗ and the prefactor U1A in (34). Since (34) represents a system inversion, the lacking terms have to be compensated by the PID part in (35) which is necessarily not as good as a direct system inversion. The control law of Cott and Macchietto [7] is derived from the methodology of generic model control (GMC). It can be written (again omitting the cooling effect of the feed (25) for the same reason) as ∗,P ID,GMC = TR∗ + Tjin

1 ˆ˙ ∗,P ID ∗ cpP mR Tjin −Q R UA

(36)

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The difference of this control law to (34) consists in the fact that in (36) the PID part is placed at the point where in (34) the feedforward part of T˙R∗ is used. Considering again that (34) inverts the system behaviour, not using the information of T˙R∗ is equal to introducing unnecessarily an exogenous perturbation. This can be proven when studying the system from the point of view of the tracking error equation, cf. [14, 15] for details. The more recent control law of Clarke-Pringle and MacGregor [6] is based on the academic Chylla-Haase benchmark [4, 5]. It is obtained by exact feedback linearization of a two-dimensional system close to (26)–(27), in which the Arrhenius’ law terms as in (14) are conserved. The heat of reaction is not observed directly, an extended Kalman filter is designed using detailed process knowledge in order to estimate the pre-exponential factor k0 nM and the heat transfer coefficient U . These values are used adaptively in the nonlinear control law based on exact feedback linearization. Since all differentially flat systems are linearizable by exact feedback linearization [9], a natural question is whether an exact feedback linearization law in the spirit of [6] based on (30) would lead to better results than (34). In [14] it has been shown that control laws derived by exact feedback linearization (without parameter estimation) are less robust against parametric uncertainties and measurement noise than a flatness-based two-degree-of-freedom one. Furthermore, for the real multi-product semi-batch process in the BASF it turned out that the use of the Arrhenius’ law terms and the respective estimation of the parameters k0 nM and U is unfeasible, because the detailed process knowledge used in the extended Kalman filter design of [6] is not available. Therefore the two-degree-of-freedom control law (34) has been chosen for application in view of robustness and reliability. Moreover, a comparison of the applied control law (34) with an NMPC as the one of Helbig, Abel, M’hamdi and Marquardt [19], which is also based on the Chylla-Haase benchmark [4, 5], is difficult. Both methodologies can be seen as model based predictive controls which differ as well in the way how the prediction is undertaken as in the way in which the tracking error is fed back (for details cf. appendix D of [14]). The advantage of the flatness-based two-degree-of-freedom control is its low complexity and that it can directly be implemented in the digital control system of the plant. Furthermore it is easily understandable by the operators because it is a combination of a wellknown PID and a model-based feedforward. On the other hand, NMPC might lead to more “optimal” results, depending on the accuracy of the used model and the quality of the respective parameter estimation. Only a fundamental comparison of both methods pointing out their respective advantages and disadvantages in as well the academic as the industrial framework could give a clear answer to the question which method to prefer in a given case.

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5 Conclusions The designed flatness-based two-degree-of-freedom control of industrial semibatch reactors (34) is implemented in the digital control system of a BASF production plant with a high output of different products. The implementation of the control law was simple due to its low complexity. The control is moreover well accepted by the operators, since it contains classic PID control and a model-based feedforward. The applied control law runs all the time throughout the whole year. The robust improved temperature control leads to improved quality and better reproducibility of the different products. A higher number of batches before necessary cleaning is now possible, which results in an even higher throughput of the plant. Acknowledgement. This contribution is dedicated to Prof. Dr.-Ing. Dr. h.c. Michael Zeitz in order to celebrate his 65th anniversary. The first author (V.H.) is deeply grateful to him for the many fruitful and vivid discussions, as well scientific as on many other topics (in particular in the “Teek¨ uche” of the ISR!), for the constant intellectual stimulation and especially for his very personal encouragement in the last years. Both authors are moreover very thankful to Prof. Dr.-Ing. Hans Schuler and Dr.-Ing. Joachim Birk, both BASF Aktiengesellschaft, Ludwigshafen (Germany), for their respective support and interest in this work.

References 1. R. Berber. Control of batch reactors: a review. Trans. IChemE, Part A, 74:3–20, 1996. 2. J. Birk. Rechnergest¨ utzte Analyse und L¨ osung nichtlinearer Beobachtungsaufgaben. Number 294 in Fortschr.-Ber. VDI Reihe 8. VDI Verlag, D¨ usseldorf (Germany), 1992. 3. J. Birk and M. Zeitz. Extended Luenberger observer for nonlinear multivariable systems. Int. J. Control, 47:1823–1836, 1988. 4. R.W. Chylla and D.R. Haase. Temperature control of semi-batch polymerization reactors. Comput. Chem. Eng., 17:257–264, 1993. 5. R.W. Chylla and D.R. Haase. Temperature control of semi-batch polymerization reactors (corrigenda). Comput. Chem. Eng., 17:1213, 1993. 6. T. Clarke-Pringle and J.F. MacGregor. Nonlinear adaptive temperature control of multi-product, semi-batch polymerization reactors. Computers chem. Engng, 21:1395–1409, 1997. 7. B.J. Cott and S. Macchietto. Temperature control of exothermic batch reactors using generic model control. Ind. Eng. Chem. Res., 28:1177–1184, 1989. 8. S. Devasia, D. Chen, and B. Paden. Nonlinear inversion-based output tracking. IEEE Trans. Automatic Control, 41:930–942, 1996. 9. M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control, 61(6):1327– 1361, 1995.

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10. M. Fliess and R. Marquez. Continuous-time linear predictive control and flatness: a module-theoretic setting with examples. Int. J. Control, 73:606–623, 2000. 11. M. Fliess, H. Sira-Ram´ırez, and R. Marquez. Regulation of non-minimum phase outputs: a flatness based approach. In D. Normand-Cyrot, editor, Perspectives in Control - Theory and Applications: a tribute to Ioan Dor´ e Landau, pages 143–164. Springer, London, 19–23 June 1998. 12. K. Graichen, V. Hagenmeyer, and M. Zeitz. Van de Vusse CSTR as a benchmark problem for nonlinear feedforward control design techniques. In Proc. (CDROM) 6th IFAC Symposium ”Nonlinear Control Systems” (NOLCOS), pages 1415–1420, Stuttgart (Germany), 1–3 September 2004. 13. K. Graichen, V. Hagenmeyer, and M. Zeitz. A new approach to inversion-based feedforward control design for nonlinear systems. Submitted. 14. V. Hagenmeyer. Robust nonlinear tracking control based on differential flatness. Number 978 in Fortschr.-Ber. VDI Reihe 8. VDI Verlag, D¨ usseldorf (Germany), 2003. 15. V. Hagenmeyer and E. Delaleau. Exact feedforward linearization based on differential flatness. Int. J. Control, 76:537–556, 2003. 16. V. Hagenmeyer and E. Delaleau. Robustness analysis of exact feedforward linearization based on differential flatness. Automatica, 39:1941–1946, 2003. 17. V. Hagenmeyer and M. Zeitz. Flachheitsbasierter Entwurf von linearen und nichtlinearen Vorsteuerungen. Automatisierungstechnik, 52:3–12, 2004. 18. V. Hagenmeyer and M. Zeitz. Internal dynamics of flat nonlinear SISO systems with respect to a non-flat output. Syst. Contr. Lett., 52:323–327, 2004. 19. A. Helbig, O. Abel, A. M’hamdi, and W. Marquardt. Analysis and nonlinear model predictive control of the Chylla-Haase benchmark problem. In Proc. UKACC Int. Conf. Control, pages 1172–1177, Exeter (England), 2–5 September 1996. 20. P. Kokotovi´c, H. K. Khalil, and J. O’Reilly. Singular perturbation methods in control: analysis and design. Academic Press, London, 1986. 21. C. Kravaris, R.A. Wright, and J.F. Carrier. Nonlinear controllers for trajectory tracking in batch processes. Computers chem. Engng, 13:73–82, 1989. 22. U. Kuhn. Eine praxisnahe Einstellregel f¨ ur PID-Regler: Die T-Summen-Regel. Automatisierungstechnische Praxis, 37:10–12, 14–16, 1995. 23. J. L´evine. Are there new industrial perpectives in the control of mechanical systems? In P.M. Frank, editor, Advances in Control (Highlights of ECC’99), pages 197–226. Springer-Verlag, London, 1999. 24. R. Mahadevan, S.K. Agrawal, and F.J. Doyle III. Differential flatness based nonlinear predictive control of fed-batch bioreactors. Control Eng. Pract., 9(2):889– 899, 2001. 25. H. Mounier and J. Rudolph. Flatness based control of nonlinear delay systems: a chemical reactor example. Int. J. Control, 71:871–890, 1998. 26. N. Petit, P. Rouchon, J.-M. Boueilh, F. Gu´erin, and P. Pinvidic. Control of an industrial polymerization reactor using flatness. J. Process Contr., 12:659–665, 2002. 27. W. Respondek. Right and left invertibility of nonlinear control systems. In H.J. Sussmann, editor, Nonlinear Controllability and Optimal Control, pages 133–176. Marcel Dekker, New York, 1990. 28. R. Rothfuß, J. Rudolph, and M. Zeitz. Flatness based control of a nonlinear chemical reactor model. Automatica, 32(10):1433–1439, 1996.

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Controlled฀Friction฀Damping฀using฀Optimally Located฀Structural฀Joints Lothar฀Gaul1 ,฀Stefan฀Hurlebaus2 ,฀Hans฀Albrecht1 ,฀and฀Jan฀Wirnitzer1 1฀

Institute฀A฀of฀Mechanics,฀University฀of฀Stuttgart,฀Pfaffenwaldring฀9,฀70550 Stuttgart,฀Germany. [email protected] 2฀ Department฀of฀Civil฀Engineering,฀Texas฀A&M฀University,฀3136฀TAMU,฀College Station,฀TX,฀77843-3136,฀USA. Summary.฀ Lightweight฀structures฀typically฀have฀low฀inherent฀structural฀damping. Effective฀vibration฀suppression฀is฀required,฀for฀example,฀in฀certain฀applications฀involving฀precision฀ positioning.฀ The฀ present฀ approach฀ is฀ based฀ on฀ friction฀ damping in฀semi-active฀joints฀which฀allow฀relative฀sliding฀between฀the฀connected฀parts.฀The energy฀dissipation฀due฀to฀interfacial฀slip฀in฀the฀friction฀joints฀can฀be฀controlled฀by varying฀the฀normal฀pressure฀in฀the฀contact฀area฀using฀a฀piezo-stack฀actuator฀.฀This paper฀focuses฀on฀the฀optimal฀placement฀of฀semi-active฀joints฀for฀vibration฀suppression.฀The฀proposed฀method฀uses฀optimality฀criteria฀for฀actuator฀and฀sensor฀locations based฀on฀eigenvalues฀of฀the฀controllability฀and฀observability฀gramians.฀Optimal฀sensor/actuator฀placement฀is฀stated฀as฀a฀nonlinear฀multicriteria฀optimization฀problem with฀discrete฀variables฀and฀is฀solved฀by฀a฀stochastic฀search฀algorithm.฀At฀optimal locations,฀conventional฀rigid฀connections฀of฀a฀large฀truss฀structure฀are฀replaced฀by semi-active฀friction฀joints.฀Two฀different฀concepts฀for฀the฀control฀of฀the฀normal฀forces in฀the฀friction฀ interfaces฀ are฀implemented.฀In฀the฀first฀approach,฀each฀semi-active joint฀has฀its฀own฀local฀feedback฀controller,฀whereas฀the฀second฀concept฀uses฀a฀global, clipped-optimal฀controller.฀Simulation฀results฀for฀a฀10-bay฀truss฀structure฀show฀the potential฀of฀the฀proposed฀semi-active฀concept.

Keywords:฀Friction฀damping,฀lightweight฀structure,฀structural฀joints,฀optimal฀placement,฀semi-active฀control.

1 Introduction Spatial truss structures are a traditional approach in lightweight construction. Such structures have large dimensions, low structural weight, and low structural damping. Vibrations pose a problem in many applications such as the deployment of antennas in space or optical interferometers. Such cases call for additional measures to improve the structure’s damping behavior. A structure can be damped using passive [1], active [2], or semi-active [3] technologies, as

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 335–350, 2005. © Springer-Verlag Berlin Heidelberg 2005

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described in [4]. The semi-active technique combines the advantages of passive and active techniques. In this work, dry friction is the mechanism used for semi-active damping of a truss structure. The present approach basically circumvents the use of additional component parts by modifying some of the nodes that link the trusses of the framework (as patented by the first author [5]). The modified nodes are used to link truss members such that frictional sliding is allowed at the contact point. The normal pressure in this frictional connection and internal truss loading determine the relative motion of the connected truss parts [6]. Since a passive device is actively controlled, this approach is called semi-active. The resulting semi-active vibration suppression system is always stable, at least in the Lyapunov sense, because of the dissipative nature of friction [7]. The friction joint is a purely passive element, where only the normal force is controlled by an active member. Since no further energy is added to the system, the system remains stable.

2 Adaptive Truss Structure The examined structure is designed in the style of lightweight structures applied in space missions such as the Shuttle Radar Topography Mission (SRTM)3 in 2002. In this mission, a radar-emitter and receiver for mapping the earth’s surface had to be positioned at a distance of about 60 m from the space shuttle. A cantilever truss structure fixed in the space shuttle was used to position one of two corresponding radar antennas. 2.1 Geometry of the Investigated Truss The 10-bay truss structure investigated is depicted in Fig. 1. The structure is assembled using aluminum tubes and joint nodes from a Meroform-M12 construction system. The joints are MERO steel ball nodes with 18 M 12 thread holes. Each of the tubes has a 22 mm outer diameter and is fitted with a screw end connector. The horizontal and vertical distance between the center of the truss nodes is 1 m. At the free end, a mass of 0.5 kg is attached to each of the five truss nodes. At the opposite end, the four end truss nodes are fixed at a large cement block. 2.2 Design of the Adaptive Joints For the implementation of the semi-active damping approach, a modified joint connection has been designed. The joint allows relative motion between the 3

For details visit the websites http://www.jpl.nasa.gov/srtm/ http://www.aec-able.com/corporate/srtm.htm

and

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Fig. 1. Experimental setup of the examined cantilever truss structure

end connector of a truss member and the truss node with the remaining members attached to it. The rigid Meroform nodes can be replaced by this adaptive joint. As shown in Fig. 2, two types of adaptive joints are considered, each with a single degree-of-freedom. In the type A joint relative displacement occurs along the longitudinal axis of the connected rod, and in the type B joint relative rotation occurs about an axis perpendicular to the longitudinal axis. y

Type A z

x

friction element Type B end connector

Fig. 2. Design types of joints

Fig. 3. Realization of the adaptive joint

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The experimental realization of the adaptive joint is shown in figure 3. Three struts 2 are fixed to the body 1 of the joint. The movable strut 3 is connected with a sliding nut 4 that can be clamped between the joint’s body and a pressure plate 5 . The frictional interfaces are between the joints body and the sliding nut as well as between the sliding nut and the pressure plate. The normal force in the frictional interface is generated by a piezoelectric stack actuator 6 and measured by a strain gauge load cell 7 . The actuator and load cell are stacked together on top of the pressure plate. The semi-active joint relies on the dry friction between the interfacial slip surfaces to dissipate energy [5]. The adjustment bolt 8 serves to fix the appropriate working point for the normal force.

!

!

!

!

!

!

!

!

3 Simulation Model Experimental determination of optimal actuator/sensor locations or optimal truss design is somewhat unpractical. Therefore a mathematical model for simulations is needed. First, the linear part of the structure is modeled. Second, the nonlinear friction behavior in an isolated joint is modelled. The linear and nonlinear models are combined into a single model which is used to predict the structure’s dynamic behavior. 3.1 FE-Model of the Structure’s Linear Part The FE discretization of the truss connections is depicted in Fig. 4. All tubes and end connectors are discretized by cubic Euler-Bernoulli beams with two nodes, each having six degrees of freedom. The Meroform nodes are considered as perfectly rigid, and the rigid portions are modelled by geometric constraints um = S nm un , where un are the independent DOFs, and um are the constrained DOFs. The mass and inertia of the Meroform connections are lumped at the centers of each truss node. The kinematics of the friction joints are described by constraint matrices, as well, which couple the degrees of freedom of the corresponding nodes according to the type of joint. The composite FE model has the form ¯ Sz ¯ ¯z˙ + S T KS ¯ z ¯ uR (v, ϕ) + S T F ¯w , ¨ ¯ = ST E ¯ + S T DS ST M

(1)

¯ , D, ¯ and K ¯ are the mass, damping ¯ is the displacement vector; M where z and stiffness matrices, respectively for the linear FE model; and S is the overall constraint matrix containing the linear node constraint matrices S nm . Equation (1) can be rewritten as ¨ + Dz˙ + Kz = E uR (v, ϕ) + F w , Mz

(2)

where the vector z contains only the independent generalized displacements zj (j = 1, . . . , N ). As will be discussed in section 3.2, the friction forces uR (v, ϕ)

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acting in the friction joints depend on the sliding velocities v = E T z˙ and the bristle deflections ϕ. The vector w represents disturbance inputs to the system. Using the modal transformation z(t) = Φη(t)

(3)

with the modal matrix Φ = [φ1 , . . . , φn ] (including n < N ’significant’ eigenmodes φi ) obtained from the undamped eigenvalue problem Kφi = ωi2 M φi ,

i = 1, . . . , n ,

(4)

and assuming proportional damping , (2) can be transformed into a set of uncoupled differential equations, ¨ + Ξ η˙ + Ωη = ΦT E uR (v, ϕ) + ΦT F w . η

(5)

The assumption of Rayleigh , or proportional damping , is justified due to the low damping in a structure with frictionless joints. Here, Ω = diag (ωi2 ), Ξ = diag (2ξi ωi ), and the vector η contains the modal coordinates ηi . The eigenfrequency and the modal damping coefficient of the i-th mode is denoted by ωi and ξi , respectively. The sliding velocities v are given by v = E T Φ η˙ .

(6)

It should be pointed out that the transformation matrix in (3) is composed of eigenvectors for the underlying linear undamped system. That means, the nonlinear state-dependent term in the excitation vector uR is neglected, and a truss structure with frictionless joint connections is considered, i.e. FNj = 0. Such an approach is often used for systems with local nonlinearities [8]. Since not all of the parameters of the FE model are known a priori, they have to be identified by means of [9] using eigenfrequencies and mode shapes obtained from an Experimental Modal Analysis (EMA) which has been performed on the structure. In particular, the unknown parameters are the axial, bending, and torsional stiffness of the end connectors. Furthermore, the material properties of the aluminum members are not exactly known. Dimensions and masses of the components are assumed to be known. The eigenfrequencies calculated with the updated FE model are compared to the measured values in Table 1. The corresponding MAC values [9] for the mode shapes are listed as well. Note that only those modes are listed whose mode shapes could clearly be identified by the EMA. The extracted modal damping ratios 0.05% ≤ ξi ≤ 0.5% confirm that the present structure is indeed a weakly damped system. In the simulations, a damping ratio of ξi = 0.1% for all modes is assumed.

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Table 1. Eigenfrequencies in [Hz] No EMA FEM error [%] MAC

end connector

truss node

rigid connection

Fig. 4. FE discretization

1

4.49 4.495

0.18

0.79

2

4.60 4.595

0.19

0.88

3 20.65 20.60

0.24

0.97

4 22.07 22.25

0.82

0.95

5 23.61 23.76

0.64

0.93

7 24.72 25.08 8 43.47 43.73

1.46 0.60

0.93 0.95

12 46.83 46.90

0.15

0.98

17 49.45 49.52

0.14

0.72

3.2 Nonlinear Joint Model For the modeling of the nonlinear behavior in the friction joints, the so-called LuGre-model [10] has been proposed [11]. This model is capable of representing relevant friction phenomena [12], such as presliding displacement, stick-slip behavior, and the Stribeck effect. The model describes the friction interface as a contact between bristles. The internal state variable ϕ representing the average deflection of the bristles is governed by the first order differential equation ϕ˙ = f (v, ϕ) = v − σ0 g(v) ϕ

with

g(v) =

|v| 2 , Fc + Fd e−(v/vs )

(7)

where v is the relative sliding velocity in the contact. The friction force transmitted in the joint FR = FN (σ0 ϕ + σ1 ϕ˙ + σ2 v) (8) =:μ (v, ϕ)

depends on the normal force FN and the dynamic friction coefficient μ(v, ϕ) . In case of a rotational joint (type B), a radius r is introduced in (8) to obtain a friction moment, i.e., MR = r FR , and v would represent the relative angular velocity. In this paper, for adaptive joints of either type, the generalized forces uRi and the generalized sliding velocities vi (i = 1... ) are contained in the vectors uR and v. The model parameters σ0k , σ1k , σ2k , Fck , Fdk and vsk are all positive and have dimensions consistent with the dimension of v . 3.3 Parameter Identification In order to identify the parameters of the LuGre-model , the transfer behavior of an isolated friction joint is measured. The sliding velocity v and the fric-

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tion force uR are recorded for different normal forces and different excitation amplitudes and frequencies. The model parameters are determined from the comparison between the simulated response and the measurements. This is done by solving the optimization problem min J (p) = p

1 2

K

k=0

(uR (tk ) − uˆR (v(tk ), FN , p))

2

(9)

with the vector p containing the parameters to be identified, the measured friction force uR (tk ), and the simulated friction force uˆR (v(tk ), FN , p), which is obtained from (8) and integration of (7). For calculation of uˆR , the measured sliding velocity v(tk ) and normal force FN , which is kept constant during the recording period, are used. The identified parameters are listed in Table 2 for both types of joints. The present joint design turned out to have quite small parameters σ1 und σ2 which have almost no influence on the dynamic behavior. Therefore, it is hard to identify their actual values. In the case of the laboratory truss structure, the system parameters are taken as constants. In real applications such as orbital truss structures, one has to consider the large changes in temperature and thus material properties. Since the system parameters greatly depend on temperature as well as displacement, further nonlinearities would have to be modelled. Table 2. Identified friction parameters Typ A B

σ0 5

σ1

1.42 · 10 1/m 6.5 · 104 1/rad

0.02 s/m 0.02 s/rad

σ2 0.01 s/m 0.01 s/rad

Fc

Fd

0.61 0.49

0.14 0.08

vs

r

0.01 m/s 0.02 rad/s

– 0.011 m

3.4 Model of the Adaptive Structure By defining the state vector of the linear subsystem as x = xT1 , . . . , xTn

T

with

xi =

η˙ i ωi ηi

(10)

where the state xi represents the i-th mode, the state space form of the friction damped truss structure is given by Ax B H x˙ = + uR (v, ϕ) + w. ˙ ϕ f (v, ϕ) 0 0

(11)

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The system matrix A as well as the matrices B and H can easily be deduced from (5) by taking the definition of the state vector x into account. The internal state variables ϕj (j = 1, . . . , ) of the semi-active joints are contained in ϕ, and the corresponding differential equations (7) are contained in f (v, ϕ). The sliding velocities are given by v = B T x. The actuator forces uR are defined by (8) as uRj = FNj μj (vj , ϕj ), where the normal forces FNj represent the controller output.

4 Actuator and Sensor Placement In the following, the open-loop system (11) associated with the measurement equation y = Cx is considered. The forces uR are regarded as control inputs, and the output matrix C is defined according to C i = [ cvi , cdi /ωi ] .

C = [C 1 , . . . , C n ] ,

(12)

For an asymptotically stable system, the controllability gramian W c and observability gramian W o are defined as [13] c

W =

∞

e

Aτ

T

BB e

AT τ

dτ

and

∞

o

W =

eA

T

τ

C T C eAτ dτ . (13)

0

0

Both gramians are symmetric, positive definite matrices and can be computed from the steady-state Lyapunov equations AW c + W c AT + BB T = 0

and

AT W o + W o A + C T C = 0 . (14)

Due to the particular form of the matrices A, B, and C, equation set (14) can be solved in closed form [14]. For structures with small damping and well separated eigenfrequencies, the gramians can be approximated as diagonal c(o) with the diagonal blocks matrices W c(o) ≈ diag wii c(o)

c(o)

wi 0

w ii

=

wic =

bi bTi , 4ξi ωi

0 c(o) wi

wio =

,

where

ωi2 cTvi cvi + cTdi cdi , 4ξi ωi3

(15) i = 1, . . . , n .

The i-th diagonal element wic (wio ) is a quantitative measure for the controllability (observability) of the i-th state xi = [η˙ i , ωi ηi ]T which represents the i-th mode. By comparing the controllability and observability gramians for collocated actuators and sensors, one can see that the actuator position with high controllability coincides with locations of high relative velocities. The crucial step in the optimization procedure is the definition of appropriate criteria for the effectiveness of actuator locations. The actuator positions,

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i.e. the positions and types of adaptive joints, should be chosen such that the controllability of the relevant modes which significantly contribute to the measurements is maximized. This demand can be formulated as minimization of the criterion np

ψ1 = ψ1max −

(ai wic )

with

wiv

ai =

np

i=1 i=1

,

(16)

wiv

is introduced to ensure ψ1 ≥ 0 . Here, the np modes where the constant included in the criterion are weighted with the scaled diagonal elements wiv of the performance measurement gramian W v , which has been built in a similar manner as W o [15]. The optimization of the above criterion only may result in an actuator configuration in which some modes can be controlled and others not. To avoid this situation, an additional objective ψ1max

np

ψ2 =

1

(17)

2

i=1

(wic )

is defined. The minimization of ψ2 forces actuator configurations which provide an uniform distribution over all modes. The sensor placement problem can be defined in a similar way as for the actuators using the observability gramian W o . Unlike the actuator case, no structural modification is involved for sensor placement. A modal reduction procedure is used to obtain a sensor layout which allows maximal observability of modes corresponding to states of the reduced control system, as well as minimum spillover from modes excluded from the reduced state vector. Although, observation spillover cannot destabilize the system due to its dissipative effects, it can degrade the control performance. 4.1 Optimization Procedure The task of selecting optimal locations for actuators out of a number of available locations is a combinatorial problem which can be stated as a nonlinear multicriteria optimization problem min ψ(q) , ¯ q∈P

ψ(q) = [ψ1 (q), ψ2 (q)]

T

,

(18)

where the objective functions ψj are defined by equations (16) and (17). The vector q = [q1 , . . . , qK ]T of design variables belongs to the admissible set P¯ = q ∈ P K | r(q) = 0, s(q) ≤ 0 which is restricted by the equality and inequality constraints. The set P = {p1 , p2 , . . . , pM } contains all M actuator locations. The locations are indicated using discrete numbers pi in order to provide an unique and complete description of each solution. In the present

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optimization problem, there are two inequality constraints. One is a design restriction which allows only one member per truss node to be connected by an adaptive joint. The other is a lower bound on the lowest eigenfrequency, ω1 ≥ ωmin , which restricts the stiffness reduction of the structure. Furthermore, the eigenvalue problem (4) is an additional equality constraint which must be solved for every design q in order to calculate the objectives ψi (q). This is due to the dependence of the constraint matrix S on the actuator locations. In the present application the following min-max problem Ψ1∗ = min Ψ1 (q) ¯ q∈P

where

Ψ1 (q) = max (wj χj (q)) , j=1,2

(19) χj (q) =

ψj (q) − ψˆj ψˆj

is solved. In addition to the standard min-max formulation, the weighting factors w1 , w2 > 0 ( wj = 1) have been introduced. Each objective is thereby assigned a different priority. The minima ψˆj must be determined for each criterion ψj separately. The scalar optimization problems are solved by a stochastic search algorithm. In the present case, Genetic Algorithms (GA) [16] are used. GA make use of the principles of natural evolution. The algorithm operates on populations which consist of a set of individuals corresponding to solutions in the design space q. The present algorithm uses overlapping populations. In each generation, the genetic operators selection, crossover, and mutation are applied to the population to create a number of new individuals (offspring) which are added to the population. Then the worst individuals are removed in order to return the population to its original size. The quality (fitness) of an individual is defined by the scalar objective function Ψ1 or Ψ2 . The stochastic nature of the genetic operators provides a search in the entire design space, so that the global optimum is more likely to be found. The characteristics of the search can be adjusted by a proper choice of the genetic parameters. The optimization procedure always converges to one of several minimum solutions which have about the same criterion value. The computation time is strongly dominated by the time needed for the solution of the eigenvalue problem. In order to save computation time, only the first 12 modes are considered in the optimization scheme. Higher modes have only little effect on the tip node motion which is of interest in this application. Figure 5 shows actuator and sensor locations. In this research, the relative velocities in the three semi-active joints and the displacement and velocity of the center node at the tip of the mast in the y− and z− directions is measured. For the local feedback controller, only the relative velocities in the three semiactive joints are used. For the cLQG controller, additional measurements are required, namely the displacement and the velocity of the center node at the tip of the mast.

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actuator 3 actuator 1 payload (0.5 kg)

excitation

sensor y

actuator 2 z

x

Fig. 5. Truss structure with actuator and sensor locations

5 Semi-Active Control In the following, two different control approaches are proposed. In the first approach, each semi-active friction joint has its own local feedback controller. The second concept uses onequasilinear global clipped-optimal controller. 5.1 Local Feedback Control Several semi-active control laws for friction dampers have been derived [17]. A bang-bang controller can be designed by inspection of the time derivative of the Lyapunov function representing the system energy. For the LuGre-model defined by (8) and (7), one arrives at the feedback law ⎧ max , μj vj ≥ εj ⎪ ⎨ FNj μj vj max min min FNj = FNj + FNj − FNj (20) , 0 < μj vj < εj εj ⎪ ⎩ min FNj , μj vj ≤ 0

which maximizes the energy dissipation in the adaptive joint by avoiding sticking. A parameter εj has been introduced to avoid chattering (quasi-sticking), an effect which degrades the energy dissipation. The dynamic friction coefficient μj is obtained from the relation μj (t) =

uRj (t − Δt) FNj (t − Δt)

(21)

where 1/Δt is the sampling rate of the controller. It is assumed that in addition to the sliding velocities v, the friction forces uR can be measured, as well. Otherwise they have to be estimated using a simple integrator model as proposed by [18], or the dynamic friction coefficients μj have to be observed using an extended Kalman filter [19, 20]. For an efficient vibration suppression, the control parameters εj of the SISO-controller are optimized [15] with respect to the system energy given by

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E(te ) =

xT x dt .

(22)

0

The objective function (22) depends also on the initial conditions. This is taken into account by considering several initial conditions. 5.2 Clipped-Optimal Control For the second approach, a so called clipped-optimal controller is used [21]. First, an LQG controller [22] is designed assuming an active control system and neglecting the actuator dynamics. The optimal actuator forces ˆ ud = −K x

(23)

are defined by the gain matrix K, which is obtained as a solution of the ˆ , which is estimated by algebraic Riccati equation [22], and the state vector x means of a Kalman filter [23]. The actuator forces ud cannot directly be applied to the structure, but instead have to be generated by the semi-active friction dampers. This is achieved by controlling the normal force in each friction joint using the following local bang-bang controller H e j u Rj , + FNmax − FNmin FNj = FNmin j j j

FNmin ≤ FNj ≤ FNmax . j j

(24)

The controller updates the normal force FNj in the j-th friction joint depending on the difference ej = udj −uRj between the desired actuator force udj and the actual friction force uRj . To avoid unwanted bang-bang behavior due to the Heavyside function H(·), a boundary layer can be introduced as has been done for the SISO controller defined by (20). In the following, the proposed controller is referred to as cLQG controller. The approximation of the desired actuator forces is limited because of the dissipativity constraint and the boundedness of the friction force. This fact shows that the cLQG is sub-optimal. If the measurement of the actual friction forces turns out to be difficult, the Kalman estimator can be extended by an integrator model [18] to estimate the friction forces, as well.

6 Simulation Results The simulation results in this section will illustrate the potential of the proposed semi-active approach for vibration suppression. Three semi-active friction joints are placed at optimal locations. Since it is desired to damp the low-order modes of vibration, all semi-active friction joints are of type A. The numerical model has a dimension of N = 2523 and is reduced to 24 states.

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0.3 E [Nm]

0.2

FNmin

FNmax

0.1 SISO

0

0

0.2

0.4

0.6

0.8 t [s]

1.0

Fig. 6. System energy for case of maximum and minimum normal force as well as for the case using a SISO controller

0.3 E [Nm]

0.2

cLQG

0.1

SISO LQG

0

0

0.2

0.4

0.6

0.8

t [s]

1.0

Fig. 7. System energy for different control concepts

The 24 states represent 9 modes and 3 additional ’correction modes’ (Krylov vectors) generated for each friction joint [24]. Figure 6 compares the decay of the system energy E (defined by (22)) for the structure with passive and semi-active friction joints. A passive joint has a constant normal force which is set to either FNmin or FNmax . For the semi-active joints the control law (20) with optimized control parameter εj is used. It should be noted that during the excitation time period, the normal forces are set to FNmax in order to enable a comparison between the different concepts. As can be seen, the damping can be significantly enhanced by controlling the normal forces in the friction joints. With a constant maximum normal force, good vibration suppression is achieved at the beginning of the response. However, as the vibration amplitude decreases, sticking occurs and no energy is dissipated. In the case of sticking, vibrations are slowly reduced via the small, inherent structural damping.

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The performance of the cLQG controller is shown in Fig. 7 together with the SISO controller. The response of an actively-damped system is also shown (denoted LQG). For this active control case, the optimal actuator forces ud are directly applied by the LQG controller to the structure. The corresponding deflections in the y- and z-directions at the tip of the structure are given in Figs. 8 and 9, respectively.

LQG

4

y [mm]

2

cLQG

SISO

0 0

0.2

0.4

0.6

0.8

t [s]

1.0

Fig. 8. Deflection of the mast tip in y-direction for different control concepts

z 4 [mm]

LQG

2

cLQG

0 -2 -4

SISO

0

0.2

0.4

0.6

0.8

t [s]

1.0

Fig. 9. Deflection of the mast tip in z-direction for different control concepts

When considering the system energy, the cLQG controller performs only slightly better than the three SISO controllers. The advantage of the cLQG controller becomes more obvious when the tip deflections are compared, as shown in Figs. 8 and 9. In order to see how a fully active system would perform, the corresponding plots are included in Figs. 7, 8, and 9 as well. As can be seen, the suppression of the tip deflections provided by the LQG controller (active system) is only slightly better as compared to the cLQG

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controller (semi-active system). However, it is important to point out that the semi-active approach requires only a fraction of the control power of the active system.

7 Conclusions A semi-active approach based on controlled energy dissipation in friction joints has been developed for suppressing vibration in a large space truss structure. A numerical model of the adaptive structure, which includes the nonlinear dynamics of the friction joints is presented. The model is updated by using data from an EMA and measurements on an isolated friction joint. The actuator/sensor placement problem is stated as a nonlinear multicriteria optimization problem with discrete variables which is solved by means of mulitcriteria optimization methods and Genetic Algorithms. Two different control approaches for semi-active damping using friction joints are introduced. Finally, the simulation results for the truss structure with three semi-active friction joints demonstrate the efficiency of the present vibration suppression approach. Acknowledgement. The authors dedicate the paper to Professor Zeitz on the occasion of his 65th birthday and thank him for the continuous and fruitful scientific cooperation. The authors gratefully acknowledge the funding of this research by the DFG (Deutsche Forschungsgemeinschaft) under grant GA209/24 “Adaptive control of mechanical joints in lightweight structures”.

References 1. H. Mitsuma, A. Tsujihata, S. Sekimoto, and F. Kuwao, “Experimental and theoretical study on damped joints in truss structure,” in Proceedings of the 8th International Modal Analysis Conference (IMAC), pp. 8–14, (Kissimmee, Florida), 1990. 2. A. Preumont, Vibration Control of Active Structures, vol. 50 of Solid Mechanics and its Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, 1997. 3. D. C. Karnopp, M. J. Crosby, and R. A. Harwood, “Vibration control using semi-active force generators,” ASME Journal of Engineering for Industry 96(2), pp. 619–626, 1974. 4. S. Hurlebaus, Smart Structures - Fundamentals and Application, Institute A of Mechanics, University of Stuttgart, 2005. lecture notes. 5. L. Gaul, “Aktive Beeinflussung von F¨ ugestellen in mechanischen Konstruktionselementen und Strukturen.” Deutsches Patent- und Markenamt, M¨ arz 2000. Patent-Nr. 19702518. 6. L. Gaul and R. Nitsche, “Friction control for vibration suppression,” Mechanical Systems and Signal Processing 14(2), pp. 139–150, 2000.

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7. R. Nitsche, Semi-Active Control of Friction Damped Systems. PhD thesis, Universit¨ at Stuttgart, 2001. Nr. 907 in Reihe 8, VDI-Fortschrittberichte, VDIVerlag. 8. A. R. Kukreti and H. I. Issa, “Dynamic analysis of nonlinear structures by pseudo-normal mode superposition method,” Computers & Structures 19(4), pp. 643–663, 1984. 9. M. I. Friswell and J. E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, 1995. 10. C. Canudas de Wit, H. Olsson, K. J. ˚ Astr¨ om, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Transactions on Automatic Control 40(3), pp. 419–425, 1995. 11. L. Gaul and R. Nitsche, “The role of friction in mechanical joints,” Applied Mechanics Reviews (ASME) 54(2), pp. 93–106, 2001. 12. B. Armstrong-H´ elouvry, P. Dupont, and C. D. Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica 30(7), pp. 1083–1138, 1994. 13. B. C. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Transactions on Automatic Control AC-26(1), pp. 17–32, 1981. 14. W. K. Gawronski and J.-N. Juang, “Model reduction for flexible structures,” Control and Dynamic Systems: Advances in Theory and Application 36, pp. 143–222, 1990. 15. J. Wirnitzer, A. Kistner, and L. Gaul, “Optimal placement of semi-active joints in large space truss structures,” in SPIE Conference on Smart Structures and Materials 2002: Damping and Isolation, G. S. Agnes, ed., No. 4697, pp. 246– 257, (San Diego, USA), 2002. 16. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, New York, 1989. 17. J. S. Lane, A. A. Ferri, and B. S. Heck, “Vibration control using semi-active friction damping,” in Friction-induced vibration, chatter, squeal, and chaos, ASME 49, pp. 165–171, 1992. 18. R. Hu and P. C. M¨ uller, “Position control of robots by nonlinearity estimation and compensation: Theory and experiments,” Journal of Intelligent and Robotic Systems 20, pp. 195–209, 1997. 19. L. Gaul and R. Nitsche, “Friction control for vibration suppression,” Mechanical Systems and Signal Processing 14(2), pp. 139–150, 2000. 20. L. Gaul and R. Nitsche, “Lyapunov design of damping controllers,” Archive of Applied Mechanics 72, pp. 865–874, 2003. 21. S. J. Dyke, B. Spencer, M. Sain, and J. Carlson, “Modelling and control of magnetorheological dampers for seismic response reduction,” Smart Material and Structures 5, pp. 565–575, 1996. 22. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley, New York, 1972. 23. R. E. Kalman and R. Bucy, “New results in linear filtering and prediction,” ASME Journal of Basic Engineering 83, pp. 95–108, 1961. 24. L. Gaul, H. Albrecht, and J. Wirnitzer, “Semi-active friction damping of large space truss structures,” Journal of Shock and Vibration 11(3–4), pp. 173–186, 2004.

Infinite-Dimensional฀Decoupling฀Control฀of฀the Tip฀Position฀and฀the฀Tip฀Angle฀of฀a฀Composite Piezoelectric฀Beam฀with฀Tip฀Mass Andreas฀Kugi฀and฀Daniel฀Thull Chair฀of฀System฀Theory฀and฀Automatic฀Control Saarland฀University,฀Saarbr¨ ucken,฀Germany. [email protected] Summary.฀ This฀ contribution฀ is฀ devoted฀ to฀ the฀ infinite-dimensional฀ control฀ of฀ a multi-layered฀ piezoelectric฀ cantilever฀ with฀ a฀ tip฀ mass.฀ Thereby,฀the฀ design฀ of฀ the layer฀structure,฀in฀particular฀of฀the฀piezoelectric฀actuator฀and฀sensor฀layers,฀is฀considered฀as฀a฀part฀of฀the฀controller฀design.฀The฀control฀objective฀is฀to฀provide฀two independently฀controllable฀ degrees-of-freedom฀ for฀ the฀tip฀ mass฀ in฀ form฀ of฀ the฀tip position฀and฀the฀tip฀angle.฀The฀control฀concept฀being฀proposed฀consists฀of฀an฀openloop฀flatness-based฀tracking฀controller฀and฀a฀linear฀dynamic฀feedback฀controller฀in order฀to฀asymptotically฀stabilize฀the฀closed-loop฀error฀system.฀A฀rigorous฀stability proof฀based฀on฀the฀C0 -semigroup฀theory฀and฀on฀LaSalle’s฀invariance฀principle฀for infinite-dimensional฀systems฀will฀be฀given.฀Finally,฀simulation฀results฀demonstrate the฀potential฀of฀this฀smart฀actuator.

Keywords:฀Piezoelectric฀structure,฀shape฀design,฀infinite-dimensional฀control,฀flatness,฀asymptotic฀stability.

1 Introduction Smart material systems are usually modeled by partial differential equations. For the purpose of a controller design these infinite-dimensional systems are traditionally approximated by a set of ordinary differential equations. There are many important contributions dealing with the controller design for distributed-parameter systems based on a discrete or modal approach, see, e.g., [2], [21] and the references cited therein. However, it is well known that in certain cases this approach leads to undesirable effects in the closed-loop system, like the so-called actuation and observation spill-over. The problem is that the control input can cause an unintentional excitation of the truncated modes and vice versa the truncated modes may have an undesired contribution to the sensor output. In both cases the performance of the closed-loop system can be degraded, or in the worst case the system can even be destabilized.

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 351–368, 2005. © Springer-Verlag Berlin Heidelberg 2005

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The number of control techniques based on a distributed-parameter approach is rather limited. One of the main reasons for this is the fact that the stability problems of infinite-dimensional systems are much more complicated than of finite-dimensional ones, see, e.g., [14], [29]. In finite-dimensional (nonlinear) systems the stability of an equilibrium solution can be easily investigated by means of the celebrated Lyapunov’s direct method, see, e.g., [8]. Thereby, the equilibrium solution is (asymptotically) stable if there exists a suitable positive definite function, the so-called Lyapunov function, such that its change along a solution of the system is negative (definite) semi-definite. In finite dimensions, the definiteness property corresponds to a local extremum at the equilibrium solution. However, since infinite-dimensional vector spaces are no longer compact, we cannot conclude that a positive definite Lyapunov functional automatically has a local extremum at the equilibrium solution, see, e.g., [14], [29]. Nevertheless, in some important contributions, see, e.g., [13], [14], [29] the notions of asymptotic and exponential stability were extended to infinite-dimensional systems by using the theory of semigroups. In addition, LaSalle’s invariance principle, well established for finite-dimensional systems and important for the stability considerations of mechanical systems, see, e.g., [8], could also be generalized to the infinite-dimensional case, see, e.g., [14]. Furthermore, new results were obtained by generalizing the so-called portHamiltonian systems approach, see, e.g., [30], to the distributed-parameter case, see, e.g., [31]. Clearly, the representation of a distributed-parameter system within the port-Hamiltonian scenario does not help to solve the stability problem itself but it reveals the underlying physical structure, which helps the control engineer to design a physics-based control law. In this context, well known concepts from (nonlinear) finite-dimensional port-Hamiltonian systems, like damping injection and energy shaping, see, e.g., [30], can be generalized to the distributed-parameter case, see, e.g., [15], [28]. For smart material systems we have an additional degree-of-freedom in form of the possibility to design the spatial distribution or shape (e.g. in form of a certain electrode pattern) of the actuators and sensors. If we consider the design of the actuators and sensors as a part of the controller synthesis task, we get totally new possibilities to optimize the performance of the overall closedloop system. The design of this spatial distribution for a specified distributed actuation and sensing is also referred to as the static or dynamic shape control problem, see [7] for a review on this topic. In the classical static shape control problem we seek for a spatial distribution of the piezoelectric actuator such that for a constant spatially distributed external disturbance there exists a constant actuator input (e.g. voltage) which makes the overall distortion of the structure from its original shape to zero, see, e.g., [5], [6]. However, if such a solution exists, it can be shown that this additional degree-of-freedom can be advantageously utilized for designing a controller, in particular also within the passivity-based scenario, see, e.g., [9], [27]. Furthermore, we may construct the actuators and sensors in such a way that the actuator inputs and sensor outputs form pairs of collocated power variables.

Infinite-Dimensional Control of a Composite Piezoelectric Beam

353

The concept of differential flatness as introduced by M. Fliess et al., see, e.g., [3], has proven to be a powerful tool to design tracking controllers. In the last years this approach was successfully extended to certain classes of infinitedimensional systems, see, e.g., [17], [23], [25] and the references cited therein. In particular, in [1], [4], [24], [25] several flatness-based tracking controllers for Timoshenko and Euler-Bernoulli beams were developed. Based on these results we extended in [12] the flatness-based controller design by the ideas of static shape control, passivity-based control and hysteresis compensation in order to render the closed-loop error system exponentially stable. This contribution is devoted to the design of an infinite-dimensional controller for a piezoelectric composite cantilever with tip mass. The objective of the controller design is to asymptotically track the position and the angle of the tip mass at the same time for given desired reference trajectories. Thereby, the design of the layer structure, in particular the shape of the surface electrodes of the actuator and sensor layers, is optimized w.r.t. the control objective. Based on this optimized design of the multi-layered composite cantilever a flatness-based open-loop trajectory controller in combination with a dynamic controller for the error system is developed. The asymptotic stability of the closed-loop system is proven by means of C0 -semigroup theory and LaSalle’s invariance principle for infinite-dimensional systems. Simulation results demonstrate the effectiveness of the proposed design.

2 Mathematical Model Subsequently, we will consider a composite piezoelectric cantilever with a tip mass as depicted in Fig. 1. Let B = {e1 , e2 , e3 } denote the orthonormal basis of a 3-dimensional Euclidean space with coordinates xj , j = 1, 2, 3. The midline of the beam is assumed to coincide with the x1 -axis in the stress-free reference state. Assuming a linear piezoelectric material and a uniaxial state of stress, we get the following constitutive relationships1 , see, e.g., [19] 3 σ 11 = c1111 ε11 − h11 3 D

E3 =

−h11 3 ε11

+ β33 D

3

(1a) (1b)

with the mechanical stress σ = σ ij ∂i ⊗ ∂j , the strain ε = εij dxi ⊗ dxj , the electric field E = Ei dxi and the electric flux density D = Di ∂i dv, dv = dx1 dx2 dx3 . Thereby, ⊗ and stand for the tensor product and the interior product, respectively. Furthermore, c1111 , h11 3 and β33 denote material parameters. They are related to the parameters usually given in a data sheet 1

Subsequently, we will use Einstein’s summation convention by summing over any index in a single term that appears both an upper and a lower index. Furthermore, ` ´k the following abbreviations ∂jk = ∂ k /∂ xj , j = 1, 2, 3, and ∂0k = ∂ k /∂tk for k = 1, 2, . . . and the time t are introduced. Note that for k = 1 we will drop the index k.

354

A. Kugi and D. Thull 

[

`

™ X   W [ 

/ D \ H UV Z LWK H OH F WUR G H V

` 

[

X  W[  [



/ IW Fig. 1. Composite piezoelectric cantilever with tip mass.

for piezoelectric materials, namely the Young’s modulus Y , the piezoelectric coefficient d311 and the permittivity coefficient κ33 , in the form c1111 =

Y κ33 κ33

−

2

(d311 )

Y

, h11 3 =

Y d311 κ33

−

2 (d311 )

Y

, β33 =

1 κ33

2

− (d311 ) Y

.

(2) Since the piezoelectric material is free of charge, the electric flux density D has to satisfy the relation ∂i Di = 0. In the geometric linearized scenario the longitudinal strain ε11 is related to the displacement uj (t, x1 ) of the mid-line of the beam in xj -direction, j = 1, 3, by the Euler-Bernoulli assumption in the form, see, e.g., [32] (3) ε11 = ∂1 u1 − x3 ∂12 u3 .

Hence the potential energy stored in the volume V of a piezoelectric structure is given by the relation Wp =

1 2

2

V

3 3 (c1111 (ε11 )2 − 2h11 )dv 3 ε11 D + β33 D

(4)

Let us assume that the piezoelectric beam of Fig. 1 consists of 2m perfectly bonded layers of laminae with and without piezoelectric properties and is built up symmetrically w.r.t. the mid-line. This symmetry assumption implies that each piezoelectric actuator layer has a symmetric counterpart. We will henceforth refer to these two symmetric actuator layers as an actuator layer couple, cf. Fig. 2. Depending on how the electric voltage U α is applied to the two actuator layers of such a couple, symmetrically or antisymmetrically w.r.t. the mid-line, either only the longitudinal motion u1 (t, x1 ) or only the bending motion u3 (t, x1 ) is actuated, see, e.g., [10]. In this contribution we will restrict ourselves to the bending motion and thus we assume all mA actuator layer couples to be supplied antisymmetrically, see also Fig. 2. Apart from the possibility to choose the sign of the voltage supply piezoelectric structures provide the additional degree-of-freedom of shaping the metallic surface electrodes of the piezoelectric layers as shown in Fig. 2. Thereby, Bα (x1 ), α = 1, . . . , mA , denotes the width of the surface electrode of the

Infinite-Dimensional Control of a Composite Piezoelectric Beam

[



% D

[ 

H OH F WUR G H

D WK S LH ] R H OH F WULF D F WX D WR U OD \ H U [  D

8

355

O

+ [

D WK S LH ] R H OH F WULF D F WX D WR U OD \ H U /

D

X

+ D



%

Fig. 2. Antisymmetrical power supply of an actuator layer couple.

actuator layer couple α as a function of x1 . Since the self-generated electric field due to the direct piezoelectric effect −h11 3 ε11 is insignificant compared with the applied electric field E3 in (1b), the integration along the component E3 of the electric field from one electrode of the α-th piezoelectric actuator layer to the other leads to u Hα l Hα

E3 dx3 = β33 Hαu − Hαl D3 = U α .

(5)

Utilizing this together with (3), (4), we get the following expression for the potential energy stored in the structure Wp =

1 2

L

Λ3 ∂12 u3

0

2

L

dx1 + U α 0

Λα (x1 )∂12 u3 dx1 ,

(6)

with 2m

Λ3 =

c1111 x3 β=1

2

dx2 dx3 , Λα (x1 ) =

Aβ

h11 3 (H u + Hαl )Bα x1 β33 α

.

(7)

Note that in (6) we have already neglected the term β33 (D3 )2 since we assume the voltage sources to be ideal. Furthermore, the material parameters may vary from layer to layer but are supposed to be constant within a single layer. Neglecting the rotational inertia of the beam, we get the kinetic energy Wk of the beam and the tip mass in the form (see Fig. 1) Wk =

μ 2

L 0

∂0 u3

2

1 dx1 + M ∂0 u3 (t, L) 2

2

1 + J ∂0 ∂1 u3 (t, L) 2

2

,

(8)

with M the mass and J the mass moment of inertia w.r.t. the x2 -axis of 2m the tip mass and μ = β=1 Aβ ρβ , where ρβ denotes the mass density and

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A. Kugi and D. Thull

Aβ the cross sectional area of the layer β. Taking into account the kinematic boundary conditions of the cantilever u3 (t, 0) = 0 and ∂1 u3 (t, 0) = 0

(9)

the equations of motion and the dynamic boundary conditions can be derived in a straightforward way by means of Hamilton’s principle μ∂02 u3 + Λ3 ∂14 u3 + ∂12 Λα (x1 )U α = 0,

(10a)

J∂02 ∂1 u3 (t, L) + Λ3 ∂12 u3 (t, L) + Λα (L)U α = 0, M ∂02 u3 (t, L) − Λ3 ∂13 u3 (t, L) − ∂1 Λα (L)U α = 0

.

(10b)

3 Shape Control Let us assume that a certain constant load, e.g. a tip force f3 , is acting on the cantilever. Within the static shape control design we are looking for a shape of the surface electrodes, in particular for the function Bα (x1 ), cf. Fig. 2, and a constant supply voltage U α such that the overall distortion of the piezoelectric structure becomes identically zero, see, e.g., [6]. We will demonstrate this shape design for the cantilever of Fig. 1 with a constant tip force f3 . Considering the kinematic boundary conditions (9), we can express the contribution of f3 to the overall energy of the system in the form L

Wf = f3 u3 (t, L) = f3

∂1 u3 dx1 .

(11)

0

If we want U α of (6) to act in the same way on the cantilever as the tip force f3 the following relation f3 =

L L ∂1 u3 dx1 = U α 0 Λα (x1 )∂12 u3 dx1 = 0 U α Λα (L)∂1 u3 (t, L) − U α Λα (0)∂1 u3 (t, 0)

− Uα

L 0

∂1 Λα (x1 )∂1 u3 dx1

=0

(12) must hold. Thus, Λα (x1 ) has to fulfill the o.d.e.-condition ∂1 Λα (x1 ) = −f3 /U α with Λα (L) = 0. A straightforward calculation shows that a solution is given by x1 f3 Lβ33 . (13) Bα x1 = B 1 − and U α = u l L Bh11 3 (Hα + Hα ) In a similar way it can be shown that the voltage U α of an antisymmetrically supplied actuator layer couple with a rectangular shape of the surface electrodes, i.e. Bα (x1 ) = B, acts in the same way on the structure as a bending moment at the tip of the beam. The sensor capabilities of the piezoelectric layer are based on the direct piezoelectric effect, i.e. the phenomenon that in response to mechanical strain

Infinite-Dimensional Control of a Composite Piezoelectric Beam

357

the piezoelectric material produces dielectric polarization. Analogous to the piezoelectric actuator layer the spatial distribution of the surface electrodes of the sensor layer serves as an additional degree-of-freedom for the design of the structure. Let us assume that the electrodes of the piezoelectric sensor layers are short-circuited. Then the constitutive equation (1b) results in D3 =

h11 3 ε11 . β33

(14)

Utilizing (3), we get the electric charge Quγ by integrating (14) over the effec¯γ (x1 ) as a function of x1 tive surface Aγ of the electrodes with the width B and the length L in the form Quγ =

D3 dx1 dx2 = Aγ

h11 3 β33

L

¯γ x1 B

0

¯ m ∂ 2 u3 dx1 . ∂1 u 1 − H γ 1

(15)

¯ γm = (H ¯ γu + H ¯ γl )/2 denotes the distance from the mid-line of the Here, H beam to the middle of the piezoelectric sensor layer γ. Since the cantilever is built up symmetrically w.r.t. the mid-line each sensor layer has a symmetric counterpart. By taking the difference of the charges measured by the two layers of such a sensor layer couple, we get the following relationship, see, e.g., [10], L

Qγ = 0

Γγ (x1 )∂12 u3 dx1 , Γγ (x1 ) =

h11 3 ¯γ x1 ¯u + H ¯l B H γ γ β33

.

(16)

Now, if we compare (16) with (11) and (12), it is easy to see that a triangular ¯γ (x1 ) = B(1 − x1 /L) of the surface electrode of the sensor layer makes shape B the measured charge Qγ being proportional to the beam deflection at the tip u3 (t, L), namely B h11 3 ¯ γu + H ¯ γl u3 (t, L) . Qγ = H (17) L β33 Analogously, the charge difference Qγ of (16) measured by a sensor layer ¯ γ (x1 ) = B, couple with a rectangular shape of the surface electrodes, i.e. B is proportional to the angle of the beam at the tip position ∂1 u3 (t, L). This result can be generalized in so far as the time derivative of the charge output, i.e. the current, measured by the sensor layer couple due to (16) and the voltage input of the actuator layer couple, see also (12), always form a socalled collocated pair1 if the spatial shape of the sensor and actuator surface ¯γ (x1 ) = Bα (x1 ). Note that this result is valid electrodes is identical, i.e. B independent of the kinematic boundary conditions and thus also holds for other beam configurations, see, e.g., [9], [11]. 1

An input u and an output ∂0 y are called collocated if the expression u∂0 y gives the power flow across the system boundaries. In the language of network theory the pair (u, ∂0 y) is also referred to as an energy port.

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4 Controller Design Let us consider the multi-layered piezoelectric beam of Fig. 1. We want to design the layer structure of the beam and a controller in such a way that we can track both the position of the tip mass u3 (t, L) and the angle of the tip mass about the x2 -axis ∂1 u3 (t, L) for given reference trajectories with asymptotically stable error dynamics. For this purpose we assume that the piezoelectric cantilever consists of mA,1 actuator layer couples with a rectangular shape and a supply voltage U 1 and mA,2 = mA − mA,1 actuator layer couples with a triangular shape and a supply voltage U 2 . Further, the beam consists of mS,1 short-circuited sensor layer couples with a rectangular shape 11 ¯ γu + H ¯ γl ) ¯ 1 with K ¯ 1 = mS,1 B h3 (H measuring the charge Q1 = ∂1 u3 (t, L)K γ=1 β33 and mS,2 = mS − mS,1 sensor layer couples with a triangular shape measuring 11 ¯u + H ¯ l ). In this ¯ 2 with K ¯ 2 = mS −mS,1 B h3 (H the charge Q2 = u3 (t, L)K γ=mS,1 +1 L β33

γ

γ

case the equations of motion and the dynamic boundary conditions of (10) simplify to μ∂02 u3 + Λ3 ∂14 u3 = 0,

(18a)

J∂02 ∂1 u3 (t, L) + Λ3 ∂12 u3 (t, L) + Un1 = 0, M ∂02 u3 (t, L) − Λ3 ∂13 u3 (t, L) + Un2 = 0 ,

(18b)

with the kinematic boundary conditions (9), Λ3 from (7) and the scaled supply mA,1 h11 B β333 (Hαu + Hαl ) and voltages Un1 = U 1 K1 and Un2 = U 2 K2 with K1 = α=1 K2 =

11 mA B h3 u α=mA,1 +1 L β33 (Hα

+ Hαl ).

4.1 Flatness-based Trajectory Planning First of all, let u ˆ3 (s, x1 ) = uˆ3 (x1 ) denote the Laplace transform of u3 (t, x1 ) w.r.t. the time t. Note that subsequently a hat on a system variable always refers to the corresponding Laplace transform. Then the solution of the resulting o.d.e. due to (18a) with zero initial conditions, i.e., u3 (0, x1 ) = 0 and ∂0 u3 (0, x1 ) = 0, is given in the following form, see [1], [24], u ˆ3 x1 = χ ˆ4 Sˆ2 x1 ˆ3 Cˆ2 x1 + χ ˆ2 Sˆ1 x1 + χ ˆ1 Cˆ1 x1 + χ

(19)

with the operator functions Cˆ1 x1 = cosh pˆx1 + cos pˆx1

/2

Cˆ2 x1 = cosh pˆx1 − cos pˆx1 /(2ˆ p2 ) Sˆ1 x1 = sinh pˆx1 + sin pˆx1 /(2ˆ p) Sˆ2 x1 = sinh pˆx1 − sin pˆx1 /(2ˆ p3 )

(20)

√ where pˆ = is(μ/Λ3 )1/4 and i denotes the imaginary unit. The operator functions have the following pleasing properties with respect to the operation of

Infinite-Dimensional Control of a Composite Piezoelectric Beam

359

differentiation, ∂1 Cˆ1 = pˆ4 Sˆ2 , ∂1 Cˆ2 = Sˆ1 , ∂1 Sˆ1 = Cˆ1 and ∂1 Sˆ2 = Cˆ2 , which can be advantageously used for symbolic computation purposes. From the kinematic boundary conditions (9) we can immediately deduce that χ ˆ1 = χ ˆ2 = 0. Since we have two control inputs, Un1 and Un2 , to satisfy the remaining dynamic boundary conditions (18b) we can make use of χ ˆ3 and χ ˆ4 in order to specify the flat outputs, see also [25] for a systematic treatment of a general class of boundary controlled linear distributed-parameter systems. For this, remember that the control objective of this contribution is the design of asymptotically stable tracking controllers for u3 (t, L) and ∂1 u3 (t, L). Due to (19) and (20) we have the following relation u ˆ3 (L) = ∂1 u ˆ3 (L)

Cˆ2 (L) Sˆ2 (L) Sˆ1 (L) Cˆ2 (L)

χ ˆ3 χ ˆ4

.

(21)

R

If we choose χ ˆ3 and χ ˆ4 in the form χ ˆ3 χ ˆ4

= adj (R)

yˆ1 yˆ2

=

Cˆ2 (L) yˆ1 − Sˆ2 (L) yˆ2 , −Sˆ1 (L) yˆ1 + Cˆ2 (L) yˆ2

(22)

with adj(R) as the adjoint matrix of R, then the deflection of the beam u ˆ3 (x1 ) can be written as follows uˆ3 x1 = Cˆ2 (L) Cˆ2 x1 − Sˆ1 (L) Sˆ2 x1

Cˆ2 (L) Sˆ2 x1 − Sˆ2 (L) Cˆ2 x1

yˆ1 + yˆ2 .

(23)

ˆ 2 can be directly calculated from (18b) in the ˆ 1 and U The control inputs U n n Laplacian domain in the form ˆn1 = χ U ˆ3

Λ3 ˆ4 Sˆ1 μ Jp

ˆ2 = χ U ˆ3 n

Λ3 ˆ4 Cˆ2 μ Mp

ˆ4 (L) − Λ3 Cˆ1 (L) + χ

Λ3 ˆ4 Cˆ2 μ Jp

(L) + Λ3 pˆ4 Sˆ2 (L) + χ ˆ4

(L) − Λ3 Sˆ1 (L)

Λ3 ˆ4 Sˆ2 μ Mp

(L) + Λ3 Cˆ1 (L) ,

(24) with χ ˆ3 and χ ˆ4 from (22). Obviously, all system variables can be expressed in terms of yˆ1 and yˆ2 and thus the lumped variables y1 (t) and y2 (t) may be used as so-called flat outputs, see also [4] for a first treatment of a flatness-based controller design for the Euler-Bernoulli beam. Up to the knowledge of the authors no physical interpretation does exist for the flat outputs in context with piezoelectric beams. However, according to (21) and (22) the relationship ˆ3 (L) and the flat output yˆ1 or yˆ2 , respectively, between the output uˆ3 (L) or ∂1 u is decoupled and takes the form u ˆ3 (L) = det (R) yˆ1 = Cˆ22 (L) − Sˆ1 (L) Sˆ2 (L) yˆ1

∂1 u ˆ3 (L) = det (R) yˆ2 = Cˆ22 (L) − Sˆ1 (L) Sˆ2 (L) yˆ2 .

(25)

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A. Kugi and D. Thull

A closer inspection of the expression det(R) shows that in the frequency domain of practical interest (25) can be fairly good approximated by the relations u ˆ3 (L) ≈

L4 yˆ1 12

and ∂1 u ˆ3 (L) ≈

L4 yˆ2 . 12

(26)

Remark 1. Clearly, instead of using u3 (t, L) and ∂1 u3 (t, L) as the desired outputs we may take two arbitrary other variables which ensure that the matrix R of (21) is regular. The design of the flatness-based controller proceeds in the same way. In order to transform the bending deflection u ˆ3 (x1 ) of (23) and the control 1 2 ˆ ˆ inputs Un and Un from (24) to the time domain we take advantage of the power series representation of the operator functions (20), see also [24], given by Cˆ1 x1 = Sˆ1 x1 =

∞

”n “ 4n − Λμ ( x1 ) 3

n=0 “ ∞ −

μ Λ3

n=0

s 4n! ”n (x1 )4n+1

2n

, Cˆ2 x1 =

s2n , Sˆ2 x1 =

(4n+1)!

∞

”n “ 4n+2 − Λμ ( x1 ) 3

n=0 “ ∞ −

n=0

(4n+2)! ”n μ (x1 )4n+3 Λ 3

(4n+3)!

s2n ,

s2n .

(27) Before we can apply (27) to (23) or (24), respectively, we must eliminate the products of operator functions by using the addition theorems of trigonometric functions. For instance, in (23) the following substitutions Sˆ1 (L) Sˆ2 x1 =

1 4pˆ4

Cˆ2 (L) Cˆ2 x1 =

1 4pˆ4

Sˆ2 (L) Cˆ2 x1 =

1 4pˆ4

Cˆ1 L + x1 − Cˆ1 L − x1 + 2 Im Cˆ1 L − ix1

Cˆ1 L + x1 + Cˆ1 L − x1 + 2 Re Cˆ1 L − ix1 Sˆ1 L + x1 + Sˆ1 L − x1 − 2 Re Sˆ1 L − ix1

(28) have to be performed. Note that on the right hand side of (28) the denominator term pˆ4 cancels out in all expressions. In a similar fashion we can transform ˆn2 of (24) into a power series representation. Since ˆn1 and U the control inputs U the calculations are rather lengthy but straightforward, in particular when using a computer-algebra program, we will just give the result in the time domain by replacing the operator sk with dk /dtk , k = 1, 2, . . ., (assuming zero initial conditions) 2

Un1 (t) = Λ3 L2 y1 (t) + Λ3 −Λ3

∞ n=1

μn Λn 3

2

∞ n=1

2n−1 4n 22n+1 L4n+3 + Jμ 2 (4n)!L (4n+3)!

Un2 (t) = −Λ3 L2 y2 (t) − Λ3 +Λ3

∞ k=1

μn Λn 3

μn 4n L4n+2 d2n 2n y1 (t) Λn 3 (4n+2)! dt

∞ n=1

3

− Λ3 L3 y2 (t) d2n y (t) dt2n 2

μn 4n L4n+2 d2n 2n y2 (t) Λn 3 (4n+2)! dt

4n L4n+1 M 22n−1 L4n (4n+1)! + μ (4n)!

(29a)

+ Λ3 Ly1 (t)

d2n y (t) dt2n 1

(29b) .

Infinite-Dimensional Control of a Composite Piezoelectric Beam

361

Thus, we have shown that the control inputs Un1 (t), Un2 (t) and the bending deflection u3 (t, x1 ) can be parametrized by the flat outputs y1 (t), y2 (t) and their time derivatives. For solving the trajectory planning problem the desired flat outputs y1d (t), y2d (t) are specified and from this the associated control 1 2 inputs Und (t), Und (t), the beam deflection u3d (t, x1 ) and the plant outputs 3 3 ud (t, L) and ∂1 ud (t, L) are calculated. Clearly, in order to be able to evaluate the infinite sums, cf. (29), y1d (t) and y2d (t) must be smooth functions with y1d (t) = 0, y2d (t) = 0 for t < 0. Furthermore, the assumption on the zero initial conditions requires that the following relations dk /dtk yjd (0) = 0, j = 1, 2, hold for all k > 0. If yjd (t) would be an analytic function in t = 0 the conditions above would imply that yjd (t) ≡ 0 and thus a change of the operating point would be impossible, see also [16], [25]. To overcome this problem it is shown in [25] that if yjd (t), j = 1, 2, is chosen as a Gevrey function of class 1 < α < 2 then the derivatives are all bounded and the convergence of all series involved in the calculation is guaranteed. 4.2 Asymptotic Stabilization of the Error System Now the desired flat outputs y1d (t) and y2d (t) are specified such that their stationary values correspond to the stationary values of the position and the angle about the x2 -axis of the tip mass, u3d (t, L) and ∂1 u3d (t, L), respectively, cf. (26). Due to the flatness property we may calculate all system variables as functions of the flat outputs and their time derivatives. Clearly, the flatness1 2 (t), Und (t)) is a solution based trajectory planning implies that (u3d (t, x1 ), Und of (9), (18). Furthermore, we have assumed at the beginning of this Section that the sensor layers measure charges which are proportional to u3 (t, L) and ∂1 u3 (t, L), respectively. Introducing the deflection error ue (t, x1 ) = u3 (t, x1 )− j j (t) = Unj (t) − Und (t), j = 1, 2, u3d (t, x1 ) and the additional control inputs Une we get the following error system μ∂02 ue + Λ3 ∂14 ue = 0,

(30a)

1 J∂02 ∂1 ue (t, L) + Λ3 ∂12 ue (t, L) + Une = 0, 2 3 2 M ∂0 ue (t, L) − Λ3 ∂1 ue (t, L) + Une = 0 ,

(30b)

¨ Morg¨ with ue (t, 0) = ∂1 ue (t, 0) = 0. Inspired by the work of O. ul, see [18], we propose the control law z˙j j Une

¯ je = Aj zj + bj ∂0 Q ¯ je ¯ je + cT zj + dj ∂0 Q = kj Q j

,

¯ 1e = ∂1 ue (t, L) Q ¯ 2e = ue (t, L) Q

(31)

with the controller state zj ∈ Rn , j = 1, 2, to asymptotically stabilize the error system (30). Thereby, the system (Aj , bj , cTj , dj ) is the minimal realization of a strictly positive real (SPR) transfer function Gj (s) = cTj (sI − Aj )−1 bj + dj which has the properties that Aj is a Hurwitz matrix and Re(Gj (ω)) ≥ dj > δj > 0 for all ω ∈ R and j = 1, 2. Furthermore, the parameters kj are

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considered positive. Now the Kalman-Yakubovich lemma, see, e.g., [8], [14], implies that there exists a positive definite matrix Pj ∈ Rn×n , a vector qj ∈ Rn and a positive scalar εj such that Pj Aj + ATj Pj = −qj qjT − εj Pj

2 (dj − δj ) .

, Pj bj = cj − qj

(32)

Remark 2. Note that the controller (30) requires the measurements of the position of the tip mass u3 (t, L) as well as of the angle of the tip mass ∂1 u3 (t, L) and the corresponding first time derivatives. As shown in Section 3 these quantities can be obtained by measuring the charge of short-circuited piezoelectric sensor layers with an appropriate shape of the surface electrodes. Remark 3. In [18] the controller contains an additional part to suppress periodic disturbances with known frequency at the system output in the stationary case. Of course, the control law (31) can also be extended in this way. Next we will sketch the proof of the asymptotic stability of the closed-loop error system (30), (31). For this purpose let us first define the function space H = {z = (ue , ve , z1 , z2 , ξ, ψ) : ue ∈ H20 (0, L), ve ∈ L2 (0, L), z1 , z2 ∈ Rn , ξ, ψ ∈ R}. Thereby, a function f : [0, L] → R is said to belong to L2 (0, L) if f L is measurable and if the (Lebesque) integral 0 |f (x)|2 dx exists. Furthermore, p the spaces Hp and H0 are defined in the form Hp (0, L) = {f : ∂1k f ∈ L2 (0, L) for k = 0, . . . , p} and Hp0 (0, L) = {f : f ∈ Hp (0, L) and f (0) = 0, ∂1 f (0) = 0}, see, e.g., [22] for a comprehensive introduction to functional analysis. The function space H is endowed with an inner product for z, z˘ ∈ H z, z˘

H

1 2 k1

=

1 2

L 0

Λ3 ∂12 ue

L 1 1 ˘ ˘ v v˘ dx1 + 2J ξ ξ + 2M ψ ψ+ 0 e e 1 1 T 1 T ˘e (L) + 2 z1 P1 z˘1 + 2 z2 P2 z˘2 2 k2 ue (L) u

∂12 u ˘e dx1 +

(∂1 ue (L)) (∂1 u ˘e (L)) +

μ 2

(33) and thus becomes a Hilbert space. It can be easily verified that for z = (ue , ∂0 ue , z1 , z2 , J∂0 ∂1 ue (L), M ∂0 ue (L)) the norm induced by the inner product (33) reads as z

2 H L 0

= z, z

2

H

2

= 21 k1 (∂1 ue (L)) + 21 k2 (ue (L)) + 12 z1T P1 z1 + 21 z2T P2 z2 + 2

2

2

2

dx1 + 12 J (∂0 ∂1 ue (L)) + 21 M (∂0 ue (L)) . (34) Note that the second line of (34) corresponds to the expressions of the kinetic and potential energy stored in the piezoelectric structure due to (6) and (8) of the free system, i.e. for U α = 0. The closed-loop error system (30), (31) can now be written as a first order evolution equation z˙ = Az, where z = (ue , ∂0 ue , z1 , z2 , J∂0 ∂1 ue (L), M ∂0 ue (L)) ∈ H and the linear operator A : D(A) ⊂ H → H is given by 1 2

Λ3 ∂12 ue

+ μ (∂0 ue )

Infinite-Dimensional Control of a Composite Piezoelectric Beam

⎡ ⎤ ⎡ ve ue Λ3 4 ⎢ ve ⎥ ⎢ − μ ∂1 u e ⎢ ⎥ ⎢ ⎢ z1 ⎥ ⎢ A z + b1 ∂1 ve (L) 1 1 ⎥ ⎢ A⎢ ⎢ z2 ⎥ = ⎢ A z + b2 ve (L) 2 2 ⎢ ⎥ ⎢ ⎣ ξ ⎦ ⎣−Λ3 ∂ 2 ue (L) − k1 ∂1 ue (L) + cT z1 + d1 ∂1 ve (L) 1 1 ψ Λ3 ∂13 ue (L) − k2 ue (L) + cT2 z2 + d2 ve (L)

363

⎤

⎥ ⎥ ⎥ ⎥ , ⎥ ⎥ ⎦

(35)

with domain D(A) = {z = (ue , ve , z1 , z2 , ξ, ψ) : ue ∈ H40 (0, L),ve ∈ H20 (0, L), z1 ,z2 ∈ Rn ,ξ = J∂1 ve (L), ψ = M ve (L)}. Following the lines of [18] or [14], the proof of asymptotic stability of (35) will be sketched in two steps: At first, we will show that the operator A generates a C0 -semigroup of contractions by utilizing the celebrated L¨ umerPhillips theorem, see, e.g., [13], [14] and in particular [20]. For this we have to show that the operator A is dissipative and there is a λ0 > 0 such that the range of (λ0 I − A) is H with I as the identity operator. A linear operatorA is said to be dissipative if < z, Az >H ≤ 0 for all z ∈ D (A). Performing several times integration by parts, utilizing (32) and completing the squares leads to the following result < z, Az >H = − 41 ε1 z1T P1 z1 − − 41 ε2 z2T P2 z2 −

1 4 1 4

z1T q1 − δ¯1 ∂1 ve (L) − 21 δ1 (∂1 ve (L)) 2 2 z T q2 − δ¯2 ve (L) − 1 δ2 ve (L) ≤ 0, 2

2

2

2

(36) with δ¯j = 2(dj − δj ) > 0, j = 1, 2. Instead of proving that the range of (λ0 I − A) is H for λ0 > 0 it is sufficient to show that A−1 exists and is bounded, see, e.g., [13], [14]. To show the existence of A−1 we assume a given ˘ ψ) ˘ ∈ H and solve the equation Az = z˘ for z ∈ D(A). z˘ = (˘ ue , v˘e , z˘1 , z˘2 , ξ, Since A1 and A2 were assumed to be Hurwitz matrices they are invertible and thus we can immediately deduce that z1 − b1 ∂1 u ˘e (L)) , z2 = A−1 z2 − b2 u˘e (L)) , ve = u˘e , z1 = A−1 1 (˘ 2 (˘ ξ = J∂1 u ˘e (L) , ψ = M u ˘e (L) .

(37)

Furthermore, from the equation −(Λ3 /μ)∂14 ue = v˘e we can extract the following relation ue x1 = − Λμ3

x1 0

σ3 0

+ 12 ∂13 ue (L)

σ2 L 1 3

σ1 L 1 3

x

v˘e (σ) dσdσ 1 dσ 2 dσ 3 + 12 ∂12 ue (L) x1

− x1

2

L

2

.

(38) Thus, if we use (38) to calculate ue (L) and ∂1 ue (L) then we can uniquely determine ∂12 ue (L) and ∂13 ue (L) as functions of z˘ by means of (37) and the ˘ We will not go into the details of this derivation here relations for ξ˘ and ψ. since the expressions become rather unwieldy. However, having ∂12 ue (L) and ∂13 ue (L) as functions of z˘ at hand, we can also express ue in terms of z˘ due to

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(38) and this proves the existence of A−1 . Furthermore, it can be easily seen that if z˘ H is bounded then this is also true for z H . Thus, the operator A−1 exists and is bounded and hence A generates a C0 -semigroup of contractions by the L¨ umer-Phillips theorem, see, [14] for many other examples. The second part of the proof is based on LaSalle’s invariance principle generalized for infinite-dimensional systems, see, e.g., [14]. Let us con2 sider the “energy-like” Lyapunov function He = z H from (34) and its continuously non-increasing change along the closed-loop error system (35), H˙ e =< z, Az >H ≤ 0, cf. (36). Then LaSalle’s invariance principle says that all solutions of (35) tend to the largest invariant subset of Z = {z ∈ H|H˙ e = 0} provided that the solution trajectories are precompact in H for t ≥ 0. As it is shown in [14], in particular Theorem 3.65, since A generates a C0 -semigroup of contractions it suffices to prove that 0 is an element of the range of A and (λ0 I − A)−1 is compact for some λ0 > 0. We have already seen that the operator A−1 exists and is bounded. Thus, due to the Sobolev Embedding theorem, see, e.g., [22], the operator A−1 is also compact in H. Clearly, this already brings about that 0 is an element of the range of A. Furthermore, in [18] and the reference cited therein it is shown that the compactness of A−1 also implies that (λ0 I − A)−1 is a compact operator. Therefore, we may apply LaSalle’s invariance principle and we may conclude that all trajectories converge to the largest invariant subset where H˙ e =< z, Az >H = 0, i.e. the set Z = {z ∈ H|z1 = 0, z2 = 0, ve (L) = ∂1 ve (L) = 0}, cf. (36). In order to find the largest invariant subset in Z we investigate the solutions of the closed-loop error system (35) which are confined to Z and thus must satisfy z˙ = Ar and

ve ue = − Λμ3 ∂14 ue ve

ue (0) = 0, ∂1 ue (0) = 0, ve (L) = 0, ∂1 ve (L) = 0 −Λ3 ∂12 ue (L) − k1 ∂1 ue (L) = 0, Λ3 ∂13 ue (L) − k2 ue (L) = 0 .

(39)

(40)

Further, it can be easily shown that the only possible solution of the system (39), (40) is given by ue = ve = 0. This also proves the asymptotic stability of the closed-loop error system (35). However, the question if the closed-loop system is also exponentially stable has not been addressed in this contribution. But we are confident that the results as presented in [18] can also be applied to our case.

5 Simulation Results Let us once again consider the multi-layered piezoelectric cantilever with tip mass as depicted in Fig. 1. Suppose that the beam has a layer structure as discussed at the beginning of Section 4 and thus can be described by the mathematical model (9) and (18). The beam has a length L = 40 mm, a width

Infinite-Dimensional Control of a Composite Piezoelectric Beam

1

0.5

0.5

−1

0 −0.5 −1

−1.5

−1.5

−2

−2 5

10

15

20 25 x1 in mm

30

35

−2.5 0

40

5

10

15

20 25 x1 in mm

30

35

40

0.5

1

0.5

0.8

0.4

0.8

0.4

0.3

0.6 0.4 0.2

u3(t,L)

∂1u3

u3d(t,L)

∂1u3d(t,L)

0.1

0.005

0.01 t in s

0.015

0.3

0.6 0.4 0.2

0

0 −0.2 0

0.2

u3(t,L) in mm

1

u3(t,L)

∂1u3

0.2

u3d(t,L)

∂1u3d(t,L)

0.1 0

0

−0.1 0.02

−0.2 0

40

∂1u3(t,L) in rad

u3(x1) in mm

0 −0.5

−2.5 0

u3(t,L) in mm

1.5

1

∂1u3(t,L) in rad

u3(x1) in mm

1.5

365

0.005

0.01 t in s

0.015

−0.1 0.02

150

30

100

10

supply voltage in V

supply voltage in V

20

0 −10 −20 −30

50 0

U1 U2

−50 −100

−40 −150

0.02

−200 0

U U2

−50 −60 0

1

0.005

0.01 t in s

0.015

0.005

0.01 t in s

0.015

0.02

Fig. 3. Simulation results for two different scenarios: on the left hand side, the angle at the tip position is held constant to zero and the tip position has to follow a reference trajectory. On the right hand side, the tip position is held constant to zero and the angle at the tip has to follow a reference trajectory.

B = 7.2 mm and a height H = 1.32 mm. Furthermore, the mass of the tip mass takes the value M = 62.8 · 10−3 kg and the mass moment of inertia about the x2 -axis is given by J = 1.05 · 10−2 kg m2 . The parameters in the mathematical model (18) are μ = 0.66 · 10−1 kg m−1 , Λ3 = 0.12 N m2 and the scaling factors for the voltages K1 = 0.26 · 10−1 N V−1 and K2 = −1.36 N m V−1 . For the controller (31) a simple PD-control law without dynamics, i.e. z1 = 0 and z2 = 0, is chosen with the parameters k1 = 103 , d1 = 2, k2 = 106 and d2 = 400.

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A. Kugi and D. Thull 1.5 1

0.5 0.4

0

∂1u3(t,L) in rad

u3(x1) in mm

0.5

−0.5 −1

0.2 0.1

−1.5

∂1u3

0

−2 −2.5 0

0.3

5

10

15

20 25 x1 in mm

30

35

−0.1 0

40

∂1u3d(t,L) 0.005

0.01 t in s

0.015

0.02

100 1

50 supply voltage in V

u3(t,L) in mm

0.8 0.6 0.4 u3(t,L) u3d(t,L)

0.2

U1 U2 −50

−100

0 −0.2 0

0

0.005

0.01 t in s

0.015

0.02

−150 0

0.005

0.01 t in s

0.015

0.02

Fig. 4. Simulation results for the case when the position and the angle of the tip mass are tracked simultaneously.

For simulation purposes in Matlab/Simulink a finite-dimensional modal approximation of the mathematical model (9), (18) was derived taking into account the first 5 modes. Further investigations show that the higher modes only have negligible contribution to the dynamic behavior. In all simulations the performance of both the tracking and the disturbance rejection behavior of the proposed control concept are examined. In all cases the disturbance is given by an impulse of the tip force f3 (t) occurring 10 ms after the start of the simulation with an amplitude of 10 N and a duration of 1 ms, see also Fig. 1. All reference trajectories are chosen as Gevrey functions of class 1.1, see, e.g., [26] for a mathematical description and a recursive calculation procedure for the derivatives. In Fig. 3 two different cases are presented: on the left hand side, the angle at the tip position ∂1 u3d (t, L) is held constant to zero and the tip position u3d (t, L) has to follow a reference trajectory with a rising time of 3 ms and a final value of 1 mm; on the right hand side, the tip position u3d (t, L) is held constant to zero and the angle ∂1 u3d (t, L) has to follow a reference trajectory with a rising time of 3 ms and a final value of 20◦ . The first picture on each side in Fig. 3 shows the shape of the beam for every 0.2 ms when the beam is moving from its initial to its final position. The reference trajectories u3d (t, L) and ∂1 u3d (t, L) as well as the actual trajectories u3 (t, L) and ∂1 u3 (t, L)

Infinite-Dimensional Control of a Composite Piezoelectric Beam

367

are depicted in the second two pictures of Fig. 3. The last two pictures present the control voltages U 1 and U 2 . It can be seen that we succeeded in obtaining a nearly completely decoupled behavior of the dynamics of the tip position and the tip angle. Furthermore, the tracking error is nearly zero all the times and the impulse disturbance can be suppressed in an excellent way. Fig. 4 shows the scenario when the tip position and the tip angle have to track reference trajectories of the tip position and the tip angle at the same time with different rising times. Thereby, u3d (t, L) has a rising time of 3 ms and a final value of 1 mm and ∂1 u3d (t, L) a rising time of 6 ms and a final value of 20◦ . Again the performance of the closed-loop system in both the tracking behavior and the disturbance rejection is excellent and corresponds to the objectives of the design.

References 1. Y. Aoustin, M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Theory and practice in the motion planning and control of a flexible robot arm using Mikusi´ nski operators. In Proc. of the 5th IFAC Symposium on Robot Control, pp. 287–293, Nantes (France), 1997. 2. R.L. Clark, W.R. Saunders, and G.P. Gibbs. Adaptive Structures, Dynamics and Control. John Wiley & Sons, New York, 1998. 3. M. Fliess M, J. L´evine, P. Martin, and P. Rouchon. Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples. Int. J. Control, 61:1327– 1361, 1995. 4. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Syst`emes lin´eaires sur les op´erateurs de Mikusi´ nski et commande d’une poutre flexible. In Proc. of the ESAIM, volume 2, pp. 183–193, 1997. 5. J.E. Hubbard and S.E. Burke. Distributed Transducer Design for Intelligent Structural Components. In H.S. Tzou and G.L. Anderson (eds), Intelligent Structural Systems. Kluwer Academic Publishers, Dordrecht, pp. 305-324, 1992. 6. H. Irschik, K. Hagenauer, and F. Ziegler. An Exact Solution for Structural Shape Control by Piezoelectric Actuation. In: U. Gabbert (ed), Smart Mechanical Systems-Adaptronics, Fortschrittberichte VDI, Reihe 11, Nr. 244, VDI-Verlag, D¨ usseldorf, pp. 93–98, 1997. 7. H. Irschik. A Review on Static and Dynamic Shape Control of Structures by Piezoelectric Actuation. Engineering Structures, 24:5–11, 2002. 8. H.K. Khalil. Nonlinear Systems. Prentice-Hall, Englewood Cliffs, New York, 3rd edition, 2003. 9. A. Kugi. Non-linear Control Based on Physical Models. LNCIS 260, Springer, London, 2001. 10. A. Kugi and K. Schlacher. Control of Piezoelectric Smart Structures. In Preprints of the 3rd -Workshop “Advances in Automotive Control”, volume 1, pp. 215-220, Karlsruhe, Germany, 2001. 11. A. Kugi and K. Schlacher. Passivit¨ atsbasierte Regelung piezoelektrischer Strukturen. Automatisierungstechnik, 50:422–431, 2002. 12. A. Kugi, D. Thull, and K. Kuhnen. An Infinite-dimensional Control Concept for Piezobender with Complex Hysteresis. Submitted to J. of Structural Control, 2005.

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13. Z. Liu and S. Zheng. Semigroups Associated with Dissipative Systems. Chapman & Hall/CRC, Boca Raton, London, 1999. ¨ Morg¨ 14. Z.-H. Luo, B.-Z. Guo, and O. ul. Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer, London, 1999. 15. A. Macchelli. Port Hamiltonian systems: A Unified Approach for Modeling and Control Finite and Infinite Dimensional Physical Systems. PhD Thesis, University of Bologna - DEIS, 2003. www-lar.deis.unibo.it/woda/data/deis-larpublications/e499.Document.pdf 16. P. Martin, R.M. Murray, and P. Rouchon. Flat Systems: Open Problems, Infinite Dimensional Extension, Symmetries and Catalog. In A. Ba˜ nos, F. LamnabhiLagarrigue, and J. Montoya (eds.), Advances in the Control of Nonlinear Systems, pp. 33–57, LNCIS 264, Springer, London, 2000. 17. T. Meurer and M. Zeitz. A Modal Approach to Flatness-based Control of Flexible Structures. Proc. Appl. Math. Mech., volume 4, issue 1, pp. 133–134, 2004. ¨ Morg¨ 18. O. ul. Stabilization and Disturbance Rejection for the Beam Equation. IEEE Trans. Automat. Cont., 46:1913–1918, 2001. 19. W. Nowacki. Dynamic Problems of Thermoelasticity. Noordhoff International Publishing, PWN-Polish Scientific Publ., Warszawa, 1975. 20. A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1992. 21. A. Preumont. Vibration Control of Active Structures. Kluwer Academic, Dordrecht, 2nd edition, 2002. 22. B.D. Reddy. Introductory Functional Analysis. Springer, New York, 1998. 23. P. Rouchon. Motion Planning, Equivalence, Infinite Dimensional Systems. Int. J. Appl. Math. Comput. Sci., 11:165–188, 2001. 24. J. Rudolph and F. Woittennek. Flachheitsbasierte Randsteuerung von elastischen Balken mit Piezoaktuatoren. Automatisierungstechnik, 50:412–421, 2002. 25. J. Rudolph. Flatness Based Control of Distributed Parameter Systems. Shaker Verlag, Aachen, 2003. 26. J. Rudolph, J. Winkler, and F. Woittennek. Flatness Based Control of Distributed Parameter Systems: Examples and Computer Exercises from Various Technological Domains. Shaker Verlag, Aachen, 2003. 27. K. Schlacher and A. Kugi. Control of Mechanical Structures by Piezoelectric Actuators and Sensors. In D. Aeyels, F. Lamnabhi-Lagarrique, and A. van der Schaft (eds), Stability and Stabilization of Nonlinear Systems. LNCIS 246, Springer, London, pp. 275–291, 1999. 28. K. Schlacher and K. Zehetleitner. Active Control of Smart Structures using Port Controlled Hamiltonian Systems. In K. Watanabe and F. Ziegler (eds), Proc. IUTAM Symposium on Dynamics of Advanced Materials and Smart Structures. Kluwer Academic Publishers, Dordrecht, pp. 357–366, 2003. 29. G.E. Swater. Introduction to Hamiltonian Fluid Dynamics and Stability Theory. Chapman & Hall/CRC, Boca Raton, 2000. 30. A.J. van der Schaft. L2 -Gain and Passivity Techniques in Nonlinear Control. Springer, London, 2000. 31. A.J. van der Schaft and B.M. Maschke. Hamiltonian Formulation of Distributedparameter Systems with Boundary Energy Flow. J. of Geometry and Physics, 42:166–194, 2002. 32. F. Ziegler. Mechanics of Solids and Fluids. Springer, New York, 2nd edition, 1995.

Nonlinear฀Flow฀Control฀Based฀on฀a฀Low Dimensional฀Model฀of฀Fluid฀Flow Rudibert฀King1 ,฀Meline฀Seibold1 ,฀Oliver฀Lehmann1,2 ,฀Bernd.฀R.฀Noack2, Marek฀Morzy´ nski3 ,฀and฀Gilead฀Tadmor4 1฀

Measurement฀and฀Control฀Group,฀Berlin฀University฀of฀Technology,฀Germany. {rudibert.king,meline.seibold}@tu-berlin.de 2฀ Hermann-F¨ ottinger-Institut,฀Berlin฀University฀of฀Technology,฀Germany. [email protected],[email protected] 3฀ Institute฀of฀Combustion฀Engines฀and฀Transportation,฀Pozna´ n฀University฀of Technology,฀Poland. [email protected] 4฀ Department฀of฀Electrical฀and฀Computer฀Engineering,฀Northeastern฀University, Boston,฀USA. [email protected] Summary.฀ Nonlinear฀ control฀ design฀ is฀ shown฀ to฀be฀a฀ critical฀ enabler฀for฀ robust model-based฀suppression฀of฀a฀flow฀instability.฀The฀onset฀of฀oscillatory฀vortex฀shedding฀ is฀ chosen฀ as฀ a฀ well฀ investigated฀ benchmark฀ problem฀ of฀ flow฀ control.฀ A฀ lowdimensional฀Galerkin฀model฀using฀a฀Karhunen-Lo`eve฀decomposition฀of฀the฀flow฀field is฀adopted฀from฀earlier฀ studies฀of฀the฀authors฀as฀a฀control-oriented฀fluid฀flow฀representation.฀Several฀strategies฀of฀nonlinear฀controller฀design฀are฀employed,฀both,฀to the฀Galerkin฀model฀and฀to฀the฀flow฀via฀a฀direct฀numerical฀simulation฀of฀the฀NavierStokes฀equations฀(NSE).฀The฀aim฀is฀to฀find฀methods฀which฀respect฀the฀validity฀of low฀order฀models.฀Examples฀are฀formal฀methods฀such฀as฀input-output฀linearization, Lyapunov-based,฀backstepping฀controllers฀etc.,฀and฀physically฀motivated฀controllers. Whereas฀the฀first฀test-bed฀is฀easily฀mastered฀by฀the฀formal฀methods,฀the฀application to฀the฀NSE฀is฀more฀critical,฀due฀to฀robustness฀issues.

Keywords: Flow฀control,฀Navier-Stokes฀equation,฀nonlinear฀control,฀Galerkin model,฀cylinder฀wake.

1 Introduction The manipulation of fluid flow has a long tradition in the engineering art of improving flow machines and transport vehicles. For instance, the drag of a truck, a plane, or a ship is reduced by aerodynamic design, i.e. shaping the geometry of the obstacle. The lift of an airplane is increased by airfoil design, and undesirable lift on a car is decreased by spoilers. Mixing enhancement is

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 369–386, 2005. © Springer-Verlag Berlin Heidelberg 2005

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a vital element of combustor design. For instance, flame-holders increase the mixing of fuel-enriched air with ignited hot air from the dead-water region behind the obstacle. Thus, the mean residence of the fluid is increased to allow for complete ignition. Many more examples of those flow control applications with passive means, i.e. without free actuation power, could be enumerated. However, passive means are typically optimized for a narrow range of operating conditions. Active methods can significantly stretch the dynamic range of flow control by exploiting additional degrees of freedom. The majority of the work done so far in the context of active flow control concentrates on open loop concepts, see [7] for a review. More and more, the benefits of closing the loop by flow sensing are realized — a trend which is enforced by the increasing affordability and reliability of actuators and sensors. This trend is reflected in a rapidly increasing number of studies devoted to the closed-loop flow control in the last couple of years. Both, for open- and closed-loop control, the flow may be affected by blowing, suction, acoustic forcing, or by magneto-hydrodynamic forces, cf. [8]. The synthesis of closed-loop flow controllers may be based on the evolution equations of incompressible viscous fluid flow. These equations consist of the continuity equation, reflecting mass conservation, ∇·U= 0

(1)

,

and the Navier-Stokes equation (NSE) as momentum balance. The nondimensionalized NSE is expressed by 1 ∂U + (U · ∇)U = −∇p + ∂t Re

U

,

(2)

where U := (U V W )T is the vector of the velocities in x-, y-, and zdirection; ∇ := (∂/∂x ∂/∂y ∂/∂z)T ; the scalar product is denoted by ’·’, i.e. (U · ∇) = UT ∇, = ∇T ∇. The Reynolds number Re represents the nondimensionalized characteristic velocity. Suitable boundary and initial conditions close the system of evolution equations. Synthesis of robust closed-loop flow control may be based on 1. a high-dimensional discretization of Eq. (2), 2. on a low dimensional models derived from Eq. (2), 3. or on experimentally identified black-box models. Approach 1 leads to high dimensional controllers which will not be applicable in real-time in the near future, cf. [5] [2]. Approach 3 has proven to be quite successful with robust and/or adaptive methods, see for example [1] [3] [4] [10] [12] [15]. However, approach 3 suffers in some respect from the limited region of validity of the employed basically linear black-box models. A promising compromise between the resolution of nonlinear physics of approach 1 and the simplicity of approach 3 is offered by approach 2 in which nonlinear low-dimensional models are employed in controller synthesis. Reduced order models can be obtained with various methods such as balanced

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truncation [19], Galerkin method [13], or vortex modeling [6] [18]. Galerkin models will be used in the following case study. This case study considers a well established benchmark problem of fluid dynamics, namely the stabilization of the flow around a circular cylinder,see Fig. 1. This flow has been under active investigation for more than one hundred $ Y R OX P H IR U F H

V H Q V R U

% F \ OLQ G H U R V F LOOD WLR Q

Fig. 1. Principal sketch of the actuated cylinder wake. The figure displays the streamlines of natural flow around a circular cylinder with diameter D = 1 (solid circle). Actuation is provided by transverse cylinder oscillation or by a transverse volume force in the grey circle. The flow state is sensed with a hot-wire anemometer, located at a typical position. Success of control is monitored in the observation region −5 < x < 15, −5 < y < 5, with x = y = 0 in the center of the cylinder.

years. Despite its simple geometry, the flow exhibits a rich kaleidoscope of an vortex street. phenomena, most predominantly the well-known von K´ arm´ That wake is considered as a prototype of so-called bluff body flows which give rise to oscillatory instabilities and has been chosen as benchmark problem in many studies. These instabilities increase drag as well as noise emission. A Galerkin model allows to describe this oscillatory flow around the cylinder with just a few dynamic states [17]. In [11], such a Galerkin model was employed to synthesize a nonlinear controller based on physical insight of the system. The obtained least-order Galerkin model describes the vortex shedding behind the circular cylinder with just three dynamical states. The controller produced a robust closed-loop performance. It could be applied as well to improved Galerkin approximations with seven or nine states, and finally to a direct numerical simulation (DNS) of the NSE. The main objective of this benchmark problem was to develop a controller, which respected the validity of the reduced order model. To this end, a physically motivated energy-based controller was proposed in [11]. The chosen volume force position did not allow to suppress vortex shedding completely. However, model-based control significantly mitigated vortex shedding without a significant change of its frequency. The present investigation focuses on more formal methods of controller synthesis which are more amenable to be used as well for higher order systems. One important objective is to see whether these methods can be tuned such

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that the region of validity of the low order models is respected in the closedloop system, i.e. the controller only slightly actuates the flow. The two-dimensional flow problem considered in [11] is used here again, in order to compare the formal methods with known results. The manuscript is organised as follows. The Galerkin modeling approach is recapitulated in section 2 and applied to the cylinder wake in section 3. Various control methods are then compared in 4 when applied to the Galerkin model. In section 5, the control is transferred to DNS studies. Finally, in Sec. 6, the main findings are summarized.

2 Galerkin Models Derived from the Navier-Stokes Equation For simplicity, a two-dimensional flow is considered, here. This flow is described in a Cartesian coordinate system x, y. The x-direction is aligned with the flow and the y-direction is orthogonal to the flow and the cylinder axis. The velocity field U(x, y, t) is approximated with a finite Galerkin expansion N

ai (t)Ui (x, y)

U[N ] (x, y, t) = U0 (x, y) +

,

(3)

i=1

with time-dependent Fourier coefficients ai and space-dependent modes Ui . The modes comprise a complete orthonormal system in a suitable Hilbert space. The spatial modes Ui , i = 1, . . . , N may conveniently be obtained from a Karhunen-Lo`eve expansion of a reference simulation [13], or from experimental data [24]. This expansion is also known as proper orthogonal decomposition. U0 represents the base flow, typically the mean flow. The evolution of the Fourier coefficients is described by the Galerkin system, obtained from a Galerkin projection of Eq. (3) on the Navier-Stokes equation, Eq. (2). For an unactuated flow, a˙ i (t) =

1 Re

N

N

qijk aj (t)ak (t) ,

lij aj (t) + j=0

i = 1, . . . , N

(4)

j,k=0

is obtained, where a0 ≡ 1. The coefficients lij and qijk are given by lij = (Ui , Uj )Ω and qijk = (Ui , (Uj · ∇)Uk )Ω , respectively, where (V, W)Ω := V · WdA represents the inner product of two solenoidal fields V and W Ω on the computational domain Ω. Incorporation of actuation leads to additional affine or nonlinear terms in the input u in the right hand side of Eq. (4), see below. For N → ∞, a Galerkin approximation, Eq. (3), describes the true velocity field U(x, y, t) with arbitrary precision [17]. The kinetic energy of the

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disturbance U around the steady solution, i.e. U = U − U0 , is then given by ∞ 1 1 a2 . (5) E = (U · U )Ω = 2 2 i=1 i A finite N th-order empirical Karhunen-Lo`eve expansion is optimal in the sense ∞ that the time-averaged unresolved energy content i=N +1 a2i /2 of the flow is the smallest for all expansions with N modes. Time averaging is denoted by ’ ’. Moreover, the energy of the ith Karhunen-Lo`eve mode is simply given by the time average of the respective squared Fourier coefficient, i.e. a2i /2.

3 Galerkin Model of the Cylinder Wake The present case study focuses on the two-dimensional laminar flow around a circular cylinder, see Fig. 1. The Reynolds number Re = 100 is chosen well above the critical Reynolds number 47 [16] for the onset of 2D vortex shedding and well below the 3D instability around 180. The control goal is to suppress the stable 2D vortex shedding at that Reynolds number. A Karhunen-Lo`eve (KL) decomposition of the unactuated flow shows that 96% of the turbulent kinetic energy E can be resolved with the first 2 KL modes [11] [17]. To describe the transient from the (unstable) steady state solution of the NSE, U0 , to the vortex shedding mode, a third so-called shift mode UΔ has to be included in the Galerkin approximation as a key enabler for a successful approximation [17]. With these 3 modes, the Galerkin approximation reads 2

U[N ] = U0 +

ai Ui + a3 UΔ

,

(6)

i=1

where UΔ accounts for the difference between the mean and steady flow. All modes, including the steady and mean flow are visualised in Fig. 2 and Fig. 3 by streamlines. A streamline is, by definition, tangent to the local velocity vector at every instant. Due to a nearly sinusoidal behavior of the Fourier coefficients a1 and a2 , the term a1 (t)U1 (x, y) + a2 (t)U2 (x, y) approximates the oscillatory fluctuation associated with the von K´arm´ an vortex street. Suppression of this flow instability can be achieved by passive means with a so-called splitter plate. If this plate is placed on the center-line behind the cylinder at y = 0 and 1.5 ≤ x ≤ 2.5, a 60% reduction in the fluctuation energy with respect to the unactuated flow has been obtained. However, instead of using a passive flow control device, active closed-loop flow control concepts are applied in the sequel. Two different actuators are sketched in Fig. 1. I) Transverse oscillations of the cylinder interact with the flow, thereby decreasing or increasing the vortex street. An opposition-like control can be

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Fig. 2. Modes of the Galerkin approximation, U1 , U2 , and the shift mode UΔ , from top to bottom. The flow field is visualized by streamlines. The closed streamlines of U1 , U2 display an sequence of alternating vortices with clockwise and counterclockwise orientation.

Fig. 3. Base flows, i.e. the mean flow (top) and the steady solution U0 (bottom). The flow is visualised like in Fig. 2.

found in [23] for this type of actuation. In [20] a PD-control of the first mode of the Karhunen-Lo`eve decomposition is applied experimentally for this configuration. II) Although of more theoretical interest, a second actuation concept will be considered here, according to [11]. A volume force in a control volume downstream of the cylinder is assumed, acting in the y-direction, see as well Fig. 1. A practical implementation may be done with a magneto-hydrodynamic force. Including this volume force in the momentum equation leads to the following modified NSE 1 ∂U + (U · ∇)U = −∇p + ∂t Re

U + bu

.

(7)

The control input u ∈ R1 describes the amplitude of the forcing on a compact support given by b in the area shown in Fig. 1. A low order model is derived by projecting Eq. (6) on the NSE and applying a Kryloff-Bogoliubov ansatz: ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ a˙ 1 g1 σr a1 −ω − γa3 −βa1 ⎣ a˙ 2 ⎦ = ⎣ ω + γa3 σr −βa2 ⎦ ⎣ a2 ⎦ + ⎣ g2 ⎦ u . (8) 0 a˙ 3 αa1 αa2 −σ3 a3 f

g

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The฀resulting฀simplification฀introduces฀an฀solution฀error฀of฀less฀than฀1%฀as compared฀to฀the฀original฀3rd฀order฀Galerkin฀model,฀but฀it฀significantly฀simplifies฀controller฀synthesis.฀Other฀simplifications,฀like฀the฀neglection฀of฀higherorder฀modes฀ or฀the฀ change฀of฀the฀ modes฀ due฀ to฀ actuation฀outweigh฀by฀ far the฀Kryloff-Bogoliubov฀simplification.฀Hence,฀this฀model฀shall฀be฀used฀in฀the following฀for฀controller฀synthesis.฀The฀model฀parameters฀are฀specified฀in฀Tab. 1,฀and฀used฀throughout฀this฀study. Fig.฀4฀shows฀the฀open-loop฀behavior฀of฀the฀low฀dimensional฀model. The฀ Galerkin฀ system฀ has฀ a฀ more฀ simple฀ structure฀ in฀ polar฀ coordinates. With a1 =฀A cos Φ, a2 = A sin Φ,฀and฀the฀parameters฀θ =฀arctan (g฀2฀/g1 )฀and

Table 1. Model parameters from the Galerkin projection. param. value σr γ

0.05439 -0.03504

param. value σ3 ω

0.05347 0.9232

param. value α g1

param. value

0.02095 -0.15402

4

β g2

0.02116 0.046387

12 10

2 2

0

1

a1

a2+a2

8 6 4 −2 2 −4

0

50

100 t

150

0

200

0

50

100 t

150

200

3.5 3

4

2

3

2

a

a

3

2.5

1.5 1

0 2

0.5 0

0 0

50

100 t

150

200

a2

−2

−2

0

2

a1

Fig. 4. Solution of Eq. (8) with a(0) = (0.1, 0.1, 0.008)T and control off (u = 0).

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gc = g1 cos θ + g2 sin θ, the low order model is transformed to ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ A˙ (σr − βa3 )A gc cos (Φ − θ) ⎣ Φ˙ ⎦ = ⎣ ω + γa3 ⎦ + ⎣ −(gc /A) sin (Φ − θ) ⎦ u αA2 − σ3 a3 0 a˙ 3

.

(9)

An input-output linearization and a LPV control, however, is more readily carried out using b1 = A cos(Φ − θ), and b2 = A sin(Φ − θ) ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ gc b˙ 1 (σr − βa3 )b1 − (ω + γa3 )b2 ⎣ b˙ 2 ⎦ = ⎣ (ω + γa3 )b1 + (σr − βa3 )b2 ⎦ + ⎣ 0 ⎦ u . (10) −σ3 a3 + α(b21 + b22 ) 0 a˙ 3

It should be pointed out that the model parameters and the modes Ui are obtained from an open-loop situation by a projection of Eq. (6) on Eq. (7). This results in three major imperfections of the Galerkin system, Eq. (8), Eq. (9), or Eq. (10) which necessitate robust controllers: I) As N = 3, Eq. (6) is an approximation of U(x, y, t) per se. II) Due to actuation, the flow field will change significantly. In the desired limit, the vortex shedding would be greatly reduced. As a result, the openloop KL modes, used in Eq. (6), will fail to capture a significant part of the instantaneous kinetic energy of the flow. III) The change of the modes is also mirrored in inaccurate growth rates of the Fourier coefficients – as predicted near the fix-point when reached in the controlled case. This could be remedied by a dynamic estimation of the model parameters according to [21]. However, this was only done partly, here, to keep the approach simple. Only σr and β have been multiplied for closed-loop studies by a factor 0.2574 to account for this. These model imperfections may impose a major challenge when a controller is employed in the Navier-Stokes simulation. The flow instability tries to profit from it, and tries to avoid the damping due to actuation. At the same time the results will show that, nevertheless, the Galerkin model can be used for controller synthesis for closed-loop, even when build up from open-loop data.

4 Controller Synthesis Based on the Galerkin Model 4.1 Physically Motivated Solutions Energy-based Control - Revisited The original work [11] was a proof-of-concept study showing that an empirical Galerkin method derived from natural flow data can also be employed for an actuated flow. One conclusion was that the system should stay in the

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region of validity of the low dimensional model, i.e. near the center manifold visualized in Fig. 4. Following this argument, more classical nonlinear methods of controller synthesis were ruled out in [11], as it was expected that these methods do not preserve the range of model validity. A simple energy-based controller was proposed, instead, with u = −u0 if gc cos (Φ − θ) > 0, and u = +u0 otherwise. With a rather complicated formula, see as well [22], the amplitude u0 was determined once every period in agreement with a desired decay rate of the amplitude A. The mean impact of the force on the phase Φ was small because of the sign-change of the angle force-term gc sin (Φ − θ), see Eq. (9), during the time of constant force direction. With this control law, the turbulent kinetic energy, expressed as N 2 i=1 ai /2, could not only be reduced for the model used in the controller synthesis itself, with N = 3. The very same control law synthesized with a third order system reduces also the energy in higher-order Galerkin modes in the real system, since the higher harmonics get their energy from the suppressed first harmonics. This was shown by applying the controller to Galerkin approximations with N = 7 and N = 9 states. Finally, the complete system was controlled in a DNS study. A nonlinear observer [25] was build up using the measurement device shown in Fig. 1 to implement the control in the DNS. The gain of the nonlinear observer was determined so that the linearized dynamics of the state-space estimation error was stable. In the following, a new and simpler version of this energy-based control is proposed. If gc cos (Φ − θ) ≥ 0, the mean influence of the control on A˙ during half a period T /2 can be approximated, using φ ≈ ωt and ωT ≈ 2π, by π/2

2gc 2π

−π/2

cos (Φ − θ)d(Φ − θ) =

2gc π

(11)

.

Replacing gc cos (Φ − θ) in the first equation of (9) by the mean influence 2gc /π, and demanding that (σr − βa3 )A + u0

2gc = −kA π

,

with a decay rate −k, yields the control u0 =

π − k − σr + βa3 A 2gc

(12)

.

For cos (Φ − θ) < 0 an opposite sign is needed. This finally leads to the new energy-based (eb) control ueb = −

πA(k + σr − βa3 ) sign{cos (Φ − θ)} 2gc

.

(13)

For e−2πk/ω = 0.95 a decay rate of −k = −0.0075 is found. Fig. 5 and Fig. 6 show the development of the Fourier coefficients, the squared amplitude

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7

2

6 5 4

2

0

1

a1

a2+a2

1

−1

2

−2 −3

3

1 0

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0

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150

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3 3

2

2 3

2.5

a

3

a

1.5

1

1

0

0.5

2 0

0

0

50

100 t

150

200

a2

−2

−2

0

2

a1

Fig. 5. Solution of Eq. (8) with a(0) = (−1.78, 1.85, 2.57)T and energy-based control ueb , with k = 0.0075.

A2 = a21 + a22 , and the input ueb , respectively, applying the controller to the low order model, Eq. (8). It should be mentioned that, for simplicity, these and all following tests are implemented without an observer. In the case of the low dimensional model, it is assumed that all three states are measured. Later, in the direct numerical simulation of the NSE, the Fourier coefficients are obtained at every sampling instant by (Ui , U )Ω . Damping Control As the region of validity of the low order model has to be respected, another physically motivated control is possible. The input u can be determined in every instant such that the quadratic difference between the right hand sides of the first two equations of (8) and ka1 − ωa2 and ωa1 + ka2 , respectively, is minimized. By this, the natural oscillation with ω is preserved. A desired damping can be obtained by an appropriate value of k. This leads to

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0.3 0.2

ueb

0.1 0

−0.1 −0.2 −0.3 −0.4

0

50

100 t

150

200

Fig. 6. ueb for Fig. 5.

A (σr − βa3 − k) cos(Φ − θ) − γa3 sin(Φ − θ) gc A Kdc1 (a3 ) cos(Φ − θ) − Kdc2 (a3 ) sin(Φ − θ) =− gc

udc = −

.

(14)

As a3 tends to zero, udc is synchronized with cos(Φ − θ). Results with udc will be given in the DNS study in section 5. With the Galerkin model, similar results are obtained as shown above with ueb . 4.2 Formally Derived Controllers It is first checked whether any output y = λ(a) exists, for which Eq. (8) has a relative degree of n = 3 at a = 0 so that an input-state linearization is possible. As the determinant of the matrix g

adf g

[g, adf g]

is 2α(g22 + g12 )2 (ω + γa3 ), its rank is 3, and, hence, unequal to the rank of g

adf g

.

∂g ∂f f− g. Therefore, an ∂x ∂x exact input-state linearization via feedback is impossible [14]. A flatness-based control [9] can not be build up for the same reason.

The Lie bracket [f , g] is defined as [f , g] = adf g =

Input-Output Linearization However, an input-output linearization is possible. If formulated for Eq. (8) with y = a3 , a control with u ∼ 1/ cos (Φ(t) − θ) results, making an input saturation function necessary, see below as well. If it is applied for the output

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y = h = b2 with Eq. (10), instead, a standard I/O-linearizing controller is found with stable zero dynamics [14] uio =

1 (−L2f h + p1 z1 + p2 z2 ) Lg Lf h

(15)

.

Vectors f and g refer to Eq. (10). In polar coordinates this control reads uio = −

A Kio1 (a3 ) cos(Φ − θ) − Kio2 (a3 ) sin(Φ − θ) gc

.

(16)

The coefficients Kioj are more intricate than Kdcj from Eq. (14). p1 and p2 are chosen as a complex conjugate pair with the same frequency as the open-loop poles and a negative real part. Again, the dynamical behavior of the Fourier coefficients looks similar to the results shown in Fig. 5, but u is larger. Lyapunov Based Synthesis Using

1 2 (a + a22 + a23 ) . 2 1 as a Lyapunov candidate, the derivative, given in polar coordinates, V (a) =

V˙ = (σr + (α − β)a3 )A2 − σ3 a23 + Agc cos(Φ − θ)u

(17)

(18)

motivates uLy =

−k A gc cos(Φ − θ) −umax sign{gc cos(Φ − θ)}

kA < umax |gc cos(Φ − θ)| otherwise . for

(19)

As A and a3 are bounded, it is easy to show that umax always can be chosen such that V˙ dt is negative, i.e. V decreases. For k = 0.0015, Figs. 7 and 8 are obtained. Note the unfavorable behaviour of u. Backstepping, LPV and Opposition Control A backstepping approach can start with the last equation in (9). Choosing A2 as input such that −σ3 a3 + αA2 = −ka3 finally yields ubs = −

A(k + σr − βa3 ) gc cos (Φ − θ)

,

(20)

following the classical procedure of a backstepping synthesis. Again, the same singularity is obtained. It has to be tackled as in the Lyapunov based control. This and the similarity with the Lyapunov based controller results in comparable behavior in simulation studies with the Galerkin model.

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1

uLy

0.5 0

−0.5 −1 0

50

100 t

150

200

Fig. 7. uLy with k = 0.015 for Fig. 8.

The first two equations in (10) set up a linear parametrically varying (lpv) model. With a state space controller, ulpv = (k1 k2 )(b1 b2 )T , the poles are placed such that in an energy optimal sense the unstable poles of the openloop system a mirrored at the imaginary axis and shifted 0.001 to the left. The controller can be written, again, as ulpv = −

A Klpv1 (a3 ) cos(Φ − θ) − Klpv2 (a3 ) sin(Φ − θ) gc

.

(21)

Finally, if in the Lyapunov based control, max|uLy | is set to 0.5, a bangbang-type control is obtained. This motivates, as a further alternative, an opposition control for which results will be given in the DNS study Γop = −kAsign{gc cos (Φ − θ)}

.

(22)

5 Application of the Controllers in DNS Studies All controllers are tested in direct numerical simulations of the NSE. The simulations have been performed on a grid with 2550 nodes. On a fine grid (7876 nodes) the suppression is changed by a few percent. Increasing the grid resolution decreases the numerical dissipation of the DNS scheme and results in an increased farwake fluctuation, both, in the natural and controlled flow. Tab. 2 shows the reduction of the turbulent kinetic energy, obtained in the converged post-transient state (see first row). For some controllers an intermediate larger reduction of the turbulent kinetic energy is observed (second row). In these cases, a restructuring of the flow field takes place. Due to this, the Galerkin system does not predict the correct phase of the flow. Actuation now feeds the instability. In the long run, a lower reduction of the turbulent kinetic energy is obtained. When the new version of the energy-based control ueb is implemented in a DNS, for instance, with k = 0.008, a maximal reduction of the turbulent

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7

2

6 5 4

2

0

1

a1

a2+a2

1

−1

2

−2 −3

3

1 0

50

100 t

150

0

200

0

50

100 t

150

200

3 3

2

2 3

2.5

a

3

a

1.5

1

1

0

0.5

2 0

0

0

50

100 t

150

200

a2

−2

−2

0

2

a1

Fig. 8. Solution of Eq. (8) with a(0) = (−1.78, 1.85, 2.57)T and Lyapunov based control uLy , with k = 0.0015 and max|uLy | = 1.0. Table 2. Reduction of the turbulent kinetic energy in the DNS studies. Grid with 2550 nodes. controller

ueb udc uio uLy ubs ulpv uop

post-transient reduction (%) 32 31 32 33 28 24 30 maximal instantaneous reduction (%) 70 44 50

kinetic energy of 70% is achieved at t = 38.4. Note that the passive reference with a splitter plate yields 60%, only. Fig. 9 shows a plot of the streamlines of the unactuated, and Fig. 10 of the actuated case with this new version of the energy-based controller at t = 38.4. A ’tamed’ von K´ arm´ an vortex street is obtained. However, at t > 38.4 the above mentioned restructuring of the flow field takes place, making it more oscillatory again. An even better result can be obtained with a completely different control law. If it would be possible to estimate the velocity V in y-direction in the center of the volume force, a simple opposition control u = −kV

,

(23)

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4 y 2 1 0 -1 -2 -3 -4 -4

-2

0

2

4

6

8

10

12 x 14

Fig. 9. Unactuated flow. The figures displays iso-contours of the streamwise velocity component U . Negative values are indicated by thinner curves and show the extend of the recirculation region. Simulation with the fine grid.

4 y 2 1 0 -1 -2 -3 -4 -4

-2

0

2

4

6

8

10

12 x 14

Fig. 10. Actuated flow with an energy-based controller ueb at t = 38.4. The flow is visualised like in Fig. 9. Simulation with the fine grid.

would yield a complete V reduction at that point. If the actuator is shifted another 2 diameters downstream, this would give rise to a complete suppression of the vortex street (data not shown).

6 Conclusions This case study shows that formal nonlinear methods of controller design are very well suited for a nonlinear Galerkin model of a benchmark fluid flow. All approaches were successfully used for the 3-dimensional synthesis model. Instead of testing the controllers first with higher order approximations of the flow, a direct implementation in the infinite system, Eq. (2), via a direct numerical simulation (DNS) was done. As an additional phase error was introduced in the DNS, some controllers had to be de-tuned slightly in theses studies compared to the simulations with the Galerkin system.

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All design methods gave rise to very similar control laws. One part of it (eb, Ly, bs, op) is synchronized with cos (Φ − θ), the rest (dc, io, lpv) with a shifted version. With the exception of the opposition controller (op), u is proportional to A/gc in all laws. Some of the formal approaches suffered from periodically singular control inputs. This necessitated the introduction of a saturation function to limit the input. The reduction of the turbulent kinetic energy was comparable with all controllers. However, it was achieved with significantly different maximal values of |u|, compare Figs. 6 and 7. Simulation studies with the Galerkin model showed that a synchronization with cos (Φ − θ), as formerly proposed in [11] on basis of physical insight, provides smaller values of |u| in the closed-loop. This finding and the varying complexity of the individual control laws motivates a careful controller design for higher dimensional Galerkin models, as well. At the same time it shows that even when such physical insight is not that obvious, due to the complexity of the model, the formal methods lead to practical solutions. None of the Galerkin model based controllers could suppress the turbulent kinetic energy of the fluctuations completely. Part of the reason lies in the physics. Even optimal suppression delays and mitigates vortex shedding only, but cannot prevent it. An additional reason is the decreasing accuracy of the Galerkin model with increasing actuation – due to uncompensated changes of the modes. This inaccuracy of the Galerkin model was the reason, why the turbulent kinetic energy of flows controlled with ueb , udc , or ubs increased again after a reduction of up to 70%. At the same time it opens up a route to further work with some kind of model-switching approach. From the encouraging results obtained for this benchmark problem it can be expected that also higher dimensional Galerkin models of fluid flows can be tackled with these approaches. This and the combinations with nonlinear observers will be part of future work. Acknowledgement. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via the Collaborative Research Center (Sfb 557) ’Control of complex turbulent shear flows’ at the Berlin University of Technology.

References 1. B. G. Allan, J.-N. Juang, D. L. Raney, A. Seifert, L. G. Pack, and D. E. Brown. Closed loop separation control using oscillatory flow excitation. In ICASE Report 2000-32, 2000. 2. J. Baker, J. Myatt, and P.D. Christofides. Drag reduction in flow over a flat plate using active feedback control. Comp. & Chem. Engng., 26:1095–1102, 2002.

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3. R. Becker, M. Garwon, C. Gutknecht, G. B¨ arwolff, and R. King. Robust control of separated shear flows in simulation and experiment. J. Process Control, 2005. Accepted for publication. 4. R. Becker, M. Garwon, and R. King. Development of model-based sensors and their use for closed-loop control of separated shear flows. In Proc. of the ECC 2003, Cambridge, 2003. 5. T. R. Bewley and S. Liu. Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech., 365:305–349, 1998. 6. A. Cottet and P. Koumoutsakos. Vortex Methods - Theory and Practice. Cambridge University Press, 2000. 7. M. Gad el Hak, A. Pollard, and J.P. Bonnet. Flow control: Fundamentals and Practice. Springer, Berlin, 1998. 8. H.-E. Fiedler and H.H. Fernholz. On management and control of turbulent shear flows. Prog. Aeronaut. Soc., 27:305–387, 1990. 9. M. Fliess, J. L´evine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control, 61:1327–1361, 1995. 10. M. Garwon, F. Urzynicok, L. H. Darmadi, G. B¨ arwolff, and R. King. Adaptive control of separated flows. In Proc. of the ECC 2003, Cambridge, 2003. 11. J. Gerhard, M. Pastoor, R. King, B.R. Noack, A. Dillmann, M. Morzy´ nski, and G. Tadmor. Model-based control of vortex shedding using low-dimensional Galerkin models. In AIAA-Paper 2003-4262, 2003. 12. L. Henning and R. King. Multivariable closed-loop control of the reattachment length downstream of a backward-facing step. In 16th IFAC World Congress 05, Prag, 2005. Accepted contribution. 13. P. Holmes, J.L. Lumley, and G. Berkooz. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, 1998. 14. A. Isidori. Nonlinear control systems. Springer, 2002. 15. R. King, R. Becker, M. Garwon, and L. Henning. Robust and adaptive closedloop control of separated shear flows. In AIAA-Paper 2004-2519, 2004. 16. M. Morzy´ nski, K. Afanasiev, and F. Thiele. Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Meth. Appl. Mech. Enrgrg., 169:161–176, 1999. 17. B.R. Noack, K. Afanasiev, M. Morzy´ nski, G. Tadmor, and F. Thiele. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid. Mech., 497:335–363, 2003. 18. M. Pastoor, R. King, B.R. Noack, A. Dillmann, and G. Tadmor. Model-based coherent-structure control of turbulent shear flows using low-dimensional vortex models. In AIAA-Paper 2003-4261, 2003. 19. C.W. Rowley. Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. on Bifurcation and Chaos, 2005. To appear. 20. S. Siegel, K. Cohen, and T. McLaughlin. Experimental variable gain feedback control of a circular cylinder wake. In AIAA-Paper 2004-2611, 2004. 21. G. Tadmor and B.R. Noack. Dynamic estimation for reduced Galerkin models of fluid flows. In The 2004 American Control Conference, WeM18.1, 2004. 22. G. Tadmor, B.R. Noack, A. Dillmann, J. Gerhard, M. Pastoor, R. King, and M. Morzy´ nski. Control, observation and energy regulation of wake flow instabilities. In 42nd IEEE Conference on Decision and Control 2003, pages 2334–2339, 2003. 23. G. Tadmor, B.R. Noack, M. Morzy´ nski, and S. Siegel. Low-dimensional models for feedback flow control. Part II: Controller design and dynamic estimation. In AIAA Paper 2004-2409, 2004.

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Flatness Based Approach to a Heat Conduction Problem in a Crystal Growth Process Joachim Rudolph1 , Jan Winkler2,1 , and Frank Woittennek1 1

2

Institut f¨ ur Regelungs- und Steuerungstheorie, Technische Universit¨ at Dresden, Germany. {rudolph, woittennek}@erss11.et.tu-dresden.de Institut f¨ ur฀Kristallz¨ uchtung฀Berlin,฀Germany. [email protected]

Summary.฀ A฀nonlinear฀heat฀conduction฀problem฀from฀crystal฀growth฀technology is฀considered.฀The฀growth฀process฀studied฀comprises฀two฀phases฀(solid฀crystal฀and liquid฀melt)฀separated฀ by฀a฀moving฀interface.฀Using฀“flatness฀based”฀methods฀solutions฀ are฀ parameterized฀ by฀ flat฀ output฀ trajectories.฀ Numerical฀ aspects฀ of฀ series convergence฀are฀discussed฀and฀flatness฀based฀feedback฀design฀is฀sketched.

Keywords:฀Boundary฀control,฀crystal฀growth,฀distributed฀parameter฀systems,฀flatness,฀free฀boundary,฀heat฀equation,฀trajectory฀planning.

1 Introduction A key step during the production of compound semi-conductors is the growth of single-crystals from melt. Especially III-V-semi-conductors like GalliumArsenide (GaAs) are grown using the so-called Vertical-Gradient-Freeze (VGF) technique. Here, a crucible is filled with molten material. The temperatures at the top and the bottom of the crucible are manipulated in such a way that the crystallization starts at the bottom, slowly moving upwards until the whole liquid is solidified. However, before crystallization can start one has to adjust a temperature profile in the melt. Both, the adjustment of the temperature profile and the crystallization, are typical representatives of processes where heat conduction problems play a crucial role. Furthermore, the crystallization can be modeled as a free boundary problem3 where the position of the boundary has to follow a prescribed trajectory. Feedback control of the VGF process is a problem still under investigation in industrial praxis. Basically, this is due to the infinite dimensional nature of 3

In contrast to a moving boundary problem here the position of the boundary results from the system dynamics.

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 387–401, 2005. © Springer-Verlag Berlin Heidelberg 2005

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the process model. Typically, the process is steered using a feedforward control which is optimized from run to run, supported by numerical simulations [33, 2]. An approach providing a solution in real time would be desirable since it would provide the opportunity to develop a feedback control. Differential flatness is a concept which is very useful in the trajectory planning and feedback design for nonlinear finite dimensional systems, i.e., systems described by ordinary differential equations. The flatness based control methods place an emphasis on trajectory design and open-loop control [7, 6, 19, 20]. This aspect gains even more importance in infinite dimension, namely for distributed parameter systems with boundary control action, the mathematical models of which comprise partial differential equations. As a consequence, flatness based methods have been developed for distributed parameter systems during the last years; for a survey see [26, 28], e.g. In the particular case of heat conduction problems the system trajectories are described by series which are parameterized by trajectories of a finite number of lumped variables, called a flat output. The flat output trajectory occurs together with all its derivatives, and it must be a Gevrey function of an order less than 2 in order to get series convergence. Such representations of solutions of partial differential equations have been known for a long time [11, 13]. Trajectory planning problems modeled with the heat equation and other second order linear or quasi-linear parabolic equations have been treated using this flatness based approach in [9, 10, 27, 18, 17, 16, 25, 5] and others4 . Similar methods have been applied on fourth order beam equations, see e.g. [8, 1, 12, 32, 15, 31].

2 Mathematical Model We assume that, due to the opaque nature of GaAs, the radiation within the crystal and the melt is negligible [3]. Furthermore we do not take convection in the melt into account, because of its small Peclet-number [3, 35, 37, 34]. Under the additional assumptions that the crucible is a cylinder of height H and inner radius R (cf. Fig. 1), and that its jacket is ideally isolated, we may neglect angular and radial gradients and can, therefore, model the process using the one-dimensional heat equation ∂ ∂ ρ T (z, t) T (z, t) = ∂t ∂z

λ(T (z, t))

∂T (z, t) . ∂z

(1)

Here ρ is the volume-specific heat capacity, λ the thermal conductivity, T is the temperature, z the position, and t is time. Since the temperature gradient in the crystal is much greater than that in the melt the temperature dependency 4

The present contribution extends a part of work electronically available at [30], see also [29].

Flatness Based Approach to a Heat Conduction Problem

389

ζ Top Heater

ζ = H − zI (t)

Crucible Melt Crystallization Front

Crystal

Bottom Heater

TM

v

I 1111111111111 0000000000000 TI 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 TC z

R

H r

r ζ = −zI (t)

Fig. 1. Sketch of a Vertical-Gradient-Freeze plant.

Table 1. Parameters of the crystal growth process model. temperature dependent volume-specific heat capacity of the solid: ρC (TC ) = ρC,0 + ρC,1 TC , (ρC,0 , ρC,1 ∈ R+ ) temperature dependent heat conduction coefλC ficient of the solid: λC (TC ) = λC,0 + λC,1 TC , (λC,0 , λC,1 ∈ R+ ) ρM,0 volume-specific heat capacity of the liquid λM,0 heat conduction coefficient of the liquid αb , αt heat transfer coefficient (bottom, top) b , t emissivity (bottom, top) Stephan-Boltzmann number σS h volume-specific crystallization enthalpy (h < 0) ρC

of the volume-specific heat capacity and of the heat conduction coefficient must be taken into account [14]. At the bottom and the top of the crucible, where the heaters reside, we have radiation. The boundary conditions resulting from the heat flow, thus, read ∂T = αb (T − ub ) + εb σS T 4 − u4b , z = 0 ∂z ∂T −λ(T ) = αt (T − ut ) + εt σS T 4 −u4t , z = H, ∂z λ(T )

(2a) (2b)

where t → ub (t) and t → ut (t) are the temperatures of the top and the bottom heater, respectively. The constant parameters given in Tab. 1 are used. All

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the preceding equations are valid for both the melt (with temperature TM ) and the crystal (with temperature TC ). They will be adapted in the following depending on the stage of the process considered. The temperature at the interface is equal to the melting (or crystallization) temperature TI : T (zI (t), t) = TM (zI (t), t) = TC (zI (t), t) = TI ,

(3a)

where zI (t) denotes the location of the interface at time t. Due to the energy release resulting from the crystallization, at the interface the temperature gradient is discontinuous ∂TM ∂TC (zI (t), t) − λC (TI ) (zI (t), t) = hvI (t). (3b) ∂z ∂z The indices M and C are used in order to distinguish the variables and parameters of the melt from those of the crystal5 . Here, vI (t) = z˙I (t) denotes the velocity of the phase interface. The initial condition for the process is given by λM (TI )

TM (z, 0) = TM,0 ∈ R,

z ∈ [0, H].

(Remember that in the beginning there is no crystal.)

3 Adjusting a Temperature Profile in the Melt As mentioned above, in the first stage we want to adjust a temperature profile in the melt from which crystallization can start. This means temperature is lowest at the bottom of the crucible where it equals the crystallization temperature TI . 3.1 Adapted Model for the Melt As the temperature in the melt is almost constant during the process, we neglect the temperature dependency of the parameters λ and ρ. Therefore, the differential equation (1) may be simplified to ∂ 2 TM ∂TM (z, t) = λM,0 (z, t), (z, t) ∈ [0, H] × R+ . ∂t ∂z 2 From (2) we obtain adapted boundary conditions ρM,0

∂TM 4 = αb (TM − ub ) + εb σS TM −u4b , z = 0 ∂z ∂TM 4 = αt (TM − ut ) + εt σS TM −u4t , z = H −λM,0 ∂z with constant parameters ρM,0 and λM,0 . λM,0

5

(4a)

(4b) (4c)

These compatibility conditions at the crystallization front will be used only during the second stage, when the crystal is present.

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3.2 Formal Power Series Parameterization of the Process The temperature profile in the melt can be expressed using the power series TM (z, t) =

∞

an (t)

n=0

zn n!

(5)

where the recursion an+2 (t) =

ρM,0 a˙ n (t), λM,0

n≥0

for the coefficients is obtained by substituting the series into the differential equation (4a) and comparing the coefficients of like powers of z. Up to now we have not yet fully specified the coefficients of the series; nothing has been said about the initialization of the recursion for the series coefficients (i.e., about a0 and a1 ). This freedom is now used to parameterize a transition by introducing a flat output. We start the process at a time t = 0 where the whole material is molten, assuming a uniform constant initial temperature TM,0 > TI . At t = t1 we want to reach another stationary profile, at which the temperature at the bottom is equal to the crystallization temperature TI and the temperature gradient has a desired value ΔTM,1 . Hence, the stationary profiles to be connected are given by TM (z, 0) = TM,0 = const., TM (z, t1 ) = TI + ΔTM,1 z,

z ∈ [0, H], z ∈ [0, H],

TM,0 > TI ΔTM,1 > 0.

(6a) (6b)

Since we have no constraints resulting from boundary conditions (Both boundary conditions contain different controls.), the first two coefficients of the power series (5) can be freely chosen and, therefore, they can be interpreted as a flat output y(t) =

y1 (t) y2 (t)

=

a0 (t) a1 (t)

=

TM (0, t) ∂TM ∂z (0, t)

.

(7)

From given trajectories for the flat output we can calculate the temperatures using the power series (5). This yields TM (z, t) =

∞ n=0

ρM,0 λM,0

n

z 2n (n) z 2n+1 (n) y1 (t) + y (t) . (2n)! (2n + 1)! 2

(8)

Now the trajectories for the controls ub and ut can be calculated using (the algebraic) equations (4b) and (4c), which are solved numerically. Since the system trajectories depend on derivatives of the flat output of arbitrary order these trajectories have to be chosen C ∞ . As an analytic function would be completely defined by these initial conditions, we must use

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non-analytic functions to parameterize the solution. The convergence of the series can be ensured using functions of Gevrey order α ≤ 2 [11, 4, 18, 29]. We may use the function [29] Φσ,T (t) =

1 2

1 + tanh

2t/T − 1 (4t/T (1 − t/T ))σ

(9)

,

which realizes a smooth transition from 0 to 1 on the interval [0, T ]. The parameter σ allows us to vary the slope of the transition. It is related to the Gevrey order α by α = 1 + 1/σ.

4 The Crystallization We will now study the second stage of the process where the crystal is growing and we have a two-phase system. 4.1 Adapted Mathematical Model Starting from equations (1) – (3), we adapt the model equations for the melt and the crystal, being valid during the growth. As in the previous section, we assume that the heat capacity and the heat conduction coefficient in the melt do not depend on the temperature. In contrast, for the solid phase we use an affine approximation for these parameters: ρC (TC ) = ρC,0 + ρC,1 TC ,

λC (TC ) = λC,0 + λC,1 TC .

Using a moving frame the origin of which is located at the interface between solid and liquid phase, we apply the transformation ζ = z − zI (t),

ϑ(ζ, t) = T (z, t)

which, with z˙I = vI , leads to ∂ ∂ − vI (t) ∂t ∂ζ

ρ ϑ(ζ, t) ϑ(ζ, t) =

∂ ∂ζ

λ ϑ(ζ, t)

∂ ϑ(ζ, t) . ∂ζ

Using the constant parameter approximations for the melt and the affine ones for the crystal we obtain ∂ϑM ∂ϑM − vI ∂t ∂ζ ∂ϑC ∂ϑC (ρC,0 +2ρC,1 ϑC ) − vI ∂t ∂ζ ρM,0

= λM,0

∂ 2 ϑM ∂ζ 2

(10a)

=

∂ 2 ϑC ∂ϑC + λC,1 (λC,0 +λC,1 ϑC ) ∂ζ 2 ∂ζ

(10b)

2

.

Flatness Based Approach to a Heat Conduction Problem

393

The boundary conditions for the second stage of the process are given by λM,0

∂ϑM = αb (ϑM − ub ) + εb σS ϑ4M − u4b ∂ζ

(11a)

∂ϑC = αt (ϑC − ut ) + εt σS ϑ4C − u4t ∂ζ

(11b)

at ζ = −zI , and −(λC,0 + λC,1 ϑC )

at ζ = H − zI . Finally, the adapted compatibility conditions read ϑC = ϑM = TI , ∂ϑM ∂ϑC − (λC,0 + λC,1 TI ) , vI h = λM,0 ∂ζ ∂ζ

at ζ = 0

(12a)

at ζ = 0.

(12b)

4.2 Formal Power Series Parameterization of the Process Again we use power series for planning trajectories of the growth process: ϑM (ζ, t) = ϑC (ζ, t) =

∞ n=0 ∞

bn (t)

ζn n!

(13a)

cn (t)

ζn . n!

(13b)

n=0

Substituting (13a) into the differential equation (10a) we obtain the recursion formula M,0 ˙ (14a) bn − vI bn+1 , n ≥ 0. bn+2 = λM,0 Furthermore, using (10b) the recursion for the coefficients cn , n ≥ 2 involved in (13b) reads6 cn+2 =

1 ρC,0 (c˙n − vI cn+1 ) + λC,0 + λC,1 c0 n

l=0

n l

2ρC,1 (cl c˙n−l − vI cl cn+1−l ) − λC,1

n+1 cl+1 cn+1−l l+1

. (14b)

The initialization of the preceding recursions are initialized as follows. 6

We assume, that the heat conduction coefficient is non-zero (λC,0 + λC,1 c0 > 0).

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J. Rudolph, J. Winkler, and F. Woittennek

First of all, we have b0 = c0 ≡ TI , because the temperature at ζ = 0 is equal to the crystallization temperature (recall that our frame is moving with the crystallization front). Moreover, the coefficients b1 , c1 , and the velocity z˙I = vI have to be chosen such that the second compatibility condition (12b) is satisfied. Thus, the process can be parameterized, for instance, by the flat output trajectory t → y˜(t) =

y˜1 (t) y˜2 (t)

=

zI (t) b1 (t)

=

zI (t) ∂ϑM ∂ζ (0, t)

.

(15)

The trajectories for the other variables can be calculated from the latter using the compatibility condition (12b), the series solutions (13) with the recursions (14a) and (14b), and the boundary conditions (11). We assume that when crystallization starts the crystallization front is stationarily located at the bottom of the crucible. The initial conditions are now the one given in (6b) completed by the one for the crystal, which due to the compatibility condition and the zero initial velocity is equal to the one for the melt. We want to realize a transition to another “stationary” solution7 of (10), to be reached at t = t2 , which satisfies the conditions z˙I (t2 ) = v2 , ∂ϑM (0, t2 ) = ΔTM,2 , ∂ζ λM,0 ΔTM,2 − v2 h ∂ϑC (0, t2 ) = . ∂ζ λC,0 + λC,1 TI

v2 > 0

(16a)

ΔTM,2 > 0

(16b) (16c)

This can be achieved using the trajectories t → y˜(t) = Φσ,t2 (t)

v2 t ΔTM,2

for the flat output where Φσ,t is the function defined in (9). After the startup of the crystallization is finished at time t2 the crystallization front moves upwards with constant velocity v2 until the whole melt is solidified. However, although a stationary solution is reached w.r.t. the moving frame, the temperatures at the top and the bottom, and, therefore, the control inputs, are not constant. So far we have shown how the start-up may be parameterized with two trajectories: First the temperature profile in the melt is adjusted, then the growth velocity is increased. These two trajectories are connected at a stationary regime. Alternatively, it is possible to avoid such an instant of rest. Instead, we may start the crystallization at t = t1 with a constant initial velocity v2 . To this end, we have to change the parameterization of the first stage. 7

Here “stationary” means that the temperature profile is constant for an observer moving with the crystallization front.

Flatness Based Approach to a Heat Conduction Problem

395

The ordinary differential equation for the melt describing the desired “stationary” regime is obtained from (10a) by setting ∂ϑ∂tM to zero: −v2 ρM,0

∂ 2 ϑ˜M ∂ ϑ˜M (ζ). (ζ) = λM,0 ∂ζ ∂ζ 2

The “stationary” solution of the second stage is given by ρM,0 λM,0 ΔTM,2 1 − exp −ζv2 ϑ˜M (ζ) = TI + ρM,0 v2 λM,0

.

(17)

In order to start crystallization with a prescribed constant velocity, we have to meet this solution at the end of the first stage of the process. Using fixed coordinates (17) reads ϑ˜M (ζ) = TM (z, t) = TI +

λM,0 ρM,0 ΔTM,2 1 − exp −(z − zI (t))v2 ρM,0 v2 λM,0

.

(18)

Since z˙I = v2 is constant and zI (t1 ) = 0, at z = 0 we have TM (0, t) = TI +

ρM,0 λM,0 ΔTM,2 1−exp (t − t1 )v22 ρM,0 v2 λM,0 =:b y1 (t)

ρM,0 ∂TM (0, t) = ΔTM,2 exp (t − t1 )v22 , ∂z λM,0 =:b y2 (t)

which define the fictitious analytic trajectories to be reached by y1 and y2 at time t = t1 . In order to connect the profiles (6a) and (18) we have to choose trajectories for y1 and y2 the derivatives of which at t = t1 are all equal to those of y1 and y2 , respectively. For instance, we may choose y1 (t) = (y1 (t) − TM,0 ) Φσ,t1 (t) + TM,0 y2 (t) = y2 (t)Φσ,t1 (t). Here Φσ,T is the function defined in (9). 4.3 Convergence Issues Again, it can be shown that the series giving the solution for the melt converges everywhere provided the trajectories are chosen Gevrey of order less than 2. However, for the series solution of the solid phase, which results from a nonlinear heat equation, convergence can be ensured by using Gevrey functions of order α < 2.

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Assume that the trajectories for vI and c1 are Gevrey of order α < 2, i.e., (l)

|vI | ≤ mv

(l!)α , γl

(l)

|c1 | ≤ m1

(l!)α , γl

(19)

where l ∈ N, 1 ≤ α ≤ 2, m1 , mv , γ ∈ R+ . Then the radius of convergence is larger than the unique positive root of the polynomial (cf. [29] for a proof) ¯ =R ¯3 p(R)

4m1 |ρ1 | ¯2 +R γ

4 |ρ0 | + 4m1 |ρ1 |mv + 3 γ ¯ 3 mv |ρ0 | + 4m1 |λ1 | − |λ0 | (20) R 2

with ρ0 := ρ0 + 2ρ1 TI ,

λ0 := λ0 + 2λ1 TI .

This additional condition bounds the maximal values admissible for v2 and ΔTC,2 (and therefore for ΔTM,2 ). Moreover, the transition time t2 must be chosen large enough. 4.4 Numerical Implementation Both, the trajectories for the temperature distributions and those for the heaters are given in terms of infinite series. Therefore, a numerical implementation of the above results requires an appropriate approximation of these series [17]. The straightforward procedure to obtain such an approximation for convergent power series is to use the truncated series (or partial sum) N

T (z, t) ≈ TN (z, t) =

cn (t) n=0

zn . n!

It is natural to ask wether there exist better approximations available from the finite sequence of coefficients c0 , . . . , cN . With this in mind in [36] a promising alternative approach has been introduced which allows to improve convergence. Instead of the partial sum TN the so called (N, ξ)-approximate k-sum j n N ξn cj (t) zj! n=0 j=0 Γ (1+nk) TˆN,k,ξ (z, t) = N ξn n=0 Γ (1+nk)

with the parameters N , k, and ξ is used to approximate the power series. As does the sequence of partial sums, with an appropriate choice of parameters k and ξ this summation method guarantees that limN →∞ TˆN,k,ξ (z, t) = T (z, t) while significantly increasing the rate of convergence.

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397

1.4 1.2 1

T(z,t)

ut(t)

Melt

tal

ys Cr

0.8 0.6

zI(t)

0.4 0.2 0

ub(t) 0.5

t

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

z

Fig. 2. Temperature profile during a crystallization process.

5 Simulation Result Fig. 2 shows the temperature profile which is obtained using the flatness based motion planning strategies for the parameterization of the crystallization process. The computations have been done using normalized coordinates and the series have been truncated at N = 15. The bold lines on the left and the right of the plot show the associated trajectories of the temperatures of the bottom heater (ub ) and the top heater (ut ), respectively, while the third bold line marks the location of the crystallization front zI (t).

6 Flatness based Feedback Control In order to make the system more robust to model inaccuracies and external disturbances the open-loop controls presented here could be completed by a flatness based feedback controller as introduced in [21, 22]. The method presented there relies on the differential flatness of any finite-dimensional approximation of the system which is obtained from the infinite series either by series truncation or via approximate k-sums. These topics will only be outlined here, for a more detailed presentation see the above cited references. One can easily see that for n odd the coefficients cn and bn in (14) depend on derivatives of y˜ up to order (n − 1)/2, while for even n derivatives of y˜1

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are needed up to order n/2, those of y˜2 up to order n/2 − 1. Moreover, the boundary conditions (2), from which the controls ub and ut are computed, also involve the spatial gradients of TC and TM . For notational simplicity, let the truncation order N be even. Involving only powers of z up to order N + 1 for the temperature distribution and up to order N for its gradient, a tracking control could be implemented by introducing the “new input” v = y˜(N/2) . A linearizing feedback is obtained by solving (2) for ub and ut . As usual, this linearized system is stabilized using the feedback N/2−1 (N/2)

v(t) = y˜d

(j)

(t) + j=0

Kj y˜d (t) − y˜(j) (t) + K−1

t t0

y˜d (τ ) − y˜(τ ) dτ

with appropriate coefficients Kj , j = −1, . . . , N/2 − 1, where the desired trajectories t → y˜i,d (t) are designed as described in the previous sections. The required derivatives of y˜ may be obtained using a nonlinear observer as proposed in [22]. Clearly, the quality of the approximation directly influences the quality of the feedback. It is, therefore, not surprising that the method based on the approximate k-sums is advantageous compared to the method of truncated power series. This has been investigated in some detail in [24].

7 Conclusion and Outlook A free boundary problem for a system with two phases (liquid and solid) has been solved using the flatness based approach leading to power series representations — see [13, 5] for a similar approach to a related problem. Approaches for improving convergence and flatness based feedback control have been sketched [22, 36, 24, 23]. Another interesting extension is the consideration of a non-planar phase boundary. Again use of a frame moving with the boundary is useful. This leads to curved coordinates. A flatness based parameterization seems being possible if distributed control is admitted, see [29] where a formal series solution is derived. However, while series convergence has been proven for the planar case [29], the question of convergence is open for the case of a bent interface. Acknowledgement. The authors would like to thank Daniel Groß for early contributions to the results presented in this paper, and in particular for writing the simulation programs.

References 1. Y. Aoustin, M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Theory and practice in the motion planning and control of a flexible robot arm using Mikusi´ nski operators. In Proc. 5th Symposium on Robot Control, pages 287–293, Nantes, France, 1997.

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2. S. Boschert, K.G. Siebert, E. B¨ ansch, K.W. Benz, G. Dziuk, and T. Keiser. Simulation of industrial crystal growth by the vertical Bridgman method. In W. J¨ ager and H.-J. Krebs, editors, Mathematics - Key Technology for the Future, pages 315–342. Springer, 2003. 3. S. Brandon and J. J. Derby. Heat transfer in vertical Bridgman growth of oxides: effects of conduction, convection, and internal radiation. J. of Crystal Growth, 121:473–494, 1992. 4. H. Chen and L. Rodino. General theory of pde and Gevrey classes. In Qi Minyou and L. Rodino, editors, General Theory of Partial Differential Equations and Microlocal Analysis, pages 6–81. Addison Wesley Longman, Harlow, Essex, 1996. 5. W. B. Dunbar, N. Petit, P. Rouchon, and P. Martin. Motion planning for a nonlinear Stefan problem. ESAIM: COCV (Control, Optimisation and Calculus of Variations), 9:275–296, 2003. 6. M. Fliess, J. L´evine, P. Martin, and P. Rouchon. A Lie-B¨ acklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, AC–44:922–937, 1999. 7. M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. Internat. J. Control, 61:1327– 1361, 1995. 8. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Syst`emes lin´eaires sur les op´erateurs de Mikusi´ nski et commande d’une poutre flexible. In ESAIM Proc., volume 2, pages 183–193, 1997. (http://www.emath.fr/proc). 9. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Controlling the transient of a chemical reactor: a distributed parameter approach. In Proc. Computational Engineering in Systems Application IMACS Multiconference, (CESA’98 ), Hammamet, Tunisia, 1998. 10. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. A distributed parameter approach to the control of a tubular reactor: a multi-variable case. In Proc. 37th IEEE Conference on Decision and Control, pages 439–442, Tampa, FL, 1998. 11. M. Gevrey. La nature analytique des solutions des ´equations aux d´eriv´ees partielles. Ann. Sci. Ecole Norm. Sup., 25:129–190, 1918. 12. W. Haas and J. Rudolph. Steering the deflection of a piezoelectric bender. In Proc. 5th European Control Conference, Karlsruhe, Germany, 1999. 13. C. D. Hill. Parabolic equations in one space variable and the non-characteristic Cauchy problem. Communications on Pure and Applied Mathematics, XX:619– 633, 1967. 14. A.S. Jordan. Some thermal and mechanical properties of InP essential to crystal growth modelling. J. of Crystal Growth, 71:559–565, 1985. 15. A. Kugi. Infinite-dimensional control of piezoelectric structures. In Proc. 3rd European Conference on Structural Control (3ECSC), 12-15 July 2004, volume 2, Vienna, Austria. 16. B. Laroche. Extension de la notion de platitude ` a des syst`emes d´ecrits par des ´ Nationale ´equations aux d´eriv´ees partielles lin´eaires. Th`ese de Doctorat, Ecole Sup´erieure des Mines de Paris, 2000. 17. B. Laroche, Ph. Martin, and P. Rouchon. Motion planning for the heat equation. Int. J. Robust and Nonlinear Control, 10:629–643, 2000.

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18. A. F. Lynch and J. Rudolph. Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems. Internat. J. Control, 2002. 19. Ph. Martin, R. M. Murray, and P. Rouchon. Flat systems, equivalence and feedback. In A. Ba˜ nos, F. Lamnabhi-Lagarrigue, and F. J. Montoya, editors, Advances in the Control of Nonlinear Systems, volume 264 of Lecture Notes in Control and Inform. Sci., pages 3–32. Springer-Verlag, 2001. 20. Ph. Martin, R. M. Murray, and P. Rouchon. Flat systems: open problems, infinite dimensional extension, symmetries and catalog. In A. Ba˜ nos, F. LamnabhiLagarrigue, and F. J. Montoya, editors, Advances in the Control of Nonlinear Systems, volume 264 of Lecture Notes in Control and Inform. Sci., pages 33–57. Springer-Verlag, 2001. 21. T. Meurer, J. Becker, and M. Zeitz. Flatness based feedback tracking control of a distributed parameter tubular reactor model. In Proc. European Control Conference (ECC), Cambridge, 2003. 22. T. Meurer and M. Zeitz. A novel design approach to flatness-based feedback boundary control for nonlinear reaction–diffusion systems with distributed parameters. In W. Kang, C. Borges, and M. Xiao, editors, New Trends in Nonlinear Dynamics and Control, Lecture Notes in Control and Information Sciences, pages 221–236. Springer-Verlag, New-York, 2003. 23. T. Meurer and M. Zeitz. Feedforward and feedback tracking control of nonlinear diffusion-convection-reaction systems using summability methods. Industrial & Engineering Chemistry Research, 44:2532–2548, 2004. 24. T. Meurer and M. Zeitz. Flatness-based feedback control of diffusion-convectionreaction systems via k-summable power series. In F. Allg¨ ower, editor, Preprints 6th IFAC-Symposium on Nonlinear Control Systems (NOLCOS 2004), volume 1, pages 191–196, Stuttgart, 2004. 25. R. Rothfuß, U. Becker, and J. Rudolph. Controlling a solenoid valve — a distributed parameter approach. In Proc. 14th Int. Symp. Mathematical Theory of Networks and Systems — mtns 2000, Perpignan, France, 2000. 26. P. Rouchon. Motion planning, equivalence, infinite dimensional systems. Int. J. Appl. Math. Comput. Sci., 11:165–188, 2001. 27. P. Rouchon and J. Rudolph. R´eacteurs chimiques diff´erentiellement plats : planification et suivi de trajectoires. In J. P. Corriou, editor, Automatique et proc´ed´es chimiques — R´eacteurs et colonnes de distillation, chapter 5, pages 163–200. Herm`es Science Publications, 2001. 28. J. Rudolph. Flatness based control of distributed parameter systems. Berichte aus der Steuerungs- und Regelungstechnik. Shaker Verlag, Aachen, 2003. 29. J. Rudolph, J. Winkler, and F. Woittennek. Flatness based control of distributed parameter systems: Examples and computer exercises from various technological domains. Berichte aus der Steuerungs- und Regelungstechnik. Shaker Verlag, Aachen, 2003. 30. J. Rudolph, J. Winkler, and F. Woittennek. Flatness based trajectory planning for two heat conduction problems in crystal growth technology. e-STA (Sciences et Technologies de l’Automatique), 1(2), 2004. 31. J. Rudolph and F. Woittennek. Motion planning for simple elastic structures. In Proc. 3rd European Conference on Structural Control (3ECSC), 12-15 July 2004, volume 1, Vienna, Austria. 32. J. Rudolph and F. Woittennek. Flachheitsbasierte Randsteuerung von elastischen Balken mit Piezoaktuatoren. at — Automatisierungstechnik, 50:412–421, 2002.

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33. R. Sheinman, S. Brandon, and D.R. Lewin. Optimal control of vertical Bridgman crystal growth. Thessaloniki, 2003. EMCC-3. 34. C. Stelian, T. Duffar, J.-L. Santailler, and I. Nicoara. Influence of temperature oscillations on the interface velocity during Bridgman crystal growth. J. of Crystal Growth, 237:1701–1706, 2002. 35. D. Vizman, I. Nicoara, and G. M¨ uller. Effects of temperature asymmetry and tilting in the vertical Bridgeman growth of semi-transparent crystals. J. of Crystal Growth, 212:334–339, 2000. 36. M. O. Wagner, T. Meurer, and M. Zeitz. Feedforward control of nonlinear diffusion-convection-reaction systems using k-summable power series. In F. Allg¨ ower, editor, Preprints 6th IFAC-Symposium on Nonlinear Control Systems (NOLCOS 2004), volume 1, pages 149–154, Stuttgart, 2004. 37. H. Weimann, J. Amon, Th. Jung, and G. M¨ uller. Numerical simulation of the growth of 2” diameter GaAs crystals by the vertical gradient freeze technique. J. of Crystal Growth, 180:560–565, 1997.

Model–based฀Nonlinear฀Tracking฀Control฀of Pressure฀Swing฀Adsorption฀Plants Matthias฀Bitzer1,2 1฀

Robert฀Bosch฀GmbH,฀CR/AEF,฀Postfach฀30฀02฀40,฀70442฀Stuttgart,฀Germany. [email protected] 2฀ The฀work฀was done while฀the author฀was with฀Institut฀f¨ ur Systemdynamik฀und Regelungstechnik,฀Universit¨ at฀Stuttgart,฀70569฀Stuttgart,฀Germany. Summary.฀ Pressure฀swing฀adsorption฀(PSA)฀processes฀are฀used฀for฀the฀separation and฀ purification฀ of฀ gas฀ mixtures.฀ PSA฀ plants฀ are฀ composed฀ of฀ several฀ fixed–bed adsorbers฀and฀are฀operated฀as฀cyclic฀multi–step฀processes,฀i.e.฀their฀plant฀structure is฀changed฀from฀one฀cycle฀step฀to฀the฀next.฀Each฀fixed–bed฀adsorber฀is฀modeled฀as a฀nonlinear฀distributed฀parameter฀system฀describing฀the฀nonisothermal฀adsorption process.฀For฀the฀control฀of฀the฀product฀purity,฀a฀process฀control฀scheme฀consisting฀of a฀nonlinear฀feedforward฀and฀a฀linear฀feedback฀control฀is฀presented.฀The฀feedforward control฀is฀set฀up฀by฀use฀of฀an฀inverse฀reduced–order฀model฀which฀is฀approximating the฀I/O฀behavior฀of฀the฀considered฀2–bed฀PSA฀plant฀and฀is฀derived฀on฀the฀basis฀of฀its rigorous฀distributed฀parameter฀model.฀The฀model฀inversion฀is฀discussed฀by฀two฀types of฀reduced–order฀models:฀a฀Hammerstein฀model฀which฀neglects฀the฀internal฀plant structure฀and฀a฀hybrid฀lumped฀parameter฀model฀which฀takes฀the฀cyclic฀multi-step process฀into฀account.฀The฀designed฀trajectory฀control฀scheme฀is฀verified฀using฀the rigorous฀simulation฀model฀of฀the฀PSA฀plant฀and฀validated฀by฀means฀of฀laboratory experiments.

Keywords:฀Pressure฀swing฀adsorption,฀distributed฀parameter฀system,฀travelling฀ concentration฀ profiles,฀ periodic฀ process฀ operation,฀ hybrid฀ multi–step process,฀nonlinear฀tracking฀control,฀model฀inversion,฀feedforward฀control,฀feedback฀control.

1฀Introduction Pressure฀swing฀adsorption฀(PSA)฀is฀a฀standard฀process฀technique฀for฀the฀separation฀of฀gas฀mixtures฀[20].฀The฀plants฀consist฀in฀general฀of฀several฀fixed–bed adsorbers฀and฀are฀operated฀as฀cyclic฀multi–step฀processes,฀i.e.฀the฀flow฀rates between฀the฀different฀adsorbers฀and฀their฀direction฀are฀changed฀by฀the฀switching฀ of฀valves฀ at฀the฀ transition฀ from฀ one฀ cycle฀ step฀ to฀ the฀ next.฀ Thereby,฀a periodic฀operation฀is฀realized฀for฀the฀adsorption฀process.฀In฀this฀contribution, a฀2–bed฀pressure฀swing฀adsorption฀plant฀for฀the฀production฀of฀oxygen฀from

T. Meurer et al. (Eds.): Control and Observer Design, LNCIS 322, pp. 403–418, 2005. © Springer-Verlag Berlin Heidelberg 2005

404

M. Bitzer Process Monitoring and Control Adjustment of Product Stream

Binary Valve Signals

Concentration

Produktion

Tank

Measured Signals (Temperature, Pressure)

T

p

#

!!

!

!" !$! ""

08

06

04

"!!

Nitrogen Exhaust Gas Compressor

Pressure Equalization I

2

Purge (Desorption)

05

Production (Adsorption)

1

03

1 2

01

01−

2

1

02

0

Air Feed Gas

!

2

!

#%)'($&( ! #%)'($&( "

Oxygen Product

Concentration Wave nim/lN ni mortsffotS % ni lietnaloM− O 07 001

p

52

T

# !"

02

# !"

51

Pres. Eq.

o

Purge Gas

C ni rutarepmeT

Q

53

T

FC

03

p

1 2

Pressure Equalization II

Fig. 1. Flowsheet of a 2–bed pressure swing adsorption unit for the production of i , i ∈ {1, 2} and the oxygen from air with travelling oxygen concentration waves yO 2 coupling schemes for the adsorbers during a 4–step cycle.

air is considered. Its flowsheet is shown in Figure 1. Each fixed–bed adsorber is described by a nonlinear model with distributed parameters [22]. The implementation of a rigorous PSA model within e.g. the simulation environment Diva [13] enables its dynamical analysis and the evaluation of new control schemes. A characteristic feature of PSA plants concerns the occurrence of travelling concentration waves which are alternating their propagation direction as a consequence of the periodic process operation. In accordance with the cyclic coupling of the fixed–bed adsorbers, the occurring waves travel back and forth within the two adsorber beds and are thereby changing their shape, see Figure 1. The cycle time as well as the duration of the cycle steps do considerably affect the product concentration, because they determine the extent of breakthrough of a concentration front at the product end of the adsorber beds. The cycle time is therefore considered to be the manipulated variable of the process.

Nonlinear Tracking Control of Pressure Swing Adsorption Plants

405

The appropriate operation of the PSA plant requires the solution of two control tasks in order to guarantee a desired purity of the product, i.e. the average concentration in the oxygen tank, see Figure 1. These control tasks comprise the stabilization of operating points as well as the tracking control for set–point changes. The main topic of this contribution is the tracking control of the product purity. Therefore, a process control scheme is presented which consists of a feedforward and a feedback control. The paper is organized as follows: in the next section, the model of the 2–bed PSA plant for the production of oxygen from air is briefly introduced and the occurring control problem is specified. Then, the plant dynamics is analyzed by means of simulation studies with the rigorous model using the simulation environment Diva [13]. The derivation of low–order control design models is exemplarily shown for a Hammerstein model [4] and a hybrid lumped parameter model which is based on a wave approach [2]. The feedforward control is designed by an inverse I/O representation of such a low–order model of the PSA plant. The model inversion is explained for the two types of reduced–order I/O models in the subsequent section. Finally, the set–up of the feedback control is discussed and the whole process control concept is evaluated by simulation studies with the rigorous model and validated by laboratory experiments.

2 Two–Bed Pressure Swing Adsorption Plant The considered 2–bed PSA plant, see Figure 1, is used for the production of oxygen from air [7]. The produced oxygen is stored in a tank from which it is taken off by the consumer. The operation cycle consists of four steps: adsorption, pressure equalization I, purge, and pressure equalization II. The related four coupling schemes of the two adsorbers are depicted at the bottom of Figure 1. 2.1 Nonlinear PSA Plant Model Each adsorber consists of a series connection of a prelayer and an adsorption layer with space ranges 0 ≤ z ≤ L1 and L1 ≤ z ≤ L2 respectively, see Figure 1. The adsorption layer model [22] considers air as a binary mixture of oxygen and nitrogen, and emanates from two phases, i.e. a gaseous phase and an adsorbed phase. The prelayer adsorbs moisture, which is not considered. Therefore, the adsorbed phase is neglected in the model of the prelayer. The distributed parameter model for the adsorption layer of each adsorber i ∈ {1, 2} consists of six quasilinear partial differential algebraic equai (z, t) in the tions (PDAEs) for the pressure pi (z, t), oxygen mole fraction yO 2 i gaseous phase, adsorbed amounts qk (z, t), k ∈ {O2 , N2 }, temperature T i (z, t), and molar flux n˙ i (z, t). The states depend on one space coordinate z and on time t. The model of the product tank is given by three ordinary differential

406

M. Bitzer Table 1. Model of 2-bed PSA plant [2, 5, 22].

Model of adsorption layer for each of the two adsorbers i ∈ {1, 2}: Mass „Balance: « X ∂q i ∂pi pi ∂T i 1 ∂ n˙ i k + ρ − = − p RT i ∂t T i ∂t ∂t Ap ∂z k Component Mass Balance: ! i i i i i pi ∂T i ∂qO yO 1 ∂ n˙ O2 i ∂p i ∂yO2 2 2 − + p =− y + ρp O2 i i RT ∂t T ∂t ∂t ∂t Ap ∂z Component Mass Balance: “ ” ∂qki = kkLDF qk∗ − qki , k ∈ {O2 , N2 } ∂t i ) calculated using IAS–Theory [24] with qk∗ = qk∗ (pi , T i , yO 2 Energy Balance: ! X 0 i ∂T i X ∂q i ∂pi 0 + ρp cˆA + c cp + ρp − ρp cp qk ΔHkst k − ∂t ∂t ∂t k k 0 “ ” i cp i ∂T 4αW n˙ = TW − T i − Dp Ap ∂z Ergun Equation: ∂pi η (1 − p )2 i M 1 − p “ i ”2 0= + 150 n˙ + 1.75 n˙ 3 2 ∂z c p dp A p c 3p dp A2p Initial Conditions (ICs): pi (z, 0) = pi0 (z) , T i (z, 0) = T0i (z) , qki

(z, 0) =

i qk0

(z) ,

k ∈ {O2 , N2 }

i i yO (z, 0) = yO (z) , 2 20

z ∈ (L1 , L2 ), t>0 z ∈ (L1 , L2 ), t>0 z ∈ (L1 , L2 ), t>0

z ∈ (L1 , L2 ), t>0

z ∈ (L1 , L2 ), t≥0 z ∈ [L1 , L2 ] z ∈ [L1 , L2 ]

The boundary conditions (BCs) depend on the connections between the adsorbers during the lth cycle step, see Figures 1 and 2. A similar model of 4 partial differential algebraic equations with BCs and ICs is given for the prelayer. Model of oxygen tank: ` ´ dxt ` ´ i (L2 , t), T i (L2 , t), n˙ tout (t) = f t xt , pi (L2 , t), yO ODE: B t xt 2 dt

t>0

xt (0) = xt0 ˆ ˜T t with state vector xt = pt , yO , Tt . 2 IC:

t equations (ODEs) for the pressure pt (t), the oxygen mole fraction yO , and 2 t t the temperature T (t). The molar flow rate n˙ out (t) of the tank output is a time–variant operational parameter which can be adjusted, see Figure 1. The model of the 2–bed PSA plant is partially given in Table 1.

Nonlinear Tracking Control of Pressure Swing Adsorption Plants

407

2.2 Cyclic Operation and Control Problem Each PSA plant is operated according to a specific cycle which determines the periodical operation mode of the plant. The considered 2–bed plant is run by a 4–step cycle which is represented by the finite state automaton [23] according to Figure 2. Thereby, the two adsorbers are operated in a phase shifted manner in order to attain a quasi–continuous production, see the coupling schemes in Figure 1. The switching from one cycle step to the next determines the periodic operation of the plant and steers its underlying continuous dynamics. Therefore, the cycle time Tk = Δt1k + Δt2k + Δt3k + Δt4k is the manipulated variable of the process, see Figure 2. It is worth noting that the cycle time Tk is a rather unconventional manipulated variable in control design, since it is contained only implicitly in the plant model. The controlled ¯˙ out ¯˙ O2 ,out /n variable is the time–averaged purity of the process, i.e. Pk+1 = n tk +Tk t tk +Tk t 1 1 t ¯ ¯ with n˙ O2 ,out = Tk tk yO2 n˙ out dt and n˙ out = Tk tk n˙ out dt. PSA plants can be classified as hybrid distributed parameter systems with a time–varying cycle time Tk which is used as the manipulated variable as well as the sampling time for the discrete–time output yk = Pk . These properties have to be considered in course of the design of the process control.

!#%&((&(% )$ !#! * '# $&" '(%& Adsorber 1 P. Equal. II

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Adsorber 2 P. Equal. I

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Adsorber 2 Production

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Adsorber 1 P. Equal. I

Adsorber 2 P. Equal. II

# """ " !"""

Fig. 2. Finite state automaton of 4–step operation cycle for the considered 2–bed PSA plant, see Figure 1. Transitions from one cycle step to the next are labeled by the switching points tlk [2].

3 Dynamical Analysis by Simulation Studies The dynamical behavior of the 2–bed PSA plant is analyzed by means of simulation studies with the rigorous plant model. For the simulation of the PSA plant, the entire model is implemented in the simulation environment Diva [13]. Thereby, the PDAEs of the distributed parameter models of the two adsorbers are spatially discretized according to the method of lines [2, 13].

408

M. Bitzer

In the following, the stationary and the transient behavior of the plant are presented and discussed with respect to process control tasks. 3.1 Cyclic Steady State PSA plants are inherently dynamic processes due to their periodic operation. If the adsorption plant is operated with fixed cycle step times and when all other input variables are held constant, the plant approaches a so–called cyclic steady state (CSS) [18], i.e. the conditions at the end of each cycle are identical to those at its start: xi (z, t) = xi (z, t + Tk ) and xt (t) = xt (t + Tk ) with i i t i , qO , qN , T i , n˙ i ]T , xt = [pt , yO , T t ]T , Tk = l Δtlk = const., xi = [pi , yO 2 2 2 2 l Δtk = const., and t → ∞. The purity Pk is constant when the plant is operated in CSS, i.e. Pk → P ∗ = const., k → ∞. The calculated purity Pk for the CSS is shown in Figure 3 in dependence of the cycle time Tk and the output molar flow rate n˙ tout . A high oxygen purity is only obtained for both moderate flow rates as well as a rather short cycle time Tk . An increase of the cycle time leads to the decrease of the purity Pk , see Figure 3. This can be explained by means of the breakthrough behavior of the nonlinear travelling concentration waves [10] which are emerging within the fixed–bed adsorbers due to the ad– and desorption phenomena on the one hand and the periodic multi–step 1 (z, t) of the operation on the other hand. In Figure 4, simulation profiles yO 2 oxygen mole fraction are given for a cycle time Tk ∈ { 10 s, 25 s, 50 s }. The 1 (z, t) which are shown for Adsorber 1 are depicted for the first profiles yO 2 cycle step (production) and a cyclic steady state operation of the plant. For large cycle times, the occurring concentration waves break through from one adsorber into the next, see the simulation result for Tk = 50 s in Figure 4. There, the concentration front located at z ≈ 0.27 m is due to a breakthrough which occurred during the preceding pressure equalization II. In course of the shown production step, this front is propagated back into Adsorber 2 and thereby also affects the concentration of the product gas. Such a breakthrough

80

k

Purity P [%]

100

60 40 20

40 T [s] k

60

80 8

6

4

2

t

nout [sl / min]

Fig. 3. Calculated purity Pk of 2–bed PSA plant for the cyclic steady state and in dependence of the cycle time Tk and the output molar flow rate n˙ tout [2].

Nonlinear Tracking Control of Pressure Swing Adsorption Plants

409

explains the decrease of the time–averaged oxygen purity as a consequence of an increase of the cycle time. For further simulation results, see [2, 5]. yO / Cycle Time 10 s

yO / Cycle Time 25 s

Production

2

yO / Cycle Time 50 s

2

2

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0 0

0.2 z [m]

0 0

0 0

0.2 z [m]

0.2 z [m]

Fig. 4. Simulation results illustrating the influence of the cycle time Tk on the propagation and breakthrough behavior of the oxygen mole fraction profile yO2 (z, t). The profiles are shown for Adsorber 1 during the first cycle step (production) [2].

3.2 Transient Behavior of 2–bed PSA Plant The propagation and shape of the concentration waves is influenced when input variables are changed or when unforeseen disturbances occur. As discussed previously, this accordingly affects the concentration in the product tank as well as the purity. The effects of three consecutive step–changes of the cycle time Tk are depicted in Figure 5, where the simulated time curve for the purity Pk is given. The output signal shows a relatively moderate transient behavior, which has even some similarities with step–responses of linear systems. 100

50 [s] Cycle Time T

k

90

Purity P

k

[%]

95

85 80 75 0

500

1000 1500 2000 2500 Time t [s]

45 40 35 30 0

500

1000 1500 2000 2500 Time t [s]

Fig. 5. Transient behavior of purity Pk subject to successive step–changes in the cycle time Tk [2].

410

M. Bitzer

4 Derivation of Simplified Models for Control Design The practicable application of advanced process control concepts is based on low–order dynamic models. Current approaches for the control design of distributed parameter systems therefore require the derivation of reduced–order models which relate the outputs of the plant with its inputs and which approximate the dominant system dynamics [6, 12]. In Section 4.1, the mathematical relation between the cycle time Tk and the purity Pk is analyzed with respect to the cyclic operation of the plant. Then, the approximation of the I/O behavior by use of a Hammerstein model [4, 16] as well as a reduced–order model [2] which is based on a wave approach [12] are discussed in Sections 4.2 and 4.3, respectively. The first approach is a simple I/O model which neglects the inner structure of the process. The latter one includes its inner dynamics and is therefore more sophisticated and highly adapted to the cyclic plant structure. 4.1 Discrete–Time I/O Behavior The correlation between the cycle time Tk and the purity Pk becomes obvious by considering the cyclic nature of the process. From Figure 2, it is evident that the periodic operation of the process is defining the two Poincar´e maps [19] Pk+1 = ψ xk , Tk ; n˙ tout

xk+1 = ϕ xk , Tk ; n˙ tout ,

(1)

for the internal state x(tk ) := xk and the purity Pk of any reduced–order model which approximates the 2–bed PSA plant together with its cyclic and variable structure [2]. The maps ϕ(·) and ψ(·) are naturally defined by passing successively the individual cycle steps according to the finite state automaton in Figure 2. It is pointed out that the cycle time Tk becomes an explicit variable in the notation (1) [2]. Since the cycle time Tk is the manipulated and the purity Pk the controlled variable, the Poincar´e maps (1) are a discrete– time representation of the global I/O–behavior of the respective reduced–order model under consideration and can be summarized according to xk+1 Pk+1

=

ϕ (xk , Tk ; n˙ tout ) ψ (xk , Tk ; n˙ tout )

,

x1 = x(0),

k = 1, 2, . . .

(2)

yk = Pk . The above notation (2) may also be utilized in order to express the CSS of the plant, see Section 3, i.e. xk → x∗ , Pk → P ∗ for k → ∞ and x∗ = ϕ (x∗ , Tk ; n˙ tout ), yk = P ∗ = ψ (x∗ , Tk ; n˙ tout ). Consequently, the CSS xk = x∗ is only a function of the cycle time Tk as well as the adjusted molar flow rate n˙ tout , and the purity Pk = P ∗ is therefore given as P ∗ = ψ x∗ (Tk ; n˙ tout ), Tk ; n˙ tout =: h∗ Tk ; n˙ tout .

(3)

The representation (3) is a very compact notation for the stationary I/O behavior of the 2–bed plant which is depicted in Figure 3.

Nonlinear Tracking Control of Pressure Swing Adsorption Plants

411

4.2 Hammerstein Model The examination of the I/O dynamics by use of the rigorous simulation model shows that the transients with regard to the purity Pk are rather moderate, see Figure 5. This indicates that the I/O behavior can roughly be approximated by a Hammerstein model [16] which consists of a series connection of a static nonlinearity h∗ (·) (3) for the stationary I/O behavior and an identified linear discrete dynamic model, see Figure 6. The static nonlinearity is calculated by use of the rigorous simulation model and then stored in a look–up table. For the approximation of the I/O dynamics, a second–order model is chosen and identified by means of step–responses of the rigorous simulation model. Its discrete–time transfer function [1] in the domain of the Z–transform holds ˜ h(z) = 1/(2.513z 2 − 1.856z + 0.343). Cycle Time

Tk

h∗(·) Stationary I/O model

uk

˜ h(·)

Purity

Pk

Identified linear I/O dynamics

Fig. 6. Approximation of I/O behavior of PSA plant by Hammerstein model.

4.3 Reduced Order Model by Means of Wave Approach In order to achieve a more precise approximation of the plant dynamics than given by the Hammerstein model, the dominant system dynamics has to be taken into account. Such a concise approximation of the I/O behavior of the PSA plant requires the approximation of the periodically travelling concentration waves together with their breakthrough behavior, see Figure 4. The approximated concentration profiles enable the calculation of the input signals of the tank and therefore of the purity. The travelling concentration profiles are approximated by the construction of a wave approach using the wave function [12] l + yO2 (z, t) = yO 2

l u − yO yO 2 2 −ρ(z−s) 1+e

(4)

l with the position s, the slope ρ, and the lower and upper asymptotes yO 2 u and yO2 . Thereby, a model reduction is achieved by introducing the position s as a new state variable of the reduced–order model [12]. For an appropriate approximation of the concentration profiles, it is necessary that the wave approach is constructed in such a way that the various operating conditions and the occurring wave patterns are captured [2]. This concerns on the one hand the complicated concentration patterns due to the cyclic operation and variable plant structure, and on the other hand the interaction of concentration fronts due to the occurrence of breakthrough phenomena from one adsorber

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into the next, see Figure 4. Overall, the construction of such a wave approach leads to a hybrid function which is flexibly designed allowing the handling of transient process situations. The detailed explanation of the set–up of the wave approach is given in [2]. Differential equations for the loci si , i ∈ {1, 2} are derived using the method of characteristics [2] in order to approximate the propagation behavior of the concentration fronts. The pressure as well as the adsorbed amounts are approximated by means of lumped mass balances. This leads to a reduced–order model of 11 ordinary differential equations for the loci si (t) and the upper u (t) of the wave approach [2], the spatially–averaged lumped asymptote yO 2 i pressures p¯ (t), the spatially–averaged adsorbed amounts q¯ki (t), k ∈ {O2 , N2 }, t and the pressure pt (t) as well as the oxygen mole fraction yO (t) of the tank [2]. 2 Due to the variable structure of the plant together with the hybrid set–up of the wave approach, the model is given in dependence of the lth cycle step and can be summarized according to x˙ = f l x, n˙ tout ,

t ∈ (tlk , tl+1 k ]

(5)

1 1 2 2 t u , p¯1 , q¯O , q¯N , p¯2 , q¯O , q¯N , pt , y O ]T with the state vector x = [ s1 , s2 , yO 2 2 2 2 2 2 l l th and the initial condition x(tk ) = xk at the beginning of the l cycle step during the k th cycle [2]. An analytical solution of the reduced–order wave model (5) cannot be calculated due to its complexity. Accordingly, the global discrete time I/O behavior (2) can only be evaluated numerically. In Figure 7, the simulated profiles yO2 for the oxygen mole fraction of the reduced–order wave model are exemplarily compared to the ones calculated with the rigorous model. The profiles are depicted for the first cycle step (production). Further simulation results of the reduced–order wave model are provided in [2].

Oxygen mole fraction yO [−] 2 Adsorber 1 Oxygen mole fraction yO [−]Adsorber 2 2

1

reduced rigorous

Production Production

0.8 0.6 0.4 0.2 0 0

0.1

0.2

0.3 ζ [m]

0.4

0.5

0.6

Fig. 7. Comparison of the reduced–order wave model and the rigorous model during the first cycle step (production). The oxygen mole fraction profiles yO2 (ζ, t) are plotted over the two adsorber beds (with ζ ∈ [0, 2L2 ], i.e. ζ = z for Adsorber 1 and ζ = 2L2 − z for Adsorber 2, respectively). The plots are depicted for a cycle time Tk = 40 s and a product stream n˙ tout = 5 sl/min [2].

Nonlinear Tracking Control of Pressure Swing Adsorption Plants

413

5 Nonlinear Model–based Tracking Control The proposed process control concept for the 2–bed PSA plant comprises a feedforward as well as a feedback control and is depicted in Figure 8. The feedforward control adjusts the cycle time in dependence of the desired purity as well as the desired product stream, and the feedback control is used in order to compensate disturbances and model uncertainties. Such a two–degree–of– freedom control structure is utilized in many practical applications and enables the independent design of the feedforward and the feedback control [11, 14, 9]. The structural organization of the section follows the design tasks which are defined by the control scheme according to Figure 8. First, the design of the feedforward as well as the feedback control are explained in Sections 5.1 and 5.2. Then, the designed process control scheme is validated by means of simulation studies with the rigorous plant model in Section 5.3 as well as with laboratory experiments in Section 5.4. 5.1 Design of Feedforward Control The set–up of a feedforward control requires the calculation of an inverse representation of a respective I/O model of the plant [14, 9]. The model inversion is explained for the reduced–order models derived in Section 4. The Hammerstein model allows an analytical inversion using a flatness-based approach [8], whereas the reduced–order wave model can only be inverted numerically [2, 3]. Flatness–based Approach by use of Hammerstein Model The feedforward control can easily be designed by exploiting the flatness of ˜ the Hammerstein model [8, 9]. The linear dynamic model h(z) = n(z)/d(z) = 2 1/(2.513z − 1.856z + 0.343) is parameterized by the flat output X(z) [8] according to U (z) = d(z)X(z) = (2.513z 2 − 1.856z + 0.343)X(z), P (z) = n(z)X(z) = X(z) [9], with U (z), P (z) representing the in– and output of Adjusted Product Stream

Desired Product Stream

n˙ t,d out Pkd

Trajectory Generator

Desired Purity

n˙ tout(t) Pkd

Feedforward Control

Feedforward Cycle Time

Tkd

Desired Trajectory + _

ek

c + Discrete Linear ΔTk Controller +

1

1

Tk

2−Bed PSA Plant

Purity

Pk

Limiter

Fig. 8. Block diagram of the feedforward and the feedback control for the purity Pk of the PSA plant [2].

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M. Bitzer

the identified linear dynamic model in the domain of the Z–transform, see Figure 6. The inverse model is then obtained in dependence of the desired d d purity Pkd as uk = 2.513Pk+2 − 1.856Pk+1 + 0.343Pkd and the cycle time Tkd with the inverse static nonlinearity according to Tkd = h−1 ˙ tout ) [4]. ∗ (uk ; n Numerical Inversion of Wave Model For the set–up of the feedforward control by use of the reduced–order wave model (5), the global I/O behavior (2) needs to be inverted, i.e. Tkd = d ψ −1 (xk , Pk+1 , n˙ tout ). The I/O relation (2) and its inverse representation cannot be evaluated analytically for the wave model (5). However, the inverse problem is identical to the solution of the zero value problem d 0 = Pk+1 − ψ xk , Tkd ; n˙ tout

(6)

which is solved numerically for Tkd by applying the shooting method [17]. The numerical solvability of the inverse problem (6) premises the existence of a solution Tkd which requires a careful motion planning for the desired trajectory Pkd and the desired product stream n˙ tout (t), see Figure 8. In order to increase the robustness and the solvability of the numerical model inversion, the zero value problem (6) is slightly reformulated and the future dynamics is taken into account [2, 3]. I.e., a shooting horizon NH > 1 is chosen and d at the end of this time horizon is considered leading to the purity Pk+N H d d d the zero value problem 0 = Pk+N ; n˙ tout . − ψ¯ xk , Tkd , Tk+1 , . . . , Tk+N H H −1 This modified end value problem requires the determination of NH cycle d d , j = 0, 1, . . . , NH − 1. For simplicity, the Tk+j are chosen as times Tk+j d d Tk+j (ΔTkd ) = Tk−1 + (j + 1)ΔTkd which leads to the new zero value problem d (7) − ψ˜ xk , ΔTkd ; n˙ tout 0 = Pk+N H

˜ represents the simulation over NH for the single variable ΔTkd . The map ψ(·) cycles and then the calculation of the purity Pk+NH (ΔTkd ) = ψ˜ xk , ΔTkd ; n˙ tout th for the NH cycle. The feedforward cycle time Tkd for the k th cycle is chosen d d as Tk = Tk−1 + ΔTkd and the Tk+j , j ≥ 1 are rejected. If no solution for ΔTkd exists, a replanning of the desired trajectory is necessary. When both the desired trajectory Pkd and the shooting horizon NH are reasonably chosen, d then Pk+1 = ψ(xk , Tkd ; n˙ tout ) ≈ Pk+1 even though (7) has been solved instead of (6). The proposed strategy therefore allows the robust numerical inversion of the reduced–order wave model3 (5) and the calculation of a feedforward control sequence Tkd for a transient set–point change [2, 3].

3

The rigorous simulation model, see Table (1), may equally be inverted numerically in order to determine the feedforward cycle time Tkd . However, a real–time calculation is not possible due to the high order of the model [2, 3].

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415

5.2 Design of Feedback Control In the open–loop mode, the desired purity Pkd cannot be exactly tracked due to model errors and disturbances. Therefore, a feedback control is necessary for the compensation of control errors and the robust performance of the PSA plant. Simulation results with the rigorous model showed that the I/O dynamics of the PSA plant are close to stable linear ones, see Figure 5. Within the vicinity of a desired trajectory, it is therefore assumed that the plant can be stabilized against disturbances by a linear control law. In the following, a discrete–time PID control is used as feedback. The parameters of the PID control are adjusted by use of simulation studies with the rigorous model. 5.3 Simulation Results The designed process control scheme is implemented in the simulation environment Diva and tested with the rigorous simulation model. The process control which is designed with the Hammerstein model shows a good tracking performance and compensation of model uncertainties as long as the planned trajectories for the desired purity Pkd are reasonably chosen and the adjusted flow rate n˙ tout is held constant, see the simulation result in Figure 9. 60

Purity P

70 60 0

Desired Purity Purity 1000 2000 Time t [s]

3000

Cycle Time Tk [s]

80

k

[%]

90

55

Feedforward Overall

50 45 40 0

1000 2000 Time t [s]

3000

Fig. 9. Closed–loop simulation of the tracking control of the purity Pk with a decrease and subsequent increase of the desired purity Pkd . The feedforward cycle time Tkd is calculated with the inverse Hammerstein model [4].

A variation of the flow rate n˙ tout leads to control errors in the purity Pk due to the unprecise adjustment of the cycle time Tkd which is calculated with the inverse Hammerstein model, see Figure 10. The reason is that only the stationary influence of the flow rate on the purity is included in the design model [4]. Since the reduced–order wave model takes also the pressure dynamics into account, the influence of a transient flow rate n˙ tout is considered by the design model. The feedforward control is then set–up according to the inverse I/O representation (7) and calculated with the reduced–order wave model. In

40

90 88 86 84 0

Desired Open−Loop Closed−Loop

Cycle Time Tk [s]

Purity P

k

[%]

92

[sl / min]

M. Bitzer

35

6.5

Flow Rate ntout(t)

416

6

5.5

30

1000 2000 Time t [s]

Feedforward Closed Loop

0

1000 2000 Time t [s]

5 0

1000 2000 Time t [s]

[s]

50

70 0

Desired Pur. Purity 1000 Time t [s]

5.5

t

40 35 0

6

Flow Rate nout(t)

80 75

45

d k

85

Cycle Time T

Purity P

k

[%]

90

[sl / min]

Fig. 10. Simulation of a variation of the output molar flow rate n(t). ˙ The feedforward cycle time Tkd is calculated with the inverse Hammerstein model [4].

1000 Time t [s]

5 0

1000 Time t [s]

Fig. 11. Open–loop simulation of set–point change scenario Pkd : 76.7% → 90.0% and n˙ tout : 5.0 sl/min → 6.0 sl/min. The feedforward cycle time Tkd is computed by inverting the reduced–order wave model and then applied to the rigorous model [2].

Figure 11, the open-loop simulation of the rigorous model is given for the set– point change scenario Pkd : 76.7% → 90.0% and n˙ tout : 5.0 sl/min → 6.0 sl/min. The simulation shows an almost negligible tracking error. 5.4 Experimental Validation The proposed process control concept is transferred [2] to a laboratory–scale 2–bed PSA plant [15] which is operated with a 6–step cycle. The plant has been constructed in order to evaluate the potential of PSA plants for the oxygen supply of fuel cell systems [15, 21]. In Figure 12, the application of the presented control scheme to the laboratory plant is shown for the set– point change scenario Pkd : 80% → 70%. This demonstrates the successful implementation of the process control scheme even in view of the measurement noise of the oxygen concentration in the product tank.

417

Desired Purity Oxygen Concentration

75 70 1000

42.5

Feedforward Cycle Time Overall Cycle Time

k

80

[s]

85

Cycle Time T

Concentration / Purity [%

O2

]

Nonlinear Tracking Control of Pressure Swing Adsorption Plants

2000 3000 Time t [s]

4000

42 41.5 41 1000

2000 3000 Time t [s]

4000

Fig. 12. Tracking control applied to laboratory plant [15] for set–point change of purity Pkd : 80% → 70% [2].

6 Conclusion The model–based nonlinear tracking control of PSA plants has been examined. The proposed control scheme consists of a nonlinear feedforward and a linear feedback control. The design of the feedforward control has been demonstrated by a Hammerstein model as well as a reduced–order wave model. The control scheme has been validated with laboratory experiments which showed that it is applicable and efficient. For PSA plants, it is desirable to study also the design of model predictive and optimization based control frameworks and to compare the various approaches. In general, further research is required in order to improve current design strategies for process control schemes of hybrid distributed parameter systems [6, 23]. This concerns the derivation of reduced–order models as well as the set–up of respective (hybrid) control laws. Acknowledgement. The author is indebted to Prof. M. Zeitz for his support during the work on his PhD–thesis. The cooperation with Prof. G. Eigenberger, W. Lengerer, and M. Stegmaier of the Institut f¨ ur Chemische Verfahrenstechnik is furthermore gratefully acknowledged. The research project has been supported by Deutsche Forschungsgesellschaft (DFG) within Sonderforschungsbereich 412. Figures 1–5, 7– 8, and 11–12 have already been published in [2], reprint with friendly permission of VDI–Verlag, D¨ usseldorf.

References 1. J. Ackermann. Abtastregelung. Band I: Analyse und Synthese. Springer, 1983. 2. M. Bitzer. Control of Periodically Operated Adsorption Processes. Fortschritt– Berichte Nr. 8/1038. VDI-Verlag, D¨ usseldorf, 2004. 3. M. Bitzer, K. Graichen, and M. Zeitz. Model-based trajectory control of pressure swing adsorption plants. In F. Allg¨ ower and Gao F., editors, Preprints IFAC Symposium on Advanced Control of Chemical Processes (ADCHEM 2003), volume II, pages 627–632, Hong Kong/China, January 11-14 2004.

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4. M. Bitzer, W. Lengerer, M. Stegmaier, G. Eigenberger, and M. Zeitz. Process control of a 2-bed pressure swing adsorption plant and laboratory experiment. In CHISA 2002, 15th International Congress of Chemical and Process Engineering (CD-ROM), Prague/Czech Republic, August 25-29, 2002. Paper No. 833. 5. M. Bitzer and M. Zeitz. Design of a nonlinear distributed parameter observer for a pressure swing adsorption plant. J. Proc. Control, 12:533–543, 2002. 6. P. D. Christofides. Control of nonlinear distributed process systems: recent developments and challenges. AIChE, 47:514–518, 2001. 7. M. Effenberger and W. Lengerer. Theoretische und experimentelle Analyse und Optimierung eines medizinischen Sauerstofferzeugers auf der Grundlage eines kommerziellen Ger¨ ates. Internal Report, Institut f¨ ur Chemische Verfahrenstechnik, Universit¨ at Stuttgart, 1998. 8. M. Fliess, J. L´evine, P. Martin, and P. Rouchon. Flatness and defect of non– linear systems: introductory theory and examples. Int. J. Control, 61:1327–1361, 1995. 9. V. Hagenmeyer and M. Zeitz. Flachheitsbasierter Entwurf von linearen und nichtlinearen Vorsteuerungen. Automatisierungstechnik, 52:3–12, 2004. 10. F. G. Helfferich and P. W. Carr. Non-linear waves in chromatography: I. Waves, shocks, and shapes. Journal of Chromatography, 629:97–112, 1993. 11. I. M. Horowitz. Synthesis of Feedback Systems. Academic Press, 1963. 12. A. Kienle. Low-order dynamic models for ideal multicomponent distillation processes using nonlinear wave propagation theory. Chemical Engineering Science, 55:1817–1828, 2000. 13. R. K¨ ohler, K. D. Mohl, H. Schramm, M. Zeitz, A. Kienle, M. Mangold, E. Stein, and E. D. Gilles. Method of lines within the simulation environment DIVA for chemical processes. In A. Vande Wouwer, P. Saucez, and W. Schiesser, editors, Adaptive Method of Lines, pages 367–402. CRC Press, Boca Raton/USA, 2001. 14. G. Kreisselmeier. Struktur mit zwei Freiheitsgraden. Automatisierungstechnik, 47:266–269, 1999. 15. W. Lengerer. Stoffanreicherung f¨ ur Brennstoffzellensysteme mittels Druckwechseladsorption. Dissertation, Universit¨ at Stuttgart, 2004. 16. L. Ljung. System Identification - Theory for the User. Prentice Hall, 1999. 17. W. Luther. Gew¨ ohnliche Differentialgleichungen – Analytische und numerische Behandlung. Vieweg, 1987. 18. S. Nilchan and C. C. Pantelides. On the optimisation of periodic adsorption processes. Adsorption, 4:113–147, 1998. 19. P. Plaschko and K. Brod. Nichtlineare Dynamik, Bifurkation und Chaotische Systeme. Vieweg Verlagsgesellschaft, 1994. 20. D. M. Ruthven, S. Farooq, and K. S. Knaebel. Pressure Swing Adsorption. VCH Publishers, New York, Weinheim, Cambridge, 1994. 21. J. St-Pierre and D. P. Wilkinson. Fuel cells: a new, efficient and cleaner power source. AIChE Journal, 47:1482–1486, 2001. 22. J. Unger. Druckwechseladsorption zur Gastrennung - Modellierung, Simulation und Prozeßdynamik. Fortschritt–Berichte Nr. 3/602. VDI-Verlag, D¨ usseldorf, 1999. 23. A. van der Schaft and H. Schumacher. An Introduction to Hybrid Dynamical Systems. Lecture Notes in control and information sciences; 251. Springer, London, Berlin, Heidelberg, New York, 2000. 24. R. T. Yang. Gas Separation by Adsorption Processes. Butterworths, Boston, 1987.

Index

a priori information, 82 accommodation, 218 Ackermann’s formula, 26 adaptive structure, 341, 349 adsorption fixed–bed, 404 layer, 405 pressure swing, 403 process, 403 advanced process control, 318 air traffic, 271 arrival manager (AMAN), 272 automatic flight pilot (AFP), 273 backstepping, 380 behaviour, 201 complete, 203 I/-, 201 non-conflicting, 202 bioprocess, 112 bottom-up design, 208 boundary conditions, 81, 238 boundary control, 387 boundary problem, free, 387 boundary value problem, 238 boundary value process, 81 Butler-Volmer kinetics, 95 calorimetric model, 325 cart-pole system, 155 chemical reactor, 59 semi-batch, 320, 322 collocated variables, 357 communication network, 4 concentration waves, travelling, 408

constrained optimal control finite time (CFTOC), 186–188 infinite time (CITOC), 190, 192, 193, 196 contractions, C0 -semigroup, 363 control, see feedback control, see feedforward control aircraft, 271 cascade, 301, 310 clipped-optimal, 335, 346 clipped-optimal, global, 345 cLQG, 348 energy-based, 376 fault-tolerant, 218 flatness-based, 359 hierarchical, 199, 204 hybrid, 199 infinite dimensional, 353 Lyapunov based, 380 nonlinear, 271 output feedback, 141 receding horizon, 189, 196 robust, 174 state feedback, 139, 188, 189 static shape, 356 controllability, 335, 342 convergent systems asymptotic properties, 135 circle criterion, 138 definitions, 133, 134 interconnection, 136 LMI conditions, 138 observer design, 140 sufficient conditions, 137

420

Index

with inputs, 134 converging-input converging state, 135 cost function, 186 crystal growth, 387 current control, 302, 310 cycle time, 404 cyclic multi–step process, 403 cyclic steady state, 408 damping friction, 335 proportional, 339 Rayleigh, 339 semi-active, 336 structural, 335 decoder dynamics, 7 degeneracy problem, 122 Demidovich condition, 137 detectability, 21, 28, 220 differential algebra, 218 differential flatness, see flatness discrete event systems, 199 discrete time control system, 69 dissipative observer design, 42 dissipative operator, 363 dissipativity, 40 distributed parameter system, 351, 387, 403 hybrid, 407 dynamic programming (DP), 187, 193 electromechanical actuators, 310 electronic commutation, 301, 307 encoder dynamics, 6 energy balance, 95 energy dissipation, 335, 345, 349 error covariance matrix, 105 error linearization approximate, 44 exact method, 37 observer, 37 error system, stabilization, 361 estimation of time derivatives, 221 of uncertain parameters, 224 Euler-Bernoulli beam, 354 evolution equation, 362 exosystem, 3 fault indicator, 224 fault variable, 219

fault-tolerant control, see control FE model, 339 feedback control, 290, 403, 413, 414 linearizing, 398 local, 335, 345 PID, 324, 327 feedback transformation, 147 feedforward control, 239, 289, 327, 413 disturbance, 324 dynamic, 289 finite automaton, 203, 407 finite volume method, 97 flat hybrid system, 223 flat output, 277, 278, 280 flat system, 220 flatness, 302, 303, 387, 413 differential, 271, 277, 316 flight management system (FMS), 273 flight path angle, 275 flow control, 369 formal power series parameterizability, 265 friction joint, 335 friction, nonlinear, 338 fuel cell, 93 molten carbonate, 94 Galerkin method, 98 Galerkin model, 372 generalized inverse, 23 Moore-Penrose inverse, 23 genetic algorithms, 344, 349 Gevrey function, 361, 388, 392, 395 Hamilton’s principle, 356 Hammerstein model, 403, 411 Hautus criterion, 102 heat conduction problem, 387 heat equation, 387 hybrid systems, 185 hybrid lumped parameter model, 403 identifiability, 221 incremental ISS, 138 inertial navigation, 62 inertial navigation system (INS), 293 infinite dimensional systems, 388 input constraints, 238 input-output linearization, 379 input-output system, 220 input-to-state convergence, 134

Index sufficient conditions, 137 integrated navigation system (INS), 290 internal dynamics, 239, 254 internal model, 4 invariance, 54 LaSalle’s principle, 364 invariant error, 56, 59 invariant frame, 55, 58 inverse I/O representation, 415 problem, 71 system, 304 inversion analytical, 413 model, 403, 413 numerical, 413 system, 235, 253, 318 isolability, 220 joint adaptive, 336, 337, 345 nonlinear model, 340 semi-active, 335, 338, 342, 344–346 Kalman criterion, 102 Kalman filter, 104, 345, 346 extended, 117, 294 Karhunen Lo`eve decomposition, 98 Karhunen-Lo`eve expansion, 372 Krylov vectors, 347 L¨ umer-Phillips theorem, 363 least squares approach, 83 lightweight structures, 335 limit solution, 133 linear matrix inequality (LMI), 43 linearizability index, 152 LPV model, 381 LuGre-model, 340, 345 Lur’e systems, 138 Lyapunov function, 345 ISS & OSS, 27, 28 method of lines, 407 modal updating, 339 modal analysis, experimental, 339 modal transformation, 339 model reduction, 98 moving frame method, 59 Multi-Parametric Toolbox (MPT), 197

421

multicriteria optimization, nonlinear, 335, 349 multiproduct batch plant, 199, 209 Navier-Stokes equation, 370 Newton flow, 72 noise triple, admissible, 83 non-anticipating systems, 202 nonlinear observer form, approx., 44 nonlinear program (NLP), 173, 174 nonlinear system, 81, 218, 238 normal form formal, 151 special strict feedforward, 155 strict feedforward, 154 observability, 70, 100, 113, 221, 335, 342, 343 map, 37 matrix, 21, 37 matrix, reduced, 21 uniform global, 71 observer, 11 asymptotic, 120 extended Luenberger, 24 high-gain, 11, 46 invariant, 55, 56, 62, 63 nonlinear, 140 observer normal form, 37 operator functions, 358 opposition control, 380 optimal estimate, 83 optimal location, 343 optimal placement, 335 output trackability, 266 output injection, 4 output regulation, 3 partial differential algebraic equation, quasilinear, 405 partial differential equation, hyperbolic & parabolic, 97 particle filter, 121 asymptotic, 125 path convergence, 271 path planning/following, 285, 291 performance index, linear, 186 periodic process operation, 403 piecewise affine systems, 183, 185, 189 piezo-stack actuator, 335 piezoelectric cantilever, 353

422

Index

Poincar´e maps, 410 Pontryagin minimum principle, 84 power series, 391 approximation, 396 by (N, ξ)-approximate k-sum, 264, 396 by partial sum, 396 proper orthogonal decomposition, 98, 372 quadratic supply rates, 40 quasi steady state assumption, 98 quasi–stationary connection, 244 realtime navigation, 294 reciprocal process, 83 reference surface, 302, 307, 311 relative guidance, 271, 273 Riccati equation, 346 rotation group, 62 sampling time, 7 separation principle, 136 servomechanism, 3 setpoint transition, 238 Shuttle Radar Topography Mission, 336 sideslip angle, 286, 298 tire, 286 simulation environment Diva, 407 SISO, 238, 346, 348 smoothing problem, 81 fixed interval, 82 solution by direct method, 86 solution by iterative method, 88, 90 spectral density matrix, 105 spillover, 343 SPR transfer function, 361 stability, 192, 194 stability boundary, 169 state estimation moving horizon, 67, 74 optimization-based, 67 problem, 69 steady state property, 27, 28 steady-state solution, 133 periodicity, 135 uniqueness, 133 steering angle, 294 stick-slip behavior, 340 strapdown navigation, 293 Stribeck effect, 340

strict feedforward form, 149 special, 152 summation method, 264 supervisor admissible, 202 control theory, 202 generically implementable, 202 least restrictive, 202 switched reluctance machine model, 303 switched reluctance servo-drives, 301 symmetry group, 54 synchronization, 15 three-tank system, 222 time scale dynamic, 206 multiple, 206 tire characteristics, 288 inverse, 289 torque characteristic, 303 torque ripple, 310 track distance, along/cross, 274 tracking, 3 error, 289 problem, 302, 303 tracking control, 359 for SR drives, 308 nonlinear, 403 trajectory planning, 235, 258, 291, 302, 306, 327, 387, 414 truss structures, 335 two-degree-of-freedom control, 235, 289, 318, 413 two-level control architecture, 204 uncertain parameters, 168, 171, 219 value function, 186, 193 Van der Pol oscillator, 15 vehicle controller, lateral, 289 vehicle model, 286 rigorous, 288 vibration suppression, 335 wave approach, 405, 411 wave model, reduced–order, 412 wheel sideforce, 289 zero dynamics, 10, 240 zero value problem, 413