Observer Design for Nonlinear Dynamical Systems: Differential Geometric Methods 3030737411, 9783030737412

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Lecture Notes in Control and Information Sciences 487

Driss Boutat Gang Zheng

Observer Design for Nonlinear Dynamical Systems Differential Geometric Methods

Lecture Notes in Control and Information Sciences Volume 487

Series Editors Frank Allgöwer, Institute for Systems Theory and Automatic Control, Universität Stuttgart, Stuttgart, Germany Manfred Morari, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, USA Advisory Editors P. Fleming, University of Sheffield, UK P. Kokotovic, University of California, Santa Barbara, CA, USA A. B. Kurzhanski, Moscow State University, Moscow, Russia H. Kwakernaak, University of Twente, Enschede, The Netherlands A. Rantzer, Lund Institute of Technology, Lund, Sweden J. N. Tsitsiklis, MIT, Cambridge, MA, USA

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Driss Boutat · Gang Zheng

Observer Design for Nonlinear Dynamical Systems Differential Geometric Methods

Driss Boutat INSA Centre Val de Loire Campus de Bourges Bourges, France

Gang Zheng Inria Lille-Nord Europe Villeneuve d’Ascq, France

ISSN 0170-8643 ISSN 1610-7411 (electronic) Lecture Notes in Control and Information Sciences ISBN 978-3-030-73741-2 ISBN 978-3-030-73742-9 (eBook) https://doi.org/10.1007/978-3-030-73742-9 Mathematics Subject Classification: 93-02, 53-02 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my family —Driss Boutat To my wife Dan, and my children Lisa, Linda and Bona —Gang Zheng

Preface

For engineers, there is always a need to know the states of the studied system to make important decisions to control the system or to reliably predict its future state. A simple and reliable method for obtaining system’s states is to measure them directly. However, in a general complex system, it is not feasible or even possible to directly measure all states due to either technical or economic reason. An alternative to get system’s states is to develop an algorithm to estimate the states. In control community, such an algorithm is named as software sensor or observer. Observer design has attracted lots of attention since the pioneer works of Luenberger and Kalman. This is due to the numerous applications in aerospace, in robotics, in image processing, in communication, in biology and many other areas. It concerns the reconstruction of states of the studied system based on the available measurements and the model of the system (which can be uncertain). To design these observers, the concept of observability is important, and it concerns the ability of estimating states by using available data of inputs and outputs of the systems. These concepts are also relevant for diagnosis or prediction of future behavior of the system. Therefore, the observer design for nonlinear dynamical systems is an important issue in the control theory. During last five decades, different approaches, including differential geometric and differential algebraic ones, have been successfully used to analyze the observability for dynamical systems. And many different types of observers have been proposed in the literature. However, those results related to observability analysis and observer design are dispersed in different articles and books. The students and researchers need to spend lots of time to understand how to deal with observer design problems. Therefore, a monograph dedicated to observer design for nonlinear dynamical systems would be very useful. This is exactly the main reason why this book is written. In order to make the reading more easy, this book starts with the linear dynamical systems case, for which different equivalent definitions of observability are discussed, and several well-known observers are then presented. After having the basic ideas of observability and observer design for linear dynamical systems, we switch to investigate those two problems for nonlinear dynamical systems. When dealing with observability analysis for nonlinear dynamical systems, we adopt the differential vii

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Preface

geometric method which is one of the most used approaches. Concerning the observer design for nonlinear dynamical systems, due to the fact that the results on this topic are vast, this book just lists several well-known nonlinear observers, and then focuses on observer design technique by using differential geometric method. The idea of this technique is to transform the studied nonlinear dynamical system into a simpler form (called as observer normal form) for which we can apply existing observers to achieve the task of state estimation. The goal of this monograph is to present how this method can be used to design observers for different types of nonlinear dynamical systems (including single/multiple outputs, fully/partially observable systems, regular/singular dynamical systems). More specifically, it will complement the existing books on observer design, by presenting some achievements in the nonlinear observer normal forms. This book contains 9 chapters, and they are organized as follows: • Chapter 1 explains the key problem of observation for dynamical systems, and recalls the basic results on observability analysis and observer design for linear and nonlinear dynamical systems. More attention is paid for the technique of observer normal forms. • Chapter 2 deals with a quick overview of the basic concepts, elements and theorems from differential geometry to ensure a clear understanding of the contents of this book. • Chapter 3 discusses and analyzes single-output nonlinear dynamical systems that admit a change of coordinates via which the studied nonlinear dynamical system can be transformed into the so-called nonlinear observer normal form with output injection. • Chapter 4 applies as well a change of coordinates to the output of nonlinear dynamical systems, compared to the result obtained in Chap. 3. It shows how to relax the geometric conditions provided in Chap. 3, by allowing nonlinear output for the desired observer normal form. • Chapter 5 provides the new concept of the extended observer normal form by adding an auxiliary dynamics to the studied nonlinear dynamical systems. It presents as well an algorithm to compute the auxiliary dynamics as well as the change of coordinates. • Chapter 6 relaxes the conditions presented in Chap. 3 in another way, by allowing output-depending matrix in the desired normal form, named as output-depending observer normal form. • Chapter 7 investigates observer design problem for partially observable systems. It shows how to treat general partial observable case when applying differential geometric method, where the notion of commutativity of Lie bracket modulo a distribution is used. • Chapter 8 extends the results on nonlinear dynamical systems with single output to the case with multiple outputs. It treats as well the partially observable case for nonlinear dynamical systems with multiple outputs. • Chapter 9 extends the differential geometric method to design observers for nonlinear singular dynamical systems, which are governed by mixing differential

Preface

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and algebraic equations. It presents the manner to regularize the singular system into a regular one, and then seek for diffeomorphism to transform the regularized system into an observer normal form with output derivative injection. Bourges, France Lille, France October 2020

Driss Boutat Gang Zheng

Contents

1 Observability and Observer for Dynamical Systems . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Observability and Observer for Linear Dynamical Systems . . . . . . . 1.2.1 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Observability and Observer for Nonlinear Dynamical Systems . . . . 1.3.1 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 5 10 13 13 18 28

2 Background on Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Vector Fields: Derivation and Dynamics . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lie Bracket of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Change of Coordinates: Diffeomorphism . . . . . . . . . . . . . . . . . . . . . . 2.5 Integrability, Involutivity and Frobenius Theorem . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 35 38 41 44 50 53

3 Observer Normal Form with Output Injection . . . . . . . . . . . . . . . . . . . . 3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Observer Normal Form with Output Injection . . . . . . . . . . . . . . . . . . . 3.3 Extension to Systems with Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Luenberger-Like Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 63 66 66 67 67 68

4 Observer Normal Form with Output Injection and Output Diffeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diffeomorphism on Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 71 xi

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Contents

4.3 Diffeomorphism for States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 High-Gain Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 85 85 87 88 89

5 Observer Normal Form by Means of Extended Dynamics . . . . . . . . . . 5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Observer Normal Form with Scalar Extended Dynamics . . . . . . . . . 5.3 High-Dimensional Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Adaptive Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 94 102 103 103 105 105 106

6 Output-Depending Observer Normal Form . . . . . . . . . . . . . . . . . . . . . . . 6.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Analysis of the Output-Depending Normal Form . . . . . . . . . . . . . . . . 6.3 Construction of New Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Step-by-Step Sliding Mode Observer . . . . . . . . . . . . . . . . . . . 6.4.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 110 111 121 121 123 124 125

7 Extension to Nonlinear Partially Observable Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Necessary and Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Diffeomorphism on the Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Homogeneous Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 129 138 141 141 142 142

8 Extension to Nonlinear Dynamical Systems with Multiple Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Construction of the Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Necessary and Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Diffeomorphism Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Equal Observability Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Unequal Observability Indices . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 144 146 150 151 151 152

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8.6 Extension to Partial Observer Normal Form with Multiple Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Properties of Δ and Δ⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Reduced-Order Luenberger-Like Observer . . . . . . . . . . . . . . . 8.7.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 156 160 169 169 171 171

9 Extension to Nonlinear Singular Dynamical Systems . . . . . . . . . . . . . . . 9.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Transformation into Regular System . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Necessary and Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Nonlinear Luenberger-Like Observer . . . . . . . . . . . . . . . . . . . 9.4.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 175 182 186 186 188 189

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Chapter 1

Observability and Observer for Dynamical Systems

Abstract Before dealing with the observer design for dynamical systems, we need firstly to analyze whether the states of the studied dynamical system can be observable or not. This property is named as observability in the literature. Therefore, this chapter aims at recalling the existing results on observability analysis approaches and observer design techniques for dynamical systems. We start with the classification of observation problems, the different approaches to analyze the observability and the categories of observers. After that, we recall several important definitions and test conditions for the observability of linear dynamical systems. Some well-known observers are then summarized, including Luenberger observer, Kalman observer and so on. Finally, we present how to analyze observability for nonlinear dynamical systems, and discuss several famous nonlinear observer design techniques.

1.1 Introduction Dynamical systems have been widely used to model different plants to be controlled in many different disciplines, ranging from biology, chemistry, to mechanics and so on. Nowadays, one of the most popular ways to model the real process is to use the state-space equations, described normally by a set of ordinary differential equations, which is named as a continuous-time system in the literature. From engineering point of view, a wide variety of information cannot directly be obtained through measurements from real systems. Due to some economical or technological reasons, we cannot place as many sensors as we want to measure the internal information, since it is expensive, or sometimes impossible. Besides, given a concrete plat, some kinds of inputs (external disturbances for example) and some parameters (such as the constants of an electrical motor, the delay in a transmission system, etc.) are unknown or are not measurable, whose estimations are sometimes needed to be used in the closed-loop controller. Similarly, more often than not, signals from sensors are © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_1

1

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1 Observability and Observer for Dynamical Systems

distorted and tainted by measurement noises. Therefore, although control is the final goal in the control theory, given a concrete model, in order to simulate, to control or to supervise processes, and to extract information conveyed by the signals, we often need to estimate parameters, internal variables, or the unknown inputs. Estimation techniques are, under various guises, presented in many parts of control, signal processing, and applied mathematics. Such an important area gave rise to a huge international literature that cannot be summarized here. Roughly speaking, in automatic control, the estimation covers at least the following topics: • Identification of uncertain parameters in the system’s model, including delays; • Estimation of state variables, which are not measured; • Observation of the fault (unknown input) and isolation. From control point of view, we know that the problem of parameters estimation can be converted into the problem of state observation, by regarding those parameters as additional states, and then extending the system’s state, provided that they are either (piece-wise) constant, or slowly time-varying. In this book, we will focus only on the state estimation problem for the following dynamical system: x˙ = f (x, u) y = h(x),

(1.1)

where x ∈ X ⊂ Rn , u ∈ U ⊂ R p , and y ∈ Y ⊂ Rm represent the state, the control input and the output of the system, respectively. It is assumed that f : X × U → X and h : X → Y are smooth. System (1.1) is said to be a single-output system if the integer m = 1, and to have multiple outputs if m ≥ 2. When dealing with state observation problem for the dynamical system, normally we need to consider the following two important issues: • With the available information (output y, input u and their successive derivatives), is it possible to reconstruct the internal information (state x) over a time interval? • With the available information, how to design a dynamics which enables us to reconstruct the required state x? The first issue is related to the concepts of Observability and Detectability, and different equivalent necessary and/or sufficient conditions are proposed in the literature to judge whether the state x of system (1.1) can be reconstructed or not. The key difference between observability and detectability is the possibility to tune the estimation velocity. Precisely, note x(t) ˆ as the estimation of x(t) of (1.1), then this system is said to be observable if the convergence rate of x(t) ˆ to x(t) can be tuned. Otherwise, it is called to be detectable. A simple example to clarify such a difference is the following linear time-invariant dynamical system: x˙1 = x1 + u x˙2 = −ax2 y = x1 ,

1.1 Introduction

3

with a > 0. It is clear that the first state x1 (t) is observable since we can have immediately its estimation as xˆ1 = y, while the second state x2 (t) is not observable due to the fact that the dynamics of x1 and x2 are completely independent. In other words, we cannot reconstruct x2 from the available information y and u. However, we notice that x2 (t) converges to 0 when t → ∞, since a > 0. This means that we can still reconstruct the state x2 (t) which will converge to 0, but we cannot tune its convergence rate which in fact depends only on the system’s parameter a. Therefore, the above simple system is not observable, but only detectable. In general, observability, defining the possibility to estimate internal states of the studied system via the available measurement and its derivatives [2], has already been widely studied for different types of dynamical systems. Using the elementary algebraic method to analyze observability can be dated back to 1960s in the work of Kalman for linear dynamical systems [20]. However, the generalization of the similar theory to nonlinear dynamical systems is not so trivial. If the studied system is observable, we can then investigate how to design the estimator x(t). ˆ Generally, there exist at least two quite different categories in the literature: • Estimation by using differentiator; • Estimation by using observer. If the studied system is fully observable, all its states can be then algebraically represented as a function of output/input and their derivatives [9]. This implies that an efficient differentiator, enabling to calculate the successive derivatives of the signal, is enough to estimate the states. Different types of differentiators have been proposed in the literature, such as digital differentiator [6], high-gain differentiator [8], tracking differentiator [14], algebraic differentiator [18], high-order sliding mode differentiator [29], and so on. The key issue is that the badly designed differentiator will amplify the high-frequency noise, if the signal is corrupted. For the second category, the list of different types of observers is quite long. From the design methodology point of view, there exist two different methods. The first method tries to directly design an observer. The second method is to transform the studied system into a simpler form (called as normal form) which enables us to apply existing observers. The first idea of normal form is due to [3] for non-autonomous nonlinear dynamical systems and [25] for autonomous nonlinear dynamical systems where the authors introduced the so-called observer canonical form with output injection with all nonlinear terms being only function of the output. Then [27] gave the associated canonical form with output injection for multi-output nonlinear dynamical systems without inputs, and the result for multi-output systems with inputs was studied in [47]. Based on the above works many algorithms have been developed to generalize the existing results, including algebraic approaches [37], geometric approaches [33]. From the convergence performance point of view, we can also classify those techniques into two different categories: asymptotic and non-asymptotic. Definition 1.1 The dynamics x˙ˆ = fˆ(x, ˆ y, u) with xˆ ∈ Rn and the user-chosen funcˆ tion f is an asymptotic observer of (1.1) if

4

1 Observability and Observer for Dynamical Systems

lim ||x(t) ˆ − x(t)|| = 0,

t→∞

where xˆ is called as the estimation of state x. It is said to be a non-asymptotic (or finite-time) observer if there exists a finite-time T > 0, such that ||x(t) ˆ − x(t)|| = 0, ∀t ≥ T. Up to now, many asymptotic observers have been widely studied, such as Kalman observer [21], Luenberger observer [32], adaptive observer [23], high-gain observer [13], unknown input observer [5] and so on. Compared to asymptotic observer, finite-time one was less studied in the literature, which, however, is well appreciated in practice. Different methods have been proposed, such as sliding mode technique [46], impulsive observer [10] and homogeneity [4]. The global finite-time observer based on homogeneity was firstly introduced by [36] for the nonlinear dynamical systems which can be transformed into a linear dynamical system with output injection. After that, [42] extended this idea and proposed a semi-global finite-time observer for the special systems with triangular structure. The global finite-time observer for such a system was studied in [34] by introducing the second gain in [43]. Generally speaking, asymptotic observers have relatively simple structures, but possess only asymptotic convergence. Sometimes, for certain reasons like safety, severe time convergence constraint is imposed for the studied system, then a nonasymptotic (convergence in a finite time) observer is desired. In this book, we will briefly investigate several types of asymptotic and non-asymptotic observers in each chapter. It is worth emphasizing again that the observability and observer design are not two equivalent concepts. A system is observable implies the existence of certain observers, but the inverse is not always true, since the system can be detectable. In other words, the deduced observability condition is only sufficient for certain types of observers, and other sorts of observers (Luenberger-like, high-gain,...) might ask for additional conditions, for example, the well-known Lipschitz condition when treating nonlinear dynamical systems. In this book, we restrict our study only on the observability and will not discuss the detectability. In order to avoid the ambiguity, in this book, when we discuss the observer design, if we do not explicitly mention the words detectable/detectability, then observer always implies the tunable asymptotic or non-asymptotic observer. The following sections will give a brief overview of observability conditions and observer design techniques which we can find in the literature. We will firstly present those results for linear time-invariant dynamical systems, since it is the simplest case, and some results are equivalent for nonlinear dynamical systems. Then we will clarify how those results for linear dynamical systems are extended to treat nonlinear dynamical systems.

1.2 Observability and Observer for Linear Dynamical Systems

5

1.2 Observability and Observer for Linear Dynamical Systems Linear time-invariant dynamical systems are commonly written in the following state-space equation: x˙ = Ax + Bu y = C x,

(1.2)

where x ∈ Rn , u ∈ R p and y ∈ Rm represent the state, the control input and the measurement, respectively, with A ∈ Rn×n , B ∈ Rn× p and C ∈ Rm×n being constant matrices. During last five decades, many different methods have been proposed for observability analysis and observer design for (1.2). Due to the fact that the studied system is linear, we can normally use the elementary algebra (vector space, linear transformation, rank, image) to analyze its properties, such as observability and detectability of systems with known or unknown inputs [35]. For such systems, several different but equivalent definitions of observability have been proposed in the literature [45]. The most common way is the distinguishability, i.e., different initial conditions yield different outputs. Since zero initial conditions produce zero outputs, the above property is equivalent to say that non-zero initial conditions imply non-zero outputs. Moreover, since the output of a system is uniquely determined by its initial conditions, we can also characterize the observability for linear dynamical systems as the ability to reconstruct initial conditions. For linear dynamical systems, this expression leads to the well-known invertibility of the observability Gramian, which can be used as a criteria to test whether the studied system is observable or not. Of course, some other checkable conditions have been proposed to test the observability, such as the famous Kalman rank condition [20], Popov–Belevich–Hautus (PBH) test condition [1], unobservable space and invariant zero set [15]. Some of those mentioned results will be recalled in the following.

1.2.1 Observability Analysis Since the objective is to estimate x(t) from available information y(t) and u(t), it is natural to consider firstly the output equation as follows: y = C x. The trivial case is that the matrix C is left invertible, i.e., m ≥ n and rank C = n. In such a case, the state can be instantaneously obtained via −1 T  C y(t). x(t) = C T C

6

1 Observability and Observer for Dynamical Systems

This trivial case is, however, not realizable in most of applications, since it explicitly implies that • the number of measurement sensors should be large enough (i.e., m ≥ n); • the placement of those sensors should be intelligent enough to satisfy rank C = n. If one of those two conditions is not valid, then instantaneous estimation of x(t) only from y(t) is not possible. We should then rely on the information provided by the dynamical equation x˙ = Ax + Bu to check whether it is possible to estimate x(t). Due to this reason, the estimation of x(t) will not any more instantaneous, but over a time interval. Precisely, for the simple linear ODE described in (1.2), its analytic solution can be written as  t e A(t−s) Bu(s)ds, (1.3) x(t) = e A(t−t0 ) x(t0 ) + t0

where t0 is the initial time, x(t0 ) is the corresponding unknown initial condition and e A(t−t0 ) is named as state transition matrix in the literature. Since the input u and the constant matrices A and B are all known, the intuitive definition of observability can be stated as follows. Definition 1.2 Observability of the state x(t) is equivalent to the reconstructibility of the initial condition x(t0 ) over the time interval [t0 , t]. The reason why the reconstruction of x(t0 ) is over the time interval [t0 , t] is that (1.3) contains the value t0 and the integration of u(t) from t0 to t. Consequently, the observability problem is to deduce the conditions under which x(t0 ) can be estimated through y(s) and u(s) for s ∈ [t0 , t]. By linking the measurable output to the initial condition x(t0 ) we have y(t) = Ce A(t−t0 ) x(t0 ) +



t

Ce A(t−s) Bu(s)ds.

(1.4)

t0

t Remark 1.1 We can observe from (1.4) that the term t0 Ce A(t−s) Bu(s)ds has no real contribution to the observability condition since the matrices A, B, C and the control u(s) for s ∈ [t0 , t] are known. In fact, we can merely redefine a virtual measurement as  t y¯ (t) = y(t) − Ce A(t−s) Bu(s)ds, (1.5) t0

which leads to

y¯ (t) = Ce A(t−t0 ) x(t0 ).

(1.6)

The observability conditions should be equivalent for both (1.4) and (1.6). Due to this observation and for the sake of simplicity, we can set B = 0 in (1.4).

1.2 Observability and Observer for Linear Dynamical Systems

7

According to Definition 1.2, the observability of a linear time-invariant dynamical system refers to estimate its initial condition x(t0 ) via available information y and u. Note that the input–output equation defined in (1.4) reveals the relations between x(t0 ) and y, u, based on which several different but equivalent definitions have been stated in the literature. For the sake of simplicity, denote x0 = x(t0 ) as the initial condition of (1.2), x(t, x0 , u) as its solution defined in (1.3) with the initial condition x0 and the input u, and y(t, x0 , u) as its associated output defined in (1.4). From (1.4), we observe that it defines a one-to-one mapping between y(t) and x0 . Given two different initial conditions x0 and x¯0 , a straightforward calculation shows that y(t, x0 , u) − y(t, x¯0 , u) = Ce A(t−t0 ) [x0 − x¯0 ] , ∀u, ∀t ≥ 0.

(1.7)

Following the above equation, different interpretations lead to different definitions of observability. One interpretation is that different initial conditions should yield different outputs, which gives the following definition. Definition 1.3 System (1.2) is said to be observable if for any initial conditions x0 ∈ Rn and x¯0 ∈ Rn with x0 = x¯0 , we have y(t, x0 , u) = y(t, x¯0 , u), ∀u, ∀t ≥ 0. The above statement enables us to define the concept of indistinguishability of the state which can produce the same output via different states. Definition 1.4 Two states x0 ∈ Rn and x¯0 ∈ Rn of system (1.2) are called indistinguishable if y(t, x0 , u) = y(t, x¯0 , u), ∀u, ∀t ≥ 0. Based on Definition 1.4 we can then have another way to define the observability. Definition 1.5 System (1.2) is said to be observable if it does not admit any indistinguishable state. We have reviewed some different but equivalent definitions of observability for linear time-invariant systems, which, however, are just conceptual, and do not provide a feasible way to check directly the observability of linear time-invariant dynamical system (1.2). Therefore, several checkable methods have been proposed in the literature to judge the observability. The first method is related to identification technique. Since the objective of observability is to estimate the state x(t), it can be viewed as the following constrained optimization problem: xˆ = arg min||y − C x||2 x∈Rn

subject to:

x˙ = Ax + Bu.

Since solving directly the above constrained minimization problem is not trivial, we can remove this constraint by replacing its analytic solution

8

1 Observability and Observer for Dynamical Systems

x(t ˆ 0 ) = arg min|| y¯ − Ce A(t−t0 ) x(t0 )||2 , x(t0 )∈Rn

(1.8)

where y¯ was defined in (1.5). Define the following matrix: 

t0 +T

W O,T =

 A(s−t0 ) T Ce Ce A(s−t0 ) ds,

(1.9)

t0

where T is positive. This matrix W O,T is conventionally named as Observability Gramian. Since the initial condition x(t0 ) is constant, according to the theory of parameter identification [31], the condition of the existence of the solution for the minimization problem (1.8) is that there exists a positive T such that the observability Gramian W O,T is invertible. Theorem 1.1 System (1.2) is observable if and only if there exists T > 0 such that rank W O,T = n.

(1.10)

Condition (1.10) is checkable, but not so direct since it depends on the choice of positive value T . In practice, when T → ∞ and A is Hurwitz, the observability Gramian is the solution of the following algebraic equation A T W O,∞ + W O,∞ A + C T C = 0, which can be easily computed. Another more direct way to judge the observability is the well-known Kalman’s observability rank condition. Define the following matrix: ⎤ C ⎢ CA ⎥ ⎥ dO = ⎢ ⎣ ... ⎦ , ⎡

(1.11)

C An−1

which is named as observability matrix. It can be noticed that the observability matrix is calculated up to order n − 1, and this is due to Cayley–Hamilton Theorem [41] which states that C Ak for k ≥ n are linearly dependent of {C, C A, . . . , C An−1 }. Based on the defined observability matrix, we have the following result. Theorem 1.2 System (1.2) is observable if and only if rank dO = n.

(1.12)

Remark 1.2 A simple way to interpret the above condition is as follows. Consider system (1.2) with B = 0, since we know the output y(t), thus its high-order derivatives are known as well. Hence we have

1.2 Observability and Observer for Linear Dynamical Systems

⎡

y(t0 ) y˙ (t0 ) .. .

⎢ ⎢ ⎣

⎤

y (n−1) (t0 )

9

⎡

⎤ C ⎥ ⎢ CA ⎥ ⎥ = ⎢ . ⎥ x(t0 ), ⎦ ⎣ .. ⎦ C An−1

where y (i) implies the ith derivative of y with respect to t. Therefore, if the observability matrix dO is non-singular, then the initial condition x(t0 ) can be reconstructed. As we will see hereafter, the same idea has been extended to deduce observability conditions for nonlinear dynamical systems. Compared to the observability Gramian W O,T , it is much easier to calculate the observability matrix dO. Another easily checkable condition is the PBH test [17] which is stated as follows. Theorem 1.3 System (1.2) is observable if and only if rank

λI − A = n, ∀λ ∈ C. C

(1.13)

Remark 1.3 The conditions (1.10), (1.12) and (1.13) are all equivalent necessary and sufficient conditions to check the observability. Although (1.13) is less checkable than (1.12), it provides another advantage: check the detectability. Notice that λ can be seen as the eigenvalues of A. Since negative eigenvalues mean the corresponding dynamical parts are stable (i.e., asymptotically converge to zero), the detectability requires only that the condition (1.13) is valid for all positive eigenvalues, i.e., rank

λI − A = n, ∀λ ∈ C+ . C

Example 1.1 Consider the following linear time-invariant dynamical system: x˙1 = −x1 − x2 x˙2 = x1 − x3 x˙3 = −x3 y = x2 − x3 . The state matrix is given by ⎡

⎤ −1 −1 0   A = ⎣ 1 0 −1 ⎦ and C = 0 1 −1 . 0 0 −1 By solving the following algebraic equation: A T W O,∞ + W O,∞ A + C T C = 0,

10

1 Observability and Observer for Dynamical Systems

⎡

we have W O,∞

⎤ 1/2 1/2 −5/6 1 −7/6 ⎦ , = ⎣ 1/2 −5/6 −7/6 5/3

which yields rankW O,∞ = 3. Thus, the observability Gramian is invertible and the studied system is observable. We can also test PBH condition to judge the observability. For this, we calculate rank

⎤ λ+1 1 0 ⎢ −1 λ 1 ⎥ λI − A ⎥ = rank ⎢ ⎣ 0 0 λ + 1 ⎦ = 3, ∀λ ∈ C C 0 1 −1

⎡

and it implies as well the studied system is observable. The simplest manner to judge the observability is no doubt the Kalman rank condition. It is easy to calculate that     C A = 1 0 0 , C A2 = −1 −1 0 , thus we have

⎡

⎤ ⎡ ⎤ C 0 1 −1 rank ⎣ C A ⎦ = rank ⎣ 1 0 0 ⎦ = 3. C A2 −1 −1 0

Therefore, the Kalman rank condition (1.12) is satisfied and the studied system is observable. 

1.2.2 Observer Design Consider the following linear time-invariant dynamical system x˙ = Ax + Bu y = C x,

(1.14)

where x ∈ Rn , u ∈ R p and y ∈ Rm . Assume that it is observable, which is equivalent to suppose that the condition rank dO = n, with dO being defined in (1.11), is satisfied which in fact depends only on the matrices A and C. Thus, when system (1.14) is observable, for simplicity we say that the pair (A, C) is observable.

1.2 Observability and Observer for Linear Dynamical Systems

11

Luenberger Observer The first type of asymptotic observers for (1.14) is the so-called Luenberger observer [32], whose dynamics is given as follows: x˙ˆ = A xˆ + Bu + K (y − C x), ˆ where xˆ ∈ Rn represents the estimation of state, K is a gain matrix to be chosen. If (A, C) is observable, then there always exists a matrix K such that (A − K C) is Hurwitz, i.e., all eigenvalues of (A − K C) are located in the left half of the complex plane. Denote the estimation error by e(t) = x(t) − x(t), ˆ then we obtain the following observation error dynamics e˙ = (A − K C)e. Due to the fact that the chosen K makes (A − K C) be Hurwitz, then we can conclude that e(t) tends to 0 as t → +∞. In other words, x(t) ˆ asymptotically converges to x(t). It is worth noting that the structure of Luenberger observer is quite simple, but the choice of the gain matrix K is not unique. Large K will place the eigenvalues of (A − K C) more negative (their real parts), thus the estimation error e will converge to 0 faster. However, to choose such a large K is not a good solution when the output is corrupted by measurement noise. In this case, e(t) will not any more converge to 0, but into a neighborhood of 0. The large K will then amplify this neighborhood. Kalman–Bucy Filter Consider again system (1.14) with noise both in the model and in the output as follows: x˙ = Ax + Bu + wx y = C x + wy ,

(1.15)

where wx ∈ Rn and w y ∈ Rm denote the noises and assumed to be Gaussian. Obviously, we can still design a Luenberger observer for such a system. In this case, it is expected that the estimation error e will not converge to 0. However, an optimal Luenberger observer should have a gain matrix K such that the square error e T e is minimal even both the model and the output are corrupted by noise. Such an optimal observer is named as Kalman–Bucy observer [21], which is of the following form: x˙ˆ = A xˆ + Bu + K (y − C x) ˆ (1.16) K = PC T R −1 P˙ = A P + P A T + Q − PC T R −1 C P, where R and Q are the covariance of the noise w y and wx , and P represents the covariance of the estimation error e(t). It has been proven in many textbooks that such a choice of K is optimal in the sense of minimizing e T e when the added model and output noises are Gaussian.

12

1 Observability and Observer for Dynamical Systems

Observer Canonical Form If system (1.14) is observable, i.e., the pair (A, C) fulfills the observability rank condition, then there always exists a change of coordinates z = T x such that it can be transformed into the following observer canonical form: ¯ z˙ = A O z + βy + Bu y = C O z, where

⎛

··· ··· ... ··· 0 ···

0 ⎜1 ⎜. AO = ⎜ ⎜ .. ⎝0

⎞ 0 0 0 0 0 0⎟ . ⎟, C = 0 ··· 0 1, ... · · · .. ⎟ ⎟ O 1 0 0⎠ 0 1 0

(1.17) (1.18)

(1.19)

which is named as the Brunovsky form [45]. Such a transformation can be achieved by applying the change of coordinates z = T x with the rows of T are determined by the following equations: Tn = C Tn−k = C Ak +

(1.20) k 

pn−i C Ak−i , for 1 ≤ k ≤ n − 1,

(1.21)

i=1

where pi for 0 ≤ i ≤ n − 1 are coefficients of the characteristic polynomial of matrix A, i.e., PA (s) = p0 + p1 s + · · · + pn−1 s n−1 + s n and

B¯ = T B

and β = [β1 , . . . , βn ]T is vector field of the following form: βn = − pn−1 βn−k = − pn−(k+1) , for 1 ≤ k ≤ n − 1. For the observable canonical form (1.18), since y and u are both known, it is easy to design a Luenberger observer of the following form: ¯ + K (y − C O zˆ ), z˙ˆ = A O zˆ + βy + Bu which gives the following observation error dynamics: e˙ = (A O − K C O )e,

1.2 Observability and Observer for Linear Dynamical Systems

13

where e = z − zˆ . Due to the special form of A O and C O , it is easy to find K such that (A O − K C O ) is Hurwitz. Once e tends to zero, since T is an invertible mapping, we can conclude that T −1 zˆ is the estimation of x.

1.3 Observability and Observer for Nonlinear Dynamical Systems Nonlinear dynamical systems are commonly written in the following state-space equation: x˙ = f (x, u) y = h(x),

(1.22)

where x ∈ X ⊂ Rn , u ∈ U ⊂ R p and y ∈ Rm represent the state, the control input and the measurement, respectively, with X and U being the neighborhood of 0 with appropriate dimension. In this book, it is assumed that f and h are smooth, i.e., f ∈ C ∞ and h ∈ C ∞ . Methodologically, two different approaches are normally used in the literature to study observability for nonlinear dynamical systems • Differential algebraic method: regard the differential as an operator, and analyze the nonlinear dynamical systems by using the theorem of module (differential ring, differential field...) [7]; • Differential geometric method: consider the nonlinear dynamical system evolving on certain manifolds, and analyze the dynamics by using the concepts of differential geometry (vector fields, Lie derivative, Lie bracket...) [19]. For nonlinear dynamical systems, till 1970s, Hermann and Krener showed in [16] that the differential geometric method was useful to analyze its properties, including observability. This method opened a door in the control domain to study many control problems for nonlinear dynamical systems, and lots of results are published by using differential geometric method. It was until the middle of 1980s that the differential algebraic method was introduced by Fliess [12] to analyze nonlinear dynamical systems. In the following, we will recall some results on observability analysis for nonlinear dynamical systems by using differential geometric method.

1.3.1 Observability Analysis Similar to the case of linear time-invariant dynamical system, the observability analysis for nonlinear dynamical system is also to estimate x(t) from available information y(t) and u(t). Generally speaking, we are seeking an injective mapping from y, u and their derivatives to x. Similar to the observability analysis for linear time-invariant

14

1 Observability and Observer for Dynamical Systems

dynamical systems, we can firstly consider the output equation y = h(x). The trivial case is that the function h defines an into mapping. It is worth noting that, due to the different nonlinear properties of the function h, this mapping might be injective globally or locally, which is the key difference compared to the linear case. To clarify this point, consider the following two different functions y = h(x) = e x and y = h(x) = sin x. It is clear that y = e x defines a global (for x ∈ R) one-to-one mapping, while y = sin x is only a local one for x ∈ [(2k − 1)π, (2k + 1)π). Given a general nonlinear function h (i.e., without specifying its nonlinearity), normally we can only characterize its local property, and this is the main reason why we consider in the most cases the local observability when studying nonlinear dynamical systems. For the trivial case where h has already defined an injective mapping, it is not necessary to consider the dynamical equation x˙ = f (x, u). The sufficient condition to judge the observability can be stated as: System (1.22) is locally observable if rank

∂h = n, ∀x ∈ X . ∂x

Note that if the above condition is satisfied, then according to Implicit Function Theorem, there exists a local inverse mapping that gives x as a unique function of y, i.e., x can be uniquely determined by y. In this sense, x is locally observable. It is worth noting that this condition implicitly requires that h(x) has n independent components. Due to the same reasons (economic or technique) for linear time-invariant dynamical systems, this trivial case is hard to be satisfied. Thus, it is not possible to use only the instantaneous value of y to achieve estimation of x. In this case, we might consider the dynamics x˙ = f (x, u) to estimate x(t) by using both the values of y and u over a time interval. As we have seen in Sect. 1.2.1 that the observability of the state x(t) for linear time-invariant dynamical system (1.2) is equivalent to the reconstructibility of its initial condition x(t0 ), this statement is also valid for nonlinear dynamical systems. We would like to remark that the observability does not require the uniqueness of solutions, which can be illustrated by the following simple example x˙ = f (x, u) y = x,

1.3 Observability and Observer for Nonlinear Dynamical Systems

15

with x ∈ Rn . It is evident that the above system is always observable even if it admits several different solutions. For the nonlinear dynamical system (1.22), denote x0 = x(t0 ) as its initial condition, x(t, x0 , u) as its solution with the initial condition x0 and the input u, and y(t, x0 , u) as its associated output. We can then define observability via the indistinguishability for nonlinear dynamical systems, as Definitions 1.4 and 1.5 for linear time-invariant dynamical systems. However, unlike linear case, the observability for nonlinear case might be global or local, and this leads to the following modified definitions. Definition 1.6 Two states x0 ∈ Rn and x¯0 ∈ Rn of system (1.22) are called indistinguishable if y(t, x0 , u) = y(t, x¯0 , u), ∀u, ∀t ≥ 0. They are X -indistinguishable if the above condition is satisfied for x0 ∈ X ⊂ Rn and x¯0 ∈ X ⊂ Rn . Based on Definition 1.6 we can then have different observability definitions for nonlinear dynamical systems [16]. Definition 1.7 System (1.22) is said to be globally observable if there is no indistinguishable state in Rn . Definition 1.8 System (1.22) is said to be weakly observable at x if there is a neighborhood Ω of x such that for every x¯ ∈ Ω, x and x¯ are distinguishable. System (1.22) is said to be weakly observable if it is so at every x ∈ X . Definition 1.9 System (1.22) is said to be locally weakly observable at x if there is a neighborhood Ω of x such that for every neighborhood V contained in Ω of x and every x¯ ∈ V , x and x¯ are distinguishable. System (1.22) is said to be locally weakly observable if it is so at every x ∈ X . Remark 1.4 Global observability means that x can be distinguished from any other states in Rn , where weak observability implies that x can distinguish any state in its neighborhood X . Local weak observability requires that x is weakly observable in its neighborhood V ⊂ X without leaving V . Obviously, the concept of local weak observability is the strongest definition, but it is the most interesting one in practice. The above different concepts of observability are due to the nonlinear property of dynamical systems. Another characteristic compared to linear time-invariant dynamical systems is the dependence of the input u. In Sect. 1.2.1, we have shown that the input u has no influence to deduce the observability conditions for linear timeinvariant dynamical systems, and here we will show that it is not true for nonlinear dynamical systems. Consider the following dynamical system: x˙1 = x1 + α(u)x2 x˙2 = x2 x1 y = x1 ,

16

1 Observability and Observer for Dynamical Systems

where α(u) is a function of the input u. It is easy to obtain that, if α(u) = 0 for all u ∈ 1 U , then the above system is globally observable since x1 = y and x2 = α(u) [ y˙ − y]. Unfortunately, if α(u) = 0 for some input u, then the above system is not observable. In other words, the observability of nonlinear dynamical systems might depend on the value of input, and we call the inputs (controls) that distinguish all pairs of states as universal inputs. Otherwise, we name them as singular inputs. Definition 1.10 The dynamical system (1.22) is uniformly observable if all inputs are universal. For linear time-invariant dynamical systems, we can obtain analytic solution of the input–output mapping, and this enables us to use the elementary algebraic approach to test the observability, such as the observability Gramian (see Theorem 1.1), the Kalman rank condition (see Theorem 1.2) and the PBH test (see Theorem 1.3). However, to deduce the observability condition from the indistinguishability for nonlinear dynamical systems is a challenging task since normally the expression of the input–output mapping cannot be obtained analytically. As indicated by the title of this book, we adopt the differential geometric method to characterize the observability condition for nonlinear dynamical systems. For this, let us introduce the following R-vector space: O = spanR {L fuk L fuk−1 ...L fu1 h i , for 1 ≤ i ≤ m and u 1 , . . . , u k ∈ U }, which is called the observation space, where f u i = f (x, u i ) and L fu1 h i is the Lie i derivative of h i in the direction of f u 1 , i.e., L fu1 h i = ∂h f , which will be detailed ∂ x u1 in Chap. 2. Define dO as the R-vector space of the differentials of the observation space O dO = spanR {dl with l ∈ O},

(1.23)

then we have the following definition of observability rank condition. Definition 1.11 System (1.22) is said to satisfy observability rank condition at x if dim(dO|x ) = n.

(1.24)

It is said to satisfy observability rank condition in X if this is true for every x ∈ X . Using the observability rank condition, we can then have the following theorem. Theorem 1.4 System (1.22) is locally weakly observable at x if the observability rank condition is satisfied at x. System (1.22) is locally weakly observable if it satisfies the observability condition for every x ∈ X . The above theorem presents a constructive way to check the observability for nonlinear dynamical system (1.22). For linear time-invariant dynamical systems, condition (1.24) is equivalent to the Kalman rank condition. However, it is too general to be used directly due to the following two reasons:

1.3 Observability and Observer for Nonlinear Dynamical Systems

17

• Firstly, unlike linear case where Cayley–Hamilton Theorem tells us to stop at the order n − 1 when checking the rank condition, there does not exist a fixed value k for nonlinear case in general; • Secondly, as we have discussed before, the observability rank condition depends on the inputs. In practice, a less general case is to consider only nonlinear dynamical system (1.22) without inputs. The reason is that the feedback controller u in (1.22) might be a smooth function of the state x, and this leads to investigate the observability for the following dynamical system: x˙ = f (x) y = h(x),

(1.25)

where f and h are both assumed to be smooth. Similar to the arguments in Remark 1.2 for linear time-invariant dynamical systems, we can define a mapping between the output and the state by calculating high-order derivatives of y, which gives ⎤ ⎤ ⎡ h(x) y(t) ⎢ y˙ (t) ⎥ ⎢ L f h(x) ⎥ ⎥ = O(x), ⎢ . ⎥=⎢ .. ⎦ ⎣ .. ⎦ ⎣ . k (k) L f h(x) y (t) ⎡

where L kf h are the so-called kth Lie derivative of h in the direction of the vector field f which will be detailed in the next chapter, y (k) represents the kth time derivative of y and O(x) is called observability mapping. For a certain integer k > 0, if O(x) is a local diffeomorphism, then we can reconstruct the state x from the output and its derivatives. Theorem 1.5 System (1.25) is locally observable at x if for a certain k such that ⎤ dh(x) ⎢ d L f h(x) ⎥ ⎥ = n. dim dO = rank ⎢ .. ⎦ ⎣ . ⎡

(1.26)

d L kf h(x)

It is locally (or globally) observable if the above rank condition is true for every x ∈ X (or for every x ∈ Rn ). We would like to emphasize that the above rank condition is only sufficient. Also, we have stated before that it is not possible to fix the value k for nonlinear dynamical systems, while k = n − 1 for linear case due to Cayley–Hamilton Theorem. This can be shown by the following example.

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1 Observability and Observer for Dynamical Systems

Example 1.2 Consider the following dynamical system: x˙ = 0 y = h(x) = x 3 . It is easy to see that dO is spanned by 3x 2 d x. By verifying the rank condition (1.26) for k = 1, we can only state that the example is locally observable since dim dO = 1 for x = 0. However, we cannot conclude that the example is not observable at x = 0 even if dim dO = 0 when x = 0. In fact, this example is obviously globally observable since x → x 3 is a bijective mapping. By calculating the high-order derivatives of y, we have ⎡

⎤ dh(x) ⎡ 2⎤ 3x ⎢ d L f h(x) ⎥ ⎢ ⎥ = ⎣ 6x ⎦ , . ⎣ ⎦ .. 6 k d L f h(x) whose rank is always equal to 1 for all x ∈ R. Therefore, the studied example is globally observable, and we found k = 3 when judging its observability property. 

1.3.2 Observer Design Consider the following nonlinear dynamical system: x˙ = f (x) y = h(x),

(1.27)

where x ∈ X ⊂ Rn , u ∈ U ⊂ R p , y ∈ Rm and the functions f and h are smooth, i.e., f ∈ C ∞ and h ∈ C ∞ . It is assumed that system (1.27) is locally observable in the sense that the observability rank condition (1.26) is satisfied for all x ∈ X . Since the objective is to reconstruct the state x(t), the estimation can be regarded as the following nonlinear constrained optimization problem xˆ = arg min||y − h(x)||2 x∈X

subject to:

x˙ = f (x).

Note that for linear case where f (x) = Ax and h(x) = C x, since its analytic solution is known, the above constrained optimization problem can be converted to the unconstrained minimization problem (1.8) where classical least-square method can be applied to identify the initial condition x(t0 ), yielding the condition on the observability Gramian. However, for nonlinear case, it is not anymore possible since it is normally difficult or impossible to find the analytic solution to remove the dynamical

1.3 Observability and Observer for Nonlinear Dynamical Systems

19

constraint. We can only use constrained optimization techniques to numerically solve this minimization problem. This book will not discuss such a method, and interested readers can refer to related literature for more details. Similar to linear case, we can also use certain dynamical systems to achieve the estimation of x(t) for (1.27). The following will outline some well-known nonlinear observers. Nonlinear Luenberger Observer For the considered nonlinear dynamical system (1.27), the corresponding nonlinear Luenberger observer is of the following form: x˙ˆ = f (x) ˆ +



∂ T (x) ˆ ∂ xˆ

−1 

 K (y) − K (h(x)) ˆ ,

where T (.) and K (.) are two functions of their arguments with appropriate dimension to be determined and T (.) is a diffeomorphism, which are the solutions of following partial differential equation: ∂ T (x) f (x) = H T (x) + K (h(x)), ∂x

(1.28) 

where H is constant matrix which should be chosen as Hurwitz one. implies the inverse of the Jacobian of the diffeomorphism T (x). Denote z = T (x) and zˆ = T (x), ˆ then we have z˙ = and

∂ T (x) ˆ ∂ xˆ

−1

∂ T (x) f (x) = H z + K (y) ∂x

ˆ ˙ ˆ z˙ˆ = ∂ T∂ (xˆx) f (x) ˆ + K (y) − K (h(x)), ˆ xˆ = ∂ T∂ (xˆx) = H zˆ + K (y)

which yields the following observation error dynamics e˙ = H e, where e = z − zˆ . Since H is chosen to be Hurwitz, e tends to zero asymptotically. Moreover, if the solution T (x) of the PDE (1.28) is a diffeomorphism, then we can concluded that xˆ tends to x asymptotically. The key issue of this method is to solve the PDE (1.28). Note that, for linear time-invariant dynamical systems, the PDE becomes the following matrix equation: T A = H T + K C, thus, if (A, C) is observable, then we can choose T = I which yields

20

1 Observability and Observer for Dynamical Systems

H = A − K C. By choosing K properly, we can always ensure that H is Hurwitz. For nonlinear systems, given f and h, the solution T (x) of this PDE always exists, but might not be injective, thus some sufficient conditions should be imposed for (1.27) (see [28]). Note that the dimension of T (x) might be equal or bigger than that of x. For the case where z and x have the same dimension, [22] gave sufficient conditions for the local solvability of the PDE (1.28). The solution is obtained by linearizing f and h around the equilibrium point 0. Due to this reason, the solution is local, so is for the proposed nonlinear Luenberger observer. However, this result is also valid for nonlinear analytic systems (not necessary to be Lipschitz). Some extensions have been done in [24] for Lipschitz continuous systems. Extended Kalman Filter Another observer for general system (1.27) is the so-called extended Kalman filter (EKF) [30], which is an extension of Kalman–Bucy filter (1.16) to treat nonlinear dynamical systems with noise in the model and in the output x˙ = f (x) + wx y = h(x) + w y ,

(1.29)

where wx ∈ Rn and w y ∈ Rm represent Gaussian noise. It is assumed that system (1.27) is locally observable in the sense that the observability rank condition (1.26) is satisfied. Note xˆ as the estimation of x, then by linearizing the nonlinear functions f and h around its estimation, we have ∂f |xˆ ∂x ∂h |xˆ . Ct = ∂x At =

Then the EKF is of the following form: ˆ x˙ˆ = f (x) ˆ + K t (y − h(x)) P˙ = At P + P AtT − K t Rt K tT + Q t K t = PCtT Rt−1 ,

(1.30)

where Q t and Rt are the driving noise covariance in the system dynamics and the measurement noise covariance, P is the estimation error covariance matrix and K t is a dynamic gain. EKF is the most widely used estimation technique and has been validated by many different applications. However, since we linearize the nonlinearity of (1.29), the resulted EKF is therefore local, i.e., only local convergence can be assured, and it is impossible to have a global one. Interested readers are referred to [26] and references therein.

1.3 Observability and Observer for Nonlinear Dynamical Systems

21

Nonlinear Luenberger observer and EKF are proposed for the general form (1.27), and the results are only local. To design global observers for nonlinear dynamical systems, we need to restrict our study to some special nonlinear forms (via some transformations). Luenberger-Like Observer The first special form of (1.27) is the following Lipschitz nonlinear dynamical system: x˙ = Ax + ϕ(x) y = C x,

(1.31)

where ϕ(x) is a Lipschitz nonlinearity with a Lipschitz constant γ , i.e., ||ϕ(x) − ϕ(x)|| ¯ ≤ γ ||x − x||. ¯

(1.32)

Then we can design an observer of the following form: x˙ˆ = A xˆ + ϕ(x) ˆ + K [y − C x], ˆ

(1.33)

where K represents the gain matrix and xˆ means the estimated state. Thus, the estimation error dynamics is governed by e˙ = (A − K C)e + [ϕ(x) − ϕ(x)], ˆ

(1.34)

ˆ The key point for such an observer is to determine K such that e where e = x − x. tends to zero asymptotically. In [44], the gain matrix K was chosen such that the following inequality is satisfied: γ
0, (0, 0) is asymptotically stable.

Exercise 2.6 Consider the following vector field: f = −x2

∂ ∂ + x1 ∂ x1 ∂ x2

on R2 . (1) Draw the integral curves of f. (2) Let ω1 =

−x2 x1 d x1 + 2 d x1 2 + x2 x1 + x22

x12

and ω2 = x 1 d x 2 + x 2 d x 1 on R2 \{(0, 0)}. • • • •

 ω1 is an isomorphism on the tangent space of R2 \{(0, 0)}. Show that ω = ω2

 1 Prove that ω( f ) = is a constant vector field. 0 Let g = x1 ∂∂x1 + x2 ∂∂x2 , compute ω(g) and deduce that there does not exist a change of coordinates φ(x) = z such that ω = dφ. Could we judge directly from the expressions ω1 and ω2 that there does not exist a change of coordinates φ(x) = z such that ω = dφ?

References 1. Abraham, R., Marsden, J., Marsden, J.: Foundations of Mechanics, vol. 36. Benjamin/Cummings Publishing Company, Reading (1978) 2. Burke, W.L.: Applied Differential Geometry. Cambridge University Press, Cambridge (1985)

54

2 Background on Differential Geometry

3. Camacho, C., Neto, A.L.: Geometric Theory of Foliations. Springer Science & Business Media, New York (2013) 4. Chen, W., Chern, S.S., Lam, K.S.: Lectures on Differential Geometry, vol. 1. World Scientific Publishing Company, Singapore (1999) 5. Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. Math. 575–620 (2003) 6. Hector, G., Hirsch, U.: Introduction to the Geometry of Foliations. Springer, Berlin (1981) 7. Hicks, N.J.: Notes on Differential Geometry, vol. 3. van Nostrand, Princeton (1965) 8. Lang, S.: Fundamentals of Differential Geometry, vol. 191. Springer Science & Business Media, New York (2012) 9. Matsumoto, S.: Measure of exceptional minimal sets of codimension one foliations. A Fête of Topology, pp. 81–94. Elsevier, Amsterdam (1988) 10. Morita, S.: Geometry of Differential Forms, vol. 201. American Mathematical Society, Providence (2001) 11. Murray, R.M., Li, Z., Sastry, S.S., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994) 12. Nijmeijer, H., Van der Schaft, A.: Nonlinear Dynamical Control Systems, vol. 175. Springer, Berlin (1990) 13. Sternberg, S.: Local contractions and a theorem of Poincaré. Am. J. Math. 79(4), 809–824 (1957) 14. Wolak, R.: Ehresmann connections for Lagrangian foliations. J. Geom. Phys. 17(4), 310–320 (1995)

Chapter 3

Observer Normal Form with Output Injection

Abstract This chapter firstly presents the pioneering work of Krener [6, 7], for the purpose of transforming a general nonlinear system into the so-called nonlinear observer normal form with output injection. As it has been already mentioned in Chap. 1, this form contains a linear part which is of Brunovsky’s canonical form, and a nonlinear part which is only function of measurable variables (i.e., input and output). A set of geometric conditions have been deduced to guarantee the existence of such a change of coordinates (a diffeomorphism) which transforms the studied nonlinear dynamical system into the proposed observer normal form with output injection. In addition, a constructive method has been proposed to facilitate the deduction of this diffeomorphism. After that, we consider how to extend such a method to treat nonlinear systems with inputs, for which the additional Lie bracket conditions need to be considered. Due to the special form of this observer normal form with output injection, a simple Luenberger-like observer [8] can be designed which yields a linear dynamics for the observation error. The concrete design procedures for this type of observer will be discussed in the last section of this chapter.

3.1 Problem Statement Consider the following single-output dynamical system: x˙ = f (x) y = h(x),

(3.1)

where X ⊆ Rn is a neighborhood of 0, x ∈ X and y ∈ R. Without loss of generality it is assumed that f and h are smooth, f (0) = 0 and h(0) = 0. By using the notations in Chap. 2, the vector field f can be written as f (x) =

n  i=1

f i (x)

∂ . ∂ xi

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_3

55

56

3 Observer Normal Form with Output Injection

For the given single-output dynamical system (3.1), we can define the so-called observability 1-forms. Definition 3.1 The observability 1-forms of dynamical system (3.1) are defined as follows: (3.2) θi = d L i−1 f h, for 1 ≤ i ≤ n, where d denotes the differential and L kf is the kth Lie derivative in the direction of the vector field f . Within this chapter, it is assumed that dynamical system (3.1) fulfills the observability rank condition in the sense that its observability 1-forms θi for 1 ≤ i ≤ n are linearly independent. In other words, the whole state of (3.1) is assumed to be observable. And we are interested in how to seek a local diffeomorphism z = φ(x) such that the studied dynamical system (3.1) can be transformed into the following nonlinear observer normal form with output injection: z˙ = A O z + β(y) y = C O z = zn ,

(3.3)

where A O and C O are of Brunovsky form, i.e., ⎛

··· ··· ... ··· 0 ···

0 ⎜1 ⎜. AO = ⎜ ⎜ .. ⎝0

and

⎞ 0 0⎟ .. ⎟ n×n .⎟ ⎟∈R ⎠ 1 00 0 10

0 0 ...

0 0 .. .



C O = 0 · · · 0 1 ∈ R1×n .

This normal form can be seen as a nonlinear version of (1.18), since it enables us to easily design a Luenberger-like observer to estimate its state (see Sect. 3.4 for the details). Such a problem was firstly investigated in [6] where necessary and sufficient conditions were stated to guarantee the existence of such a change of coordinates z = φ(x). However, no constructive approach was presented to calculate such a φ(x). This chapter aims at revisiting the result stated in [6], and presents the explicit formulas to compute such a diffeomorphism.

3.2 Observer Normal Form with Output Injection For the dynamical system (3.1), we can define a frame τ = [τ1 , . . . , τn ] where τi for 1 ≤ i ≤ n are vector fields defined according to the following criteria:

3.2 Observer Normal Form with Output Injection

57

• the first vector field τ1 is the unique solution of the following algebraic equations: θk (τ1 ) = 0, for 1 ≤ k ≤ n − 1 θn (τ1 ) = 1;

(3.4)

• the rest vector fields are calculated via the Lie bracket by the following induction:  τi = τi−1 , f , for 2 ≤ i ≤ n.

(3.5)

The following recalls the result stated in [4, 6]. Theorem 3.1 There exists a local diffeomorphism z = φ(x) that transforms system (3.1) into the nonlinear observer normal form (3.3) if and only if  τi , τ j = 0,

(3.6)

where τi and τ j for 1 ≤ i, j ≤ n are defined in (3.4)–(3.5). The proof of the above theorem can be found in [5]. Theorem 3.1 presents only the conditions on the existence of such a diffeomorphism, but does not provide a constructive way to calculate it. In what follows, we will present an explicit formula to compute such a diffeomorphism. For this, let us consider the following matrix:



Λ = θ τ = θi τ j 1≤i, j≤n = Λi, j 1≤i, j≤n , where

(3.7)

⎤ θ1 θ = ⎣ ... ⎦ , τ = [τ1 , . . . , τn ] . θn ⎡

The following proposition gives the algorithm to compute Λ’s expression. Proposition 3.1 The coefficients of the matrix Λ, defined in (3.7), satisfy the following relation:

(3.8) Λi, j+1 = Λi+1, j − L f Λi, j . Particularly, if Λi, j is a constant function, then Λi, j+1 = Λi+1, j . Proof Recall that





dθi τ j , f = L τ j θi ( f ) − L f θi τ j − θi [τ j , f ] .

(3.9)

As θi are exact differential 1-forms, then dθi = 0. Moreover, for 1 ≤ j ≤ n − 1, by definition we know that τ j+1 = [τ j , f ]. Therefore, Eq. (3.9) becomes



θi τ j+1 = L τ j θi ( f ) − L f θi τ j .

58

3 Observer Normal Form with Output Injection

Recall that the Lie derivative can be written as L = dı + ıd where ı is the antiderivation operator (or interior product) and d the differential. Then, we deduce that





θi τ j+1 = θi+1 τ j − L f θi τ j . Thus, according to the definition of Λi, j , the above equality can be written as that in (3.8).

Moreover,

if θi τ j is a constant function, then its Lie derivative vanishes and we have θi τ j+1 = θi+1 τ j , i.e., Λi, j+1 = Λi+1, j .  Following the above result, we know that θi (τ1 ) = 0 for 1 ≤ i ≤ n − 1 and θn (τ1 ) = 1, then we deduce that θi (τ2 ) = 0 for 1 ≤ i ≤ n − 2 and θn−1 (τ2 ) = 1. By repeating the same procedure, a straightforward calculation shows that Λ has the following form: ⎞ ⎛ 0 0 ··· 0 1 ⎜0 0 ··· 1 Λ2,n ⎟ ⎜. . . .. ⎟ . ⎜ . . . . (3.10) Λ=⎜. . . . . ⎟ ⎟. ⎠ ⎝0 1 ··· Λ n−1,n−1 Λn−1,n 1 Λn,2 · · · Λn,n−1 Λn,n It is clear that this matrix is invertible, thus we can define the following 1-forms: ω = Λ−1 θ := (ωi )1≤i≤n ,

(3.11)

where (ωi )1≤i≤n are given by the following equations [1, 2]: ωn = θ1  ωn−r = θr +1 −

n  i=n−r +1

 Λr +1,i ωi , for 1 ≤ r ≤ n − 1.

(3.12)

Theorem 3.2 The following three assertions are equivalent [3]: (1) There exists a local change of coordinates z = φ(x) that transforms system (3.1) into the nonlinear observer normal form (3.3); (2) The vector fields τi and τ j for 1 ≤ i, j ≤ n defined in (3.4)–(3.5) satisfy the commutativity condition (3.6); (3) The 1-forms ωi defined in (3.12) for 1 ≤ i ≤ n are closed, i.e., dωi = 0.

(3.13)

Furthermore, the change of coordinates z = φ(x) is determined by  z i = φi (x) =

γ

ωi ,

(3.14)

3.2 Observer Normal Form with Output Injection

59

where γ : [0, 1] → X is any smooth curve on contractile neighborhood of 0 such that γ (0) = 0 and γ (1) = x. Proof The equivalence between (3.6) in the second assertion and (3.13) in the third assertion was proven in Lemma 2.2 of Sect. 2.4. The rest is to prove the equivalence between the first assertion and the second one. For this, let φ be the diffeomorphism given in (3.14), which fulfills φ∗ := dφ = ω. The following shows how the Jacobian of this diffeomorphism φ transforms the vector field f , which is, in fact, equivalent to check ∂z∂ i φ∗ ( f ). Using Eq. (2.16) with f 1 = τi and f 2 = f , we obtain ∂ ∂ φ∗ ( f ) = φ∗ [τi , f ] = φ∗ (τi+1 ) = ∂z i ∂z i+1 for all 1 ≤ i ≤ n − 1. The integration of the above equation leads to z˙ = φ∗ ( f ) = A O z + β(z n ), where z n is determined by (3.14) as  zn =

γ

ωn .

Based on (3.12) and θ1 = dh = dy, we finally have y = z n . Therefore, we proved that the diffeomorphism z = φ(x) transforms dynamical system (3.1) into the following observer normal form: z˙ = φ∗ ( f ) = A O z + β(z n ) y = zn .  In order to clearly understand the above result, let us firstly apply Theorem 3.2 to linear dynamical systems. Example 3.1 Consider the following linear dynamical system: x˙ = Ax y = Cx with

⎛

⎞ 1 −1 0

A = ⎝ 1 2 0 ⎠ and C = 0 1 −1 . 0 1 1

Let us apply the proposed procedure to calculate the observability 1-forms θ , defined in (3.2), and this yields

60

3 Observer Normal Form with Output Injection



θ1 = C = 0 1 −1 =

d x2 − d x3 θ2 = C A = 1 1 −1 = d x1 + d x2 − d x3 θ3 = C A2 = 2 0 −1 = 2d x1 − d x3 . Obviously, these 1-forms are linearly independent, thus the dynamical system fulfills the observability rank condition. Then, we need to calculate the frame τ = [τ1 , τ2 , τ3 ]. According to (3.4), the first vector field τ1 is the unique solution of the following algebraic equations: Cτ1 = 0 C Aτ1 = 0 C A2 τ1 = 1. A straightforward calculation gives ⎛

⎞ 0 ∂ ∂ τ1 = ⎝ −1 ⎠ = − − . ∂ x2 ∂ x3 −1 Then, according to (3.5), the second and the third vector fields are ⎛

⎞ 1 ∂ ∂ ∂ τ2 = [τ1 , Ax] = Aτ1 = ⎝ −2 ⎠ = −2 −2 ∂ x ∂ x ∂ x3 1 2 −2 and

⎛

⎞ 3 ∂ ∂ ∂ τ3 = [τ2 , Ax] = Aτ2 = ⎝ −3 ⎠ = 3 −3 −4 . ∂ x ∂ x ∂ x3 1 2 −4

According to (3.7), we can then calculate Λ as ⎛

⎞ ⎛ ⎞ Cτ1 Cτ2 Cτ3 00 1 Λ = θ τ = ⎝ C Aτ1 C Aτ2 C Aτ3 ⎠ = ⎝ 0 1 4 ⎠ , 1 4 10 C A2 τ1 C A2 τ2 C A2 τ3 which yields

⎛

Λ−1 Thus

⎞ 6 −4 1 = ⎝ −4 1 0 ⎠ . 1 0 0

3.2 Observer Normal Form with Output Injection

61

⎛

⎞ ⎛ ⎞ ⎛ ⎞ C −2 2 −3 −2d x1 + 2d x2 − 3d x3 ω = Λ−1 θ = Λ−1 ⎝ C A ⎠ = ⎝ 1 −3 3 ⎠ = ⎝ d x1 − 3d x2 + 3d x3 ⎠ . C A2 0 1 −1 d x2 − d x3 As a result, by integration of ω, the following diffeomorphism is obtained: ⎛

⎞ ⎛ ⎞ z1 −2x1 + 2x2 − 3x3 z = ⎝ z 2 ⎠ = ⎝ x1 − 3x2 + 3x3 ⎠ , z3 x2 − x3 which transforms the studied example into the following observer normal form: z˙ 1 = β1 (y) z˙ 2 = z 1 + β2 (y) z˙ 3 = z 2 + β3 (y) y = z3, with β1 (y) = 3y, β2 (y) = −6y and β3 (y) = 4y.



The next example highlights the proposed result on a nonlinear dynamical system. Example 3.2 Consider the following nonlinear dynamical system: x˙1 = x2 x3 x˙2 = x1 + x32 x˙3 = x2 − x3 y = h(x) = x3 . The observability 1-forms are as follows: θ1 = d x3 θ2 = d x2 − d x3 θ3 = d x1 − d x2 + (2x3 + 1) d x3 . Obviously, these 1-forms θ = [θ1 , θ2 , θ3 ]T are linearly independent, thus the dynamical system satisfies the observability rank condition. Similar to the linear case, we can then calculate the frame τ = [τ1 , τ2 , τ3 ]. The solution of the following algebraic equations θ1 τ1 = 0 θ2 τ1 = 0 θ3 τ1 = 1 yields

62

3 Observer Normal Form with Output Injection

τ1 =

∂ . ∂ x1

Therefore, by induction we obtain τ2 = [τ1 , f ] = and τ3 = [τ2 , f ] =

∂ ∂ x2

∂ ∂ + x3 . ∂ x3 ∂ x1

The above calculations give the matrix Λ as ⎛

⎞ ⎛ ⎞ θ1 τ1 θ1 τ2 θ1 τ3 0 0 1 −1 ⎠ , Λ = θ τ = ⎝ θ2 τ1 θ2 τ2 θ2 τ3 ⎠ = ⎝ 0 1 1 −1 3x3 + 1 θ3 τ1 θ3 τ2 θ3 τ3 which yields the 1-forms ω as ⎛

⎞ ⎛ ⎞ ω1 θ1 ω = ⎝ ω2 ⎠ = Λ−1 ⎝ θ2 ⎠ , ω3 θ3 ⎛

where Λ−1 Thus we have

⎞ −3x3 1 1 = ⎝ 1 1 0⎠. 1 00

⎛

⎞ ⎞ ⎛ 1 0 −x3 d x1 − x3 d x3 ⎠. d x2 ω = ⎝0 1 0 ⎠ = ⎝ 00 1 d x3

By integrating the above equation, the desired change of coordinates is as follows: ⎛

⎞ x1 − 21 x32 ⎠ z=⎝ x2 x3 via which the studied system can be transformed into the following normal form with output injection: ⎧ z˙ 1 = 0 ⎪ ⎪ ⎨ z˙ 2 = z 1 + 23 y 2 z˙ = z 2 + 0 ⎪ ⎪ ⎩ 3 y = z3. 

3.3 Extension to Systems with Inputs

63

3.3 Extension to Systems with Inputs The result presented in Sect. 3.2 only treated dynamical systems without inputs, while in practice most of systems contain the control inputs. In order to extend this result to systems with inputs, let us consider the following single-input system: x˙ = f (x) + g(x)u y = h(x),

(3.15)

where x ∈ X ⊆ Rn , u ∈ R, y ∈ R, f : X → Rn , g : X → Rn . It is assumed that f , g and h are smooth, f (0) = 0, g(0) = 0 and h(0) = 0. Theorem 3.3 System (3.15) can be transformed, via a local diffeomorphism, into the following normal form with output injection: z˙ = A O z + β(y) + η(y)u y = C O z,

(3.16)

where A O and C O are of Brunovsky form, β(y) and η(y) are functions of y with appropriate dimension, if and only if (1) one of the conditions in Theorem 3.2 is fulfilled; (2) for 1 ≤ i ≤ n − 1, the conditions [g, τi ] = 0 are satisfied. Proof Based on the results presented in Theorem 3.2, we can state that there exists a local diffeomorphism z = φ(x) such that φ∗ ( f ) = A O z + β(y) and φn (x) = z n if and only if one of the conditions in Theorem 3.2 is fulfilled. Thus, it remains to prove that the Jacobian of this diffeomorphism φ(x) will transform g(x) of (3.15) g := φ∗ (g). into η(y) of (3.16). For this, we need to investigate the value of ∂φ(x) ∂x Note that the deduced diffeomorphism φ(x) satisfies φ∗ (τi ) = ω(τi ) = ∂z∂ i for 1 ≤ i ≤ n − 1, where ω is defined in (3.11). Then we have   ∂ ∂ = [φ∗ (g), φ∗ (τi )] = φ∗ ([g, τi ]), φ∗ (g) = φ∗ (g), ∂z i ∂z i which implies that

∂ φ∗ (g) = 0, ∂z i

64

3 Observer Normal Form with Output Injection

if and only if [g, τi ] = 0 for 1 ≤ i ≤ n − 1. Note that ∂z∂ i φ∗ (g) = 0 is equivalent to state that φ∗ (g) is only a function of y, noted as η(y). Finally, we proved g(x) in (3.15) can be transformed into η(y) in (3.16) if and only if [g, τi ] = 0 for 1 ≤ i ≤ n − 1.  The following simple example illustrates how to apply this result to dynamical systems with single input and single output. Example 3.3 Consider the following nonlinear dynamical system: x˙1 = −x1 x3 − x2 x32 − x22 + (x3 − x2 )u x˙2 = x1 + x2 x3 x˙3 = x2 + u y = x3 . A straightforward calculation leads to θ1 = d x3 θ2 = d x2 θ3 = d x1 + x3 d x2 + x2 d x3 . Therefore, we have

τ1 = τ2 = τ3 =

∂ ∂ x1 ∂ ∂ x2 ∂ ∂ x3

− x3 ∂∂x1 − x2 ∂∂x1 .

It is easy to verify that [τ1 , τ2 ] = [τ1 , τ3 ] = [τ2 , τ3 ] = 0. Since g = (x3 − x2 )

∂ ∂ + , ∂ x1 ∂ x3

then we can check that [τ1 , g] = [τ2 , g] = 0. Thus, according to Theorem 3.3, there exists a change of coordinates that transforms the above dynamical system into the normal form (3.16). Indeed, a simple calculation yields ω1 = dz 1 = d(x1 + x2 x3 ) ω2 = dz 2 = d x2 ω3 = dz 3 = d x3 .

3.3 Extension to Systems with Inputs

65

By integrating the above equation, we can then obtain the following diffeomorphism: ⎡

⎤ ⎛ ⎞ z1 x1 + x2 x3 ⎠, x2 z = ⎣ z2 ⎦ = ⎝ z3 x3 which transforms the above dynamical system into the following normal form: z˙ 1 = yu z˙ 2 = z 1 z˙ 3 = z 2 + u y = z3.  Theorem 3.3 gave the necessary and sufficient conditions to transform a singleinput single-output nonlinear system to an observer normal form with output injection. This result can be easily generalized to treat multiple-input nonlinear systems as well. Consider now the following system: x˙ = f (x) + y = h(x),

m i=1

gi (x)u i

(3.17)

where u i ∈ R, gi : X → Rn for 1 ≤ i ≤ m is assumed to be smooth with gi (0) = 0. Then, the following corollary can be stated. Corollary 3.1 System (3.17) can be transformed, via a local diffeomorphism, into the following normal form with output injection: z˙ = A O z + β(y) + y = C O z,

m i=1

ηi (y)u i

(3.18)

where A O and C O are of Brunovsky form, β(y) and ηi (y) are functions of y with appropriate dimension, if and only if 1. one of conditions in Theorem 3.2 is fulfilled; 2. for 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ m, the condition [g j , τi ] = 0 is satisfied. Proof The proof of the above corollary is quite similar to that of Theorem 3.3.



Remark 3.1 Up to now, we have treated the nonlinear dynamical systems without inputs (Theorem 3.2), with single input (Theorem 3.3) and with multiple inputs (Corollary 3.1). It can be seen that the drift term f and the control term g have been treated separately. More precisely, the desired diffeomorphism z = φ(x) is mainly determined by the drift term f . Then, additional conditions on the control term g can be added to treat system with single input or multiple inputs. In other words, the control term g has no influence on the deduction of the desired change of coordinates

66

3 Observer Normal Form with Output Injection

z = φ(x). Consequently, in order to focus on the different techniques we are going to apply to deduce the diffeomorphism, the rest of this book will only treat nonlinear dynamical systems without inputs. As we have explained, those results can be then easily adapted for systems with inputs by simply adding complementary conditions on the control terms.

3.4 Observer Design 3.4.1 Luenberger-Like Observer Since the normal forms (3.3), (3.16) and (3.18) have the similar structures, containing both linear terms (A O z and C O z) and nonlinear terms of known variables (y and u), the designed observers for these three normal forms possess also similar structures. For the sake of generality, let us consider the normal form (3.18) for a general nonlinear dynamical systems with multiple inputs. For this normal form, different types of observers can be designed, such as those presented in Chap. 1. Here, we will design the simplest asymptotic observer: Luenberger-like observer. Consider the following dynamics: z˙ˆ = A O zˆ + β(y) +

m 

ηi (y)u i + K (y − C O zˆ ).

(3.19)

i=1

Note e = z − zˆ , then according to (3.18) and (3.19) we obtain e˙ = (A O − K C O )e. Due to the fact that the pair (A O , C O ) is observable, there exists a K such that (A O − K C O ) is Hurwitz. To seek such a K , we can place directly the poles of the matrix (A O − K C O ). Consequently, e is asymptotically stable, and we have lim zˆ (t) = z(t).

t→+∞

Finally, we can obtain the estimation of x by inverting the diffeomorphism φ(x). In summary, the following dynamics m ηi (y)u i + K (y − C O zˆ ) z˙ˆ = A O zˆ + β(y) + i=1 xˆ = φ −1 (ˆz ) is an asymptotic Luenberger-like observer of (3.17).

(3.20)

3.4 Observer Design

67

3.4.2 Design Procedure The following summarizes the procedure to design observer based on the proposed normal form with output injection: According to (3.2), calculate the associated 1-forms θ for the studied system; Determine the family of vector fields τ defined in (3.4) and (3.5); Calculate Λ according to (3.7); Compute ω according to (3.11), then integrate it to obtain the diffeomorphism z = φ(x); Step 5: Apply this diffeomorphism to calculate φ∗ ( f ) and φ∗ (g) where f and g are given by (3.17); Step 6: Design the proposed Luenberger-like observer (3.20) to estimate x of (3.17).

Step 1: Step 2: Step 3: Step 4:

Exercises Exercise 3.1 Consider the following dynamical system: ⎧ x˙1 = −x33 ⎪ ⎪ ⎨ x˙2 = x1 + 2x2 x3 x˙3 = x2 + x32 ⎪ ⎪ ⎩ y = h(x) = x3 and note f = −x33

∂ ∂ ∂ + (x1 + 2x2 x3 ) + x2 + x32 ∂ x1 ∂ x2 ∂ x3

as the vector field that generates the above dynamics. (1) Compute L f h, L 2f h and compare them to y˙ and y¨ . (2) Show that θ1 = dh, θ2 = d L f h and θ3 = d L 2f h are independent 1-forms. (3) Compute τ1 as the solution of the following algebraic equation: 

θ1 (τ1 ) = θ2 (τ1 ) = 0 θ3 (τ1 ) = 0.

(4) Compute τ2 = [τ1 , f ] and τ3 = [τ2 , f ] . (5) Show that those vector fields commute, i.e., [τ1 , τ2 ] = [τ1 , τ3 ] = [τ2 , τ3 ] = 0. What can we deduce from this fact? (6) Construct the following matrix:

68

3 Observer Normal Form with Output Injection

⎛

⎞ θ1 (τ1 ) θ1 (τ2 ) θ1 (τ3 ) Λ = ⎝ θ2 (τ1 ) θ2 (τ2 ) θ2 (τ3 ) ⎠ . θ3 (τ1 ) θ3 (τ2 ) θ3 (τ3 ) ⎛

⎞ θ1 (7) Compute dz = ω = Λ−1 ⎝ θ2 ⎠ and deduce the diffeomorphism z = φ(x). θ3 (8) Check that in the new coordinates we obtain an observer normal form of the following form: z˙ = Az + β(y) y = z3. Exercise 3.2 Consider the following dynamical system: ⎧ x˙1 = −x33 ⎪ ⎪ ⎨ x˙2 = x1 x˙3 = x2 ⎪ ⎪ ⎩ y = h(x) = sin(x3 ). Note f = −x33

∂ ∂ ∂ + x1 + x2 ∂ x1 ∂ x2 ∂ x3

as the vector field that generates the above dynamics. (1) Answer the same questions (1)–(7) in Exercise 3.1. (2) What can we deduce?

References  1. Boutat, D.: Geometrical conditions for observer error linearization via 0, 1, ..., (n − 2). In: Proceedings of 7th IFAC Symposium on Nonlinear Control Systems (2007) 2. Boutat, D.: Extended nonlinear observer normal forms for a class of nonlinear dynamical systems. Int. J. Robust Nonlinear Control 25(3), 461–474 (2015) 3. Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. Math. 575–620 (2003) 4. Guay, M.: Observer linearization by output-dependent time-scale transformations. IEEE Trans. Autom. Control 47(10), 1730–1735 (2002) 5. Isidori, A.: Nonlinear Control Systems. Communications and Control Engineering, 3rd edn. Springer, London (1995) 6. Krener, A., Isidori, A.: Linearization by output injection and nonlinear observers. Syst. Control Lett. 3(1), 47–52 (1983) 7. Krener, A., Respondek, W.: Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim. 23(2), 197–216 (1985) 8. Luenberger, D.: An introduction to observers. IEEE Trans. Autom. Control 16(6), 596–602 (1971)

Chapter 4

Observer Normal Form with Output Injection and Output Diffeomorphism

Abstract For certain nonlinear dynamical systems, the geometric conditions provided in Chap. 3 might not be satisfied, because the desired observer normal form was imposed to have a linear output. This chapter aims at relaxing this constraint by proposing new geometric conditions [1]. Those conditions enable us to deduce a change of coordinates that transforms a given nonlinear dynamical system into an observer normal form with nonlinear output [3]. For such a normal form, a high-gain observer will be designed, and the related design procedure will be given at the end of this chapter.

4.1 Problem Statement Consider the following single-output dynamical system: x˙ = f (x) y = h(x),

(4.1)

where x ∈ X ⊆ Rn with X being a neighborhood of 0, and y ∈ R. Without loss of generality it is assumed that f and h are smooth with f (0) = 0 and h(0) = 0. As we have already discussed in Chap. 3, system (4.1) can be transformed into the observer normal form with output injection (3.3) if and only if the commutativity condition (3.6) of Theorem 3.1 is satisfied. In general, this commutativity condition is restrictive, which can be shown by the following example. Example 4.1 Consider the following Lotka–Volterra model [2] by assuming per capita growth rates are linear: x˙1 = x1 (a − bx2 ) x˙2 = x2 (−c + ex1 ) y = x1 ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_4

(4.2)

69

70

4 Observer Normal Form with Output Injection …

where x1 and x2 are the states which are assumed to be positive, y is the output and the parameters a, b, c and e are all positive with the following meanings: • • • •

a is the growth rate of the prey in the absence of interaction with predator; b measures the impact of predation; c is the natural death of predator in the absence of food; e denotes the efficiency of the predator in interaction with prey.

Following Definition 3.1 in Chap. 3, the observability 1-forms for the studied system (4.2) are given by θ1 = dy = d x1 θ2 = (a − bx2 ) d x1 − bx1 d x2 , and it is clear that θ1 is independent of θ2 . Therefore, the dynamical system fulfills the observability rank condition. Now, let us follow the procedure presented in Chap. 3 to construct the family of vector field τ . By solving the algebraic equation defined in (3.4), we obtain τ1 = −

1 ∂ . bx1 ∂ x2

Then τ2 can be computed by the Lie bracket defined in (3.5), and this yields τ2 = [τ1 , f ] =

∂ a − bx2 − c + ex1 ∂ − , ∂ x1 bx1 ∂ x2



 x1 (a − bx2 ) . x2 (−c + ex1 ) Then, it is easy to check that

where f =

[τ1 , τ2 ] =

2 τ1 = 0, x1

therefore the condition (3.6) of Theorem 3.1 is not satisfied, and we cannot find a change of coordinates z = φ(x) that transforms dynamical system (4.2) into the nonlinear observer normal form (3.3). However, if we apply the following change of coordinates 

z1 z2



 =

c ln x1 − ex1 − bx2 ln x1



to system (4.2), a simple calculation yields z˙ 1 = dtd (c ln x1 − ex1 − bx2 ) = ac − eax1 = ac − eay z˙ 2 = (a − bx2 ) = z 1 − c ln y + ey y¯ = z 2 .

4.1 Problem Statement

71

It is worth noting that the new output y¯ = ln(y) is linear in the new coordinates. And the above transformed system can be written into the following nonlinear observer normal form with nonlinear output: z˙ 1 = β1 (y) z˙ 2 = z 1 + β2 (y) y = ϕ(z 2 ), where ϕ is a diffeomorphism on the output, and β1 (y) = ac − eay β2 (y) = −c ln y + ey ϕ(z 2 ) = e z2 .  The above example clearly shows that, even if the commutativity condition (3.6) of Theorem 3.1 is not satisfied, it is still possible to transform a nonlinear dynamical system into the normal form (3.3) with nonlinear output, and this is achieved by applying a diffeomorphism z = φ(x) to the state, which implicitly applies as well a diffeomorphism ϕ(y) to the output. Therefore, this chapter focuses on how to calculate such a diffeomorphism ϕ(y) on the output, together with a local change of coordinates z = φ(x), such that the studied dynamical system (4.1) can be transformed into the following nonlinear observer normal form with nonlinear output diffeomorphism: z˙ = A O z + β( y¯ ) y¯ = C O z = z n , ⎛

··· ··· ... ··· 0 ···

0 ⎜1 ⎜. where y¯ = ϕ(y), A O = ⎜ ⎜ .. ⎝0

(4.3)

⎞ 0 0 0 0 0 0⎟ . ⎟ and C = 0 · · · 0 1 . ... O · · · .. ⎟ ⎟ 1 0 0⎠ 0 1 0

4.2 Diffeomorphism on Output For the studied system (4.1), let us follow the same procedure presented in Chap. 3 to calculate firstly the 1-forms ⎡ ⎤ θ1 ⎣ θ = ... ⎦ θn

72

4 Observer Normal Form with Output Injection …

with θi = d L i−1 f h, for 1 ≤ i ≤ n, which is assumed as well to be linearly independent. Therefore, all states of system (4.1) are observable since the observability rank condition is satisfied. Based on the above 1-forms, in Chap. 3, the family of vector fields τ was determined by (3.4) and (3.5). Then an invertible matrix Λ = θ τ has been deduced with the special form described in (3.10), i.e., ⎛

··· 0 ⎜ · · · θ2 τn−1 ⎜ . Λ = θτ = ⎜ . . . .. ⎜ ⎝ 0 θ τ ··· ··· n−1 2 θn τ1 ∗ · · · · · · 0 0 .. .

0 0 .. .

⎞ θ1 τn ∗ ⎟ .. ⎟ . ⎟ ⎟, ∗ ⎠

(4.4)

∗

where θn τ1 = · · · = θ1 τn = 1. This is due to the construction of τ1 by fulfilling the algebraic equation (3.4). Then, a simple calculation shows that its inverse can be written as the following form: ⎛

Λ−1

∗ ∗ .. .

∗ ··· ⎜ ∗ ··· ⎜ .. . . ⎜ =⎜ . . ⎝ ∗ θ τ ··· n 1 θn τ1 0 · · ·

⎞ ∗ θn τ1 θn τ1 0 ⎟ .. .. ⎟ . . ⎟ ⎟, ··· 0 ⎠ ··· 0

(4.5)

where ∗ just implies any values which satisfy Λ−1 Λ = I . In Theorem 3.1, it has been proven that the differential of the desired diffeomorphism z = φ(x) was defined as dz = dφ(x) = Λ−1 θ. Therefore, the last component of the diffeomorphism satisfies dz n = [θn (τ1 )] θ1 . Recall that the vector field τ1 in Chap. 3 was determined by (3.4), satisfying θn (τ1 ) = 1 and θk (τ1 ) = 0 for 1 ≤ k ≤ n − 1. Moreover, according to Definition 3.1, we have θ1 = dh. Consequently, this construction of τ1 implicitly implies that the deduced diffeomorphism has z n = h(x). Equivalently, it implies that the transformed system will yield a linear output y = z n . The above analysis explained why the output of the transformed system is not affected by the deduced diffeomorphism, and more importantly it shows the clue

4.2 Diffeomorphism on Output

73

how to introduce a diffeomorphism on the output, i.e., modify the algebraic equation (3.4) which was used to determine τ1 in Chap. 3. Following this thought, let us determine the first vector field σ1 by the following algebraic conditions 

θk (σ1 ) = 0, for 1 ≤ k ≤ n − 1 θn (σ1 ) = l(y),

(4.6)

where l(y) is a nowhere-vanishing function of y which will be determined later. It is clear that (4.6) is equivalent to (3.4) when l(y) = 1. In this case, we have σ1 = τ1 . For the case l(y) = 1, σ1 can be seen as a generalization of τ1 by relaxing the condition θn (τ1 ) = 1, allowing such a nowhere-vanishing function l(y). Denote τ = [τ1 , . . . , τn ] as the family of vector fields determined by (3.4) and (3.5). It is clear that the new constructed vector field σ1 satisfies σ1 = l(y)τ1 .

(4.7)

Similar to (3.5), from the vector field σ1 , we can then define the following family of independent vector fields by induction: σi = [σi−1 , f ], for 2 ≤ i ≤ n.

(4.8)

As the function l(y) is assumed to be nowhere-vanishing in its domain and τi are independent, then the vector fields σi are independent as well.   Suppose now the commutativity conditions are not satisfied for τ , i.e., τi , τ j = 0 for 1 ≤ i, j ≤ n, then the key point is to seek such a nowhere-vanishing function l(y)  to make the commutativity conditions be satisfied for the vector fields of σ , i.e., σi , σ j = 0 for 1 ≤ i, j ≤ n. Before stating the differential geometric conditions on the existence of such a diffeomorphism on the output, it is useful to understand the relations between the new family of vector fields σ and the former family of vector fields τ deduced in Chap. 3. Lemma 4.1 The vector field τ and the vector field σ are linked by the following equation:   k−1  k−1  j  L f l τk− j , (−1) j (4.9) σk = j j=0 j+1 j where k−1 are the binomial coefficients, and L f l = L f L f l for j ≥ 0 with j L 0f l = l. Proof The proof will be performed by induction. Before that, let us check the first three equations with k = 1, 2, 3. For k = 1, according to (4.9), it is obvious that σ1 = L 0f l(y)τ1 = l(y)τ1 ,

74

4 Observer Normal Form with Output Injection …

which is always true by the definition of σ1 in (4.6). When k = 2, (4.9) reads as σ2 =

1  j=0

(−1) j

1  j

 j L f l τ2− j

(4.10)

= l(y)τ2 − (L f l)τ1 . On the other hand, σ2 is constructed by following (4.8), and we have σ2 = [σ1 , f ] = [lτ1 , f ] = L lτ1 f − L f (lτ1 ) 1 = ∂∂ xf lτ1 − ∂∂lx τ1 f − l ∂τ f ∂x = l[τ1 , f ] − (L f l)τ1 = lτ2 − (L f l)τ1 .

(4.11)

Therefore, from (4.10) and (4.11) we can conclude that (4.9) is satisfied for k = 2. For the case k = 3, (4.9) becomes σ3 = lτ3 − 2L f (l)τ2 + L 2f (l)τ1 . By following (4.8), we have σ3 = [σ2 , f ] = [lτ2 − L f (l)τ1 , f ] = [lτ2 , f ] − [L f (l)τ1 , f ] = l[τ2 , f ] − L f τ2 + L 2f τ1 − L f (l)[τ1 , f ] = lτ3 − 2L f (l)τ2 + L 2f (l)τ1 . Thus, the relation (4.9) is true as well for k = 3. For the general case, we are going to proceed by induction. For this, it is assumed that the formula (4.9) is true for k ≥ 1, and we will show that it is true as well for k + 1. Suppose that σk satisfies (4.9), then σk+1 can be calculated by following (4.8) as σk+1 = [σk , f ] ⎡ ⎤   k−1   k−1 j L f l τk− j , f ⎦ = ⎣ (−1) j j j=0

⎞ ⎛     k−1    k−1 k−1 j j = L f l [τk− j , f ] − L f ⎝ (−1) j L f l ⎠ τk− j . (−1) j j j j=0 j=0 k−1 

j

j+1

Replacing [τk− j , f ] in the above equation by τk− j+1 and L f L f by L f , we obtain

4.2 Diffeomorphism on Output

σk+1 =

k−1 

(−1) j

j=0

75

    k−1  k−1  j  k − 1  j+1  L f l τk− j+1 − L f l τk− j . (−1) j j j j=0

Rewriting the index in the second sum of the above equation, we obtain σk+1 =

k−1 

(−1) j

j=1

 k−1 j

+

k−1   j−1

 j L f l τk− j + (−1)k (L kf l)τ1 + lτk+1 .

(4.12)

Finally, with the following binomial expansions using Pascal’s triangle       k k−1 k−1 . + = j j j −1 Equation (4.12) can be read as σk+1

   k j L f l τk− j , = (−1) j j=0 k 

j

which is exactly Eq. (4.9) for k + 1, and we proved Lemma 4.1 by induction.



The relation between σi and τi has been exploited in (4.9) of Lemma 4.1, which depends on the expression of function L if l(y). Therefore, before computing the Lie   bracket σi , σ j for 1 ≤ i, j ≤ n, we should firstly know how to compute L if l(y). To do so, we adopt the following result for high-order derivative of the composition of functions given by Faà di Bruno’s Formula [7]. To state it, we denote Pm = {1, 2, . . . , m}, Pmm = Pm × · · · × Pm and  Um = k = (k1 , . . . , km ) ∈

Pmm

 m  : iki = m . i=1

Based on the above definitions, we denote then by k! = k1 ! · · · km ! and |k| =

m 

ki .

i=1

Hence, the successive Lie derivative of l(y) can be achieved by combinatorial calculus, which is stated in the following lemma. Lemma 4.2 The mth Lie derivative of the function l(y) = l(h(x)) in the direction of the vector field f is given by the following formula: L mf l

=

m  m! k∈Um

where l (k) =

d k l(y) dy k

k!

l

(|k|)

m  i=1



L if h i!

ki ,

denotes the kth derivative of l(y) with respect to y.

(4.13)

76

4 Observer Normal Form with Output Injection …

To give an idea how the above formula works, let us calculate the first three high-order derivatives of l in the direction of f • L f l = l  (y)L f h • L 2f l = l  (y)(L f h)2 + l  (y)L 2f h • L 3f l = l (3) (y)(L f h)3 + 3l  (y)(L f h)L 2f h + l  (y)L 3f h. Based on the result stated in Lemma 4.2, we would like to highlight the following facts. Remark 4.1 If k1 = k2 = · · · = km−1 = 0 and km = 1, i.e., k = 1, then it is easy to see from (4.13) that the expression L mf l contains the term l  (y)L mf h. Hence, we have (1) L τk L mf l = 0 if m < n − k, since L τk L if h = 0 for i ≤ m < n − k; (2) L τk L mf l = l  (y) if m = n − k, because L τk L if h = 0 for i < m. To check that the commutativity of vector fields σi for 1 ≤ i ≤ n, it is natural to verify step by step. Thus, we will firstly check the commutativity of σ1 with the other vector fields, and then pass to σ2 and so on. Let us start with the vector field σ1 . Lemma 4.3 The following items are equivalent: (1) [σ1 , σk ] = 0, for 2 ≤ k ≤ n − 1; (2) [τ1 , τk ] = 0, for 2 ≤ k ≤ n − 1. Moreover, we have

dl [σ1 , σn ] = l 2 [τ1 , τn ] + (−1)n−1 − 1 l τ1 . dy

(4.14)

Proof Using the result stated in Lemma 4.1, we have   k−1 (L if l)τk−i , for 2 ≤ k ≤ n. σk = (−1) i i=0 k−1 

i

From Remark 4.1, we know that L τ1 L if l = 0 for i < n and L τk l = 0 for k < n. Therefore, for k < n we have [σ1 , σk ] =

k−1  i=0

(−1)i

  k−1 l(L if l)[τ1 , τk−i ]. i

The following will use the above results to prove Lemma 4.3 by induction. For this, consider firstly k = 2. • For the case n = 2 and k = 2, then from the expression of σ2 in Lemma 4.1, we have σ1 = lτ1 σ2 = lτ2 − (L f l)τ1 .

4.2 Diffeomorphism on Output

77

A straightforward calculation gives   [σ1 , σ2 ] = lτ1 , lτ2 − (L f l)τ1 = l 2 [τ1 , τ2 ] − ll  dh(τ2 )τ1 − ll  d L f h (τ1 ) τ1 = l 2 [τ1 , τ2 ] − 2ll  τ1 because dh(τ2 ) = 1 and d L f h (τ1 ) = 1, which is exactly the formula (4.14) for n = 2; • For the case n > 2 and k = 2, we obtain [σ1 , σ2 ] = l 2 [τ1 , τ2 ]. Therefore, as l(y) = 0, we can state that [σ1 , σ2 ] = 0 if and only if [τ1 , τ2 ] = 0. With the last equation obtained above, we can then step forward to consider the case k = 3. By following the same way, it is easy to show that [σ1 , σ3 ] = 0 if and only if [τ1 , τ3 ] = 0. This process is iterated up to k = n − 1. Now, assume that [τ1 , τk ] = 0 for k ≤ n − 1, then [σ1 , σn ] = l 2 [τ1 , τn ] − L τn (l)τ1 + (−1)n−1 L τ1 (L n−1 f l)τ1 , which yields, thanks to Remark 4.1, the following equation:

[σ1 , σn ] = l 2 [τ1 , τn ] + (−1)n−1 − 1 ll  τ1 , 

and it is exactly formula (4.14). The following corollary is a direct result of Lemma 4.3.

Corollary 4.1 Assume that the equivalent items (1) and (2) of Lemma 4.3 are fulfilled. (1) If n is an even integer, i.e., n = 2 p for some p ∈ N, then [σ1 , σn ] = 0, if and only if [τ1 , τn ] = μ1 (y)τ1 . In this case, we obtain

1

l(y) = e 2

y 0

μ1 (s)ds

,

which is one solution of the differential equation l 2 μ1 − 2ll  = 0, deduced from (4.14); (2) If n is an odd integer, i.e., n = 2 p + 1 for some p ∈ N, then [σ1 , σn ] = 0,

78

4 Observer Normal Form with Output Injection …

if and only if [τ1 , τn ] = 0. Example 4.2 Let us consider again Example 4.1 to show how to compute the function l. From the previous calculations, we know that [τ1 , τ2 ] =

1 τ1 , x1

thus in the above differential equation μ = x21 = 2y . Therefore, according to Corollary 4.1, we get  y ds l(y) = e 0 ys = y. Now set σ1 = lτ1 , then

σ2 = [σ1 , f ] = lτ2 − L f lτ1 .

Therefore, [σ1 , σ2 ] = [lτ1 , lτ2 − L f lτ1 ] = l 2 [τ1 , τ2 ] − l L τ2 lτ1 − l L τ1 (L f l)τ1 . As l = y = x1 , then

L f l = x1 (a − bx2 ),

which implies that L τ1 (L f l) = 1. We also have L τ2 l = 1 and [τ1 , τ2 ] = 2l , then we deduce [σ1 , σ2 ] = 0.  We saw that, if n is an odd integer, i.e., n = 2 p + 1 for some p ∈ N, then Eq. (4.14) is reduced to [σ1 , σn ] = l 2 [τ1 , τn ]. Therefore, [σ1 , σn ] = 0 if and only if [τ1 , τn ] = 0. In this case, since the Lie bracket [σ1 , σn ] does not give us information about l(y), we should study other Lie brackets, such as [σ2 , σn ], to deduce l(y). Inspired by this thought, the following lemma is to investigate the commutativity of σ2 with σn for the case where n is odd. Lemma 4.4 Assume n = 2 p + 1 and [τ1 , τk ] = 0 for all 2 ≤ k ≤ n, then the following items are equivalent: (1) [σ2 , σk ] = 0 for all 2 ≤ k ≤ n − 1; (2) [τ2 , τk ] = 0 for all 2 ≤ k ≤ n − 1. Moreover, we have [σ2 , σn ] = l 2 [τ2 , τn ] − nl

dl τ2 + γ (x)τ1 . dy

(4.15)

4.2 Diffeomorphism on Output

79

Proof Recall that the expressions of σ2 and σ3 are σ2 = lτ2 − L f (l)τ1 and σ3 = lτ3 − 2L f (l)τ2 + L 2f (l)τ1 . If n = 3, then from the above two expressions we obtain

[σ2 , σ3 ] = l 2 [τ2 , τ3 ] + −2l L τ2 L f (l) − l L τ3 l τ2 + γ τ1 , where

γ = −3l  L f (l) + l L τ2 L 2f (l) + l L τ3 L f (l).

A straightforward calculation leads to [σ2 , σ3 ] = l 2 [τ2 , τ3 ] − 3ll  τ2 + γ τ1 . Now, we assume that n = 2 p + 1 > 3, then we have the following cases: • If 2 ≤ k ≤ n − 2, then [σ2 , σk ] = 0 if and only if [τ2 , τk ] = 0. Indeed, recall from above that   k−1  k−1  j  L f l τk− j . (−1) j σk = j j=0 As σ2 = lτ2 − L f (l)τ1 , then we have   [σ2 , σk ] = [lτ2 , σk ] − L f (l)τ1 , σk . We also know that, for 0 ≤ j ≤ k ≤ n − 2, j

L τ2 L f l = 0,

j

L τ1 L f l = 0,

L τk l = 0

and L τk L f l = 0. As [τ1 , τk ] = 0, we obtain    k − 1  j  L f l τ2 , τk− j . (−1) [σ2 , σk ] = l [τ2 , σk ] = j j=0 k−1 

j

Therefore, for k = 3 ≤ n − 2 we have [σ2 , σk ] = l [τ2 , σk ] = l [τ2 , τ3 ] = 0.

80

4 Observer Normal Form with Output Injection …

Finally, we can prove the result by induction as follows: If [σ2 , σs ] = [τ2 , τs ] = 0 for all s ≤ k − 1, then a straightforward calculation leads to σ2 , σs+1 = l τ2 , τs+1 . Thereby, [τ2 , τk ] = 0 for all k ≤ n − 2 if and only if [σ2 , σk ] = 0 for all k ≤ n − 2. • If k = n − 1, then we obtain     

 σ2 , σn−1 = l 2 τ2 , τn−1 + (−1)n−2 L τ2 L n−2 f l τ1 + L σn−1 L f l τ1  

= l 2 τ2 , τn−1 + (−1)n−2 ll  + ll  τ1 . Thus, we have [σ2 , σn−1 ] = l 2 [τ2 , τn−1 ] + ll  (−1 + (−1)n−1 )τ1 . As n is odd, this leads to [σ2 , σn−1 ] = 0 if and only if [τ2 , τn−1 ] = 0. Therefore, for 2 ≤ k ≤ n − 1, we proved that the item (1) is equivalent to the item (2) of Lemma 4.4. Finally, consider σk defined in (4.9). Taking k = n, by keeping only the non-zero terms, we have    n−1  n−2  L f l τ2 + (−1)n−1 L n−1 [σ2 , σn ] = l [τ2 , τn ] + lτ2 , (−1)n−2 n−2 f l τ1

−l L τn l τ2 . Therefore, after some simplifications we obtain [σ2 , σn ] = l [τ2 , τn ] − nll  τ2 + γ τ1 ,  

where γ = −nl  L f l + l L τn L f l + l L τ2 L n−1 f l .



Like the case of even dimension (i.e., n = 2 p) where the function l(y) can be obtained via [σ1 , σn ] = 0, the similar result can be stated for the case of odd dimension. Corollary 4.2 Assume that the conditions stated in Lemma 4.4 are fulfilled, then [σ2 , σn ] = 0 only if [τ2 , τn ] = μ2 (y)τ2 + μ1 (x)τ1 . In this case, we obtain 1

l(y) = e n

y 0

μ2 (s)ds

,

which is one solution of l 2 μ2 (y) − nll  = 0. Remark 4.2 Corollary 4.2 provides only a necessary condition to ensure [σ2 , σn ] = 0. Precisely, substituting this necessary condition [τ2 , τn ] = μ2 (y)τ2 + μ1 (x)τ1 back into (4.15), we have dl τ + γ (x)τ1 [σ2 , σn ] = l 2 [τ2 , τn ] − nl dy

2 2 = l μ2 (y) − nll  τ2 + l 2 μ1 (x)τ1 .

4.2 Diffeomorphism on Output

81

It is clear that, if l(y) is chosen as the solution of l 2 μ2 (y) − nll  = 0, we have [σ2 , σn ] = l 2 μ1 (x)τ1 , which shows that such a choice of l(y) can only ensure [σ2 , σn ] = 0 when μ1 (y) = 0. This issue will be highlighted in the following example. Example 4.3 Consider the following dynamical system: x˙1 = x3 x1 x˙2 = x1 − 3x3 x˙3 = x2 y = x3 . A straightforward calculation provides θ1 = d x3 θ2 = d x2 θ3 = d x1 − 3d x3 . Therefore, the vector fields of the frame τ are given by τ1 = τ2 = τ3 =

∂ ∂ x1 ∂ ∂ x2 ∂ ∂ x3

+ x3 ∂∂x1

+ x3 ∂∂x2 + x32 − x2 ∂∂x1 .

These vector fields fulfill the following equations: [τ1 , τ2 ] = 0 [τ1 , τ3 ] = 0 [τ2 , τ3 ] = −τ1 . Comparing to the necessary condition stated in Corollary 4.2, i.e., [τ2 , τ3 ] = μ2 (y)τ2 + μ1 (x)τ1 , we have

μ2 (y) = 0 μ1 (x) = −1.

Solving the equation l 2 μ2 (y) − nll  = 0, we obtain l(y) = dconst . Thus we have

82

4 Observer Normal Form with Output Injection … 2 [σ2 , σn ] = l 2 μ1 (x)τ1 = −d 2const τ1 = −dconst

∂ = 0. ∂ x1

This example shows that the stated problem cannot be solved by means of a diffeomorphism on the output. We will see later in Chap. 5 that this kind of systems can be transformed into another observer normal form, thanks to the so-called immersion technique. 

4.3 Diffeomorphism for States It is clear now that the information of the existence of such a function l(y) is hidden in two Lie brackets, either [τ1 , τn ] or [τ2 , τn ], depending on the dimension (odd or even) of the studied system. In the case where the state dimension n is even, then the information of l(y) is hidden in [τ1 , τn ]. For n is odd, then l(y) is contained in [τ2 , τn ]. These two Lie brackets enable us to determine the nowhere-vanishing function l(y). After having determined the function l(y), we can then calculate the family of vector fields σ = [σ1 , . . . , σn ] via (4.7) and (4.8). With the 1-forms θ = [θ1T , . . . , θnT ]T where θi = d L i−1 f h for 1 ≤ i ≤ n, we can then calculate the following matrix: ⎛

0 ⎜ 0 ⎜ . . Λ¯ = θ σ = ⎜ ⎜ . ⎝ 0 l(y)

0 ··· 0 0 · · · l(y) .. .. . . .. . l(y) · · · Λ¯ n−1,n−1 Λ¯ n,2 · · · Λ¯ n,n−1

⎞ l(y) Λ¯ 2,n ⎟ .. ⎟ . ⎟ ⎟. Λ¯ n−1,n ⎠ Λ¯ n,n

(4.16)

It is clear that this matrix is invertible for nowhere-vanishing function l(y), thus we can define the following 1-forms: ω¯ = Λ¯ −1 θ := (ω¯ i )1≤i≤n .

(4.17)

Note that the determination of function l(y) enables us to compute uniquely the frame σ , which can then be used to calculate the diffeomorphism ϕ(y) on the output, according to the results stated in Corollaries 4.1 and 4.2. Based on the matrix Λ¯ defined in (4.16), we can then state the result on the deduction of the diffeomorphism z = φ(x) in a similar way as Theorem 3.2. Theorem 4.1 The following three assertions are equivalent: (1) There exists a local change of coordinates z = φ(x) that transforms dynamical system (4.1) into the nonlinear observer normal form (4.3); (2) The vector fields σi and σ j for 1 ≤ i, j ≤ n defined in (4.7)–(4.8) commute, i.e.,   σi , σ j = 0;

(4.18)

4.3 Diffeomorphism for States

83

(3) The 1-forms ω¯ i defined in (4.17) for 1 ≤ i ≤ n are closed, i.e., d ω¯ i = 0.

(4.19)

Furthermore, the change of coordinates is determined by ! z i = φi (x) =

γ

ω¯ i ,

(4.20)

where γ : [0, 1] → X is any smooth curve on contractile neighborhood of 0 such that γ (0) = 0 and γ (1) = x. Proof The proof is similar to that of Theorem 3.2 in Chap. 3 by replacing τi by σi . The only difference is the diffeomorphism on the output. Therefore, the following will focus only on the deduction of this diffeomorphism ϕ(y). For the matrix Λ¯ defined in (4.16), a simple calculation shows that its inverse is of the following form: ⎛

Λ¯ −1

Since

∗ ∗ .. .

∗ ··· ∗ 1 ⎜ ∗ · · · l(y) ⎜ ⎜ .. . . .. =⎜ . . . ⎜ 1 ⎝ ∗ l(y) · · · · · · 1 0 ··· ··· l(y)

1 l(y)

⎞

⎟ ⎟ ⎟ ⎟. ⎟ 0 ⎠ 0 0 .. .

(4.21)

dz = dφ(x) = Λ¯ −1 θ,

thus the last component of the diffeomorphism satisfies dz n =

1 1 θ1 = dy, l(y) l(y)

where we use the definition of θ1 with θ1 = dh = dy. Finally, we obtain the diffeomorphism on the output as ! y ds . (4.22) z n = ϕ(y) = 0 l(s)  Theorem 4.1 shows the constructive way to calculatethe diffeomorphism z = y ds φ(x) whose last component is of the form z n = ϕ(y) = 0 l(s) . Precisely, we first ¯ build the matrix Λ defined in (4.16) by evaluating the 1-forms θi on the vector fields σi . Then, we compute the components of the change of coordinates as the integral of the components of Λ¯ −1 ω¯ where the components of ω¯ are given in (4.17). The following example is to highlight the above result.

84

4 Observer Normal Form with Output Injection …

Example 4.4 Consider the following system: x˙1 = α1,1 (x2 )x1 + α1,2 (x2 )x12 + β1 (x2 ) x˙2 = α2,2 (x2 )x1 + β2 (x2 ) y = x2 ,

(4.23)

where α2,2 (x2 ) = 0. A straightforward calculation gives us the observability 1-forms as follows: θ1 = d x2 

θ2 = α2,2 (x2 )d x1 + α2,2 (x2 )x1 + β2 (x2 ) d x2 . Thereby, we deduce the frame τ as 1 ∂ α2,2 ∂ x1 ∂ + α12,2 ∂ x2

τ1 = τ2 =



 α2,2 (α2,2 x1 α2,2

+ β2 ) + α1,1 (x2 ) + 2α1,2 (x2 )x1



∂ . ∂ x1

It can be checked that these two vector fields do not commute. Indeed, we have       α α2,2 α1,2 α1,2 ∂ τ1 . + 2 = 2 + [τ1 , τ2 ] = 2 α2,2 2 ∂ x1 α2,2 α2,2 α2 2,2

2,2

According to Corollary 4.1, we obtain y

l(y) = e

 (s)  y α1,2 (s) α2,2 0 α2,2 (s) ds+ 0 α 2 (s) ds 2,2

Hence σ1 = l(y)τ1 = e

y 0

= α2,2 (y)e α1,2 (s)ds α2,2 (s)

y 0

α1,2 (s)ds α2,2 (s)

.

∂ , ∂ x1

and we obtain σ2 = [σ1 , f ] =e

y 0

α1,2 (s)ds α2,2 (s)

  α2,2 ∂∂x2 + α1,1 −

α1,2 α2,2

 

α2,2 x1 + β2 + 2α1,2 x1 ∂∂x1 .

Then, it is easy to check that [σ1 , σ2 ] = 0. This last condition ensures the existence of the diffeomorphism z = φ(x). To construct it, we first build Λ¯ as follows:     0 l(y) θ1 (σ1 ) θ1 (σ2 ) , = Λ¯ = θ2 (σ1 ) θ2 (σ2 ) l(y) Λ¯ 22

4.3 Diffeomorphism for States

where l(y) = α2,2 (y)e

y 0

85

α1,2 (s)ds α2,2(s)

and

  

α1,2 ¯ Λ22 = l α1,1 − α2,2 x1 + β2 + 2α1,2 x1 + l α2,2 x1 + β2 . α2,2 Since the inverse of Λ¯ is Λ¯ −1 = then we obtain

and



¯

− Λl 222 1 l

ω2 = 1l θ1 =

1 l



d x2 l(x2 )

¯

ω1 = − Λl 222θ1 + 1l θ2

1,2 = − 1l α1,1 − αα2,2 β2 + α1,2 x1 y 1

= d l α2,2 x1 + 0 ς (s)ds ,

where ς =−

,

0



d x2 + 1l α2,2 d x1

  α1,2 1 α1,1 − β2 . l α2,2

Finally, we obtain the following diffeomorphism: z = φ(x) =

1 l

α2,2 x1+

y

0 ς (s)ds y ds 0 l(s)



via which the studied system (4.23) can be transformed into the sought normal form. 

4.4 Observer Design 4.4.1 High-Gain Observer By considering the normal form (4.3), similar to the normal form (3.18) in Chap. 3, different types of observers can be designed. Here, we will design a high-gain asymptotic observer [4, 5] for such a normal form. Consider the following dynamics: z˙ˆ = A O zˆ + β( y¯ ) + Δε K (ϕ(y) − C O zˆ ),

(4.24)

86

4 Observer Normal Form with Output Injection …

where y¯ = ϕ(y) and

⎡

⎤ 0 ··· 0 εn−1 · · · 0 ⎥ .. . . . .. ⎥ . .⎦ 0 0 ··· ε

εn ⎢0 Δε = ⎢ ⎣ ...

with ε being large positive constant and K = [K 1 , . . . , K n ]T with K i being chosen such that (A O − K C O ) is Hurwitz, which is always possible since the pair (A O , C O ) is observable. Note ei = z i − zˆ i , then according to (4.3) and (4.24), we obtain e˙i = ei−1 − εn−i+1 K i en .

(4.25)

Denote e¯ = [e¯1 , . . . , e¯n ]T and e¯i =

1

ei εn−i

for 1 ≤ i ≤ n, then we have   1 1 ei−1 − εn−i+1 K i en e˙i = εn−i e˙¯i = εn−i 1 = εn−i ei−1 − εK i en = εe¯i−1 − εK i e¯n  = ε e¯i−1 − K i e¯n which is equivalent to

(4.26)

¯ e˙¯ = ε [A O − K C O ] e.

Due to the fact that A O − K C O is Hurwitz and ε is positive, e¯ is asymptotically stable, so is e. Consequently, we have lim zˆ (t) = z(t).

t→+∞

Therefore, we can obtain the estimation of x by inverting the diffeomorphism φ(x). In summary, the following dynamics zˆ˙ = A O zˆ + β( y¯ ) + Δε K (ϕ(y) − C O zˆ ) xˆ = φ −1 (ˆz )

(4.27)

is an asymptotic observer of (4.1). Compared to Luenberger-like observer designed in (3.20), an additional diagonal matrix Δε is adopted when designing observer (4.27). The main reason is to attenuate the influence of noise in the model. In such a situation, the observation error dynamics of (4.25) can be rewritten as

4.4 Observer Design

87

e˙i = ei−1 − εn−i+1 K i en + di , where di represents the noise (or uncertainties which cannot be modeled). Following the same procedure, we can then obtain   1 1 n−i+1 K i en + di e˙¯i = εn−i  e˙i = εn−i ei−1 −1 ε = ε e¯i−1 − K i e¯n + εn−i di which is equivalent to

(4.28)

e˙¯ = ε [A O − K C O ] e¯ + Δ¯ ε d,

where d = [d1 , . . . , dn ]T and ⎡

1 εn−1

⎢0 Δ¯ ε = ⎢ ⎣ ... 0

0 1 εn−2

.. . 0

··· ··· ... ···

⎤ 0 0⎥ . .. ⎥ .⎦ 1

Due to the fact that ε is chosen to be large positive constant, the influence of the 1 , defined in Δ¯ ε [6]. noise di is attenuated by multiplying εn−i

4.4.2 Design Procedure The following summarizes the procedure to design observer based on the proposed normal form (4.3): Step 1: According to (3.2), calculate the associated 1-forms θ for the studied system; Step 2: Determine the family of vector fields τ defined in (3.4) and (3.5); Step 3: Calculate the new family of vector fields σ according to (4.7) and (4.8) with unknown function l(y) to be determined; Step 4: Apply Corollaries 4.1 and 4.2 to compute l(y); Step 5: Calculate Λ¯ according to (4.16) and ω according to (4.17), then compute the diffeomorphism φ(x) according to (4.20); Step 6: Apply this diffeomorphism to calculate φ∗ ( f ) where f is given by (4.1); Step 7: Design the proposed high-gain observer (4.27) to estimate x of (4.1).

88

4 Observer Normal Form with Output Injection …

Exercises Exercise 4.1 Consider the dynamical system ⎧ ⎨ x˙1 = x1 x2 x˙2 = x1   ⎩ y = h(x) = sin x2 , with x2 ∈ 0, π2 and note f = x1 x2

∂ ∂ + x1 ∂ x1 ∂ x2

as the vector field that generates the above dynamics. (1) (2) (3) (4) (5)

Compute L f h. Compute τ2 = [τ1 , f ]. Show that [τ1 , τ2 ] = ϕ(y) ∂∂x1 = 0. What can we deduce from this fact? Set σ1 = α(y)τ1 . Compute σ2 = [σ1 , f ] and deduce α(y) such that [σ1 , σ2 ] = 0. Construct the matrix   θ1 (σ1 ) θ1 (σ2 ) . Λ= θ2 (σ1 ) θ2 (σ2 ) 

 θ1 and deduce the diffeomorphism z = φ(x). θ2 (7) Check that in this new coordinate we obtain the following observer normal form:

(6) Compute dz = ω = Λ−1

z˙ = Az + β(y) y = z2 .

(4.29)

Exercise 4.2 Consider the following dynamical system: ⎧ x˙1 = −x33 ⎪ ⎪ ⎨ x˙2 = x1 x˙3 = x2 ⎪ ⎪ ⎩ y = h(x) = sin(x3 ). Use the same procedure as in the above exercise to find an observer normal form similar to (4.29). Exercise 4.3 Consider the following dynamical system: ⎧ x˙1 = x1 x3 ⎪ ⎪ ⎨ x˙2 = x1 x˙3 = x1 ⎪ ⎪ ⎩ y = h(x) = x3 .

Exercises

89

Show that it is not possible to find the diffeomorphism for the above dynamical system by using the previous algorithm to transform it into the observer normal form (4.29).

References 1. Boutat, D.: Extended nonlinear observer normal forms for a class of nonlinear dynamical systems. Int. J. Robust Nonlinear Control 25(3), 461–474 (2015) 2. Boutat, D., Saif, M.: Observer normal forms for a class of Predator-Prey models. J. Frankl. Inst. 353(10), 2178–2198 (2016) 3. Boutat, D., Benali, A., Hammouri, H., Busawon, K.: New algorithm for observer error linearization with a diffeomorphism on the outputs. Automatica 45(10), 2187–2193 (2009) 4. Busawon, K., Farza, M., Hammouri, H.: A simple observer for a class of nonlinear systems. Appl. Math. Lett. 11(3), 27–31 (1998) 5. Gauthier, J.P., Kupka, I.: Deterministic Observation Theory and Applications. Cambridge University Press, Cambridge (2001) 6. Hammouri, H., de Leon Morales, J.: Observer synthesis for state-affine systems. In: Proceedings of the 29th IEEE Conference on Decision and Control, pp. 784–785 (1990) 7. Johnson, W.: The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109(3), 217–234 (2002)

Chapter 5

Observer Normal Form by Means of Extended Dynamics

Abstract This chapter provides a new geometric algorithm to overcome some obstructions that cannot be handled by those results proposed in Chaps. 3 and 4. This result is based on those presented in Chaps. 3 and 4, by adding an auxiliary dynamics into the studied nonlinear systems. Necessary and sufficient geometric conditions are deduced to guarantee the existence of such an auxiliary dynamics and a change of coordinates to transform the nonlinear dynamical system together with that added auxiliary dynamics into an observer normal form [9]. Unlike the normal form (3.3) and (4.3), the desired observer normal form in this chapter will be a function of the output and the auxiliary variables which are used to define the auxiliary dynamics. Moreover, this algorithm enables us to compute the auxiliary dynamics as well as the change of coordinates. This chapter only deals with the case of scalar extended dynamic, and gives an example to illustrate the possibility to do as well high-dimensional extension.

5.1 Problem Statement Consider the following single-output dynamical system x˙ = f (x) y = h(x),

(5.1)

where x ∈ X ⊆ Rn with X being a neighborhood of 0, and y ∈ R. Without loss of generality, it is assumed that f and h are smooth with f (0) = 0 and h(0) = 0. We have shown in Chap. 3 that such a nonlinear system might be transformed, via a diffeomorphism on the state, into the observer normal form with output injection (3.3) if and only if τ , defined in (3.5), satisfies the commutativity condition (3.6). If not, then we may introduce a diffeomorphism on the output, and transform system (5.1) into the observer normal form (4.3). If unfortunately this solution does not work as well, we can try to introduce an extended dynamics into the studied system to solve such a problem. The following example highlights this idea. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_5

91

92

5 Observer Normal Form by Means of Extended Dynamics

Example 5.1 Consider the following dynamical system x˙1 = γ (y)x1 x˙2 = x1 x˙3 = x2 y = x3 with γ  (y) = to

dγ (y) dy

(5.2)

= 0, i.e., γ (y) is not constant. A straightforward calculation leads

It is easy to see that

τ1 = τ2 = τ3 =

∂ ∂ x1 ∂ ∂ x2 ∂ ∂ x3

+ γ (y)τ1 + γ (y)τ2 − x2 γ  (y)τ1 .

[τ2 , τ3 ] = −2γ  (y)τ1 = 0,

therefore Theorem 3.2 in Chap. 3 cannot be applied to transform the studied system into the normal form (3.3) via a local diffeomorphism. Alternatively, as proposed in Chap. 4, we can try to seek firstly a diffeomorphism on the output, and then construct another diffeomorphism on the state to transform the studied system into (4.3). However, since n = 3 and γ  (y) = 0, therefore [τ2 , τ3 ] = 0τ2 − 2γ  (y)τ1 . By using Corollary 4.2, we have μ1 (x) = −2γ  (y) = 0 μ2 (y) = 0. In order to satisfy the equation l 2 μ2 (y) − nll  = 0, while l(y) should be a constant not equal to 0. According to Remark 4.2, we have [σ2 , σn ] = l 2 μ1 (x)τ1 = 0, and we can draw the conclusion that Theorem 4.1 cannot be applied as well to transform (5.2) into (4.3). Nevertheless, if we increase the system’s dimension by adding an auxiliary dynamics as follows: x˙1 = γ (y)x1 x˙2 = x1 x˙3 = x2 (5.3) w˙ = κ(w)γ (y) y = x3 ,

5.1 Problem Statement

93

where w ∈ R is an auxiliary variable and κ(w) = 0 which can be chosen freely such that the dynamics of w is bounded. Then it can be seen that, with the following change of coordinates ⎞ ⎛ z1 w ⎜ z2 ⎟ ⎜ e− 0 ⎜ ⎟=⎜ ⎝ z3 ⎠ ⎜ ⎝ ξ ⎛

 w ds ⎞ e− 0 κ(s) x1 w ds  ds ⎟ y κ(s) x + e − 0 κ(s) ⎟ 2 0 γ (s)ds ⎟  w ds − 0 κ(s) ⎠ e x3 w

system (5.3) can be transformed into the following observer normal form: z˙ 1 = 0 − z˙ 2 = z 1 − γ (y)e  −

z˙ 3 = z 2 − e ξ˙ = κ(ξ )γ (y) y¯ = z 3 ,

ξ ds 0 κ(s)

ξ

ds 0 κ(s)

 y 0

y 0

γ (s)ds

γ (s)ds − γ (y)y



where the former output y is linked to the new output by y¯ = e−

(5.4)

ξ

ds 0 κ(s)

y.



In summary, if the commutativity condition (4.18) of Theorem 4.1 in Chap. 4 cannot be fulfilled, then we might construct an auxiliary dynamics w˙ = η(w, y) with w ∈ R such that the dynamical system (5.1), together with this auxiliary dynamics, admits a diffeomorphism (z T , ξ )T = φ(x, w) which transforms the extended dynamical system x˙ = f (x) w˙ = η(w, y) (5.5) y = h(x) into the following nonlinear observer form: z˙ = A O z + β( y¯ , w) ξ˙ = μ(ξ, y¯ ) y¯ = z n = C O z,

(5.6)

where the new output y¯ is linked to the former one y via a diffeomorphism ϕ(y, w), i.e., y¯ = ϕ(y, w). Sometimes, this solution is named as immersion technique in the literature, and the rest of this chapter aims at deducing such a diffeomorphism to transform (5.5) to (5.6).

94

5 Observer Normal Form by Means of Extended Dynamics

5.2 Observer Normal Form with Scalar Extended Dynamics As usual, it is assumed that system (5.1) fulfills the observability rank condition, i.e., the 1-forms θi = d L i−1 f h for 1 ≤ i ≤ n is linearly independent. Denote τ = [τ1 , . . . , τn ] as the frame determined by (3.4)–(3.5). In order to introduce the extended dynamics, denote w ∈ R as an auxiliary variable and assume that its dynamics is governed by w˙ = η(w, y), which will be determined hereafter. Note that w ∈ R can be considered as a known variable since it contains only measurable variable y, and the initial condition of w can be chosen freely by us. With this extended dynamics, system (5.1) can be rewritten as   x˙ f (x) = F(x, w) = . w˙ η(w, y) y = h(x) Note F in the following form of vector field F = f +η

∂ ∂w

(5.7)

and let l(w) = 0 be a nowhere-vanishing function of w which will be determined hereafter. Following the same procedure presented in Chap. 4, we define σ1 = l(w)τ1 .

(5.8)

Thereby, we can construct the following family of vector fields by induction: σi = [σi−1 , F].

(5.9)

Recall that a result on the relationship between the frame σ and τ has been stated in Lemma 4.1 of Chap. 4 for the dynamical system without extension. The following result is just an adaptation for the extended system. Lemma 5.1 The vector field τ and the vector field σ are linked by the following equation:   k−1 

j j k −1 L F l τk− j . (−1) (5.10) σk = j j=0 Proof The proof of this lemma is quite similar to that of Lemma 4.1 in Chap. 4, thus it is left to the readers. 

5.2 Observer Normal Form with Scalar Extended Dynamics

95

The main difference between the above result and the one stated in Lemma 4.1 in Chap. 4 is the successive derivatives of L kF l, where the vector field F is defined in (5.7). Moreover, since l(w) in (5.10) is a function of the auxiliary variable w, therefore it is necessary to understand how to calculate the high-order derivatives of the composition of those functions given in Lemma 5.1. Proposition 5.1 For F and τi defined in (5.7) and (3.5), the following equations hold (1) L τi L kF l = 0 for 1 ≤ i ≤ n and 0 ≤ k ≤ n − i; l = l  (w) ∂η for 1 ≤ i ≤ n where l  (w) = (2) L τi L n−i+1 F ∂y

dl . dw

Proof According to the definition of F since F = f + η(y, w) then we have

∂ , ∂w

L F l = l  (w)η(y, w).

As the above function depends on the output y = h(x), therefore L 2F l = ηL ∂w∂ (L F l) + l  (w)

∂η L f h. ∂y

By induction, for 1 ≤ k ≤ n − i, it can be shown that L kF l contains a term of the form l  (w) ∂η L k−1 h. Due to the definition of τi in (3.5), we have ∂y f L τi L kf h = 0 for 0 ≤ k ≤ n − i and L τi L n−i f h = 1. Using the above two equations, we can then obtain the expected result, and prove l = l  (w) ∂η for 1 ≤ i ≤ n and 0 ≤ k ≤ n − i.  that L τi L kF l = 0 and L τi L n−i+1 F ∂y Based on Proposition 5.1, the following will show how to deduce a differential equation to determine the function l(w). Before this, let us make the following assumption. Assumption 5.1 For system (5.1) with dimension n, (1) if n is even, i.e., n = 2 p for some p ∈ N, then it is assumed that [τ1 , τk ] = 0 for all 2 ≤ k ≤ n; (2) if n is odd, i.e., n = 2 p + 1 for some p ∈ N, then it is assumed that [τ1 , τk ] = 0 for all 1 ≤ k ≤ n, [τ2 , τk ] = 0 for all 2 ≤ k ≤ n − 1 and [τ2 , τn ] = μ1 (y)τ1 where μ1 (y) is a function of the output y.

96

5 Observer Normal Form by Means of Extended Dynamics

The above assumption just means that we have already tried those two methods presented in Chaps. 3 and 4 which allow us to reach the stage where the above assumption is satisfied. Similar to the approach used in Chap. 4, we can then determine the nowhere-vanishing function l(w) by calculating the Lie brackets between σi and σ j . Theorem 5.1 Under Assumption 5.1, the following assertions hold. (1) If n = 2 p + 1 and [τ2 , τn ] = μ1 (y)τ1 , then [σ2 , σn ] = 0 if and only if there exist l(w) and η(y, w) that fulfill the following differential equation: lμ1 + 2l  η y = 0,

(5.11)

where η y = ∂η ; ∂y (2) If n = 2 p and [τ3 , τn ] = μ2 (y)τ2 + μ1 τ1 , then [σ3 , σn ] = νσ1 if there exists l(w) that fulfills the following differential equation: lμ2 + (n + 1)l  η y = 0, where η y =

(5.12)

∂η . ∂y

Proof Consider firstly the case n = 2 p + 1. We know from Lemma 5.1 that σ2 = lτ2 − L F lτ1 , thus we have 

n−1 









τn−i [σ2 , σn ] = σ2 ,

i=0 2 = l L τn (L F l) + (−1)n−1 L τ2 L n−1 F l τ1 + l [τ2 , τn ] , (−1)i n−1 j

L iF l

which, after simplification, yields

[σ2 , σn ] = l 2 [τ2 , τn ] + l l  η y + (−1)n−1 l  η y τ1 , where η y = ∂η . ∂y Now, if [τ2 , τn ] = μ1 (y)τ1 , then [σ2 , σn ] = 0 if and only if there exist η(y, w) and l(w) such that the following equation is fulfilled lμ1 + 2l  η y = 0. For the case n = 2 p, since [σ2 , σn ] = 0 if and only if [τ2 , τn ] = 0, therefore, we need to calculate the Lie bracket [σ3 , σn ] to get the information about l(w). From Lemma 5.1, we obtained σ3 = lτ3 − 2L F lτ2 + L 2F lτ1 ,

5.2 Observer Normal Form with Scalar Extended Dynamics

97

thus its Lie bracket with σn reads as  [σ3 , σn ] = σ3 ,

n−1 

i

L F l τn−i (−1)i n−1 j



i=0  n−1



 L τ3 L n−2 l τ2 + μτ1 . = l 2 [τ3 , τn ] + 2l L τn (L F l) + l (−1)n−2 n−2 F

A straightforward calculation leads to

[σ3 , σn ] = l 2 [τ3 , τn ] + 2ll  η y + (n − 1) ll  η y τ2 + μτ1 , which can be simplified as follows: [σ3 , σn ] = l 2 [τ3 , τn ] + (n + 1) ll  η y τ2 + μτ1 . In order to eliminate the term containing τ2 in the above equation, if we assume that [τ3 , τn ] = μ2 (y)τ2 + μ1 τ1 , then it leads to the following equation: lμ2 (y) + (n + 1)l  η y = 0, 

and it ends the proof of Theorem 5.1.

Remark 5.1 Theorem 5.1 states that the expected solutions of l(w) and η(y, w) should satisfy either (5.11) or (5.12), depending on the dimension of the studied system. It is evident that those solutions might not be unique. In general, (1) for the case n = 2 p + 1, we might take the solution of (5.11) as η(y, w) = κ(w)ψ(y) + κ1 (w), where κ(w) = 0,

dψ(y) dy

= μ1 (y) and l(w) = e− 2

1

w 0

ds κ(s)

;

(2) for the case n = 2 p, we might take the solution of (5.12) as η(y, w) = κ(w)ψ(y) + κ1 (w), where κ(w) = 0,

dψ(y) dy

= μ2 (y) and l(w) = e− n+1 1

w 0

ds κ(s)

.

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5 Observer Normal Form by Means of Extended Dynamics

Remark 5.2 As we have seen in the proof of Theorem 5.1 for the case n = 2 p that the auxiliary dynamics is introduced to eliminate the term containing τ2 when calculating the Lie bracket [σ3 , σn ]. Then, we obtain [σ3 , σn ] = μ1 σ1 . If μ1 = 0, we need to check other Lie brackets, such as [σ4 , σn ]. Remark 5.3 As we did in Chap. 4, after having determined the nowhere-vanishing function l(w), we can then compute the new vector field σ1 via (5.8). Following (5.9), we obtain σi for 2 ≤ i ≤ n. If their Lie brackets commute, i.e.,   σi , σ j = 0, for 1 ≤ i, j ≤ n, then there always exists another vector field, noted as σn+1 , such that the following properties hold: (1) σn+1 =

∂ + T, ∂w

where T ∈ span{σ1 , . . . , σn }. This implies that σn+1 does not depend on other σi for 1 ≤ i ≤ n; (2) σn+1 commutes with other σi , i.e., [σi , σn+1 ] = 0, for 1 ≤ i ≤ n. The next step is then to evaluate the 1-forms θi = d L i−1 f h for 1 ≤ i ≤ n, together with θn+1 = dw, on the vector fields σi for 1 ≤ i ≤ n + 1. For this, note ⎡

⎤ θ1 ⎢ .. ⎥ ⎢ ⎥ θ =⎢ . ⎥ ⎣ θn ⎦ dw and

  σ = σ1 , . . . , σn , σn+1 .

Then, we can calculate the following matrix ⎛

0 0 ⎜ . ⎜ 0 .. ⎜ ⎜ .. ⎜ Λ¯ = θ σ = ⎜ . ... ⎜ ⎜0 1 ⎜ ⎝ 1 Λ¯ n,2 0 0

... 0

1 Λ¯ 1,n+1

⎞

⎟ ... 1 Λ¯ 2,n Λ¯ 2,n+1 ⎟ ⎟ . .. ⎟ .. ... .. . ⎟ ⎟ .. .. ⎟ ... ... . . ⎟ ⎟ ... ... Λ¯ n,n Λ¯ n,n+1 ⎠ ... ... 0 1

(5.13)

5.2 Observer Normal Form with Scalar Extended Dynamics

99

with Λ¯ i, j = θi σ j for 1 ≤ i, j ≤ n + 1. It is clear that this matrix is invertible, since its n + 1 columns are linearly independent. Thus, we can define the 1-forms ω¯ i for 1 ≤ i ≤ n + 1, which are components of the following 1-forms: ⎤ ω¯ 1 ⎥ ⎢ ω¯ = ⎣ ... ⎦ = Λ¯ −1 θ. ω¯ n+1 ⎡

(5.14)

Similar to Theorem 4.1, we can state the following result for the dynamical system with extended dynamics. Theorem 5.2 The following three assertions are equivalent: (1) There exists a local change of coordinates (z T , ξ )T = φ(x, w) that transforms dynamical system (5.5) into the nonlinear observer normal form (5.6); (2) The vector field σi and σ j for 1 ≤ i, j ≤ n + 1 defined in (5.8)–(5.9) commute, i.e.,   (5.15) σi , σ j = 0; (3) The 1-forms ω¯ i defined in (5.14) for 1 ≤ i ≤ n + 1 are closed, i.e., d ω¯ i = 0.

(5.16)

Furthermore, the change of coordinates is determined by  z i = φi (x, w) = γ ω¯ i  ξ = φn+1 (x, w) = γ ω¯ n+1 ,

(5.17)

where γ : [0, 1] → X is any smooth curve on contractile neighborhood of 0 such that γ (0) = 0 and γ (1) = x. Proof The proof is similar to that of Theorem 3.2 in Chap. 3 and that of Theorem 4.1 in Chap. 4.  Example 5.2 (Example 5.1 continued) For system (5.2), we have obtained τ1 = τ2 = τ3 =

∂ ∂ x1 ∂ ∂ x2 ∂ ∂ x3

+ γ (y)τ1 + γ (y)τ2 − x2 γ  (y)τ1 ,

which yields, since γ  (y) = 0, the following equation: [τ2 , τ3 ] = −2γ  (y)τ1 = 0. Therefore, we have concluded that neither Theorem 3.2 nor Theorem 4.1 can be applied to transform (5.2) into the presented form (3.3) or (4.3).

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5 Observer Normal Form by Means of Extended Dynamics

The following shows how to transform (5.2) into (5.6) by means of the extended dynamics. To do so, let us consider system (5.2) with the extended dynamics as follows: x˙1 = γ (y)x1 x˙2 = x1 (5.18) x˙3 = x2 w˙ = κ(w)ψ(y) + κ1 (w), where w ∈ R is an auxiliary and known variable. Since n = 3, according to Theorem 5.1, as [τ2 , τ3 ] = −2γ  (y)τ1 , thus μ1 (y) = −2γ  (y). Following the discussions in Remark 5.1, we can choose η(y, w) = κ(w)ψ(y) + κ1 (w), where κ(w) = 0 and

ψ(y) = −2γ (y),

which gives

η y = −2κ(w)γ  (y).

Then, (5.11) can be read as lμ1 + 2l  η y = −2lγ  (y) − 4κ(w)l  γ  (y) = 0. Due to the fact that γ  (y) = 0, then one solution of the above equation is l(w) = e− 2

1

w 0

ds κ(s)

.

With the determined nowhere-vanishing function l(w), according to (5.8), we can then define the first vector field σ1 as follows: σ1 = lτ1 = l

∂ . ∂ x1

Following (5.9), a straightforward calculation gives σ2 = lτ2 − lγ τ1 = l and

∂ ∂ x2

σ3 = lτ3 − 2lγ τ2 + l γ  x2 + γ 2 τ1 = l ∂∂x3 − lγ ∂∂x2 .

5.2 Observer Normal Form with Scalar Extended Dynamics

101

It can be then checked that [σ2 , σ3 ] = 0. Since the dimension of the extended system is n + 1 = 4, we need to construct the 4th vector field that commutes with the other three vector fields σ1 , σ2 and σ3 . Following the discussion in Remark 5.3, we can see that the following vector field σ4 =

l ∂ ∂ ∂ l ∂ l + x1 + (x2 − γ x3 + ψ) + x3 ∂w l ∂ x1 l ∂ x2 l ∂ x3

with

dψ(y) = γ (y) dy

satisfies this requirement. Recall that the observability 1-forms for (5.2) is θ1 = d x3 , θ2 = d x2 and θ3 = d x1 . Thus, we have ⎤ ⎡ ⎤ ⎡ d x3 θ1 ⎢ θ2 ⎥ ⎢ d x2 ⎥ ⎥ ⎢ ⎥ θ =⎢ ⎣ θ3 ⎦ = ⎣ d x1 ⎦ dw dw and

⎡

l ⎢0 σ = [σ1 , σ2 , σ3 , σ4 ] = ⎢ ⎣0 0

0 l 0 0

0, −lγ , l, 0,



l l

⎤

l x l 1

(x2 − γ x3 + ψ) ⎥ ⎥. l ⎦ x 3 l 1

Next, we can compute Λ¯ = θ (σ ) as follows: ⎛

0 ⎜0 Λ¯ = ⎜ ⎝l 0

0 l 0 0

l, −lγ , 0, 0,

l x l 3 l x l 2 l x l 1

⎞ −

l l

 (γ x3 ) + ll ψ ⎟ ⎟, ⎠

1

and we have ⎛

⎞⎛ ⎞ ⎞ ⎛ 0 0 1l , − l 12 l  x1 dz 1 θ1

⎜ dz 2 ⎟ ⎜ 1 γ 1 0, 12 −ψl  − l  x2 ⎟ ⎜ θ2 ⎟ ⎜ ⎟⎜ ⎟ ⎟ ⎜l l l ¯ −1 ⎝ dz 3 ⎠ = Λ θ = ⎝ 1 0 0, − 12 l  x3 ⎠ ⎝ θ3 ⎠ . l l dξ dw 0 0 0, 1

This leads to the following change of coordinates:

102

5 Observer Normal Form by Means of Extended Dynamics w

z 1 = xl1 = e− 0 κ(s)x1 w w ds − 0 κ(s) − 0 z 2 = xl2 + ψl = e x + e 2  w ds z 3 = xl3 = e− 0 κ(s) x3 ξ = w, ds

ds κ(s)

y 0

γ (s)ds

which transforms the dynamical system (5.2) into the form (5.4).



5.3 High-Dimensional Extension Section 5.2 presented how to introduce a scalar auxiliary dynamics to relax the commutativity condition of Lie brackets. If such a scalar extension fails to relax the commutativity condition, then we might try to add high-dimensional auxiliary dynamics. The following presents only an example of two-dimensional extension, and a more general setting for such a method can be found in [1, 2, 4] and references therein. Consider the following dynamical system x˙1 = γ (y)x2 , x˙2 = x1 x˙3 = x1 , x˙4 = x3 y = x4 .

(5.19)

A straightforward calculation leads to θ1 = d x4 , θ2 = d x3 θ3 = d x2 , θ4 = d x1 and

τ1 = τ3 = τ4 =

∂ , τ2 = ∂∂x2 ∂ x1 ∂ + γ ∂∂x1 ∂ x3 ∂ − x3 γ  ∂∂x1 . ∂ x4

Thereby, we obtain [τ1 , τ2 ] = [τ1 , τ3 ] = [τ1 , τ4 ] = [τ2 , τ3 ] = 0 and

(5.20)

[τ3 , τ4 ] = −2γ  τ1 .

We would like to remark that a simple scalar dynamic extension for (5.19) cannot solve the problem to transform the studied system into the proposed nonlinear observer normal form (5.6), and this is due to (5.20) where the Lie brackets are 0 which prevent us from computing l(w). Therefore, we need to seek high-dimensional extension.

5.3 High-Dimensional Extension

103

For this, let us consider now the following auxiliary dynamics w˙ 1 = w2 w˙ 2 = −γ . It can be checked that the following change of coordinates   y γ (s)ds z 1 = e−w1 x1 + w2 x2 + w22 x3 + w23 x4 + 2w2 0   y −w1 2 x2 + 3w2 x4 + 2w2 x3 + 2 γ (s)ds z2 = e 0

−w1

z3 = e (x3 + 3w2 x4 ) z 4 = e−w1 x4 brings the studied example into the following normal form:   y  y 4 2 2 γ (s)ds + 3w2 γ y + 2γ γ (s)ds z˙ 1 = −e w2 y + 2w2 0  0  y z˙ 2 = z 1 − e−w1 4w23 y + 4w2 γ (s)ds + 6γ w2 y 0   y −w1 2 6w2 y + 3γ y + 2 γ (s)ds z˙ 3 = z 2 − e −w1

0

z˙ 4 = z 3 − 4e−w1 w2 y .

w˙ 1 = w2 , w 2 = −γ y¯ = z 4 = e−w1 y.

5.4 Observer Design 5.4.1 Adaptive Observer The idea of adaptive observer was initialized in [5] to treat the problem of simultaneous state and parameter estimations [8]. In [7], a Kalman-like adaptive observer was presented for state-affine systems with linear time-varying matrix and a high-gain observer was proposed in [6] for the nonlinear system with triangular form. After that, other adaptive observers are studied to estimate of the states for more general normal forms, including the output-depending normal form [3, 10]. Consider the normal form (5.6), this section will design such an adaptive observer for this normal form. Let us consider the following dynamics:

104

5 Observer Normal Form by Means of Extended Dynamics

z˙ˆ = A O zˆ + β( y¯ , w) + K t (ϕ(y) − C O zˆ ),

(5.21)

where y¯ = ϕ(y) and w = ξ are regarded as known measurements, and K t satisfies the following equation: S˙ = −ρ S − S A O − A TO S + C OT C O K t = S −1 C OT ,

(5.22)

which ρ > 0. Unlike the Luenberger-like and high-gain observers, (5.21) uses a time-varying gain K t . This observer works as well even the matrices A O and C O are time-varying, under the condition that the pair (A O , C O ) is observable. In this case, (5.21) is similar to the so-called Kalman–Bucy filter without noises in the model and the output. When A O and C O are constant matrices of the Brunovsky form, it has been proven that the matrix S defined in (5.22) is symmetric positive definite, and S∞ (the value of matrix S when t → ∞) is the unique solution of the following Riccati equation 0 = ρ S∞ + S∞ A O + A TO S∞ − C OT C O whose components have the similar structure of Δε of the high-gain observer (4.27). To show the convergence of the proposed observer, let us note e = z − zˆ . Then according to (5.6) and (5.21), we obtain e˙ = A O e − K t C O e with K t being determined by (5.22). To prove the convergence of e to 0, we can choose the following Lyapunov function: V = e T Se, then its time derivative equals T T ˙ V˙ = e˙ T Se  + e S e˙ T+ e Se T = e −ρ S − C C e = −ρV − ||C O e||22 .

Finally, we can state that there exists a positive ρ such that V˙ ≤ −ρV, and this implies that e will exponentially converge to 0, i.e., lim zˆ (t) = z(t).

t→+∞

5.4 Observer Design

105

Consequently, we can obtain the estimation of x by inverting the diffeomorphism φ(x). In summary, the following dynamics zˆ˙ = A O zˆ + β( y¯ , w) + K t (ϕ(y) − C O zˆ ) S˙ = −ρ S − S A O − A TO S + C OT C O K t = S −1 C OT xˆ = φ −1 (ˆz )

(5.23)

is an asymptotic observer of (5.1).

5.4.2 Design Procedure The following summarizes the procedure to design observer based on the proposed normal form (5.6). According to (3.2), calculate the associated 1-forms θ for (5.1); Determine the family of vector fields τ defined in (3.4) and (3.5); Extend system (5.1) with an auxiliary dynamics (5.7); Calculate the new family of vector fields σ according to (5.8) and (5.9); Calculate l(w) and η(y, w) according to the results stated in Remark 5.1; Compute Λ¯ and ω¯ according to (5.13) and (5.14), and deduce the diffeomorphism φ(x) according to (5.17); Step 7: Apply this diffeomorphism to calculate φ∗ ( f ), with f being defined in (5.1); Step 8: Design the proposed adaptive observer (5.23) to estimate x of (5.1). Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:

Exercises Exercise 5.1 Consider the following dynamical system: ⎧ x˙1 = x1 x3 ⎪ ⎪ ⎨ x˙2 = x1 + x32 x˙3 = x2 ⎪ ⎪ ⎩ y = h(x) = x3 . Compute the observability 1-forms θ1 , θ2 and θ3 . Compute the vector fields τ1 , τ2 and τ3 . Show that [τ2 , τ3 ] = ϕ(y)τ1 = 0. What can we deduce from this fact? Add an auxiliary dynamics w˙ = −y and set σ1 = l(w)τ1 , then compute σ2 and σ3 . 5. Determine l(w) such that [σ2 , σ3 ] = 0. 1. 2. 3. 4.

106

5 Observer Normal Form by Means of Extended Dynamics

6. Find a new vector field σ4 which is independent of and commute with σ1 , σ2 and σ3 . 7. Set θ4 = dw and compute the matrix Λ¯ = (θi (σ j )) to deduce the change of coordinates. 8. Show that this diffeomorphism can transform the studied system into the extended observer normal form (5.6). Exercise 5.2 Consider the following dynamical system: ⎧ x˙1 = x1 sin(x3 ) − x32 ⎪ ⎪ ⎨ x˙2 = x1 x˙3 = x2 ⎪ ⎪ ⎩ y = h(x) = x3 . Follow the same procedure as detailed in the above exercise to calculate the extended observer normal form.

References 1. Boutat, D.: Extended nonlinear observer normal forms for a class of nonlinear dynamical systems. Int. J. Robust Nonlinear Control 25(3), 461–474 (2015) 2. Boutat, D., Busawon, K.: On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form. Int. J. Control 84(1), 94–106 (2011) 3. Busawon, K., Farza, M., Hammouri, H.: A simple observer for a class of nonlinear systems. Appl. Math. Lett. 11(3), 27–31 (1998) 4. Califano, C., Moog, C.: The observer error linearization problem via dynamic compensation. IEEE Trans. Autom. Control 59(9), 2502–2508 (2014) 5. Carroll, R., Lindorff, D.: An adaptive observer for single-input single-output linear systems. IEEE Trans. Autom. Control 18(5), 428–435 (1973) 6. Gauthier, J.P., Hammouri, H., Othman, S.: A simple observer for nonlinear systems applications to bioreactors. IEEE Trans. Autom. Control 37(6), 875–880 (1992) 7. Hammouri, H., de Leon Morales, J.: Observer synthesis for state-affine systems. In: Proceedings of the 29th IEEE Conference on Decision and Control, pp. 784–785 (1990) 8. Ioannou, P.A., Sun, J.: Robust Adaptive Control, vol. 1. PTR Prentice-Hall, Upper Saddle River (1996) 9. Tami, R., Boutat, D., Zheng, G.: Extended output depending normal form. Automatica 49(7), 2192–2198 (2013) 10. Yu, L., Zheng, G., Boutat, D.: Adaptive observer for simultaneous state and parameter estimations for an output depending normal form. Asian J. Control 19(1), 356–361 (2017)

Chapter 6

Output-Depending Observer Normal Form

Abstract In the previous three chapters, the desired observer normal forms (3.3), (4.3) and (5.6) have the common point, i.e., the linear part is a constant matrix of Brunovsky form and the nonlinear part is a function of known (measured) variables. In this chapter, we enlarge the desired observer normal form by allowing outputdepending matrix for its linear part, and it is named as output-depending observer normal form [11, 12]. Like previous chapters, a set of geometric conditions will be deduced to guarantee the existence of a change of coordinates that brings the studied nonlinear dynamical system into the proposed output-depending observer normal form. As what has been done in the previous three chapters, the proposed algorithm enables us to seek a new frame of vector fields such that their Lie brackets are vanished. For such a normal form, this chapter synthesizes a non-asymptotic observer, called as step-by-step sliding mode observer, and the associated design procedure will be given at the end of this chapter.

6.1 Problem Statement Consider the following single-output dynamical system x˙ = f (x) y = h(x),

(6.1)

where x ∈ X ⊆ Rn with X being a neighborhood of 0, and y ∈ R. Without loss of generality it is assumed that f and h are smooth with f (0) = 0 and h(0) = 0. We have shown in Chaps. 3–5 that such a nonlinear system might be transformed into the observer normal form with output injection (3.3), (4.3) or (5.6), either via a diffeomorphism on the state, on the output, or via the technique of immersion. However, there do exist certain systems for which all the mentioned techniques fail, i.e., the commutativity condition of Lie brackets cannot be satisfied. This can be shown by the following example. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_6

107

108

6 Output-Depending Observer Normal Form

Example 6.1 Consider the following dynamical system: x˙1 = γ (y) x˙2 = α(y)x1 x˙3 = x2 y = x3 ,

(6.2)

where α(y) = 0. Thus, following Definition 3.1, we obtain the following observability 1-forms: θ1 = d x3 θ2 = d x2 θ3 = α(y)d x1 + α  (y)x1 d x3 . Since α(y) = 0, therefore those 1-forms are linearly independent, which implies that (6.2) is observable. Then, following (3.4) and (3.5), a straightforward calculation gives τ1 = α1 ∂∂x1  τ2 = ∂∂x2 + x2 αα τ1     α α ∂ 2 α τ1 . τ3 = ∂ x3 + x2 α τ2 − αx1 α + x2 α Based on the above vector fields, we can check that            τ1 [τ2 , τ3 ] = αα τ2 − x2 αα τ1 − x2 αα αα + 2x2 αα         2   τ1 , = αα τ2 − x2 αα + 3x2 αα where α  (y) = dα(y) . dy According to Corollary 4.2, the above relationship suggests to seek a nowherevanishing function l(y) from [τ2 , τ3 ]. By applying the result stated in Corollary 4.2, we have        α 2 α + 3x2 μ1 (x) = − x2 α α and μ2 (y) =

α . α

Consequently, by solving l 2 μ2 (y) − nll  = 0, as stated in Corollary 4.2, we obtain 1

l(y) = e n

y 0

μ2 (s)ds

1

= α3.

6.1 Problem Statement

109

With this l(y), according to (4.7) and (4.8), we can calculate the new vector fields σ as follows: σ1 = lτ1 σ2 = lτ2 − x2 l  τ1

σ3 = lτ3 − 2x2 l  τ2 + αx1 l  + x22 l  τ1 . It can be then checked that [σ2 , σ3 ] contains a term of the μ(x2 , y)τ1 where 

 α  2 μ(x2 , y) = l μ1 (x) + −x2 α(l ) + x2 l l + x2 l l − 2x2 l . α  2

2



As μ(x2 , y) depends on the variable x2 = y˙ , thus the results presented in Chaps. 4 and 5 cannot be applied. For such a case, Chap. 5 provides a solution by introducing an auxiliary dynamics, which increases the dimension of the studied system. If we want to keep the same dimension without the introduction of auxiliary dynamics, an alternative is to modify the conventional structure of normal forms (3.3) and (4.3). In fact, if we take a look at system (6.2), we can notice that it can be written into the following form: x˙ = A(y)x + β(y) (6.3) y = Cx with

⎛

⎞ 0 00 A(y) = ⎝ α(y) 0 0 ⎠ 0 10 β(y) = [γ (y), 0, 0]T

and C = [0, 0, 1]. It can be checked that the pair (A(y), C) is observable if α(y) = 0, and different types of observers have already been proposed in the literature for such a special form, named as output-depending observer normal form since the matrix A(y) is now a function of output. Therefore, it is natural to set this general form as a new target to relax the commutativity condition of Lie brackets, without augmenting the system’s dimension.  Motivated by the above example, this chapter presents how to seek a local diffeomorphism z = φ(x), such that the studied dynamical system (6.1) can be transformed into the following output-depending normal form: z˙ = A O (y)z + β(y) y = C O z, where the matrix A O (y) has the following form:

(6.4)

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6 Output-Depending Observer Normal Form

⎛

0 ⎜ α1 (y) ⎜ . A O (y) = ⎜ ⎜ .. ⎝ 0 0 and

⎞ ··· 0 0 0 ··· 0 0 0⎟ .⎟ ... ... · · · .. ⎟ ⎟ 0 0⎠ · · · αn−2 (y) ··· 0 αn−1 (y) 0



C0 = 0 · · · 0 1 .

In general, the target normal form needs to be observable such that conventional observers proposed in the literature can be reused. Therefore, for the proposed outputdepending normal form (6.4), it is assumed that αi (y) = 0 for 1 ≤ i ≤ n − 1 such that the observability rank condition is satisfied.

6.2 Analysis of the Output-Depending Normal Form Before deducing the necessary and sufficient conditions to transform (6.1) into the proposed output-depending observer normal form (6.4), it is worth analyzing in which frame (6.4) is written and how to obtain such a frame using the Lie bracket. Let us rewrite the vector field f that governs the dynamical system (6.4) as follows:  ∂ ∂ + (αi−1 xi−1 + βi ) . ∂ x1 ∂ xi i=2 n

f = β1

Now, let τi for 1 ≤ i ≤ n be the vector fields associated with f which are defined in (3.4) and (3.5). It is clear that this frame does not commute, as shown in Example 6.1. To build the correct frame which is commutative in the sense of Lie brackets, denote θi = d L i−1 f h for 1 ≤ i ≤ n as the observability 1-forms, then according to (3.4), the first vector field τ1 is τ1 =

∂ 1 . α1 × · · · × αn−1 ∂ x1

Hence, if we construct a new vector field σ1 as σ1 = α1 × · · · × αn−1 τ1 and generate the following vector fields σi = then have ∂ σi = ∂ xi

1 [σ , αi−1 i−1

f ] for 2 ≤ i ≤ n, we can

6.2 Analysis of the Output-Depending Normal Form

111

for all 1 ≤ i ≤ n. It is evident that the constructed new frame σ is commutative in the sense of Lie brackets, which is exactly the canonical basis where f is expressed. The above analysis reveals a way how to construct a commutative frame σi from τi , provided that αi is not equal to 0. The following will detail how to determine those nowhere-vanishing functions αi which will be used to generate such a new frame σi .

6.3 Construction of New Vector Fields Consider the dynamical system (6.1), according to Definition 3.1, let us denote ⎡

⎤ θ1 θ = ⎣ ... ⎦ , θn with θi = d L i−1 f h for 1 ≤ i ≤ n. Since (6.1) is assumed to be observable in the sense that the observability rank condition is satisfied, thus θi for 1 ≤ i ≤ n are linearly independent. Note now the frame τ as τ = [τ1 , . . . , τn ] , where τ1 and τi for 2 ≤ i ≤ n are determined via (3.4) and (3.5). Recall that if the commutativity conditions of Lie brackets fail to be fulfilled, then we want to seek a set of nowhere-vanishing functions αi (y) for 1 ≤ i ≤ n − 1. For the sake of simplicity, denote n−1 αi πn−1 = α1 × α2 × · · · × αn−1 = Πi=1

(6.5)

as the product of functions αi for 1 ≤ i ≤ n − 1, and τ1 =

∂ . πn−1 ∂ x1 1

To build the correct frame which is commutative in the sense of Lie brackets, define the first new vector field as σ1 = πn−1 τ1

(6.6)

and calculate the following vector fields by induction: σi =

1 [σi−1 , f ]. αi−1

(6.7)

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6 Output-Depending Observer Normal Form

The key problem is how to determine those functions αi (y). To achieve this goal, similar to the approaches used in the previous chapters, we need first to establish the relationship between the frame τ and the frame σ with σ = [σ1 , . . . , σn ]. Lemma 6.1 The vector fields of τ and the vector fields of σ are linked by the following equation:  σ1 = πn−1 τ1  (6.8) σi = ij=1 ai, j τ j , for 2 ≤ i ≤ n, where πn−1 is defined in (6.5), and ai,k are given by induction as follows: ⎧ 1 ai,1 = − αi−1 L a , for 2 ≤ i ≤ n ⎪ ⎪ f i−1,1

⎨ 1 ai,k = αi−1 −L f ai−1,k + ai−1,k−1 , for 2 ≤ k ≤ i − 1 ⎪ ⎪ ⎩ ai,i = αi+1 × · · · × αn−1 for 1 ≤ i ≤ n − 1 an,n = 1.

(6.9)

Proof We will complete the proof by induction. Assume that (6.8) is true for the ith order, then the following will show that it is also true the (i + 1)th order. By definition, we know that σi+1 = α1i [σi , f ]. Since (6.8) is satisfied for the ith order, thus we have ⎤ ⎡ i 1 ⎣ ai, j τ j , f ⎦ . σi+1 = αi j=1 Using the properties of Lie bracket and the fact τi+1 = [τi , f ], we get σi+1

⎛ ⎞ i i 

1 ⎝ = − L f ai, j τ j + ai, j τ j+1 ⎠ . αi j=1 j=1

By separating the first and the last terms, we obtain σi+1

⎛ ⎞ i i−1  



1 1 ⎝ 1 = − L f ai,1 τ1 + − L f ai, j τ j + ai, j τ j+1 ⎠ + ai,i τi+1 . αi αi j=2 α i j=1

A change on the index j leads to σi+1

⎛ ⎞ i i 



1 1 ⎝ ai,i = − L f ai,1 τ1 + − L f ai, j τ j + ai, j−1 τ j ⎠ + τi+1 αi αi j=2 αi j=2

and we proved that (6.8) is also true for the (i + 1)th order.



From (6.8) and (6.9), we can see that σi contains the term L kf h. Hence, it is necessary to know the coefficients of the highest Lie derivative L kf h in each σi . To

6.3 Construction of New Vector Fields

113

better understand this relation, let us take a look at the first three vector fields σi for 1 ≤ i ≤ 3. (1) For i = 1, according to (6.8), we have σ1 = πn−1 τ1 . Let us note the coefficient as γ1,1 = πn−1 which is a function of the output y; (2) For the second vector field, we have σ2 = −

 πn−1 πn−1 L f hτ1 + τ2 , α1 α1

and we can note a2,1 = γ2,1 L f h where the coefficient γ2,1 = − πn−1 . α1 (3) For i = 3, we obtain  σ3 = α1 2

 πn−1 2 α1 L f h +



 πn−1 α1

 In the function a3,1 =

1 α2





 πn−1 α1

Lfh

2



L 2f h +

 τ1 − α1

2



 πn−1 α1

   πn−1 πn−1  α1 + α1



Lfh

2

 πn−1 α1



and γ2,2 =

π

L f hτ2 + αn−1 τ . 2 α1 3

 , the coefficient of inter-

π

est is γ3,1 = α2n−1 before the highest Lie derivative L 2f h, a3,2 = γ3,2 L f h with  α1     πn−1 πn−1 1 , and γ3,3 = απ2n−1 γ3,2 = − α2 α1 + α1 . α1 The above discussion shows a clue on how to relate αi to σi through γi,i , which is the coefficient before the highest Lie derivative L kf h. This relation will be summarized by the following lemma. Lemma 6.2 Note σi =

i−1  i− j γi, j L f hτ j + γi,i τi + Ri j=1 i− j

for 4 ≤ i ≤ n where γi, j is the coefficient before L f h, then those coefficients satisfy the following properties: (1) the coefficient before L if h in the direction of vector field τ1 equals γi,1 = −

γi−1,1 ; αi i− j

(2) for 2 ≤ j ≤ i − 2, the coefficient before L f h in the direction of τ j satisfies γi, j =

1 γi−1, j − γi−1, j−1 ; αi−1

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6 Output-Depending Observer Normal Form

(3) the coefficient before the Lie derivative L f h in the direction of τi−1 fulfills γi,i−1 =

1  γi−1,i−2 − γi−1,i−1 ; αi−1

(4) for j = i, we have γi,i =

γi−1,i−1 . αi−1

Proof According to (6.7), for 4 ≤ i ≤ n, we have σi = i−1  write σi−1 = ai−1, j τ j as

1 αi−1

  σi−1 , f . Therefore,

j=1

σi−1 =

i−1  i−( j+1) γi−1, j L f hτ j + Ri−1 , j=1

where the Ri−1 is a vector field that contains the low-order Lie derivatives of h. Note that the above equality is equivalent to the following one σi−1 =

i−2  i−( j+1) γi−1, j L f hτ j + γi−1,i−1 τi−1 + Ri−1 . j=1

Then a straightforward calculation leads to i−2

1  σi = − γαi−1,1 γi−1,k − γi−1,k−1 L i−k L if hτ1 + αi−1 f hτk i−1 k=2

1  γi−1,i−2 − γi−1,i−1 L f hτi−1 + γi−1,i−1 + αi−1 τi + Ri , αi−1

where Ri contains all the low-order Lie derivatives of h. Finally, we proved these 4 properties stated in Lemma 6.2.  Recall that the objective to seek this new frame σ is that its vector fields should satisfy the commutativity condition of Lie brackets. Therefore, the following concerns the calculation of Lie brackets for σ . Lemma 6.3 The following statements hold: (1) [σ1 , σi ] = 0 for 1 ≤ i ≤ n − 1 if and only if [τ1 , τ j ] = 0 for 1 ≤ j ≤ n − 1;  τ1 + γ1,1 γn,n [τ1 , τn ]. (2) [σ1 , σn ] = γ1,1 γ1,n − γn,n γ1,1 Proof According to (3.4), we have L τ1 L if h = 0 for 1 ≤ i ≤ n − 2 and L τ1 L n−1 f h = 1. Since i−1  i− j γi, j L f hτ j + γi,i τi + Ri σi = j=1

6.3 Construction of New Vector Fields

115

for 1 ≤ i ≤ n − 1 where Ri contains low-order Lie derivatives of h, then we have [σ1 , σi ] =

i−1 

i− j

j=1

γi, j L f hπn−1 [τ1 , τ j ] + γi,i πn−1 [τ1 , τi ] + [τ1 , Ri ].

Therefore, if we assume that the first item is true up to i − 1, then [σ1 , σi ] = 0 if and only if [τ1 , τi ] = 0 because [τ1 , Ri ] = 0. Consequently, we have [σ1 , σi ] = 0 for 1 ≤ i ≤ n − 1 if and only if [τ1 , τ j ] = 0 for 2 ≤ j ≤ i. Also, a similar calculation can prove the item (2) of Lemma 6.3.  According to Lemma 6.2, we have γn,n = 1 γ1,1 = πn−1 and γ1,n = (−1)n−1

 πn−1 . πn−1

Then, following the item (2) of Lemma 6.3, we have  τ1 + πn−1 [τ1 , τn ], [σ1 , σn ] = −2πn−1

if n is even, and [σ1 , σn ] = πn−1 [τ1 , τn ], if n is odd. It is clear that, for the case of odd dimension, we have [σ1 , σn ] = 0 if and only if [τ1 , τn ] = 0. However, for the case of even dimension, if we have [τ1 , τn ] = μ(y)τ1 , then [σ1 , σn ] = 0 is equivalent to the following differential equation  + μ(y)πn−1 = 0. −2πn−1

The above equation leads to the following solution: 1

πn−1 = e 2

y 0

μ(s)ds

.

n−1 αi (y) which contains n − 1 variables to be determined, thereNote that φn−1 = Πi=1 fore, the above equation can solve the problem treated in this chapter for the case n = 2. In this case, we have 1

α1 (y) = φ1 = e 2

y 0

μ(s)ds

.

For the case n ≥ 3, we need to seek more equations of αi (y), and this can be obtained by calculating other Lie brackets than [τ1 , τn ]. To clearly explain the idea,

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6 Output-Depending Observer Normal Form

let us consider the simple case with n = 3. Since the dimension is odd, therefore, we must have [τ1 , τ3 ] = 0 and we are wondering what we can get from [σ2 , σ3 ]. A straightforward calculation gives  α1  + 3α2 τ2 + α2 [τ2 , τ3 ] + ν(x)τ1 . [σ2 , σ3 ] = − α1 

In this case, we have two functions α1 and α2 to be determined. We would like to point out that the function α2 (y) is in fact given by a diffeomorphism on the output. Indeed, in the normal form z˙ 1 = β1 (y) z˙ 2 = α1 (y)z 1 + β2 (y) z˙ 3 = α2 (y)z 2 + β3 (y) y = z3, 

if we set

y

ξ3 = y¯ = 0

then we have ξ˙3 = z 2 +

ds , α2 (s)

β3 (y) . α2 (y)

The above fact means that we can always choose αn−1 = 1 for any dimensional nonlinear system. Consequently, we have in fact only n − 2 unknown functions αi (y) for 1 ≤ i ≤ n − 2 to be determined. Applying the above remark to the case n = 3, [σ2 , σ3 ] = 0 only if [τ2 , τ3 ] = μ(y)τ2 with α1 (y) and μ(y) satisfying the following differential equation:  −

 α1 + 3α2 + α2 μ(y) = 0, α1

where α2 = 1. Hereafter, we assume that αn−1 = 1 and we want to seek n − 2 unknown functions αi for 1 ≤ i ≤ n − 2. Note σi =

i−1  i− j γi, j L f hτ j + γi,i τi + Ri , j=1

6.3 Construction of New Vector Fields

117

then the following provides a result to determine those αi (y). Lemma 6.4 Assume that [σi , σs ] = 0 for all 1 ≤ s ≤ n − i, then (1) for n − i + 1 ≤ s ≤ n − 1, we have

[σi , σs ] = γi,i γs,s+i−n − γs,s γi,s+i−n τs+i−n + γi,i γs,s [τi , τs ] + Ri,s ; (6.10) (2) for s = n, we have

[σi , σn ] = γi,i γn,i − γn,n γi,i τi + γi,i γn,n [τi , τn ] + Ri,n . Proof The proof of this lemma is similar to that of Lemma 6.3.

(6.11) 

The assumption imposed in Lemma 6.4 is consistent with the fact that L τi L sf h = 0 for 1 ≤ s ≤ n − i. Another important fact is that, except γi,i , all other terms γi, j for i = j contain the derivatives of αi (y). Finally, [σi , σs ] = 0 only if [τi , τs ] = μi,s (y)τs+i−n + R and the following equations:

(6.12) γi,i γs,s+i−n − γs,s γi,s+i−n + μi,s (y)γi,i γs,s = 0 and

γi,i γn,i − γn,n γi,i + μi,n (y)γi,i γn,n = 0

(6.13)

are satisfied, which enable us to deduce the unknown functions αk (y) for 1 ≤ k ≤ n − 2. After having determined the function αi (y), we can then calculate the family of vector fields σ = [σ1 , . . . , σn ] via (6.6) and (6.7). With the 1-forms θ = [θ1T , . . . , θnT ]T where θi = d L i−1 f h for 1 ≤ i ≤ n, we can then calculate the following matrix: ⎛ ⎞ 0 0 ··· 0 1 ⎜ 0 0 · · · πn−1 (y) Λ¯ 2,n ⎟ ⎜ . .. .. .. ⎟ . ⎜ ¯ . (6.14) Λ = θσ = ⎜ . . .. . . ⎟ ⎟. ⎝ 0 π (y) · · · Λ¯ ⎠ ¯ Λ 2 n−1,n−1 n−1,n π1 (y) Λ¯ n,2 · · · Λ¯ n,n−1 Λ¯ n,n It is clear that this matrix is invertible for nowhere-vanishing functions αi (y) with 1 ≤ i ≤ n − 1, thus we can define the following 1-forms: ω¯ = Λ¯ −1 θ := (ω¯ i )1≤i≤n .

(6.15)

Similar to previous chapters, we can state the following result that guarantees the change of coordinates. Theorem 6.1 There exists a local change of coordinates z = φ(x) that transforms dynamical system (6.1) into the nonlinear observer normal form (6.4) if and only

118

6 Output-Depending Observer Normal Form

if there exist nowhere-vanishing functions αi (y) for 1 ≤ i ≤ n − 1 such that the [σi , σ j ] = 0 for all 1 ≤ i, ≤ j ≤ n. Furthermore, the change of coordinates is determined by  z i = φi (x) =

γ

ω¯ i ,

(6.16)

where γ : [0, 1] → X is any smooth curve on contractile neighborhood of 0 such that γ (0) = 0 and γ (1) = x. Proof The proof of necessity is similar to that of Theorems 4.1 and 5.2. Therefore, the following gives only proof of sufficiency. Suppose that such a change of coordinates φ(x) exists, then it will act on σi as follows: ∂ . φ∗ (σi ) = ∂z i To determine how f is transformed by means of φ∗ , let us compute σi+1 =

∂φ∗ ( f ) . ∂z i

As

1 [σi , f ] αi

for 1 ≤ i ≤ n − 1, then we have  ∂φ∗ ( f ) ∂ = , φ∗ ( f ) ∂z i ∂z i because

∂ ∂z i

is a constant vector field. Therefore, we have ∂ ∂φ∗ ( f ) = φ∗ [σi , f ] = αi−1 φ∗ (σi+1 ) = αi ∂z i ∂z i+1

for 1 ≤ i ≤ n − 1. Thus, the component of φ∗ ( f ) in the direction of form ∂ ∂ + β(z n ) . αi (y) ∂z i+1 ∂z i+1

∂ ∂z i+1

has the

Now, if all the vector fields σi for 1 ≤ i ≤ n commute, then the change of coordinates is given by its differential as follows: dz = Λ¯ −1 θ, where the components of the matrix Λ¯ are given by (6.14). Example 6.2 Consider the following single output nonlinear dynamical system



6.3 Construction of New Vector Fields

119

x˙1 = x2 e x3 x˙2 = e−2x3 x1 x˙3 = x2 y = x3 . It is clear that the above system is observable because the observability 1-forms are as follows: θ1 = d x3 θ2 = d x2 θ3 = e−2x3 d x1 − 2e−2x3 x1 d x3 . A straightforward calculation shows that τ1 = e2x3 ∂∂x1 τ2 = ∂∂x2 − 2x2 τ1

τ3 = ∂∂x3 − 2x2 τ2 + 2e−2x3 x1 + e−x3 τ1 . Thus, we have [τ1 , τ2 ] = [τ1 , τ3 ] = 0 and [τ2 , τ3 ] = −2

∂ = 0. ∂ x2

Following the presented procedure, let us compute the new frame by setting σ1 = α1 τ1 then we get σ2 = and σ3 = [σ2 , f ] = τ3 −

1 α  x2 [σ1 , f ] = τ2 − 1 τ1 α1 α1 α1 x2 τ2 + α1



α1 −2x3 e x1 + α1



α1 α1



 x22 τ1 .

The objective now is to seek α1 (y) such that [σ2 , σ3 ] = 0. For this, we compute    2     α1 α1 α1 α1 +2 x2 τ1 . [σ2 , σ3 ] = [τ2 , τ3 ] − τ2 + −2x2 − x2 α1 α1 α1 α1 As [τ2 , τ3 ] = −2τ2 , we have

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6 Output-Depending Observer Normal Form



α [σ2 , σ3 ] = − 2 + 1 α1





α τ2 + −2x2 1 − x2 α1



α1 α1

2



α1 +2 α1



 x2 τ1 .

In order to eliminate the τ2 direction, we need to impose 2+

α1 = 0, α1

which implies that α1 = e−2y . Then we obtain σ1 =

∂ ∂ , σ2 = ∂ x1 ∂ x2

σ3 =

∂ ∂ + e x3 . ∂ x3 ∂ x1

and

Now, it can be checked that [σ1 , σ2 ] = [σ1 , σ3 ] = [σ2 , σ3 ] = 0, which implies that there exists a diffeomorphism to transform the studied system into the presented observer normal form (6.4). In order to deduce such a diffeomorphism, let us compute ⎞ ⎛ ⎞ 0 0 1 θ1 (σ1 ) θ1 (σ2 ) θ1 (σ3 ) ⎠ 0 Λ = ⎝ θ2 (σ1 ) θ2 (σ2 ) θ2 (σ3 ) ⎠ = ⎝ 0 1 −2x3 −x3 −2x3 e 0 e − 2e x1 θ3 (σ1 ) θ3 (σ2 ) θ3 (σ3 ) ⎛

whose inverse is ⎛ Λ−1 = ⎝

1 e−2x3

−x

⎞ 1 −e 3 + 2x1 e−2x3 0 e−2x 3 0 1 0 ⎠. 1 0 0

Hence, we obtain the differential of the change of coordinates as follows: ⎛

⎞ ⎛ ⎞ ⎛ ⎞ dz 1 θ1 x 1 + e x3 ⎝ dz 2 ⎠ = Λ−1 ⎝ θ2 ⎠ = d ⎝ x2 ⎠ , dz 3 θ3 x3 which implies that the diffeomorphism is of the following form:

6.3 Construction of New Vector Fields

121

z 1 = x 1 + e x3 z 2 = x2 z 3 = x3 with which the studied system can be transformed into z˙ 1 = 0 z˙ 2 = e−2y z 1 − e−y z˙ 3 = z 2 y = z3. 

6.4 Observer Design 6.4.1 Step-by-Step Sliding Mode Observer Compared to asymptotic observer, finite time one was less studied in the literature, which however is well appreciated in practice. Different methods have been proposed, such as sliding mode technique [6, 9], impulsive observer [3, 7], algebraic methods [1, 4]. This section presents a simple non-asymptotic observer for (6.1): step-by-step sliding mode observer [5, 10], based on the sliding mode technique. For this, suppose that the state of (6.1) is bounded, so the state of (6.4) is also bounded. Then, let us rewrite (6.4) into the following form z˙ 1 = β1 (y) z˙ 2 = α1 (y)z 1 + β2 (y) .. .

z˙ n−1 = αn−2 (y)z n−2 + βn−1 (y) z˙ n = αn−1 (y)z n−1 + βn (y) y = zn .

(6.17)

Consider the following dynamics z˙ˆ 1 = β1 (y) + λ1 sgn(˜z 1 − zˆ 1 ) z˙ˆ 2 = β2 (y) + λ2 sgn(˜z 2 − zˆ 2 ) .. .

z˙ˆ n−1 = βn−1 (y) + λn−1 sgn(˜z n−1 − zˆ n−1 ) z˙ˆ n = βn (y) + λn sgn(y − zˆ n ) with

(6.18)

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6 Output-Depending Observer Normal Form

z˜ 1 = z˜ 2 = .. .

z˜ n−2 = z˜ n−1 =

1 λ sgn(˜z 2 α1 (y) 2 1 λ sgn(˜z 3 α2 (y) 3

− zˆ 2 ) − zˆ 3 ) (6.19)

λ sgn(˜z n−1 − zˆ n−1 ) αn−2 (y) n−1 1 λ sgn(y − zˆ n ), αn−1 (y) n 1

where λi > 0. The step-by-step sliding mode observer works as follows. Firstly, consider the dynamics z˙ n and z˙ˆ n , and note ei = z n − zˆ n , we have e˙n = αn−1 (y)z n−1 − λn sgn(y − zˆ n ).

(6.20)

Choose the following Lyapunov function Vn = we get

en2 , 2

V˙n = en αn−1 (y)z n−1 − λn en sgn(en ) = en αn−1 (y)z n−1 − λn |en |.

Since the term αn−1 (y)z n−1 is bounded, if we choose λn such that λn > sup αn−1 (y(t))z n−1 (t), t>0

then we have

√ 1  V˙n < − λn − αn−1 (y)z n−1 2Vn2 ,

which implies that Vn will converge to 0 in a finite time. Consequently, zˆ n will converge to z n in a finite time. Suppose that when t > T1 , we have zˆ n = z n , thus en = 0 and e˙n = 0. According to (6.20), we obtain the exact estimation of z n−1 , noted as z˜ n−1 which has the following form: 1 λn sgn(y − zˆ n ), ∀t > T1 . z n−1 = αn−1 (y) According to (6.19), we can then deduce that z˜ n−1 = z n−1 , ∀t > T1 and

z˙ˆ n−1 = βn−1 (y) + λn−1 sgn(˜z n−1 − zˆ n−1 ) , ∀t > T1 . = βn−1 (y) + λn−1 sgn(z n−1 − zˆ n−1 )

6.4 Observer Design

123

The next step is then to define en−1 = z n−1 − zˆ n−1 , and Vn−1 =

2 en−1 . 2

By choosing

λn−1 > sup αn−2 (y(t))z n−2 (t) t>0

similarly, we can prove that Vn−1 converges to 0 in a finite time. Therefore, there exists T2 > T1 > 0 such that zˆ n−1 = z˜ n−1 = z n−1 , ∀t > T2 . By iterating the same procedure, it is easy to prove that, there exist Tn > · · · > T1 such that zˆ i = z i , ∀t > Tn for 1 ≤ i ≤ n. From the procedure of the proof, it is clear that the observation errors converge to 0 in a step-by-step way, i.e., ei converges to 0 in a finite time before ei−1 . For this reason, the proposed observer is named as a step-by-step sliding mode observer. Finally, we can obtain the estimation of x by inverting the diffeomorphism φ(x). In summary, the following dynamics: z˙ˆ 1 = β1 (y) + λ1 sgn(˜z 1 − zˆ 1 ) z˙ˆ 2 = β2 (y) + λ2 sgn(˜z 2 − zˆ 2 ) .. .

z˙ˆ n−1 = βn−1 (y) + λn−1 sgn(˜z n−1 − zˆ n−1 ) z˙ˆ n = βn (y) + λn sgn(y − zˆ n ) xˆ = φ −1 (ˆz ),

(6.21)

where z˜ i is defined in (6.19), is a non-asymptotic observer of (6.1).

6.4.2 Design Procedure The following summarizes the procedure to design observer based on the proposed normal form (6.4). Step 1: According to (3.2), calculate the associated 1-forms θ for (6.1); Step 2: Determine the family of vector fields τ defined in (3.4) and (3.5); Step 3: Calculate the new family of vector fields σ according to (6.6) and (6.7); Step 4: Deduce αk (y) for 1 ≤ k ≤ n − 2 according to (6.12) and (6.13); Step 5: Compute Λ¯ and ω¯ according to (6.14) and (6.15), then deduce the diffeomorphism φ(x) according to (6.16);

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6 Output-Depending Observer Normal Form

Step 6: Apply this diffeomorphism to calculate φ∗ ( f ), with f being defined in (6.1); Step 7: Design the proposed observer (6.21) to estimate x of (6.1).

Exercises Exercise 6.1 Consider the following dynamical system: ⎧ ⎨ x˙1 = x1 x2 − x12 x˙2 = x1 ⎩ y = h(x) = x2 . Find its output-depending observer normal form. Exercise 6.2 Consider the following output depending observer normal form: ⎧ x˙1 = β1 (y) ⎪ ⎪ ⎨ x˙2 = x1 + β2 (y) x˙3 = α(y)x2 + β3 (y) ⎪ ⎪ ⎩ y = h(x) = x3 . Show that it can be transformed into an observer normal form. Exercise 6.3 Consider the nonlinear dynamical system described by the following equations [2, 8] ⎧ χ˙ 1 = ϕ1,1 (y)χ1 + ψ1,n−1 (y)χn−1 + ϕ1,n (y) ⎪ ⎪ ⎪ ⎪ χ˙ i = ϕi,1 (y)χi−1 + ϕi,i (y)χi + ψi,n−1 (y)χn−1 + ϕi,n (y) for 2 ≤ i ≤ n − 3 ⎪ ⎪ ⎪ 1 ⎪ (y)χn−2 + ψn−2,n−1 (y)χn−1 ⎨ χ˙ n−2 = ϕn−2,1 (y)χn−3 + ϕn−2,n−2 1 2 +ψn−2,n−1 (y)χn−1 + ψn−2,n (y) ⎪ 1 2 ⎪ (y)χn−1 + ψn−1,n (y) χ˙ n−1 = ψn−1,1 (y)χn−2 + ψn−1,n−1 (y)χn−1 + ψn−1,n−1 ⎪ ⎪ ⎪ ⎪ = ψ (y)χ + ϕ (y) χ ˙ ⎪ n n,1 n−1 n,n ⎪ ⎩ y = χn , (6.22) where χ = (χ1 , . . . , χn )T ∈ U ⊂ Rn represents the state in a neighborhood of 0. It is assumed that dynamical system (6.22) satisfies the observability rank condition, i.e., for 2 ≤ i ≤ n − 2 we have ϕi,1 (y) = 0, ψn−1,1 (y) = 0 and ψn,1 (y) = 0 for all y. Seek a change of coordinates together with an auxiliary dynamics that transforms the above dynamical system into the following observer normal form: x˙ = A(y, w)x + β(y, w) y = zn ,

Exercises

125

where A(y, w) is a matrix of Brunovsky canonical form, depending on the output and the auxiliary variables. Exercise 6.4 Consider the following affine dynamical system: ⎧ ⎨ x˙1 = x1 x2 + x1 u x˙2 = x1 ⎩ y = x2 . By increasing the dimension of the above dynamical system with the auxiliary dynamics w˙ = −u, show that the following change of coordinates z 1 = − 21 ew x22 + x1 ew z 2 = ew x 2 will transform the studied system into the following observer normal form: ⎧ 1 −w 2 ⎪ ⎪ z˙ 1 = 2 e 1y2 u 2 ⎨ z˙ 2 = z 1 + 2 e−w y2 − yew u w˙ = −u ⎪ ⎪ ⎩ y = z2 .

References 1. Barbot, J.P., Fliess, M., Floquet, T.: An algebraic framework for the design of nonlinear observers with unknown inputs. In: Proceedings of the 46th IEEE Conference on Decision and Control, pp. 384–389 (2007) 2. Boutat, D.: Extended nonlinear observer normal forms for a class of nonlinear dynamical systems. Int. J. Robust Nonlinear Control 25(3), 461–474 (2015) 3. Engel, R., Kreisselmeier, G.: A continuous-time observer which converges in finite time. IEEE Trans. Autom. Control 47(7), 1202–1204 (2002) 4. Fliess, M., Sira-Ramırez, H.: Reconstructeurs d’état. Comptes Rendus Mathematique 338(1), 91–96 (2004) 5. Floquet, T., Barbot, J.P., Perruquetti, W.: Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems. Automatica 39(6), 1077–1083 (2003) 6. Fridman, L., Levant, A., Davila, J.: Observation of linear systems with unknown inputs via high-order sliding-modes. Int. J. Syst. Sci. 38(10), 773–791 (2007) 7. Raff, T., Allgower, F.: An impulsive observer that estimates the exact state of a linear continuoustime system in predetermined finite time. In: Proceedings of Mediterranean Conference on Control and Automation, pp. 1–3 (2007) 8. Tami, R., Boutat, D., Zheng, G.: Extended output depending normal form. Automatica 49(7), 2192–2198 (2013) 9. Utkin, V.: Sliding Modes in Control and Optimization. Springer Science & Business Media, New York (2013) 10. Xiong, Y., Saif, M.: Sliding mode observer for nonlinear uncertain systems. IEEE Trans. Autom. Control 46(12), 2012–2017 (2001)

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11. Zheng, G., Boutat, D., Barbot, J.P.: Output dependent observability linear normal form. In: Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, pp. 7026–7030 (2005) 12. Zheng, G., Boutat, D., Barbot, J.P.: Single output-dependent observability normal form. SIAM J. Control Optim. 46(6), 2242–2255 (2007)

Chapter 7

Extension to Nonlinear Partially Observable Dynamical Systems

Abstract In the previous chapters, we have applied differential geometric method to design observers for nonlinear dynamical systems whose states are supposed to be fully observable. However, in practice some systems might be only partially observable, i.e., only a part of states are observable. For this topic, several works have been done by using different techniques, such as reduced-order observer and LMI technique [12], differential geometric method [3, 7] where the desired normal form is quite special. The objective of this chapter is to show how to treat general partial observable case when applying differential geometric method. In this chapter, we will use the notion of commutativity of Lie bracket modulo a distribution, and at the same time apply a diffeomorphism on the output [10, 11]. Also, necessary and sufficient geometric conditions have been established to guarantee the existence of a diffeomorphism such that the studied nonlinear system can be transformed into the desired general partial observer normal form. A homogeneous finite-time observer is then designed for such a normal form at the end of this chapter.

7.1 Problem Statement Consider the following nonlinear dynamical system x˙ = f (x) y = h(x),

(7.1)

where x ∈ X ⊂ Rn with X being a neighborhood of 0, and y ∈ R. Without loss of generality it is assumed that f and h are smooth with f (0) = 0 and h(0) = 0. For system (7.1), as we have presented in Chap. 3, if the pair ( f (x), h(x)) locally satisfies the observability rank condition on X , i.e.,   rank dh, d L f h, . . . , d L nf h (x) = n, for x ∈ X , then the following 1-forms

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_7

127

128

7 Extension to Nonlinear Partially Observable …

θ1 = dh and θi = d L i−1 f h, for 2 ≤ i ≤ n are independent on X . Therefore, there exists a family of vector fields τ = [τ1 , . . . , τn ], where the first vector field τ1 is the solution of (3.4) and other vector fields are obtained by (3.5). According to Theorem 3.2, if 

 τi , τ j = 0 for 1 ≤ i, j ≤ n,

(7.2)

then system (7.1) can be transformed, by means of a local change of coordinates z = φ(x), into the nonlinear observer normal form (3.3). Obviously, for the nonlinear dynamical system (7.1), if   rank dh, d L f h, . . . , d L nf h (x) = r < n for x ∈ X , which implies that only a part of states of the studied system are observable, the proposed method in Chap. 3 could not be applied. This chapter will deal with the geometric conditions that enable us to transform a partial observable dynamical system into the following normal form: z˙ = A O z + β(y) ζ˙ = η(z, ζ ) y = C O z,

(7.3)

where z ∈ Rr , ζ ∈ Rn−r , y ∈ R, β : R → Rr , η : Rr × Rn−r → Rn−r , and C O = (0, . . . , 0, 1) ∈ R1×r and A O is the r × r Brunovsky matrix ⎛

⎞ 0 .. ⎟ .⎟ r ×r 0 0⎟ ⎟∈R . 0 0⎠ 0 ··· ··· 1 0

0 0 ··· ⎜ 1 0 ... ⎜ AO = ⎜ ⎜0 1 ··· ⎝0 ··· ···

0 .. .

For (7.3), we can easily design different types of observers to estimate the part of observable state z, which will be presented in Sect. 7.4. The following will discuss how to deduce a change of coordinates that transforms (7.1) into the target normal form (7.3).

7.2 Necessary and Sufficient Conditions

129

7.2 Necessary and Sufficient Conditions As we require only the partial observability for the studied system (7.1), thus we assume that   rank dh, d L f h, . . . , d L nf h (x) = r < n for x ∈ X ⊂ Rn . As for the previous chapters of this book, denote the observability 1-forms for 1 ≤ i ≤ r by θi = d L i−1 f h and note Δ = span{θ1 , θ2 , . . . , θr } as the co-distribution spanned by the observability 1-forms. The partial observability rank condition implies that d L if h ∈ Δ for i ≥ r + 1. In the same way as for the full observability, we define the following distribution Δ⊥ as follows: Δ⊥ = kerΔ = {X : θk (X ) = 0, for 1 ≤ k ≤ r }.

(7.4)

The following result provides some useful properties of distribution Δ⊥ defined in (7.4) that will be used within this chapter. Lemma 7.1 The distribution Δ⊥ has the following properties: C1: Δ⊥ is involutive in the sense that for any two vector fields H1 ∈ Δ⊥ and H2 ∈ Δ⊥ , we have [H1 , H2 ] ∈ Δ⊥ . C2: there exist (n − r ) vector fields {τ¯r +1 , . . . , τ¯n } that span Δ⊥ such that [τ¯i , τ¯ j ] = 0. for r + 1 ≤ i, j ≤ n, i.e., {τ¯r +1 , . . . , τ¯n } is the commutative basis of Δ⊥ ; C3: Δ⊥ is f -invariant in the sense that for any vector field H ∈ Δ⊥ , we have [ f, H ] ∈ Δ⊥ . Proof Let us recall from (2.16) the following equality: dθi (H1 , H2 ) = L H1 θi (H2 ) − L H2 θi (H1 ) − θi ([H1 , H2 ]) .

(7.5)

Since θi = d L i−1 f h for 1 ≤ i ≤ r , then they are exact 1-forms (thus closed), and the above formula becomes

130

7 Extension to Nonlinear Partially Observable …

θi ([H1 , H2 ]) = L H1 θi (H2 ) − L H2 θi (H1 ).

(7.6)

Now, as H1 ∈ Δ⊥ and H2 ∈ Δ⊥ , we have [H1 , H2 ] ∈ Δ⊥ because θi ([H1 , H2 ]) = 0. Thus Δ⊥ is involutive, and C1 is fulfilled in Lemma 7.1. In regard of the second property C2 in Lemma 7.1, it is a consequence of C1 thanks to the well-known Frobenius’ Lemma 2.3 stated in Chap. 2. In order to prove the third property C3 in Lemma 7.1, we need to show that [ f, H ] ∈ Δ⊥ . For this, we will use again (7.6) by replacing H1 by f and H2 by H . As θi for 1 ≤ i ≤ r are exact, then according to the Eq. (7.6), we have θi ([ f, H ]) = L f θi (H ) − L H θi ( f ). Since H ∈ Δ⊥ , then θi (H ) = 0 which implies that L f θi (H ) = 0. Therefore, θi ([ f, H ]) = 0 is equivalent to L H θi ( f ) = 0. Moreover, based on the Cartan’s identity given in (2.13) and the fact that dθi ( f ) = θi+1 for 1 ≤ i ≤ r − 1, we have L H θi ( f ) = dı H θi ( f ) + ı H dθi ( f ) = θi+1 (H ) = 0, which implies that [ f, H ] ∈ Δ⊥ . Therefore Δ⊥ is f -invariant.



Now, if τ1 is one of the vector fields that fulfills the following under-determined algebraic equations: θk (τ1 ) = 0 for 1 ≤ k ≤ r − 1 (7.7) θr (τ1 ) = 1, then, for any H ∈ Δ⊥ , τ1 + H is also a solution of (7.7). As in Chap. 3, we build by induction a family of vector fields from any chosen τ1 satisfying (7.7) as follows:   τi = τi−1 , f for 2 ≤ i ≤ r.

(7.8)

Let τ˜1 be another solution of (7.7) and construct another family of (r − 1) vector fields   (7.9) τ˜i = τ˜i−1 , f for 2 ≤ i ≤ r. The following lemma provides the relationship between the above two families of vector fields for the different chosen τ1 and τ˜1 . Lemma 7.2 There exists a family of Hi ∈ Δ⊥ for 1 ≤ i ≤ r such that τ˜i = τi + Hi ,

7.2 Necessary and Sufficient Conditions

131

where τi and τ˜i are defined in (7.8) and (7.9), respectively. Proof Let us prove it for τ˜2 and the same method works for the rest of the vector fields. For this, we have τ˜2 = [τ˜1 , f ] = [τ1 + H1 , f ] = [τ1 , f ] + [H1 , f ] = τ2 + H2 in the above we used the property that Δ⊥ is f -invariant, i.e., [H1 , f ] = H2 ∈ Δ⊥ . Then, by induction, we can deduce that for 1 ≤ i ≤ r there exists Hi ∈ Δ⊥ such  that τ˜i = τi + Hi . The above Lemma 7.2 stated that the family of vector fields τi for 1 ≤ i ≤ r are defined modulo Δ⊥ . The next result shows other important properties of vector fields τi . Lemma 7.3 Given a solution τ1 of (7.7) based on which a family of τi for 2 ≤ i ≤ r are obtained from (7.8), then it satisfies the following properties: (1) Δ⊥ is τi -invariant, thus for any H ∈ Δ⊥ , we have [τi , H ] ∈ Δ⊥ ; (2) The Lie bracket of vector fields of the family of τi is defined modulo Δ⊥ , thus [τi + Hi , τ j + H j ] = [τi , τ j ] modulo Δ⊥ ; for any Hi ∈ Δ⊥ and H j ∈ Δ⊥ with 1 ≤ i, j ≤ r , we have (3) For 1 ≤ i ≤ r and r + 1 ≤ j ≤ n, we have   τi , τ¯ j ∈ Δ⊥ ,

(7.10)

where τ¯ j was defined in C2 of Lemma 7.1, which is the commutative basis of Δ⊥ . Proof (1) Note that dθk = 0 since θk = d L k−1 f h for 1 ≤ k ≤ r is closed, then for H ∈ Δ⊥ , we obtain θk ([τi , H ]) = L τi θk (H ) − L H θk (τi ) = −L H θk (τi ), where we used the fact that H ∈ Δ⊥ , then it belongs to the kernel of θk . According to the definition of τ1 , then the function θk (τ1 ) is constant, thus its Lie derivative is 0. Therefore, we obtain θk ([τ1 , H ]) = −L H θk (τ1 ) = 0, which yields [τ1 , H ] ∈ Δ⊥ . Now, set i = 2 and use Jacobi’s identity defined in (2.15) to get

132

7 Extension to Nonlinear Partially Observable …

[τ2 , H ] = [[τ1 , f ], H ] = −[[H, τ1 ], f ] − [[H, f ], τ1 ]. Since Δ⊥ is f and τi -invariant we obtain [τ2 , H ] ∈ Δ⊥ . Using repeatably Jacobi’s identity, the invariance and the involutivity of Δ⊥ , by induction we get the following result: [τi + Hi , τ j + H j ] = [τi , τ j ] modulo Δ⊥ . (3) Due to the fact that Δ⊥ is τi -invariant and the fact that τ¯ j for r + 1 ≤ j ≤ n are the basis of Δ⊥ , we obtain   τi , τ¯ j ∈ Δ⊥ .  Lemma 7.4 For a givenτi for2 ≤ i ≤ r there exists a family of vector fields τ¯i = τi modulo Δ⊥ such that τ¯i , τ¯ j = 0 for 1 ≤ i, j ≤ r , if and only if [τi , τ j ] ∈ Δ⊥ . Proof Necessity: Suppose that we have τ¯i = τi + Hi , τ¯ j = τ j + H j for 1 ≤ i, j ≤ r and Hi ∈ Δ⊥ and H j ∈ Δ⊥ such that τ¯i , τ¯ j = 0, then from the second item of Lemma 7.3, we obtain   0 = τ¯i , τ¯ j = [τi , τ j ] modulo Δ⊥ , which leads to [τi , τ j ] ∈ Δ⊥ . Sufficiency: Let us assume that [τi , τ j ] ∈ Δ⊥ for 1 ≤ i, j ≤ r . We will prove that there exists Hi ∈ Δ⊥ for 1 ≤ i ≤ r such that, for τ¯k = τk + Hk ,we have [τ¯i , τ¯ j ] = 0. For this, let us consider a family of vector fields {τ¯r +1 , . . . , τ¯n } that is a commutative basis of Δ⊥ . It has been stated in [5] that we can complete this basis by r independent commutative vector fields {X 1 , . . . , X r }, such that they commute with {τ¯r +1 , . . . , τ¯n } and form together a basis of the tangent fiber bundle T X of X . Then, on this basis, the vector fields τi 1 ≤ i ≤ r can be expressed as follows: τi =

r

μi,l (x)X l +

n

μi,l (x)τ¯l .

l=r +1

l=1

By setting Hi = −

n

μi,l (x)τ¯l ∈ Δ⊥

l=r +1

and Hj =

n l=r +1

we obtain

μ j,l (x)τ¯l ∈ Δ⊥ ,

7.2 Necessary and Sufficient Conditions

133

τ¯i = τi + Hi =

r

μi,l (x)X l

l=1

and τ¯ j = τ j + H j =

r

μ j,l (x)X l .

l=1

Now, from Lemma 7.3, we know that [τi , τ j ] = [τ¯i , τ¯ j ] modulo Δ⊥ .

(7.11)

As we have [X k , X s ] = 0 for 1 ≤ k, s ≤ r , then Eq. (7.11) becomes

[τi , τ j ] =

r 

μi,l L X l μ j,s X s −

l,s=1

r 

 μ j,s L X s μi,l X l

modulo Δ⊥ .

s,l=1

Therefore, if [τi , τ j ] ∈ Δ⊥ , then, there exist Hi ∈ Δ⊥ and H j ∈ Δ⊥ defined above, such that r r μi,l L X l μ j,s X s − μ j,s L X s μi,l X l = 0 [τ¯i , τ¯ j ] = l,s=1

s,l=1

thanks to the fact that X k ∈ / Δ⊥ .



Let us provide an example to highlight the above results. Example 7.1 Consider the following dynamical system z˙ 1 = 0 z˙ 2 = z 1 + y 2 η˙ = −η + νz 12 + y y = z2 . A straightforward calculation gives the following observability 1-forms θ1 = dz 2 θ2 = dz 1 + 2z 2 dz 2 . The intersection of their kernel defines the following co-distribution Δ = span{θ1 , θ2 } and the associated distribution  ∂ . Δ = span ∂η ⊥



(7.12)

134

7 Extension to Nonlinear Partially Observable …

From (7.7), we obtain τ1 =

∂ ∂ + r1 (z 1 , z 2 , η) , ∂z 1 ∂η

where r1 (z 1 , z 2 , η) is a function of its arguments. From (7.8), we get τ2 = [τ1 , f ] =

∂ ∂ + r2 , ∂z 2 ∂η

∂r1 1 where r2 = 2μz 1 − r1 − ∂z (z 1 + z 22 ) − ∂r (−η + νz 12 + z 2 ). The Lie bracket of the ∂η 2 above two vector fields leads to   ∂ ∂r2 ∂r2 ∂r1 ∂r1 r1 − r2 , + − [τ1 , τ2 ] = ∂z 1 ∂η ∂z 2 ∂η ∂η

which is not equal to 0 if r1 = 0 and ν = 0. However, we have [τ1 , τ2 ] ∈ Δ⊥ . The following will find two other vector fields which are commutative, according ∂ ∂ and H2 = −r2 ∂η , and then we obtain to Lemma 7.4. For this, let us take H1 = −r1 ∂η ∂ ∂  the desired result τ¯1 = ∂z1 and τ¯2 = ∂z2 which yields [τ¯1 , τ¯2 ] = 0. Now as we did in (3.10) in Chap. 3, let us recall the procedure to compute the change of coordinates. For this, we consider r independent vector fields τi for 1 ≤ i ≤ r , and define the following matrix: ⎞ 0 0 ··· 0 1 ⎜0 0 ··· 1 Λ2,r ⎟ θ1 ⎜. . . . .. ⎟ . ⎜ ⎠ ⎝ . . . . . Λ1 = . (τ1 , . . . , τr ) = ⎜ . . . . . ⎟ ⎟, ⎝ θr 0 1 · · · Λr −1,r −1 Λr −1,r ⎠ 1 Λr,2 · · · Λr,r −1 Λr,r ⎛

⎛

⎞

(7.13)

where its entries are Λi, j = θi (τ j ) for 1 ≤ i, j ≤ r . Due to the fact that this matrix is invertible, we can then define the following 1-forms: ⎞ ω1 T ⎝ ... ⎠ = Λ−1 1 [θ1 , . . . , θr ] , ωr ⎛

(7.14)

where its components are linear combination of the 1-forms θi for 1 ≤ i ≤ r and given by ωr = θ1   n  (7.15) ωr −k = θk+1 − Λk+1,i ωi for 1 ≤ k ≤ r − 1. i=r −k+1

7.2 Necessary and Sufficient Conditions

135

From (7.14) and (7.15), we can deduce that 1-form ωk for 1 ≤ k ≤ r are independent of the choice of τi modulo Δ⊥ . From this, we can build the change of coordinates as state in the following theorem. Theorem 7.1 There exists a local change of coordinates (z T , ζ T )T = φ(x) on X which transforms system (7.1) into the target normal form (7.3), if and only the following conditions are fulfilled (1) there exists a family of vector fields τ¯i = τi modulo Δ⊥ such that [τ¯i , τ¯l ] = 0 for  r and 1 ≤ l ≤ r ; 1 ≤ i ≤ (2) τ¯i , τ¯ j = 0 for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n where τ¯ j was defined in C2 of Lemma 7.1. Under the above conditions, the change of coordinates φ is locally determined by its differential φ∗ := dφ = ω, i.e., φ(x) = γ ω, where γ is any path on X with γ (0) = 0 and γ (1) = x. Proof Necessity: The proof is the same as those for the normal forms obtained in the previous chapters. Let us just specify the main difference with the normal form (7.3). Let us denote the first partial observability 1-form for the normal form (7.3) by nf θ1 = dzr and the others have the following form nf

θi

= dzr −i+1 +

i−1

nf

ljθj

j=1

for 2 ≤ i ≤ r . Therefore the associated distribution is given by Δ

nf ⊥

 = span

∂ ∂ ,..., ∂ζ1 ∂ζn−r

 . ⊥

Hence, the first vector field is τ1 = we build the rest vector fields as

∂ ∂z 1

+ H1 where H1 ∈ Δn f , and by induction

nf

=

∂ nf + Hi , ∂z i

nf

τi nf

where Hi ∈ Δn f vector fields

⊥

nf

nf

for 2 ≤ i ≤ r . From this, we deduce the following family of nf

τ¯i

=

∂ ⊥ nf = τi modulo Δn f ∂z i

  nf nf = 0 for 1 ≤ i ≤ r and 1 ≤ l ≤ r . such that τ¯i , τ¯l As we want them to commute, we can then choose τ¯ j = ∂ζ∂j−r , thus we have   nf nf τ¯i , τ¯ j = 0 for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n. To complete the proof, it is enough nf

136

7 Extension to Nonlinear Partially Observable …

to set ω = dφ = φ ∗ [dz, dζ ]T , where φ ∗ is the pullback defined from φ on the space of 1-forms. Sufficiency: Hereafter, we will state only the main difference with the full observability case stated in Chap. 3, and we will just give the key steps which enable us to adapt the proof of Theorem 3.2. For this, we consider the following two cases: • for 1 ≤ i ≤ r − 1, we have   ∂ ∂ φ∗ ( f ) = , φ∗ ( f ) = [φ∗ (τ¯i ), φ∗ ( f )] = φ∗ [τ¯i , f ] = φ∗ (τ¯i+1 + Hi ) ∂z i ∂z i for some Hi ∈ Δ⊥ . Thus, we have ∂ ∂ φ∗ ( f ) = φ∗ (τ¯i+1 ) + φ∗ (Hi ) = + φ∗ (Hi ); ∂z i ∂z i+1

(7.16)

• for r + 1 ≤ j ≤ n, we have   ∂ ∂ φ∗ ( f ) = , φ∗ ( f ) = [φ∗ (τ¯ j ), φ∗ ( f )] = φ∗ [τ¯ j , f ]. ∂ζ j ∂ζ j Since τ¯ j span the distribution Δ⊥ which is f -invariant (see C3 of Lemma 7.1), then there exists H j ∈ Δ⊥ such that H j = [τ¯ j , f ]. Thus, we have ∂ φ∗ ( f ) = φ∗ (H j ). ∂ζ j

(7.17)

As for any H ∈ Δ⊥ , we know that  φ∗ (H ) = ω(H ) ∈ span

∂ ∂ ,..., ∂ζ1 ∂ζn−r

 .

The remainder of the proof can be obtained in the same way as that of Theorem 3.2 in Chap. 3.  Let us present an example to show how we deal with those conditions stated in Theorem 7.1. Example 7.2 Consider the following nonlinear system: x˙1 = −x32 + x13 x3 − 21 x33 + x25 x˙2 = x1 − 21 x32 x˙3 = −x3 + x13 − 21 x32 y = x2 . A simple calculation leads to the following partial observability 1-forms:

(7.18)

7.2 Necessary and Sufficient Conditions

137

θ1 = d x2 θ2 = d x1 − x3 d x3 . Therefore, we have

  rank dh, d L f h, d L 2f h = 2,

thus r = 2 because d L 2f h = 5x24 θ1 . From this, we have the co-distribution Δ = span {θ1 , θ2 }, and the associated distribution   ∂ ∂ ⊥ . Δ = span x3 + ∂ x1 ∂ x3 Due to the fact that τ¯3 needs to be a commutative basis of Δ⊥ , we can then choose τ¯3 = x3

∂ ∂ + . ∂ x1 ∂ x3

As a solution of (7.7), we obtain τ1 =

∂ + r1 (x)τ¯3 . ∂ x1

By removing all terms in the direction of τ¯3 in τ1 , we get τ¯1 = Now, we compute [τ¯1 , f ] =

∂ . ∂ x1

∂ + 3x12 τ¯3 , ∂ x2

and we cancel all terms in the direction of τ¯3 to obtain τ¯2 =

∂ . ∂ x2

As a result, we obtain the following vector fields τ¯1 = ∂∂x1 τ¯2 = ∂∂x2 τ¯3 = x3 ∂∂x1 +

∂ ∂ x3

that fulfill all conditions of Theorem 7.1, therefore there exits a change of coordinates which can transform the studied system into the target form (7.3). For the deduction of this diffeomorphism, we use the obtained family of vector fields τ¯ = [τ¯1 , τ¯2 , τ¯3 ] which gives

138

7 Extension to Nonlinear Partially Observable …

 Λ1 =

   θ1 01 . (τ¯1 , τ¯2 ) = θ2 10

Thus, we have 

ω1 ω2

 =

Λ−1 1



θ1 θ2



 =

dz 1 dz 2



 =d

x1 − 21 x32 x2

 .

Therefore, we get the following change of coordinates: ⎛

⎞ x1 − 21 x32 ⎠, φ (x) = ⎝ x2 x3 which transforms (7.18) into the following normal form: z˙ 1 = 0 z˙ 2 = z 1 ζ˙1 = z 13 − z 3 + y = z2 .

z 32 2

 2 3 2 1 4  3z 1 + 2 z 1 z 3 + 4 z 3 − 1 

7.3 Diffeomorphism on the Output Let us remark that the deduced change of coordinates φ(x) in Sect. 7.2 does not modify the output. This is due to the fact that θr (τ1 ) = 1. In Chap. 4, we have seen that if the commutativity conditions of Lie brackets are not fulfilled, one way to relax this constraint is to seek a new vector field, noted as σ¯ 1 , such that θr (σ¯ 1 ) becomes a function of the output, and this will introduce a diffeomorphism on the output. For this purpose, we will follow the same algorithm as in Chap. 4. Indeed will adapt the normal form (4.3) obtained in Chap. 4 together with the normal form (7.3) as follows: z˙ = A O z + β( y¯ ) (7.19) ζ˙ = η(z, ζ ) y¯ = C O z = ϕ(y). As in Sect. 7.2, we calculate the observability 1-forms θi for 1 ≤ i ≤ r where r is the rank of observability matrix, which defines the co-distribution Δ = span{θ1 , θ2 , . . . , θr }

7.3 Diffeomorphism on the Output

139

and the kernel (or the annihilator) of the co-distribution Δ, noted as Δ⊥ . We assume that we have already obtained a family of vector fields τ¯ j for r + 1 ≤ j ≤ n which are a commutative basis of Δ⊥ , and τ¯i for 1 ≤ i ≤ r without any component in the direction of τ¯ j for r + 1 ≤ j ≤ n. Thus, hereafter, we assume that there exist 1 ≤ i, l ≤ r such that [τ¯i , τ¯l ] = 0, hence Theorem 7.1 cannot be applied. As it was explained in Chap. 4, in order to introduce a diffeomorphism on the output, let l(y) = 0 be a function of the output which will be determined later. Then we can define a new vector field σ1 from τ¯1 as follows: σ¯ 1 = l(y)τ¯1 .

(7.20)

Like what we did in the previous chapters, we define by induction the following vector fields:   (7.21) σ¯ i = σ¯ i−1 , f modulo Δ⊥ for 2 ≤ i ≤ r . As in the proof of Lemma 4.1 of Chap. 4, a straightforward calculation leads to i k−1 i−k L f l(y)τ¯k modulo Δ⊥ , (7.22) σ¯ i = (−1)i−k Ci−1 k=1 k−1 where Ci−1 are the binomial coefficients and L 0f l(y) = l(y). By canceling all terms in the directions of τ¯ j for r + 1 ≤ j ≤ n to get the commutative basis of Δ⊥ , we obtain

σi =

i

k−1 i−k L f l(y)τ¯k , for 1 ≤ i ≤ r. (−1)i−k Ci−1

(7.23)

k=1

Now, let us set σ j = τ¯ j , for r + 1 ≤ j ≤ n,

(7.24)

where {τ¯r +1 , . . . , τ¯n } are commutative basis of Δ⊥ . Then, as in Chap. 4, let us define the following matrix: ⎛ ⎞ θ1 ⎜ Λ˜ 1 = ⎝ ... ⎠ (σ1 , . . . , σr ) = ⎜ ⎝ θr ⎛

⎞ 0 · · · 0 l(y) .. . · · · l(y) ∗ ⎟ ⎟, 0 ··· ∗ ∗ ⎠ l(y) · · · ∗ ∗

(7.25)

which is invertible because l(y) = 0, and therefore we can define the following 1-forms: ⎛ ⎞ ω˜ 1 T ⎝ ... ⎠ = Λ˜ −1 (7.26) 1 [θ1 , . . . , θr ] ω˜ r

140

7 Extension to Nonlinear Partially Observable …

and using the same method in Sect. 7.2, we can uniquely determine ω˜ i for r + 1 ≤ i ≤ n by solving the following equations: j

ω˜ i (σ j ) = δi

(7.27)

for 1 ≤ j ≤ n. Finally, we have the following theorem. Theorem 7.2 There exists a local change of coordinates (z T , ζ T )T = φ(x) on X which transforms system (7.1) into the target normal form (7.19) if and only if the following conditions are fulfilled: (1) [σi , σl ]= 0 for 1 ≤ i ≤ r and 1 ≤ l ≤ r where σi defined in (7.23); (2) σi , σ j = 0 for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n where σ j defined in (7.24). Under the above conditions, the change of coordinates φ is locally given by its differential φ∗ := dφ = ω, ˜ i.e.,  φ(x) =

γ

ω, ˜

(7.28)

where γ is any path on X γ (0) = 0 and γ (1) = x. Moreover, we have zr =  y with 1 dc is a change of coordinates on the output. y¯ = ϕ(y) where ϕ(y) = 0 l(c) Proof The proof of Theorem 7.2 is similar to that of Theorem 7.1, thus the following just gives hints to determine l(y) and some different elements with respect to the result presented in Chap. 4. According to (7.20), we define σ1 = l(y)τ¯1 where τ¯1 is obtained in Sect. 7.2 by canceling all terms in the directions of τ¯ j for r + 1 ≤ j ≤ n. We obtain by induction   σi = σi−1 , f modulo Δ⊥ for 2 ≤ i ≤ r . Then we can deduce [σ1 , σi ] = =

i  k=1 i  k=1

+

k−1 i−k L f l(y) [σ1 , τ¯k ] (−1)i−k Ci−1

  k−1 i−k L f l(y) l(y) [τ¯1 , τ¯k ] − L τ¯k l(y)τ¯1 (−1)i−k Ci−1

i 

k=1

(7.29)

k−1 L σ1 L i−k (−1)i−k Ci−1 f l(y)τ¯k .

Now, we compute l(y), by imposing [σ1 , σi ] = 0. More details on this issue can be founded in Chap. 4. After having deduced l(y), the rest of the proof for Theorem 7.2 is similar to that of Theorem 7.1. 

7.4 Observer Design

141

7.4 Observer Design 7.4.1 Homogeneous Observer In this section, we will present a homogeneous non-asymptotic observer for (7.1). The global finite-time observer based on homogeneity [1, 2] was firstly introduced by [6] for the nonlinear dynamical systems which can be transformed into a linear dynamical system with output injection. After that, [8] extended this idea and proposed a semiglobal finite-time observer for the special systems with triangular structure. The global finite-time observer for such a system was studied in [4] by introducing the second gain in [9]. Let us suppose that there exists a diffeomorphism (z T , ζ T )T = φ(x) on X which transforms system (7.1) into the partial observer normal form (7.19). To design nonasymptotic observer of z in (7.19), we can consider the following dynamics: z˙ˆ = A O zˆ + β( y¯ ) + K ϕ(y) − C O zˆ α ,

(7.30)

where K = [K 1 , . . . , K n ] is a constant matrix to be chosen such that the matrix (A O − K C O ) is Hurwitz, and ϕ(y) − C O zˆ α is defined as follows: ⎤ |ϕ(y) − C O zˆ |α1 sgn(ϕ(y) − C O zˆ ) .. ⎦,

ϕ(y) − C O zˆ α = ⎣ . αn |ϕ(y) − C O zˆ | sgn(ϕ(y) − C O zˆ ) ⎡

(7.31)

where α j for 1 ≤ j ≤ n are defined as α j = jγ − ( j − 1)

(7.32)

with γ ∈ (1 − n1 , 1). Note e = z − zˆ , then according to (7.19) and (7.30), we obtain ⎤ e˙1 ⎢ e˙2 ⎥ ⎢ . ⎥= ⎣ .. ⎦ ⎡

e˙n

⎡

⎤ ⎤⎡ ⎤ ⎡ ··· 0 0 K 1 |en |α1 sgn(en ) e1 α · · · 0 0 ⎥ ⎢ e2 ⎥ ⎢ K 2 |en | 2 sgn(en ) ⎥ ⎥ ⎢ . ⎥ − ⎢. .. . . .. ⎥ ⎦ . . ⎦ ⎣ .. ⎦ ⎣ .. . αn en K n |en | sgn(en ) 0 0 ··· 1 0

0 ⎢1 ⎢. ⎣ ..

0 0 .. .

and it has been proven in [6] that e will converge to zero in a finite time T (z(0)), i.e., zˆ (t) = z(t), ∀t ≥ T (z(0)). In conclusion, the dynamics (7.30) is a non-asymptotic homogeneous observer of (7.19).

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7 Extension to Nonlinear Partially Observable …

7.4.2 Design Procedure The following summarizes the procedure to design observer based on the proposed normal form (7.19). Step 1: Determine the observability 1-forms θi for 1 ≤ i ≤ r ; Step 2: Pick a solution τ1 of (7.7), then cancel all its terms in the directions of {τ¯r +1 , . . . , τ¯n } which yields τ¯1 ; Step 3: Iteratively, for 2 ≤ i ≤ r , compute [τ¯i−1 , f ] in which eliminate all terms in the directions of {τ¯r +1 , . . . , τ¯n } which gives τ¯i ; Step 4: Calculate the new family of vector fields σ¯ according to (7.20) and (7.21); Step 5: Deduce l(y) according to (7.29); Step 6: Compute Λ˜ and ω˜ according to (7.25) and (7.26), then deduce the change of coordinates φ(x) according to (7.28); Step 7: Apply this change of coordinates to calculate φ∗ ( f ), with f being defined in (7.1); Step 8: Design the proposed homogeneous observer (7.30) to estimate z of (7.19).

References 1. Bhat, S., Bernstein, D.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000) 2. Bhat, S., Bernstein, D.: Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 17(2), 101–127 (2005) 3. Jo, N.H., Seo, J.H.: Observer design for non-linear systems that are not uniformly observable. Int. J. Control 75(5), 369–380 (2002) 4. Menard, T., Moulay, E., Perruquetti, W.: A global high-gain finite-time observer. IEEE Trans. Autom. Control 55(6), 1500–1506 (2010) 5. Nijmeijer, H., Van der Schaft, A.: Nonlinear Dynamical Control Systems, vol. 175. Springer, Berlin (1990) 6. Perruquetti, W., Floquet, T., Moulay, E.: Finite-time observers: application to secure communication. IEEE Trans. Autom. Control 53(1), 356–360 (2008) 7. Röbenack, K., Lynch, A.: Observer design using a partial nonlinear observer canonical form. Int. J. Appl. Math. Comput. Sci. 16, 333–343 (2006) 8. Shen, Y., Xia, X.: Semi-global finite-time observers for nonlinear systems. Automatica 44(12), 3152–3156 (2008) 9. Shen, Y., Huang, Y., Gu, J.J.: Global finite-time observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 56(2), 418–424 (2011) 10. Tami, R., Zheng, G., Boutat, D., Aubry, D.: Partial observability normal form for nonlinear functional observers design. In: Proceedings of the 32nd IEEE Chinese Control Conference, pp. 95–100 (2013) 11. Tami, R., Zheng, G., Boutat, D., Aubry, D., Wang, H.: Partial observer normal form for nonlinear system. Automatica 64, 54–62 (2016) 12. Trinh, H., Fernando, T., Nahavandi, S.: Partial-state observers for nonlinear systems. IEEE Trans. Autom. Control 51(11), 1808–1812 (2006)

Chapter 8

Extension to Nonlinear Dynamical Systems with Multiple Outputs

Abstract Till now, all the observer normal forms presented in Chaps. 3–7 are focused on nonlinear dynamical systems with single output. The extension of the previous results to the case with multiple outputs is not trivial [3, 13]. The first attempt was made by [6, 10], and this chapter will firstly summarize their ideas, and then discuss the main differences between single output and multiple outputs. After that, some special cases related to observability indices will be discussed for nonlinear dynamical systems with multiple outputs, for which necessary and sufficient geometric conditions will be presented. Another extension treated in this chapter is the partially observable case, which has been analyzed in Chap. 7 for systems with single output. By following a similar procedure as that in Chap. 7, a desired partial observer normal form with multiple outputs is then proposed [8]. Of course, we give as well the geometric conditions which enable us to compute a diffeomorphism to transform a nonlinear system with multiple outputs into such a normal form, for which a reduced-order Luenberger-like observer is proposed to estimate its observable states.

8.1 Problem Statement Consider the following multi-output nonlinear dynamical system x˙ = f (x) y = h(x) = (h 1 (x), . . . , h m (x))T ,

(8.1)

where x ∈ X ⊆ Rn is the state vector, X is an open domain of Rn , f : X → Rn and h : X → Rm are both assumed to be smooth. Without loss of generality, it is assumed that 0 ∈ X , f (0) = 0 and h(0) = 0. Moreover, it is assumed that all components of h: yi = h i for 1 ≤ i ≤ m, are independent. Otherwise, we can just drop the dependent components and keep only the maximal independent components. For the case with multiple outputs, this chapter investigates firstly the problem how to seek a change of coordinates z = φ(x) with z = (z 1T , z 2T , . . . , z mT )T and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_8

143

144

8 Extension to Nonlinear Dynamical Systems …

 T m z i = z i,1 , z i,2 , . . . , z i,ri , for 1 ≤ i ≤ m with i=1 ri = n, which can transform the dynamical system (8.1) into the following observer normal form: z˙ i = A Oi z i + βi ( y¯ ) y¯i = C Oi z i

(8.2)

for 1 ≤ i ≤ m, where A Oi and C Oi are (ri × ri ) and (1 × ri ) constant matrices, respectively, of the following form: ⎡

A Oi

⎤ 0 ··· 0 0 ⎢1 ··· 0 0⎥   ⎢ ⎥ = ⎢ . . . . ⎥ , C Oi = 0, . . . , 0, 1 , . . . . ⎣. . . .⎦ 0 ··· 1 0

(8.3)

i.e., the pairs (A Oi , C Oi ) are of the well-known Brunovsky form. Moreover, y¯ = ( y¯1 , . . . , y¯m ) = ϕ(y), where ϕ is a local diffeomorphism of the original outputs,  T and βi = βi,1 , βi,2 , . . . , βi,ri , for 1 ≤ i ≤ m, are vectors of functions of the new outputs y¯ .

8.2 Construction of the Frame Note that in Chap. 3, for the single-output case, we had constructed a family of vector fields τ in (3.4) and (3.5) for the purpose of producing the desired change of coordinates. For the multiple outputs case studied in this chapter, we will follow the same procedure. Unlike the single-output case where the observability rank condition depends only on the single output and its high-order derivatives, it becomes more complicated for the multiple-output case since the high-order derivatives of some outputs might be dependent. To characterize this dependence, the so-called observability indices have been introduced in the literature when analyzing the dynamical systems with multiple outputs. For this, denote D0 = span {dh i , for 1 ≤ i ≤ m} , which is spanned by the differential of the outputs, and it satisfies rankD0 = m. Set m 0 = m, then for k ≥ 1, define the following family of co-distributions  j−1 Dk = span d L f h i , for 1 ≤ i ≤ m, 1 ≤ j ≤ k + 1 .

8.2 Construction of the Frame

145

Hence, for j ≥ 1, we have D j−1 ⊆ D j , and denote the difference of their ranks as m j = rankD j − rankD j−1 . Using these notations, the observability indices can be then defined as follows. Definition 8.1 ([6]) The observability indices of the pair ( f, h) are defined as ri := card{m j | i ≤ m j , 0 ≤ j ≤ m}

(8.4)

for 1 ≤ i ≤ m. With a possible reordering of the output indices, it can be assumed that r1 ≥ r2 ≥ · · · ≥ rm . Based on the concept of observability indices, the following assumption is imposed for the studied system (8.1). Assumption 8.1 For the pair ( f, h) defined in system (8.1), it is supposed that the observability rank condition is satisfied in the sense that the rank of the following co-distribution  (8.5) Δ = span d L k−1 f h i , 1 ≤ i ≤ m, 1 ≤ k ≤ ri m ri = n. Without loss of generality, assume as well that the is equal to n, i.e., i=1 observability indices {ri }1≤i≤m of (8.1) are constant on X . Like in the previous chapters, we define θi,k = d L k−1 f h i , then we can construct the following multi-valued 1-form: ⎛

θ = (θi,k )1≤i≤m,1≤k≤ri

⎞ θ1,1 ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ θ1,r1 ⎟ ⎜ ⎟. =⎜ ⎟ ⎜ θ2,1 ⎟ ⎜ .. ⎟ ⎝ . ⎠ θm,rm

(8.6)

  Now, let us define a family of vector fields τi,1 1≤i≤m by solving the following algebraic equations:   θ j,k τi,1 = 0, for 1 ≤ k ≤ ri − 1, if j ≤ i − 1    θ j,r j  τi,1 = 1 if j = i, θ j,k τi,1 = 0, for 1 ≤ k ≤ r j − 1   if j ≥ i + 1 θ j,k τi,1 = 0, for 1 ≤ k ≤ r j , for 1 ≤ i ≤ m and 1 ≤ j ≤ m.

(8.7)

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8 Extension to Nonlinear Dynamical Systems …

With the above solution, by applying the Lie bracket [τi,k−1 , f ] between vector fields τi,k−1 and f , we can define the following vector fields: τi,k = [τi,k−1 , f ]

(8.8)

for 1 ≤ i ≤ m and 2 ≤ k ≤ ri . Since it is assumed that Assumption 8.1 is fulfilled, then the family of vector fields {τi,1 }1≤i≤m defined in (8.7) and τi,k defined in (8.8) yield the following frame   τ = τi, j 1≤i≤m, 1≤ j≤ri = (τ1,1 , . . . , τ1,r1 , . . . , τm,1 , . . . , τm,rm ) of the tangent fiber bundle T X of X . It is clear that, for 1 ≤ i ≤ m and 2 ≤ k ≤ ri , we always have   θ j,k τi,s = 0, if rk > rs . We would like emphasize that the solution of (8.7) is generally not unique. Multiple solutions mmight exist even for the case when the dynamical system is fully observri = r = n. In fact, the first vector field (i.e., τ1,1 ) should satisfy r able, i.e., i=1 is then unique), and equations (for the fully observable case with r = n, τ1,1 mτi,1 for m r = r − (r − r ) + 2 < i < m should satisfy only iri + m i k=i+1 k j i with ri > r j and 1 ≤ k ≤ ri − r j such that r −1+k

d L fj

hj ∈ / span {£i ∩ £} .

(8.12)

Due to the fact the studied system is fully observable, we get r −1+k d L fj hj

=

i−1  s=1

where μ =

ri i  



r s −ri

 λs,ri +t θs,ri +t

+ λi,ri θi,ri + μ,

(8.13)

t=1,rs >ri

βs,t θs,t with βi,ri = 0, which contains all the terms with the indices

s=1t=1 rs ≤ ri .

satisfying It is obvious that μ ∈ span {£i ∩ £} and θs,ri +t ∈ £ \ £i ∩ £. Particularly we obtain μ(τ¯i,1 ) = 0 and μ(τ¯s,rs −(ri +t)+1 ) = 0 for 1 ≤ s ≤ i − 1 and 0 ≤ t ≤ rs − ri . Concerning Eq. (8.12), the following result presents an alternative to verify it. Proposition 8.1 For 1 ≤ i < j ≤ m, the following three statements are equivalent: (1) for 1 ≤ k ≤ ri − r j ,

148

8 Extension to Nonlinear Dynamical Systems … r −1+k

d L fj

h j ∈ span {£i ∩ £} ;

(2) for 1 ≤ s ≤ i − 1, 0 ≤ t ≤ rs − ri and 1 ≤ k ≤ ri − r j , 

r −1+k

d L fj h j (τ¯i,1 ) = 0 r j −1+k h j (τ¯s,rs −(ri +t)+1 ) = 0; dL f

(3) for 1 ≤ s ≤ i − 1, 0 ≤ t ≤ rs − ri and 1 ≤ k ≤ ri − r j , 

dh j (τ¯i,r j +k ) = 0 dh j (τ¯s,rs −(ri +t)+r j +k ) = 0.

Proof For τ j,s defined in (8.7)–(8.8) and £i defined in (8.9), we can see that the distribution determined via the kernel of the co-distribution span {£i ∩ £} is spanned by τ¯i,1 and τ¯s,rs −(ri +t)+1 for 1 ≤ s ≤ i − 1 and 0 ≤ t ≤ rs − ri . Hence, according to Eq. (8.13), it is easy to conclude that the first statement of Proposition 8.1 is equivalent to the second one. For the purpose of proving the equivalence between the second statement of Proposition 8.1 and the third one, we should firstly show that if d L lf h j (τ¯ p,q ) = 0, then l (8.14) d L l+1 f h j (τ¯ p,q ) = −d L f h j (τ¯ p,q+1 ). In order to prove this, let us recall a basic result stated in Chap. 2, i.e., for a differential 1-form ν, its differential dν is a differential 2-form, defined by its evaluation on a pair of vector fields X and Y as follows: dν(X, Y ) = L X ν(Y ) − L Y ν(X ) − ν([X, Y ]).

(8.15)

In (8.15), if we choose ν = d L lf h j , X = f and Y = τ¯ p,q , due to the fact that dν = 0, i.e., an exact 1-form, then (8.15) can be rewritten as 0 = L f d L lf h j (τ¯ p,q ) − L τ¯ p,q d L lf h j ( f ) − d L lf h j ([ f, τ¯ p,q ]). Clearly, we can see that if d L lf h j (τ¯ p,q ) = 0, then l 0 = −d L l+1 f h j (τ¯ p,q ) + d L f h j (τ¯ p,q+1 )

and this is exactly the formulae (8.14) which we want to prove. Since d L lf h j (τ¯ p,q ) = 0 for 1 ≤ l ≤ r j − 2 + k therefore, by induction we get r −1+k

d L fj

(τ¯ p,q ) = dh j (τ¯ p,q+r j +k−1 ).

8.3 Necessary and Sufficient Conditions

149

Finally, since d L lf h j (τ¯ p,q ) = 0 for 0 ≤ l ≤ r j − 2 + k, if we choose τ¯ p,q = τ¯i,1 or τ¯s,rs −(ri +t)+1 , then we can prove that the second statement of Proposition 8.1 is equivalent to the third one. We would like to remark that the third statement of Proposition 8.1 is very useful in practice due to the fact that it only contains the terms which have already been calculated (i.e., dh j , τ¯i,r j +k and τ¯s,rs −(ri +t)+r j +k ). In other words, it is not necessary r −1+k to compute high-order derivatives d L fj h j for k ≥ 1. Besides, this shows that (8.11) in Theorem 8.1 is fundamental for the case where the outputs are not function of their derivatives in the proposed observer normal form (8.2). The following example is to highlight the above discussion. Example 8.1 Consider the following dynamical system: x˙1 x˙3 x˙4 x˙6 y1

= 0, x˙2 = x1 = x2 + x3 x4 = x3 , x˙5 = x5 x6 = x5 = x4 , y2 = x3 + x6 .

Let us firstly calculate the corresponding observability 1-forms θ1,1 θ1,3 θ1,4 θ2,1 θ2,2

= d x4 , θ1,2 = d x3 = d x2 + x4d x3 + x3 d x4  = d x1 + d x4 x2 + x3 x42 + x32 = d x3 + d x6 = d x5 + d x2 + x4 d x3 + x3 d x4 .

By following Eqs. (8.7)–(8.8) and (8.9), we can have τ¯1,1 = τ¯1,3 = τ¯2,1 = and

∂ , τ¯1,2 ∂ x1 ∂ , τ¯1,4 ∂ x3 ∂ , τ¯2,2 ∂ x5

= = =

∂ ∂ x2 ∂ ∂ x4 ∂ ∂ x6

+ x4 ∂∂x3 + x6 ∂∂x5

 £1 = θ1,1 , θ1,2 , θ1,3 , θ2,1 , θ2,2 , d L 2f h 2 , d L 3f h 2   £ = θ1,1 ,θ1,2 , θ1,3 , θ1,4 , θ2,1 , θ2,2  £1 ∩ £ = θ1,1 , θ1,2 , θ1,3 , θ2,1 , θ2,2 .

For the obtained vector fields τ¯i, j , we can check that they commute. Therefore (8.10) of Theorem 8.1 is satisfied. Consequently, we can find the following diffeomorphism:

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8 Extension to Nonlinear Dynamical Systems …

z 1,1 z 1,2 z 1,3 z 1,4

= = = =

x1 x2 x3 − 21 x42 x4 .

Nevertheless, condition (8.11) of Theorem 8.1 is violated, since a straightforward computation yields     d L 2f h 2 = θ1,4 + x6 θ2,2 − θ1,3 + x5 θ2,1 − θ1,3 ∈ / £1 − (£1 ∩ £) . In fact, we get 1 z 2,1 = x5 − x62 2 and z 2,2 = x6 = y2 − y˙1 , which shows that z 2,2 , deduced via the applied diffeomorphism, contains the first derivative   of y1 . This is indeed coherent with the fact that dh 2 (τ¯1,3 ) = 0 or d L 2f h 2 τ¯1,1 = 0.

8.4 Diffeomorphism Deduction Suppose that all conditions of Theorem 8.1 are satisfied, then we can easily extend the result on diffeomorphism deduction for dynamical systems with single output to treat the case with multiple outputs. For this, let us evaluate the observability 1-forms defined in θ with the vector fields belonging to frame τ , which enables us to calculate the following matrix:   Λ = θ j,k (τi,l )1≤i≤m, 1≤l≤ri 1≤ j≤m, 1≤k≤r j ⎛ θ1,1 (τ1,1 ) · · · θ1,1 (τ1,r1 ) θ1,1 (τ2,1 ) ⎜ θ1,2 (τ1,1 ) · · · θ1,2 (τ1,r1 ) θ1,2 (τ2,1 ) ⎜ ⎜ .. .. .. .. ⎜ . . . . ⎜ (τ ) · · · θ (τ ) θ (τ θ =⎜ 1,r1 1,r1 1,r1 2,1 ) ⎜ 1,r1 1,1 ⎜ θ2,1 (τ1,1 ) · · · θ2,1 (τ1,r1 ) θ2,1 (τ2,1 ) ⎜ ⎜ .. .. .. .. ⎝ . . . .

· · · θ1,1 (τm,rm ) · · · θ1,2 (τm,rm ) .. .. . . · · · θ1,r1 (τm,rm ) · · · θ2,1 (τm,rm ) .. .. . .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠

(8.16)

θm,rm (τ1,1 ) · · · θm,rm (τ1,r1 ) θm,rm (τ2,1 ) · · · θm,rm (τm,rm )

Since we have assumed that the observability rank condition imposed in Assumption 8.1 is satisfied, Λ defined in (8.16) is invertible. Denote Λ−1 as the inverse of Λ, then the following multi-valued 1-form can be computed

8.4 Diffeomorphism Deduction

151

ω = Λ−1 θ,

(8.17)

where ω := (ωi, j )1≤i≤m, 1≤ j≤ri . Finally, if the components of the multi-valued 1-form ω are closed, i.e., dωi, j = 0 for 1 ≤ i ≤ m, 1 ≤ j ≤ ri , then, according to Poincaré’s Lemma presented in Chap. 2, there exists a local diffeomorphism φ := (φi, j )1≤i≤m, 1≤ j≤ri such that ωi, j = dφi, j

(8.18)

for 1 ≤ i ≤ m, 1 ≤ j ≤ ri . Consequently, for the case with multiple outputs, the similar requirement that the components of the multi-valued 1- f or m ω are closed is a sufficient condition for the existence of the local diffeomorphism φ. As what we have seen in the previous chapters, if such a condition is satisfied, then we can construct a change of coordinates using (8.17).

8.5 Special Cases In this section, we will state some special results that are similar to the single-output case. The first case is that the observability indices are the same and the second case is that their differences do not exceed 1 [2].

8.5.1 Equal Observability Indices The first special case we analyze here is the case where the observability indices are equal. Theorem 8.2 ([10]) Assume that the observability indices of the dynamical system (8.1) are equal, i.e., the following condition holds: r1 = r2 = · · · = rm . Then, the following two assertions are equivalent. (1) The vector fields of the frame τ commute, i.e.,   τi,k , τ j,s = 0 for all 1 ≤ i, j ≤ m, 1 ≤ k ≤ ri and 1 ≤ s ≤ r j ;

(8.19)

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8 Extension to Nonlinear Dynamical Systems …

(2) There exists a local diffeomorphism z = φ(x) which transforms the dynamical system (8.1) into the normal form (8.2) where ϕ is the identity mapping on the output. The result stated in Theorem 8.2 is just a special case of Theorem 8.1, thus we can easily prove it by simply adapting the proof of Theorem 8.1. Consequently, according to Theorem 8.1, the item (1) can be considered as a criterion to verify the existence of a change of coordinates for system (8.1).

8.5.2 Unequal Observability Indices In this section, we study the case where the observability indices are not equal, but satisfy the following inequality |ri − r j | ≤ 1 for any 1 ≤ i, j ≤ m. In particular, we assume that Assumption 8.2 For system (8.1), it is assumed that there exists an integer ν ∈ {1, . . . , m − 1} such that its observability indices satisfy the following properties: • r1 = r2 = · · · = rν = ρ1 ; • rν+1 = · · · = rm = ρ2 ; • ρ1 = ρ2 + 1. Under Assumption 8.2, in the following theorem, we study the existence of a change of coordinates for the dynamical system (8.1). Theorem 8.3 Assume that the observability indices of the dynamical system (8.1) satisfy Assumption 8.2. Then the following two assertions are equivalent (A1 ) : The vector fields of frame τ commute, i.e., [τi, j , τk,s ] = 0 for 1 ≤ i ≤ m, 1 ≤ j ≤ ri , 1 ≤ k ≤ m and 1 ≤ s ≤ rk ; (A2 ) : The multi-valued 1-form ω given in (8.17) is closed. Moreover, if the assertion (A1 ) or (A2 ) is fulfilled, then the following assertion is valid. (A3 ) : There exists a local diffeomorphism which transforms (8.1) into the normal form (8.2), whose the outputs are given by 

for 1 ≤ i ≤ ν, y¯i = z i,r j = yi , y¯i = z i,ri = yi + γ j (y1 , . . . , yν ), for ν + 1 ≤ i ≤ m.

(8.20)

8.5 Special Cases

153

Proof Using the same argument in the previous section, it is easy to prove that (A1 ) is equivalent to (A2 ). In the following, we focus on proving that (A3 ) can be deduced from (A2 ). For this, suppose that (A2 ) is fulfilled, then from (8.18), we have φ∗ := dφ = ω, and we can deduce the following change of coordinates z = φ(x) with z = (z 1T , z 2T , . . . , z mT )T and

 T z i = z i,1 , z i,2 , . . . , z i,ri

for 1 ≤ i ≤ m, where dz i, j = ωi, j for 1 ≤ i ≤ m and 1 ≤ j ≤ ri . Now, let us calculate the transformation of the dynamical system (8.1) by using this change of coordinates. To this aim, compute the partial derivatives of φ∗ ( f ) in ∂ the direction of as follows: ∂z i, j   ∂φ∗ ( f ) ∂ ∂ = φ∗ [ f, τi, j ] = ω(τi, j+1 ) = = φ∗ ( f ), ∂z i, j ∂z i, j ∂z i+1 for 1 ≤ i ≤ m and 1 ≤ j ≤ ri − 1. Then, by integration, we obtain φ∗ ( f ) =

m  j=1

⎛

⎞

r j −1

⎝

 k=1

z j,k

∂ ∂z j,k+1

⎠ + β (z) ,

where β is the obtained function of z. Then it is clear that the linear part of φ∗ ( f ) is of the form given in (8.2). Finally, we seek the new outputs by calculating   the Lie derivatives of the outputs h k , for 1 ≤ k ≤ m, along the vector fields τi, j 1≤i≤m, 1≤ j≤ri defined in (8.7)–(8.8). For 1 ≤ k ≤ ν, we obtain 

L τi, j h k = 1, if i = k, j = rk L τi, j h k = 0, otherwise.

Thus, we get the new outputs as y¯k = yk = z k,rk . For ν + 1 ≤ k ≤ m, we obtain ⎧ ⎨ L τi, j h k = 1, if i = k and j = rk L τ h k = 0, if 1 ≤ i ≤ ν and j < ri − 1 ⎩ i, j L τi, j h k = 0, if ν + 1 ≤ i ≤ m and j = k.

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8 Extension to Nonlinear Dynamical Systems …

If 1 ≤ i ≤ ν, then L τi,ri h k might be different to 0. Hence, h k can depend on yi , for 1 ≤ i ≤ ν. Thus, we get the new outputs as y¯k = z k,rk + γ (y1 , . . . , yν ).  Different from the first special case where the observability indices are equal, we can have ϕ different from the identity mapping for the unequal case. This implies that we allow a diffeomorphism on the outputs. However, this does not influence the design of observers, as it contains only known variables. The following example is to highlight the above discussion. Example 8.2 Consider the following dynamical system x˙1 x˙2 x˙3 x˙4 y1

= = = = =

−k1 x1 + k2 x3 sin(Nr x4 ) + u 1 −k1 x2 − k2 x3 cos(Nr x4 ) + u 2 −k4 x3 − k3 x1 sin(Nr x4 ) − k3 x2 cos(Nr x4 ) x3 x4 , y2 = x2 , y3 = x1

(8.21)

which describes the movement of a PM stepper motor [1]. For such a system, a straightforward calculation gives the following observability 1-forms: θ1,1 = d x4 , θ1,2 = d x3 θ2,1 = d x2 , θ3,1 = d x1 . Secondly, based on (8.7)–(8.8), the vector fields forming the frame τ are given as follows: ∂ ∂ x3 ∂ ∂ ∂ ∂ = + k2 sin(Nr x4 ) − k2 cos(Nr x4 ) − k4 ∂ x4 ∂ x1 ∂ x2 ∂ x3 ∂ ∂ = , τ3,1 = . ∂ x2 ∂ x1

τ1,1 = τ1,2 τ2,1

Thirdly, an additional calculation gives ⎛

0 1 0 ⎜1 0 −k 4 Λ = θτ = ⎜ ⎝ 0 −k2 cos(Nr x4 ) 1 0 k2 sin(Nr x4 ) 0

⎞ 0 0⎟ ⎟. 0⎠ 1

We can verify that τ1,1 , τ1,2 , τ2,1 and τ3,1 commute each other with respect to the Lie bracket, i.e., (A1 ) holds. Moreover, the control directions g1 (x) = ∂∂x1 and g2 (x) =

8.5 Special Cases

155

∂ ∂ x2

do not depend on y˙1 , since [τ1,1 , gk ] = 0, for k = 1, 2. Please refer to Corollary 3.1 in Chap. 3 for more details. Consequently, by calculating ω = Λ−1 θ defined in (8.17), we get ⎞ k4 d x 4 + d x 3 ⎟ ⎜ d x4 ⎟ ω=⎜ ⎝ k2 cos(x4 Nr )d x4 + d x2 ⎠ , −k2 sin(x4 Nr )d x4 + d x1 ⎛

which gives the following diffeomorphism: ⎛

⎞ ⎛ ⎞ z 1,1 k4 x 4 + x 3 ⎜ z 1,2 ⎟ ⎜ ⎟ x4 ⎜ ⎟ ⎜ ⎟ ⎝ z 2,1 ⎠ = ⎝ x2 + k2 sin x4 Nr ⎠ . z 3,1 x1 + k2 cos x4 Nr Finally, using Theorem 8.3 we obtain the following form: z˙ 1,1 z˙ 1,2 z˙ 2,1 z˙ 3,1 y¯1 y¯2 y¯3

= = = = = = =

−k3 y1 sin(Nr y3 ) − k3 y2 cos(Nr y3 ) z 1,1 − k4 y3 −k1 y2 + u 2 −k1 y1 + u 1 y1 + Nk2r cos(Nr y3 ) = z 3,1 y2 + Nk2r sin(Nr y3 ) = z 2,1 y3 = z 1,2 .

8.6 Extension to Partial Observer Normal Form with Multiple Outputs This section1 is dedicated to analyzing the partial observer normal form for system (8.1). For this, the following assumption is imposed. Assumption 8.3 For system (8.1), it is assumed that the observability indices {ri }1≤i≤m are constant on the open set X , and satisfy m 

ri = r < n.

i=1

1 Section 8.6 contains excerpts from [8]. © 2020 IEEE. Reprinted, with permission, from Saadi, W.,

Boutat, D., Zheng, G., Sbita, L., Yu, L.: Algorithm to compute nonlinear partial observer normal form with multiple outputs. IEEE Transactions on Automatic Control 65(6), 2700–2707 (2020).

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8 Extension to Nonlinear Dynamical Systems …

Under Assumption 8.3 where the studied system (8.1) is only partially observable, the targeted observer normal form has to be adapted into the following one z˙ i = A Oi z i + βi ( y¯ )

(8.22)

η˙ = ξ(z, η) y¯i = C Oi z i = z i,ri = yi + ϕ(y1 , . . . , yi−1 ),

(8.23) (8.24)

where A Oi and C Oi are respectively the (ri × ri ) and (1 × ri ) well-known Brunovsky forms defined in (8.3). The objective is then to deduce necessary and sufficient geometric conditions that guarantee the existence of a diffeomorphism φ(x) = T T   (z T , η T )T where z = z 1T , . . . , z mT with z i = z i,1 , . . . , z i,ri for 1 ≤ i ≤ m and η = (η1 , . . . , ηn−r )T such that (8.1) might be transformed into (8.24). j−1 For (8.1), as we have presented in Sect. 8.2, we can firstly calculate θi, j = d L f h i , and then construct the corresponding co-distribution Δ as follows:   Δ = span θi, j , for 1 ≤ i ≤ m and 1 ≤ j ≤ ri .

(8.25)

Under Assumption 8.3, the above co-distribution is of dimension r where r = rank =

m 

ri ≤ n,

i=1

which implies that d L kf h j ∈ Δ for any positive integer k. In this case, we can follow the same method when treating partial observer normal form for systems with single output in Chap. 7 to determine the annihilator (or kernel) of Δ as follows: (8.26) Δ⊥ = {X : ν(X ) = 0, for any ν ∈ Δ}, which is a distribution of dimension n − r . In order to construct the change of coordinates for the partially observable case, like what we did for the fully observable case, we need as well to introduce a family of vector fields τ . Here, we follow the same way for the full observable case to build such a frame, i.e., the first m vector fields τ¯i,1 for 1 ≤ i ≤ m are chosen as one solution of the algebraic equations (8.7), and calculate the rest of vector fields by using (8.8).

8.6.1 Properties of Δ and Δ⊥ When treating partial observer normal form for systems with single output, we have stated in Chap. 7 that the co-distribution Δ and the distribution Δ⊥ process certain properties (see Lemma 7.1). The following will investigate similar properties for multiple-output case.

8.6 Extension to Partial Observer Normal Form with Multiple Outputs

157

Lemma 8.1 Consider the co-distribution Δ defined in (8.25) and the distribution Δ⊥ defined in (8.26) for system (8.1), we have the following properties: (1) Δ and Δ⊥ are involutive; (2) Δ and Δ⊥ are invariant with respect to f of (8.1) in the sense that L f ν ∈ Δ, ∀ν ∈ Δ and

[ f, H ] ∈ Δ⊥ , ∀H ∈ Δ⊥ ;

(3) there exist (n − r ) vector fields, noted as {τr +1 , . . . , τn }, that span Δ⊥ and commute, i.e., [τi , τ j ] = 0 for r + 1 ≤ i, j ≤ n. Proof Let us firstly prove the first item of Lemma 8.1. Since the co-distribution Δ defined in (8.25) is spanned by the exact forms θ j,k , therefore it is involutive. Besides, for any two vector fields X, Y ∈ Δ⊥ , according to (8.15), we can then get dθi, j = 0 and θi, j (Y ) = θi, j (X ) = 0 for 1 ≤ i ≤ m and 1 ≤ j ≤ ri . This yields θi, j ([X, Y ]) = 0, which is equivalent to [X, Y ] ∈ Δ⊥ . Hence, Δ⊥ is also involutive, and we proved (1) of Lemma 8.1. Concerning the second item of Lemma 8.1, by applying the well-known Cartan’s identity presented in Chap. 2, we get L f θi, j = ı f dθi, j + dı f θi, j = dı f θi, j because dθi, j = 0. Moreover, since Assumption 8.3 is supposed to be fulfilled, i.e., the partial observability rank condition is satisfied, then we obtain dı f θi, j = dθi, j ( f ) = θi+1 for 1 ≤ j ≤ ri − 1 and

dθi,ri ( f ) = d L rfi +1 θi,ri ∈ Δ.

Thus, we always have the following equation: L f θi, j ∈ Δ.

158

8 Extension to Nonlinear Dynamical Systems …

Now, let us choose H ∈ Δ⊥ . Due to the fact that θi, j for 1 ≤ i ≤ m and 1 ≤ j ≤ ri are closed, by applying (8.15), we can get θi, j ([ f, H ]) = L f θi, j (H ) − L H θi, j ( f ) = −L H θi, j ( f ). Note that the interior product of a function by a vector field is equal to zero, thus i H θi, j ( f ) = 0. Based on this result, and by applying again the Cartan’s identity, we get L H θi, j ( f ) = i H dθi, j ( f ). As dθi, j ( f ) = L f θi, j ∈ Δ⊥ , we obtain θi, j ([ f, H ]) = 0, thus [ f, H ] ∈ Δ⊥ , and we proved (2) of Lemma 8.1. Finally, since Δ⊥ is involutive, it is well-known that we can find (n − r ) vector fields which form a basis {τr +1 , . . . , τn }, such that [τi , τ j ] = 0 for all r + 1 ≤ i, j ≤ n. Therefore, we proved (3) of Lemma 8.1.  Lemma 8.2 For system (8.1) with a family of vector fields τi, j deduced by (8.7)– (8.8) for 1 ≤ i ≤ m and 1 ≤ j ≤ ri , Δ⊥ defined in (8.26) is invariant with respect to τi, j in the sense that (8.27) [τi, j , H ] ∈ Δ⊥ for any H ∈ Δ⊥ . Proof In fact, we need only to prove (8.27) is true for τi,1 with 1 ≤ i ≤ m. To show this, let us calculate τi, j = [τi, j−1 , f ] for 1 ≤ i ≤ m and 1 ≤ j ≤ ri , and then use the well-known Jacobi’s identity (2.15), which enables us to obtain [τi, j , H ] = [[τi, j−1 , f ], H ] = −[[H, τi, j−1 ], f ] − [[ f, H ], τi, j−1 ]. Based on (2) of Lemma 8.1 (i.e., [ f, H ] ∈ Δ⊥ ), it is clear to see that [τi, j , H ] belongs to Δ⊥ if and only if [τi, j−1 , H ] ∈ Δ⊥ . Hence, if we can prove (8.27) is true for τi,1 with 1 ≤ i ≤ m, then it will be satisfied as well for τi, j with 1 ≤ i ≤ m and 2 ≤ j ≤ ri . Thus, the following is to prove (8.27) is true for τi,1 with 1 ≤ i ≤ m. For this, we need to prove   θs,t ( τi,1 , H ) = 0   for 1 ≤ i ≤ m, 1 ≤ s ≤ m and 1 ≤ t ≤ rs , and this is equivalent to τi,1 , H ∈ Δ⊥ . As θs,t for 1 ≤ s ≤ m and 1 ≤ t ≤ rs are closed and θs,t (H ) = 0 for any H ∈ Δ⊥ , then from (8.15) we have   θs,t ( τi,1 , H ) = L H θs,t (τi,1 )

8.6 Extension to Partial Observer Normal Form with Multiple Outputs

159

for which the following two different cases should be distinguished. Case 1: i = 1 According to (8.7), we know that θs,t (τ1,1 ) is a constant (either 0 or 1 for different indices), therefore, its Lie derivative is 0. Thus, we obtain   θs,t ( τ1,1 , H ) = L H θs,t (τ1,1 ) = 0,   which implies that τ1,1 , H ∈ Δ⊥ . Case 2: 2 ≤ i ≤ m For i = 1, the function θs,t (τi,1 ) is not necessary to be constant for all θs,t . In what follows, we will prove the result for i = 2, and the same argument can be used to prove the result for 3 ≤ i ≤ m. According to (8.7), it can be seen that θs,t (τ2,1 ) = 0 for 2 ≤ s ≤ m and 1 ≤ t ≤ rs , except when s = 2 and t = r2 for which θ2,r2 (τ2,1 ) = 1. On the other hand, we have θ1,t (τ2,1 ) = 0 for 1 ≤ t ≤ r2 . Consequently, for all θs,t , we have   θs,t ( τ2,1 , H ) = 0 for any H ∈ Δ⊥ . It remains us to prove that the result is true as well for θ1,r2 +1 , . . . , θ1,r1 . As we have proved in the above that it is always true if θ1,k (τ2,1 ) for r2 + 1 ≤ k ≤ r1 is constant. In the following, we will show that it is true as well even if θ1,k (τ2,1 ) is not constant. For this, for any θ1,k (τ2,1 ) = λ which is not constant, we can rewrite it as follows: θ1,k = λθ2,r2 + ν,

(8.28)

where ν is 1-form such that ν(τ2,1 ) = 0 since θ1,k and θ2,r2 belong to Δ which implies ν ∈ Δ. Therefore, we have  (8.29) ν= qi, j θi, j , which is a combination of some θi, j that vanish on τ¯2,1 and qi, j are functions. As θ2,r2 ([τ2,1 , H ]) = 0 because of (8.15), due to the fact that θ2,r2 (τ2,1 ) = θ2,r2 (H ) = 0, then θ2,r2 ([τ2,1 , H ]) = 0 is equivalent to ν([τ2,1 , H ]) = 0. To show this, consider again (8.15) for ν as follows: dν(τ2,1 , H ) = L τ2,1 ν(H ) − L H ν(τ2,1 ) − ν([τ2,1 , H ]). As ν(H ) = ν(τ2,1 ) = 0, this equation becomes dν(τ2,1 , H ) = −ν([τ2,1 , H ]). On the other hand, by differentiation of (8.29), we obtain

(8.30)

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8 Extension to Nonlinear Dynamical Systems …

dν =



dqi, j ∧ θi, j +



qi, j dθi, j =



dqi, j ∧ θi, j

since dθi, j = 0. Therefore, dν(τ2,1 , H ) =



dqi, j (τ2,1 )θi, j (H ) − dqi, j (H )θi, j (τ2,1 ) = 0

since θi, j (H ) = θi, j (τ2,1 ) = 0. Finally, we obtain θ1,k ([τ2,1 , H ]) = ν([τ2,1 , H ]) = 0. Thus, we proved that [τ2,1 , H ] ∈ Δ⊥ for any H ∈ Δ⊥ .



Based on the above lemma, we can then state the following result. Corollary 8.1 For system (8.1) with a family of vector fields τi, j deduced by (8.7)– (8.8) for 1 ≤ i ≤ m and 1 ≤ j ≤ ri , for any Hi ∈ Δ⊥ and H j ∈ Δ⊥ we have     τi,k + Hi , τ j,s + H j = τi,k , τ j,s modulo Δ⊥ .

(8.31)

Proof Given a family of vector fields τi, j deduced by Eqs. (8.7)–(8.8) for 1 ≤ i ≤ m and 1 ≤ j ≤ ri , let us calculate the following Lie bracket           τi,k + Hi , τ j,s + H j = τi,k , τ j,s + τi,k , H j + Hi , τ j,s + Hi , H j . As we have proved in Lemma 8.1 that Δ⊥ is involutive, therefore 

 Hi , H j ∈ Δ⊥ .

    From Lemma 8.2, we deduced that τi,k , H j and Hi , τ j,s both belong to Δ⊥ , thus we proved the result stated in Corollary 8.1. 

8.6.2 Transformation This section deals with the geometric condition that guarantees the existence of a change of coordinates transforming system (8.1) into the proposed observer normal form (8.22)–(8.24). Theorem 8.4 There exists a local diffeomorphism (z T , η T )T = φ(x) via which (8.1) can be transformed into (8.22)–(8.24) if and only if the following conditions are satisfied (1) there exists a family of vector fields τi,1 for 1 ≤ i ≤ m which are solutions of algebraic equations (8.7), we have 

 τi,k , τ j,s ∈ Δ⊥ ,

(8.32)

8.6 Extension to Partial Observer Normal Form with Multiple Outputs

161

where τi,k and τ j,s are defined by (8.8) for all 1 ≤ i, j ≤ m, 1 ≤ k ≤ ri and 1 ≤ s ≤ rj; (2) for all 1 ≤ i ≤ m, 1 ≤ k ≤ ri we have   τi,k , τ¯l ∈ Δ,

(8.33)

where the vector fields τ¯l , for r + 1 ≤ l ≤ n, are deduced in the item (3) of Lemma 8.1; (3) for 1 < i < j ≤ m and ri > r j we have dh j (τi,r j +k ) = 0 and dh j (τs,rs −(ri +t)+r j +k ) = 0,

(8.34)

with 1 ≤ k ≤ ri − r j , 1 ≤ s ≤ i − 1 and 0 ≤ t ≤ rs − ri . Before to prove Theorem 8.4, the following statements can be made by applying the results of Lemma 8.2 and Corollary 8.1. Remark 8.1 We have the following equivalences: • The items (1), (2) and (3) of Theorem 8.4 are fulfilled by at least one family of vector fields τi, j . • The items (1), (2) and (3) of Theorem 8.4 are fulfilled for the family of vector fields τ¯i, j obtained from any τi, j by removing the components in τ¯l directions.  Especially, this  family fulfills the following commutativity conditions τ¯i,k , τ¯ j,s =  0 and τ¯i,k , τ¯l = 0. Let us show an example to highlight the first two items in Remark 8.1. Example 8.3 Consider the following dynamical system: x˙1 x˙2 x˙3 x˙4 y1

= x3 x1 = x1 = −x3 + x3 x22 = −x4 + x1 x32 + x42 = x2 , y2 = x3 .

Let us firstly calculate the corresponding 1-forms θ1,1 = d x2 θ1,2 = d x1 θ2,1 = d x3 . It is easy to see that the above obtained 1-forms are linearly independent. Hence, according to (8.5), we get   Δ = span θ1,1 , θ1,2 , θ2,1 ,

162

8 Extension to Nonlinear Dynamical Systems …

which is of rank 3. Based on Definition 8.1, we obtain the related observability indices as r1 = 2 and r2 = 1. Moreover, according to (8.26), we get ! ∂ . Δ = span ∂ x4 

⊥

Moreover, by solving (8.7), we can choose the following solution: τ11 = τ12 = τ13 =

∂ ∂ x1 ∂ ∂ x2 ∂ ∂ x3

and it is clear that [τ12 , τ13 ] = − However, if we choose τ13 =

+ x3 ∂∂x1 + x32 ∂∂x4

∂ ∂ − 2x3 ∈ / Δ⊥ . ∂ x1 ∂ x4 ∂ ∂ + x2 , ∂ x3 ∂ x1

then we obtain the following Lie bracket: [τ12 , τ13 ] = −2x3

∂ ∈ Δ⊥ . ∂ x4

As τ¯4 = ∂∂x4 , then (8.33) in Theorem 8.4 is satisfied. In fact, the second item of Remark 8.1 implies that we should take τ12 =

∂ ∂ + x3 modulo Δ⊥ . ∂ x2 ∂ x1

The last condition in Theorem 8.4 is obviously satisfied since we have r1 = 2 and r2 = 1. In summary, the above discussions are to highlight the fact that if we take the following diffeomorphism z 1 = x1 − x3 x2 z 2 = x2 , z 3 = x3 , z 4 = x4 , then we obtain the following partial observable normal form: z˙ 1 = z 2 z 3 (1 − z 22 ) z˙ 2 = z 1 + z 2 z 3 z˙ 3 = −z 3 + z 3 z 22 z˙ 4 = −z 4 + z 1 z 32 + z 3 z 42 + z 42 y1 = z 2 , y2 = z 3 .

8.6 Extension to Partial Observer Normal Form with Multiple Outputs

163

The proof of Theorem 8.4 is based on the following lemma. Lemma 8.3 For a family of vector fields τ1, j satisfying (8.7) and the associated τi, j for 2 ≤ j ≤ ri deduced from (8.8), there exists a family of vector fields τ¯i, j = τi, j modulo Δ⊥   such that τ¯i, j , τ¯s,t = 0 for 1 ≤ j ≤ ri and 1 ≤ t ≤ rs , if and only if [τi, j , τs,t ] ∈ Δ⊥ . Proof Necessity: Suppose that there exist Hi, j ∈ Δ⊥ and Hs,t ∈ Δ⊥ , which determine τ¯i, j = τi, j + Hi, j and τ¯s,t = τs,t + Hs,t for 1 ≤ j ≤ ri and 1 ≤ t ≤ rs such that  τ¯i, j , τ¯s,t = 0, then we have   0 = τ¯i, j , τ¯s,t = [τi, j , τs,t ] modulo Δ⊥ , which implies that [τi, j , τs,t ] ∈ Δ⊥ . Sufficiency: Assume that for one choice of the family of vector fields τ1, j satisfying (8.7), we have [τi, j , τs,t ] ∈ Δ⊥ for 1 ≤ j ≤ ri and 1 ≤ t ≤ rs defined in (8.8). Then we need to prove that there exist Hi, j ∈ Δ⊥ , Hs,t ∈ Δ⊥ for 1 ≤ j ≤ ri and 1 ≤ t ≤ rs such that [τ¯i, j , τ¯s,t ] = 0 where τ¯i, j = τi, j + Hi, j and τ¯s,t = τs,t + Hs,t . Let {τ¯r +1 , . . . , τ¯n } be (n − r ) determined commutative basis of Δ⊥ , it is well known that there exist r commutative vector fields, noted as {X 1 , . . . , X r }, such that they commute with {τ¯r +1 , . . . , τ¯n }, thus form together a basis of the tangent fiber bundle T X of X . Therefore, we can rewrite τi, j as τi, j =

r 

i, j

γl (x)X l +

n 

i, j

μl (x)τ¯l ,

l=r +1

l=1

where functions γl (x) are constant on the leaves of Δ⊥ , thus L τ¯i γl = 0 because of the invariance of Δ⊥ by means of τi, j thanks to Lemma 8.2, i.e., [τi, j , τ¯i ] ∈ Δ⊥ . By choosing n  i, j Hi, j = − μl (x)τ¯l ∈ Δ⊥ , i, j

i, j

l=r +1

we have τ¯i, j = τi, j + Hi, j =

r 

i, j

γl (x)X l .

l=1

For the same reason, there exists Hs,t =

n

s,t l=r +1 κl (x)τ¯l

∈ Δ⊥ yielding

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8 Extension to Nonlinear Dynamical Systems …

τ¯s,t = τs,t + Hs,t =

r 

α j,l (x)X l .

l=1

Moreover, for 1 ≤ j ≤ ri and 1 ≤ s ≤ rt , due to the invariance of Δ⊥ by means of τi, j , we have [τi, j , τs,t ] = [τ¯i, j , τ¯s,t ] modulo Δ⊥ . (8.35) i, j

As [X k , X s ] = 0 for 1 ≤ k, s ≤ r and the fact that functions γl on the leaves of Δ⊥ , then (8.35) becomes " [τi , τ j ] =

r 

l,m=1

i, j γl L X l αms,t X s

r 

−

m,l=1

i, j

and αl are constant

# modulo Δ⊥ .

i, j αls,t L X m γl X l

Thus, if [τi, j , τs,t ] ∈ Δ⊥ , then there exist Hi, j ∈ Δ⊥ and Hs,t ∈ Δ⊥ defined above, such that [τ¯i, j , τ¯s,t ] =

r 

i, j

γl L X l αms,t X s −

l,m=1

r 

i, j

αls,t L X m γl X l = 0,

m,l=1

which is due to the fact that X k ∈ / Δ⊥ .



Using the result stated in Lemma 8.3, the proof of Theorem 8.4 is detailed in the following. Proof (of Theorem 8.4.) Necessity: Suppose that there exists a local change of coordinates (z T , η T )T = φ(x) which transforms (8.1) into (8.22)–(8.24). Note ω = dφ(x), then ω is the pullback of dz and dη for (8.22)–(8.23). Therefore, we need only to check the necessity for (8.22)–(8.23). It is clear that, for (8.22)–(8.23), there exist ∂ τ¯i,k = ∂z i,k for 1 ≤ i ≤ m and 1 ≤ k ≤ ri and τ¯ j =

∂ ∂η j

for 1 ≤ j ≤ n − r such that all conditions in Theorem 8.4 are satisfied for (8.22)– (8.23). Then, according to the property of pullback, we can conclude that those conditions are necessary for (8.1). Sufficiency: Suppose that all conditions in Theorem 8.4 are satisfied. Then we can construct the following multi-valued matrix Λ

8.6 Extension to Partial Observer Normal Form with Multiple Outputs

⎛

θ1,1 (τ1,1 ) · · · θ1,1 (τ1,r1 ) ⎜ .. .. .. ⎜ . . . ⎜ ⎜ θ1,r1 (τ1,1 ) · · · θ1,r1 (τ1,r1 ) Λ=⎜ ⎜ θ2,1 (τ1,1 ) · · · θ2,1 (τ1,r1 ) ⎜ ⎜ .. .. .. ⎝ . . . θm,rm (τ1,1 ) · · · θm,rm (τ1,r1 )

165

⎞ θ1,1 (τ2,1 ) · · · θ1,1 (τm,rm ) ⎟ .. .. .. ⎟ . . . ⎟ θ1,r1 (τ2,1 ) · · · θ1,r1 (τm,rm ) ⎟ ⎟. θ2,1 (τ2,1 ) · · · θ2,1 (τm,rm ) ⎟ ⎟ ⎟ .. .. .. ⎠ . . . θm,rm (τ2,1 ) · · · θm,rm (τm,rm )

As the 1-forms θi j are independent, so are the vector fields. As τst ∈ / Δ⊥ where Δ⊥ is the kernel of θi j , then Λ is invertible. Thus, we can define r differential 1-forms as follows: ⎛ ⎞ ⎛ ⎞ ω1,1 θ1,1 ⎜ .. ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ω1,r1 ⎟ ⎜ ⎟ −1 ⎜ θ1,r1 ⎟ ⎜ ⎟ =Λ ⎜ (8.36) ω=⎜ ⎟ ⎟. ω θ ⎜ 2,1 ⎟ ⎜ 2,1 ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎝ . ⎠ ⎝ . ⎠ ωm,rm θm,rm From the above formula, we see that 1-forms ωi, j are linear combinations of 1-forms θi, j . Thus, they span the same co-distribution Δ and have the annihilator as Δ⊥ . Therefore, they are well defined from any family τi,1 modulo Δ⊥ . By the definition we have ωi, j (τs,t ) = δis δ tj , where δis is the Kronecker delta, therefore if the condition (8.32) of Theorem 8.4 is fulfilled, then we can find n − r differential 1-forms ω j for 1 ≤ j ≤ n − r such that j

• ω j (τ¯k ) = δk for r + 1 ≤ k, j ≤ n; • ω j (τs,t ) = 0 thus the annihilator of these 1-forms is spanned by τs,t . Note that, by using (8.15), the above 1-forms ω j together with ωi,k are independent and closed. Then, according to the construction procedure, we have ω j (τ¯ j ) = and ωi,k (τi,k ) =

∂ ∂η j ∂ . ∂z i,k

Therefore, thanks to Poincaré’s Lemma, there exists a change of coordinates φ(x) = (z T , η T )T such that the differential of φ is φ∗ = ω. Let us show how the vector field f in the dynamical system (8.1) is transformed by φ. To do so, we compute ∂η∂ j φ∗ ( f ), and we obtain

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8 Extension to Nonlinear Dynamical Systems …

  ∂ ∂ φ∗ ( f ) = , φ∗ ( f ) . ∂η j ∂η j Then we have

  ∂ φ∗ ( f ) = φ∗ ( τ¯ j , f ) ∈ φ∗ (Δ⊥ ) ∂η j

because τ¯ j ∈ Δ⊥ and Δ⊥ is f -invariant according to Lemma 8.1. Therefore we get  ! ∂ ∂ φ∗ ( f ) ∈ span , for r + 1 ≤ k ≤ n . ∂η j ∂ηk Now we compute

∂ ∂z i, j

∂ ∂z i, j

φ∗ ( f ) for 1 ≤ i ≤ m and 1 ≤ j ≤ ri − 1, and we obtain

  φ∗ ( f ) = φ∗ ( τi, j , f ) = φ∗ (τi, j+1 + H ) = φ∗ (τi, j+1 ) + φ∗ (H ),

where H ∈ Δ⊥ which gives ∂ ∂ φ∗ ( f ) = modulo Δ⊥ . ∂z i, j ∂z i, j+1 Therefore, by integration, we have φ∗ ( f ) =

m  j=1

⎛

⎞

r j −1

⎝

 k=1

z j,k

∂ ∂z j,k+1

  ⎠ + β zr1 , . . . , zrm modulo Δ⊥ .

  It rests to find the relation between zr1 , . . . , zrm and the outputs y1 , . . . , ym . For this, it should be noted that from (8.34), together with (8.7) for 1 ≤ i, j ≤ m and ri > r j + 1, we have d L kf h j (τi,1 ) = 0 for 0 ≤ k ≤ ri − 1. Now, using (8.15) for ν = d L kf h j , X = f and Y = τi,1 , we obtain 0 = L f d L kf h j (τi,1 ) − L τ¯i,1 d L kf h j ( f ) − d L kf h j [ f, τi,1 ]. If d L kf h j (τi,1 ) = 0, then we obtain k d L k+1 f h j (τi,1 ) = d L f h j (τi,2 ).

Therefore, by induction, we deduce that dh j (τi,ri −1 ) = · · · = dh j (τi,1 ) = 0.

8.6 Extension to Partial Observer Normal Form with Multiple Outputs

167

Thus, the output h j does not depend on y˙i , . . . , yi(ri −1) . However, dh j (τi,ri ) may not be equal to 0. Therefore, by integration, we obtain y j = zr j + ϕ j (zr1 , . . . , zr j−1 ). As zr1 = y1 , zr2 = y2 + ϕ(y1 ) and so forth zrm = ym + ϕ(y1 , . . . , ym−1 ), finally, we obtain (8.24).  In order to highlight the proposed result, let us consider an example. Example 8.4 Consider the following nonlinear system: x˙1 x˙2 x˙4 x˙5 x˙6 y1

= x2 x5 + x3 x4 − x33 + x3 x5 = x1 , x˙3 = x2 = −2x3 x4 + 2x33 − 2x2 x5 = x4 − x32 = −x6 + x2 x5 = x3 , y2 = x5 .

(8.37)

According to the presented scheme, let us firstly calculate the corresponding 1-forms θ1,1 = d x3 , θ1,2 = d x2 θ1,3 = d x1 , θ2,1 = d x5 θ2,2 = d x4 − x3 d x3 . It is easy to see that the above obtained 1-forms are linearly independent. Hence, according to (8.5), we get   Δ = span θ1,1 , θ1,2 , θ1,3 , θ2,1 , θ2,2 , which is of rank 5. Based on Definition 8.1, we obtain the related observability indices as r1 = 3 and r2 = 2. Moreover, according to (8.26) we get Δ⊥ = span {τ6 } , where τ6 =

∂ . ∂ x6

Based on the proposed method, we solve (8.7) to get one following solution τ¯1,1 =

∂ ∂ and τ¯2,1 = ∂ x1 ∂ x4

based on which we can calculate the other vector fields as follows:

168

8 Extension to Nonlinear Dynamical Systems … ∂ ∂ x2 ∂ ∂ x3 ∂ ∂ x5

τ 1,2 = τ 1,3 = τ 2,2 =

+ x5 ∂∂x1 − 2x5 ∂∂x4 + x5 ∂∂x6 + x3 ∂∂x1 − 2x3 ∂∂x4 .

Then we can check that [τ¯i, j , τ¯k,s ] = 0 except 

 ∂ ∈ Δ⊥ . τ¯1,3 , τ¯2,2 = ∂ x6

However, if we choose the following vector fields τi, j = τ¯i, j for j = 3 τ1,3 = τ¯1,3 − x5 τ6 , then all vector fields commute, i.e., [τi, j , τk,l ] = 0, which enables us to compute the change of coordinates. For this, let us firstly calculate ⎛

0 ⎜0 ⎜ Λ=⎜ ⎜1 ⎝0 0

0 1 1 0 0 x5 0 0 0 −2x5 − 2x3

⎞ 0 0 0 0 ⎟ ⎟ 0 x3 ⎟ ⎟ 0 1 ⎠ 1 −2x3

based on which we get dz 1,1 = ω1,1 = −x5 θ1,1 + θ1,3 − x3 d x5 = d (x1 − x3 x5 ) dz 1,2 = ω1,2 = d x2 dz 1,3 = ω1,3 = d x3 dz 2,1 = ω2,1 = (2x3 + 2x5 ) θ1,1 + 2x3 θ2,1 + θ2,2 = d(x4 + 2x3 x5 ) z 2,2 = ω2,2 = d x5 . Moreover, as we have chosen τ6 = take

∂ , ∂ x6

then for the complementary basis, we might

η = x6 . Finally, we obtain the following diffeomorphism:

8.6 Extension to Partial Observer Normal Form with Multiple Outputs

169

z 1,1 = x1 − x3 x5 z 1,2 = x2 , z 1,3 = x3 z 2,1 = x4 + 2x3 x5 z 2,2 = x5 , η = x6 , with which the studied system can be transformed into the following normal form: z˙ 1,1 = y12 + y1 y2 z˙ 1,2 = z 1,1 + y1 y2 z 1,3 = z 1,2 , z˙ 2,1 = 2y13 z˙ 2,2 = z 2,1 − 2y1 y2 − y12 η˙ = −η + z 1,2 z 2,2 y1 = z 1,3 , y2 = z 2,2 .

8.7 Observer Design 8.7.1 Reduced-Order Luenberger-Like Observer The concept of reduced-order observer was firstly introduced in [7], which needs only estimate unmeasurable states of the studied system. It has been generalized for nonlinear dynamical systems by imposing the Lipschitz conditions for nonlinear terms [9, 11] and invariant manifold [5]. Besides, such a technique has been used to design observer for descriptor system [4] and chaotic synchronization system [12]. To design an observer for the normal form (8.2) or (8.22)–(8.24), we consider in this chapter a reduced-ordered observer. For this, let us decompose the observable state z of the normal form (8.2) or (8.22)–(8.24) by applying the following linear transformation     z z = = T z, z y with which (8.2) or (8.22)–(8.24) can be then decomposed as z˙ = A11 z + A12 z + β¯1 ( y¯ ) z˙ = A21 z + A22 z + β¯2 ( y¯ ) y¯ = z,

(8.38)

where z ∈ Rn−m and z ∈ Rm , and 

A11 A12 A21 A22



= T A O T −1 , [0, I ] = C O T −1 ,



 β¯1 ( y¯ ) = Tβ ( y¯ ) . β¯2 ( y¯ )

170

8 Extension to Nonlinear Dynamical Systems …

Since the pair (A O , C O ) in (8.2) or (8.22)–(8.24) is observable, i.e.,  rank

λI − A O CO

 = n, ∀λ ∈ C.

Due to the fact that T is invertible, thus     λI − A O λI − T A O T −1 rank = rank CO C O T −1 ⎡ ⎤ λI − A11 −A12 = rank ⎣ −A21 λI − A22 ⎦ 0 ⎡ ⎤I λI − A11 0 = rank ⎣ −A21 0 ⎦ I  0 λI − A11 . = m + rank A21 Therefore, we obtain  rank

λI − A11 A21

 = n − m, ∀λ ∈ C,

which implies that the observability of (A O , C O ) ensures that (A11 , A21 ) in (8.38) is observable as well. Consequently, there exists a vector K such that (A11 − K A21 ) is Hurwitz. Consider now the following dynamics: ζ˙ = (A11 − K A21 )ζ + β¯1 ( y¯ ) + (A11 K + A12 − K A22 − K A21 K ) y¯ − K β¯2 ( y¯ ) zˆ = ζ + K y¯ . (8.39) Note e = z − zˆ , then according to (8.38) and (8.39), we obtain e˙ = = = =

z˙ − z˙ˆ A11 z + A12 z + β¯1 ( y¯ ) − ζ˙ − K z˙ A11 z + (A11 − K A21 )ˆz − K A21 z (A11 − K A21 )e.

Due to the fact that there exists a K such that (A11 − K A21 ) is Hurwitz, therefore, e is asymptotically stable, and we have lim zˆ (t) = z(t).

t→+∞

In summary, the dynamics (8.39) is an asymptotic reduced-order observer of (8.2) or (8.22)–(8.24).

8.7 Observer Design

171

8.7.2 Design Procedure The following summarizes the complete procedure when applying the proposed method for a given multi-output nonlinear dynamical system of the form (8.1): Step 1: Determine the observability indices ri for 1 ≤ i ≤ m and compute the associated observability 1-forms θi, j for 1 ≤ i ≤ m and 1 ≤ j ≤ ri ; Step 2: Determine Δ, Δ⊥ and seek a commutative basis {τ¯r +1 , . . . , τ¯n } that spans Δ⊥ ; Step 3: Choose any solution τi,1 of (8.7) for 1 ≤ i ≤ m, and eliminate all its terms modulo Δ⊥ which yields τ¯i,1 ; Step 4: Iteratively, for 2 ≤ j ≤ ri , compute [τ¯i, j , f ] and eliminate all its terms modulo Δ⊥ which yields τ¯i, j ; Step 5: Calculate the multi-valued matrix Λ and the associated r differential 1-forms ω; Step 6: Integrating ω yields z = φ1 (x), and then we can choose η = φ2 (x), complementary to φ1 (x) such that φ(x) = [φ1T (x), φ2T (x)]T forms a change of coordinates; Step 7: For (8.1), applying φ(x) = (z T , η T )T to obtain the normal form (8.22)– (8.24), and then design the proposed reduced-order Luenberger-like observer (8.39) to estimate z of (8.2) or (8.22)–(8.24).

References 1. Bodson, M., Chiasson, J., Novotnak, R.: High-performance induction motor control via inputoutput linearization. IEEE Control Syst. Mag. 14(4), 25–33 (1994) 2. Boutat, D., Liu, D.Y.: Observer design for a class of non-linear systems with linearisable error dynamics. IET Control Theory Appl. 9(15), 2298–2304 (2015) 3. Boutat, D., Zheng, G., Barbot, J.P., Hammouri, H.: Observer error linearization multi-output depending. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 5394– 5399 (2006) 4. Darouach, M., Zasadzinski, M., Hayar, M.: Reduced-order observer design for descriptor systems with unknown inputs. IEEE Trans. Autom. Control 41(7), 1068–1072 (1996) 5. Karagiannis, D., Carnevale, D., Astolfi, A.: Invariant manifold based reduced-order observer design for nonlinear system. IEEE Trans. Autom. Control 53(11), 2602–2614 (2008) 6. Krener, A., Respondek, W.: Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim. 23(2), 197–216 (1985) 7. Luenberger, D.: An introduction to observers. IEEE Trans. Autom. Control 16(6), 596–602 (1971) 8. Saadi, W., Boutat, D., Zheng, G., Sbita, L., Yu, L.: Algorithm to compute nonlinear partial observer normal form with multiple outputs. IEEE Trans. Autom. Control 65(6), 2700–2707 (2020)

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9. Sundarpandian, V.: Reduced order observer design for nonlinear systems. Appl. Math. Lett. 19(9), 936–941 (2006) 10. Xia, X.H., Gao, W.B.: Nonlinear observer design by observer error linearization. SIAM J. Control Optim. 27(1), 199–216 (1989) 11. Xu, M.: Reduced-order observer design for one-sided Lipschitz nonlinear systems. IMA J. Math. Control Inf. 26(3), 299–317 (2009) 12. Zheng, G., Boutat, D.: Synchronisation of chaotic systems via reduced observers. IET Control Theory Appl. 5(2), 308–314 (2011) 13. Zheng, G., Boutat, D., Barbot, J.P.: Multi-output dependent observability normal form. Nonlinear Anal.: Theory Methods Appl. 70(1), 404–418 (2009)

Chapter 9

Extension to Nonlinear Singular Dynamical Systems

Abstract This chapter aims at extending the differential geometric method to design observers for nonlinear singular dynamical systems. Singular systems widely exist in engineering systems, such as chemical system, biological system, electrical circuit and so on. These systems are governed by mixing differential and algebraic equations, which is the key difference with respect to regular systems [5, 6]. Due to this reason, many well-defined concepts relative to observation problem for regular (non-singular) systems have to be reconsidered for singular ones. For such systems, observability has been analyzed in [3, 10], and different types of observers have been proposed in [4, 9, 11]. In this chapter, we will show how to apply differential geometric method to design observers for such a nonlinear singular system [12]. The basic idea is to regularize the studied singular system into a nonlinear regular system with the injection of the output derivative, then seek a diffeomorphism to transform the regularized system into an observer normal form with output derivative injection, based on which a Luenberger-like observer is designed.

9.1 Problem Statement Consider the following multi-output nonlinear singular dynamical system E x˙ = f (x) y = h(x),

(9.1)

where x ∈ X ⊆ Rn is the state vector, X is an open domain of Rn , f : X → Rn and h = [h 1 , . . . , h m ]T : X → Rm are both assumed to be smooth, with E ∈ Rn×n being singular, i.e., rankE < n. Without loss of generality, it is assumed that 0 ∈ X , f (0) = 0 and h(0) = 0. Note that if the matrix E has full row rank, i.e., rankE = n, then system (9.1) is equivalent to x˙ = E −1 f (x) y = h(x), © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9_9

173

174

9 Extension to Nonlinear Singular Dynamical Systems

which is of the general form (8.1), and we can apply the results developed in Chap. 8 to seek the desired change of coordinates. Note that singular system contains not only differential equations, but also algebraic ones [1, 2]. Since differential geometric approach has been developed for dynamical system governed by an ordinary differential equation, we are wondering whether it is still possible to apply this approach to seek a diffeomorphism for the singular systems, with which system (9.1) can be transformed into a simpler observer normal form? In order to answer this natural question, the following example presents an idea of how to treat the singular case. Example 9.1 Consider the following nonlinear singular system: ⎤ 1 ⎤⎡ ⎤ ⎡ x˙1 010 x1 − 23 x32 + 23 x32 x2 3 ⎥ ⎣ 0 0 1 ⎦ ⎣ x˙2 ⎦ = ⎢ ⎣ x2 − x32 ⎦ . 000 x˙3 x +x ⎡

y = x1

3

(9.2)

1

⎡

⎤ 010 It is clear that E = ⎣ 0 0 1 ⎦ is singular, and this explains as well that the dynamics 000 x˙1 is missing. By applying the following change of coordinates, ⎤ ⎡ 3 ⎤ z1 x2 − x32 ⎦, ⎣ z2 ⎦ = ⎣ x 3 z3 x1 ⎡

it can be checked that the studied system (9.2) can be transformed into the following form: z˙ 1 = y z˙ 2 = z 1 (9.3) z˙ 3 = z 2 + y + y˙ y = z3. In fact, the above form can be written in a more compact one as z˙ = A O z + β(y, y˙ ) y = C O z, which can be seen as a generalization of the observer normal form with output injection (3.3). Since this general form contains both y and y˙ , therefore we call it the observer normal form with output derivative injection.

9.1 Problem Statement

175

This example reveals at least two important facts: (1) Even no dynamics has been defined for x1 by a differential equation, if certain properties are satisfied, it is possible to regularize this singular system into a regular one. For example, in (9.3), the third equation is in fact an algebraic equation, but it has been regularized by using y˙ ; (2) Due to the effect of regularization, the transformed normal form will contain both y and y˙ .  From the above analysis, for the nonlinear singular system (9.1), this chapter investigates how to seek a diffeomorphism z = φ(x) with z = (z 1T , z 2T , . . . , z mT¯ )T and T 

m¯ ri = n, via which system (9.1) z i = z i,1 , z i,2 , . . . , z i,ri , for 1 ≤ i ≤ m¯ with i=1 can be transformed into the following observer normal form with output derivative injection z˙ i = A Oi z i + αi ( y¯ ) + βi ( y¯ ) y˙¯ , for 1 ≤ i ≤ m¯ (9.4) y¯i = C Oi z i = z i,ri with A Oi and C Oi being of Brunovsky form ⎛

A Oi

0 ⎜1 ⎜ ⎜ = ⎜ ... ⎜ ⎝0 0

⎞ ··· 0 0 0 ··· 0 0 0⎟ ⎟ . ⎟ , C = (0, . . . , 0, 1) .. .. . . · · · .. ⎟ ⎟ Oi ··· 1 0 0⎠ ··· 0 1 0

and αi and βi being, respectively, ri and ri × m¯ matrix.

9.2 Transformation into Regular System Since the objective of this book focuses on the observer design by applying differential geometric method, before presenting the procedure to transform nonlinear singular system (9.1) into a regular one, it is worth understanding what is the condition to design an observer for linear singular system. For this, we consider the following linear singular system without input ¯ E x˙ = Ax y = Cx

(9.5)

with x ∈ Rn , rankE < n. In [8], it has been stated that there exists a simple Luenberger-like observer

176

9 Extension to Nonlinear Singular Dynamical Systems

ξ˙ = N ξ + L y xˆ = ξ + K y

(9.6)

with properly chosen constant matrices N , K and L, if the following two conditions: ⎡

⎤ E A rank ⎣ 0 E ⎦ = n + rank E 0 C 

and rank

 sE − A = n, ∀s ∈ C, Re(s) ≥ 0 C

(9.7)

(9.8)

are satisfied. For the purpose of generalizing the above two conditions for nonlinear singular system (9.1), we can impose the following similar condition. Assumption 9.1 It is assumed that, for E, f (x) and h(x) defined in (9.1), the following condition: ⎡ ∂ f (x) ⎤ E ∂x rank ⎣ 0 E ⎦ = n + rank E (9.9) 0 ∂h(x) ∂x is satisfied. It is worth highlighting that (9.9) in Assumption 9.1 is an extension of (9.7) to deal with nonlinear singular case. Indeed, if we set f (x) = Ax and h(x) = C x, then (9.9) is equivalent to (9.7), since ⎡

E rank ⎣ 0 0

⎤ E A E ⎦ = rank ⎣ 0 E ⎦ = n + rank E, ∂h(x) 0 C ∂x

∂ f (x) ∂x

⎤

⎡

which has been stated as the sufficient and necessary observability condition for linear singular systems. Note rank E = q < n, therefore we can always find two elementary matrices with appropriate dimension, noted as T and S, such that  SET =

 Iq 0 . 0 0

(9.10)

Hence, by introducing the new change of variables ζ = T −1 x, (9.1) can be written as S E T ζ˙ = S f (T ζ ) y = h(T ζ )

9.2 Transformation into Regular System

or equivalently

177

E¯ ζ˙ = f˜(ζ ) ˜ ), y = h(ζ

(9.11)

 Iq 0 ˜ ) = h(T ζ ). Then the above system can , f˜(ζ ) = S f (T ζ ) and h(ζ 0 0 be decomposed into the following two subsystems: where E¯ =



ζ˙1 = f˜1 (ζ1 , ζ2 ) 0 = f˜2 (ζ1 , ζ2 ) ˜ 1 , ζ2 ), y = h(ζ  with ζ =

  ζ1 , ζ1 ∈ Rq , ζ2 ∈ Rn−q , f˜ = ζ2

(9.12)

 f˜1 , and we have the following result. f˜2

Lemma 9.1 The following condition  rank

∂ f˜2 (ζ ) ∂ζ2 ˜ ) ∂ h(ζ ∂ζ2

 =n−q

(9.13)

is true if and only if (9.9) is satisfied. Proof Due to the fact that S and T in (9.10) are both invertible, therefore, we obtain ⎡

E rank ⎣ 0 0

⎤ ⎡ ∂ f (x) ⎤   S0 0 E ∂x T 0 ⎦ ⎣ ⎦ ⎣ ⎦ E 0 E = rank 0 S 0 0 T ∂h(x) 0 0 I 0 ∂h(x) m ∂x ∂x ⎤ ⎡ T S E T S ∂ f∂(x) x SET ⎦ . = rank ⎣ 0 ∂h(x) T 0 ∂x

∂ f (x) ∂x

⎤

⎡

According to the definition x = T ζ , we get S and

Hence, we have

∂ f˜(ζ ) ∂ f (x) T = ∂x ∂ζ

˜ ) ∂ h(ζ ∂h(x) T = . ∂x ∂ζ

178

9 Extension to Nonlinear Singular Dynamical Systems

⎤ f˜(ζ ) S E T ∂ ∂ζ ⎥ ⎢ E ⎦ = rank ⎣ 0 S E T ⎦ ∂h(x) ˜ ) 0 ∂ h(ζ ∂x ∂ζ ⎡ ⎤ ˜ ) ∂ f˜1 (ζ ) Iq 0 ∂ f∂ζ1 (ζ ∂ζ 1 2 ⎢ ⎥ ⎢ 0 0 ∂ f˜2 (ζ ) ∂ f˜2 (ζ ) ⎥ ⎢ ∂ζ1 ∂ζ2 ⎥ = rank ⎢ 0 ⎥ ⎢ 0 0 Iq ⎥ ⎢0 0 0 0 ⎥ ⎣ ⎦ ˜ ) ∂ h(ζ ˜ ) h(ζ 0 0 ∂ ∂ζ  ˜ 1 ∂ζ2

⎡

∂ f (x) ∂x

E rank ⎣ 0 0

⎡

⎤

= 2q + rank

∂ f 2 (ζ ) ∂ζ2 ˜ ) ∂ h(ζ ∂ζ2

 Consequently, the condition (9.9) is equivalent to rank

. ∂ f˜2 (ζ ) ∂ζ2 ˜ ) ∂ h(ζ ∂ζ2

 = n − q.



In the following, we show how to transform the studied nonlinear singular system (9.1) into the associated nonlinear regular one and such a procedure is called regularization.   ∂ f˜2 (ζ )

2 It is clear that if rank ∂∂ζ = n − q, then we can find an invertible matrix ˜ ) h(ζ ∂ζ 2   Q 1 (ζ ) ∈ R(m+n−q)×(m+n−q) which yields Q(ζ ) = Q 2 (ζ )

 Q(ζ )  and Q 1 (ζ )

∂ f˜2 (ζ ) ∂ζ2 ˜ ) ∂ h(ζ ∂ζ2

∂ f˜2 (ζ ) ∂ζ2 ˜ ) ∂ h(ζ ∂ζ2



 =

In−q 0m

 (9.14)

 = In−q . Now let us calculate the derivative of the algebraic

equation in (9.12), which gives  ˜   ˜     ∂ f˜2 (ζ )  ∂ f 2 (ζ ) ∂ f 2 (ζ ) 0 ∂ζ ∂ζ1 ∂ζ2 ˜ ˙ (ζ ) + = ∂ h(ζ ζ = f ζ˙2 . 1 ˜ ) ˜ ) ˜ ) ∂ h(ζ ∂ h(ζ y˙ ∂ζ

∂ζ1

∂ζ2

For the above equation, by applying (9.14), we can obtain  ζ˙2 = −Q 1 (ζ )

∂ f˜2 (ζ ) ∂ζ1 ˜ ) ∂ h(ζ ∂ζ1



  0 f˜1 (ζ ) + Q 1 (ζ ) . y˙

This implies that we can regularize the algebraic equation in (9.12) by adding the obtained artificial dynamics, and this leads to the following regular system with

9.2 Transformation into Regular System

output derivatives



with

179

ζ˙ = f¯(ζ ) + g(ζ ) y˙ ¯ ) y¯ = h(ζ

⎡ ⎢ f¯(x) = ⎣

f˜ (ζ ) 1 ˜ 

 g(ζ ) = and

∂ f 2 (ζ ) ∂ζ1 ˜ ) ∂ h(ζ ∂ζ1

−Q 1 (ζ )

0 Q 1 (ζ )

(9.15) ⎤ ⎥ ⎦ f˜1 (ζ )

(9.16)



    0 f˜2 (ζ ) ¯ y¯ = , h(ζ ) = ˜ . y h(ζ )

(9.17)

(9.18)

We would like to highlight that y¯ (which can be regarded as a new output) contains two parts: measurable variable y and a deduced algebraic equation from (9.1). With the obtained new output y¯ , for the sake of simplicity and without loss of generality, based on the regularized form (9.15), we can assume that the first m¯ outputs are linearly independent with m¯ ≤ n + m − q, i.e., {d h¯ 1 , . . . , d h¯ m¯ } are linearly independent. If this assumption is not satisfied, we can then perform certain algebraic manipulations to remove the dependent outputs and reorder them. To investigate the observability of the regularized system, the following assumption is needed. Assumption 9.2 For the regularized system (9.15), it is assumed that there exist ri m¯ for 1 ≤ i ≤ m¯ such that i=1 ri = n and ⎡

d h¯ 1 .. .

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ d L ri −1 h¯ 1 ⎥ ¯ ⎥ ⎢ f ⎥ ⎢ . ⎥ = n, ⎢ .. rank ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ d h¯ m¯ ⎥ ⎢ .. ⎥ ⎢ ⎦ ⎣ . rm¯ −1 ¯ d L f¯ h m¯

(9.19)

where L kf¯ h¯ i for 1 ≤ k ≤ ri . Clearly, (9.19) implies that the regularized system is locally observable. Moreover, we would like to show that the condition (9.19) is a natural generalization of the condition (9.8) when studying nonlinear singular systems.

180

9 Extension to Nonlinear Singular Dynamical Systems

Proposition 9.1 Consider system (9.1) with f (x) = Ax and h(x) = C x. If Assumption 9.1 is fulfilled, then (9.19) implies (9.8) . Proof For system (9.1) with f (x) = Ax and h(x) = C x, we can always find two invertible matrices T and S such that (9.1) might be transformed via the diffeomorphism x = T ζ into the following system: ˜ E¯ ζ˙ = Aζ ˜ y = Cζ,

(9.20)

  I 0 ˜ , A = S AT and C˜ = C T . The sake of simplicity denote where E¯ = S E T = q 0 0     A˜ A˜ 1 ¯ ˜ ¯ E q = [Iq , 0], A = ˜ and C = ˜2 , then (9.20) is equivalent to A2 C E¯ q ζ˙ = ζ˙1 = A˜ 1 ζ ¯ y¯ = Cζ,   0 . where ζ1 ∈ R and y¯ = y Now let us rewrite C¯ as

(9.21)

q

  C¯ = C¯ 1 , C¯ 2 .

Then, based on the result stated in Lemma 9.1, if Assumption 9.1 is fulfilled, we can obtain rank C¯ 2 = n − q. Therefore, we can find an invertible matrix Q 1 such that Q 1 C¯ 2 = In−q . Besides, due to the fact that 

E¯ rank ¯q C





I 0 = rank ¯q ¯ C1 C2 

then there exists an invertible matrix P =  P

E¯ q C¯

 = n,

 Iq 0 such that −Q 1 C¯ 1 Q 1

 = In .

For system (9.21), let us consider its output derivative which can be written as E¯ q ζ˙ = A˜ 1 ζ C¯ ζ˙ = y˙¯ .

9.2 Transformation into Regular System

181

 E¯ q = In , then the above dynamics can be transformed, by C¯ multiplying both side the invertible matrix P, into the following normal form with output derivative injection   ¯ +P 0 ζ˙ = Aζ y˙ ¯ y¯ = Cζ, 

Note that we have P

where A¯ =



 A˜ 1 . −Q 1 C¯ 1 A˜ 1

Therefore, for system (9.1) with f (x) = Ax and h(x) = C x, the condition (9.19) is indeed equivalent to the condition for linear time-invariant dynamical  observability ¯ C¯ is observable. According to the results stated in Chap. 1, systems, i.e., the pair A, this is equivalent to the following condition: 

 s In − A¯ rank = n, ∀s ∈ C. C¯ Consequently, we obtain 

s In − A¯ rank C¯



⎤ ⎡ ⎤ s E¯ q − A˜ 1 s E¯ q − A˜ 1   = rank ⎣ s 0, In−q + Q 1 C¯ 1 A˜ 1 ⎦ = rank ⎣ s 0, Q 1 C¯ 2 + Q 1 C¯ 1 A˜ 1 ⎦ C¯ C¯ ⎡ ⎤ s E¯ q − A˜ 1 = rank ⎣ s Q 1 C¯ − s Q 1 C¯ 1 E¯ q + Q 1 C¯ 1 A˜ 1 ⎦ C¯ ⎡ ⎤⎡ ⎤ Iq 0 0 s E¯ q − A˜ 1 ⎦⎣ ⎦ = rank ⎣ −Q 1 C¯ 1 Q 1 0 s C¯ ¯ 0 0 In−q+m C ⎡ ⎤ ¯ q − A˜ 1     s E ¯ ˜ s E¯ q − A˜ 1 ⎣ − A˜ 2 ⎦ = rank s E − A = rank = rank C¯ C˜ C˜ = n.

Since E¯ = S E T =

⎡



 Iq 0 , A˜ = S AT and C˜ = C T , then we get 0 0 

rank

   s E¯ − A˜ sE − A = rank =n C C˜

for all s ∈ C. Finally, we have proved that, for system (9.1) with f (x) = Ax and h(x) = C x, if Assumption 9.1 is fulfilled, then (9.19) implies (9.8). 

182

9 Extension to Nonlinear Singular Dynamical Systems

9.3 Necessary and Sufficient Conditions For the regularized system, this section gives necessary and sufficient condition to transform it into the output derivative injection form (9.4). Recall that if Assumptions 9.1 and 9.2 for nonlinear singular dynamical system (9.1) are both fulfilled, then this singular system might be regularized into ζ˙ = f¯(ζ ) + g(ζ ) y˙ ¯ ), y¯ = h(ζ

(9.22)

where f¯, g and h¯ were given in (9.16)–(9.18), respectively, where the output components {d h¯ 1 , . . . , d h¯ m¯ } are assumed to be linearly independent. In order to transform the regularized system (9.22) into (9.4), by following the procedure introduced in Chap. 8, we need firstly calculate the following 1-forms h¯ i , for 1 ≤ i ≤ m¯ and 1 ≤ k ≤ ri , θi,k = d L k−1 f¯ which enables us to define the following distribution:   Δ = span θ j,k , 1 ≤ j ≤ m¯ and 1 ≤ k ≤ r j .

(9.23)

Like what we have done in Chap. 8 for nonlinear dynamical systems with multiple outputs, the family of vector fields τi,1 1≤i≤m¯ can be calculated via the following equations:  θi,ri  τi,1 = 1, for 1 ≤ i ≤ m¯ θi,k τi,1 = 0, for 1 ≤ k ≤ ri − 1 (9.24) θ j,k τi,1 = 0, for j < i and 1 ≤ k ≤ r j θ j,k τi,1 = 0, for j > i and 1 ≤ k ≤ r j . Since Assumption 9.2 is supposed to be fulfilled for the regularized system (9.22), then the vector fields (τi,1 )1≤i≤m¯ satisfying (9.24) can generate the following frame  τ = . . . , τi, j , . . . 1≤i≤m,1≤ ¯ j≤ri with

  τi,k = τi,k−1 , f , for 1 ≤ i ≤ m¯ and 2 ≤ k ≤ ri .

Following the similar procedure in Chap. 8, denote the 1-forms as T θ = (θ1,1 , . . . , θ1,r1 , . . . , θm,1 ¯ , . . . , θm,r ¯ m¯ )

and calculate Λ = θ (τ ).

(9.25)

9.3 Necessary and Sufficient Conditions

183

Since the regularized system (9.22) is supposed to be observable in the sense that the observability rank condition (9.19) is satisfied, therefore Λ is invertible, which enables us to compute the following matrix: ω = Λ−1 θ.

(9.26)

Therefore, we can extend the result stated in Theorem 8.1 of Chap. 8 to treat the regularized system (9.22). Theorem 9.1 For system (9.22), suppose that Assumptions 9.1 and 9.2 are fulfilled. Then there exists a local diffeomorphism z = φ(ζ ) transforming (9.22) into the output derivative injection normal form (9.4), if and only if (1) [τi, j , τs,l ] = 0, for 1 ≤ i ≤ m, ¯ 1 ≤ j ≤ ri , 1 ≤ s ≤ m¯ and 1 ≤ l ≤ rs ; ¯ where Δ is defined in (9.23) and Δi is (2) dim [Δi ] = dim [Δi ∩ Δ] for 1 ≤ i ≤ m, defined in (8.9);   ¯ 1 ≤ j ≤ ri − 1 , 1 ≤ k ≤ m. ¯ (3) τi, j , gk = 0 for 1 ≤ i ≤ m, Proof In Chap. 8, it has been stated in Theorem 8.1 that the first two conditions in Theorem 9.1 are necessary and sufficient to guarantee that we can find a local diffeomorphism z = φ(ζ ) such that (9.22) without the term g(ζ ) y˙ can be transformed into (8.2). Therefore, it remains to prove that the third condition of Theorem 9.1 is necessary and sufficient such that the Jacobian of the deduced diffeomorphism z = φ(ζ ) can ¯ transform gk into βk ( y¯ ) for 1 ≤ k ≤ m. For this, let us recall the results stated in Chap. 3. In Theorem 3.3, we have studied the nonlinear dynamical systems with single-input single-output where the input term g(x)u plays the same role as the term g(ζ ) y˙ . For multiple inputs case, a similar result has been stated in Corollary 3.1 where the similar Lie bracket conditions (only for single-output case) have been proven to be necessary and sufficient. Therefore, the following just gives the sketch of the proof when treating multiple-output case. Firstly, let us consider the 1-forms ω defined in (9.26), then we have ω(τ ) = In×n . The above equation means that ω(τi, j ) for 1 ≤ i ≤ m¯ and 1 ≤ j ≤ ri are constant. Therefore, like what we have proceeded before in the previous chapters, we can obtain dω(τi, j , τk,s ) = L τi, j ω(τk,s ) − L τk,s ω(τi, j ) − ω([τi, j , τk,s ]) = −ω([τi, j , τk,s ]). The above equation implies that [τi, j , τk,s ] = 0 is equivalent to dω = 0 since ω is an isomorphism.

184

9 Extension to Nonlinear Singular Dynamical Systems

Then, based on the result stated in Poincaré Lemma (see Lemma 2.1 in Chap. 2), dω = 0 implies that we can find a local diffeomorphism z = φ(ζ ) such that ω = dφ. and ∂ φ∗ (τi, j ) = ∂z i, j for 1 ≤ i ≤ m¯ and 1 ≤ j ≤ ri . Consequently, for 1 ≤ k ≤ m, ¯ we obtain ∂ φ∗ (gk ) = φ∗ ([τi, j , gk ]) = 0 ∂z i, j if [τi, j , gk ] = 0. By integration of the above equation, it is clear that φ∗ (gk ) = βk ( y¯ ). Therefore, we proved that (9.22) can be transformed into (9.4).



The following gives an academic example to show how to apply the aboveproposed method. Example 9.2 Consider the following nonlinear singular system E x˙ = f (x) y = h(x), ⎡

where

−4 ⎢ 4 E =⎢ ⎣ −4 0

3 −1 2 1

0 0 0 0

(9.27) ⎤ 0 0⎥ ⎥ 0⎦ 0

⎤ 4x1 − 4x2 − 3x4 + (3x2 − x3 + 2x4 ) (x3 + x4 ) ⎢ 4x2 − 4x1 + x4 + (3x3 − x2 + 2x4 ) (x3 + x4 ) ⎥ ⎥ f (x) = ⎢ ⎦ ⎣ 4x3 + 6x4 + 2 (x2 − x3 ) (x3 + x4 ) −x4 + (x2 + x3 + 2x4 ) (x3 + x4 ) ⎡

and

 x3 + x4 + 2x2 x4 + x22 + x42 . h(x) = x2 + x4 

Note rankE = q = 2 < n = 4, then we can find the following two elementary matrices: ⎡ ⎤ 1 0 0 0 ⎢ 1 1 0 0 ⎥ ⎥ T =⎢ ⎣ 1 1 1 −1 ⎦ −1 −1 0 1

9.3 Necessary and Sufficient Conditions

185

⎡

and

0

⎢1 4 S=⎢ ⎣0 1 4

1 4 1 4 1 4 1 4

0

1 4

⎤

0 ⎥ ⎥ − 41 ⎦ 0 − 21 1 4 1 4

with which the matrix E can be transformed into ⎡ 100 ⎢0 1 0 E¯ = S E T = ⎢ ⎣0 0 0 000

⎤ 0 0⎥ ⎥. 0⎦ 0

Hence, we can introduce the following change of variables: ⎤ x1 ⎢ x2 − x1 ⎥ ⎥ ζ = T −1 x = ⎢ ⎣ x3 + x4 ⎦ x2 + x4 ⎡

via which the studied nonlinear singular system can be transformed into ζ˙1 ζ˙2 0 y¯1 y¯2

= ζ2 + ζ32 = ζ3 + ζ3 ζ4 = ζ3 − ζ1 + ζ4 = ζ3 + ζ42 = ζ4 .

(9.28)

Besides, we can check the following condition: ⎡

E ⎢ rank ⎣ 0 0

∂ f (ζ ) ∂ζ

⎤

⎥ E ⎦ = 6 = 4 + rankE,

∂h(ζ ) ∂ζ

which implies that Assumption 9.1 is fulfilled. Consequently, according to the result stated in Sect. 9.2, the studied singular dynamical system can be regularized into the following one: ⎡ ⎤ ⎡ ⎤ ζ2 + ζ32 0 ⎢ ζ3 + ζ3 ζ4 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ ζ˙ = ⎢ ⎣ ζ2 + ζ32 ⎦ + ⎣ −1 ⎦ y˙3 0 ⎡ ⎤ 1 ζ3 − ζ1 + ζ4 y = ⎣ ζ3 + ζ42 ⎦ . ζ4

186

9 Extension to Nonlinear Singular Dynamical Systems

Now, in order to deduce the diffeomorphism z = φ(ζ ) to transform the regularized system into the output derivative injection observer normal form, we need firstly check Assumption 9.2, which yields r1 = r3 = 1 and r2 = 2. So, we can calculate ⎤ ⎡ d h¯ 1 −1 ⎢ 0 ⎢ d h¯ 2 ⎥ ⎥ ⎢ rank ⎢ ⎣ d L f¯ h¯ 2 ⎦ = rank ⎣ 0 0 d h¯ 3 ⎡

0 0 1 0

1 1 2ζ3 0

⎤ 1 2ζ4 ⎥ ⎥ = 4, 0 ⎦ 1

which implies that Assumption 9.2 is fulfilled as well. Finally, by following the procedure presented in Sect. 9.3, we obtain ⎤ d h¯ 1 ⎢ d h¯ 2 ⎥ ⎥ θ =⎢ ⎣ d L f¯ h¯ 2 ⎦ d h¯ 3 ⎡

and the corresponding τ defined in (9.25). This enables us to obtain the following diffeomorphism ⎤ ⎡ ⎤ ⎡ ζ3 − ζ1 + ζ4 z 1,1 ⎢ z 2,1 ⎥ ⎢ ζ3 + ζ 2 ⎥ 4 ⎥ ⎢ ⎥ ⎢ ⎣ z 2,2 ⎦ = ⎣ ζ2 + ζ32 ⎦ . z 3,1 ζ4 With this deduced diffeomorphism, the studied nonlinear singular dynamical system might be transformed into the following form: z˙ 1,1 z˙ 2,1 z˙ 2,2 z˙ 3,1 y1

=0 = z 2,2 + (2y3 − 1) y˙3 = y2 − y32 + y2 y3 − y33 + 2(y2 − y32 ) y˙2 + 4(y32 − y2 y3 ) y˙3 = y˙3 = z 1,1 , y2 = z 2,1 , y3 = z 3,1 .

9.4 Observer Design 9.4.1 Nonlinear Luenberger-Like Observer Consider the transformed normal form (9.4), this section will design a nonlinear Luenberger-like asymptotic observer for such a normal form. For this, we rewrite (9.4) into the following compact form: z˙ = A O z + α( y¯ ) + β( y¯ ) y˙¯ y¯ = C O z,

(9.29)

9.4 Observer Design

187

where A O = diag[A O1 , . . . , A Om¯ ] α = (α1T , . . . , αmT¯ )T β = (β1T , . . . , βmT¯ )T and C O = diag[C OT 1 , . . . , C OT m¯ ]T . Due to the special form of A O and C O , it is easy to show that the pair (A O , C O ) is observable. Therefore, we can always find a constant matrix G such that (A O − GC O ) is Hurwitz. Before designing the nonlinear Luenberger-like observer, let us make the following assumption. Assumption 9.3 It is assumed that β( y¯ ) in (9.29) is integrable, i.e., ∃K ( y¯ ) such that

∂ K ( y¯ ) = β( y¯ ). ∂ y¯

(9.30)

It is worth noting that the above condition for β( y¯ ) (named as integrability) is a restrictive condition. For linear time-invariant systems where β( y¯ ) in this case is constant (thus it is always integrable), the required integrability condition is satisfied. When considering nonlinear dynamical systems with single output, such an integrability is implicitly fulfilled, since in this case, we have y¯ ∈ R. However, such a condition is not always fulfilled for nonlinear dynamical systems with multiple outputs, and that is why such a condition has been imposed when designing Luenberger-like observer for nonlinear dynamical systems with multiple outputs [7, 12]. Now, let us suppose that Assumption 9.3 is satisfied and consider the following dynamics: ξ˙ = N ξ + L( y¯ ) (9.31) zˆ = ξ + K ( y¯ ), where

∂ K ( y¯ ) ∂ y¯

= β( y¯ ) N = A O − GC O L( y¯ ) = G y¯ + N K ( y¯ ) + α( y¯ ).

To show the convergence, let us define e = z − zˆ , then we have the following observation error: e˙ = A O z + α( y¯ ) + β( y¯ ) y˙¯ − N zˆ + N K ( y¯ ) − L( y¯ ) −

∂ K ( y¯ ) ˙ y¯ . ∂ y¯

188

9 Extension to Nonlinear Singular Dynamical Systems

As it is assumed that Assumption 9.1 is fulfilled, then we have which the observation error reads as

∂ K ( y¯ ) ∂ y¯

= β( y¯ ), with

e˙ = (A O − GC O )e + α( y¯ ) + G y¯ − GC zˆ + A O zˆ − N zˆ + N K ( y¯ ) − L( y¯ ). In order to simplify the above equation, let N = A O − GC O , and L( y¯ ) = α( y¯ ) + G y¯ + N K ( y¯ ). Hence, the above equation is equivalent to e˙ = (A O − GC O )e, where the matrix G can be freely chosen. In order to have the convergence, it is sufficient to choose a G such that (A O − GC O ) is Hurwitz. Consequently, we proved that (9.31) is an exponential observer of (9.29). Based on the above result, we can state that T φ −1 (ˆz ) exponentially converges to x. In conclusion, the following dynamics ξ˙ = N ξ + L(y) zˆ = ξ + K (y) ζˆ = φ −1 (ˆz ) xˆ = T ζˆ

(9.32)

is an exponentially nonlinear Luenberger-like observer of (9.29).

9.4.2 Design Procedure Finally, for a given nonlinear singular system (9.1), the procedure to design a Luenberger-like observer can be summarized as follows: Step 1: Calculate the elementary matrices T and S to transform (9.1) to semi-explicit form (9.12); Step 2: Regularize (9.12) into a nonlinear regular system (9.15), if Assumption 9.1 is satisfied; Step 3: If Assumption 9.2 and all conditions of Theorem 9.1 are satisfied, then deduce a diffeomorphism z = φ(ζ ) to transform the regularized system (9.15) into observer normal form (9.29); Step 4: Finally, if Assumption 9.3 is satisfied, then construct the proposed Luenberger-like observer (9.32) to estimate x of (9.1).

References

189

References 1. Bejarano, F.J., Perruquetti, W., Floquet, T., Zheng, G.: State reconstruction of nonlinear differential-algebraic systems with unknown inputs. In: Proceedings of the 51st IEEE Conference on Decision and Control, pp. 5882–5887 (2012) 2. Bejarano, F.J., Floquet, T., Perruquetti, W., Zheng, G.: Observability and detectability of singular linear systems with unknown inputs. Automatica 49(3), 793–800 (2013) 3. Bejarano, F.J., Perruquetti, W., Floquet, T., Zheng, G.: Observation of nonlinear differentialalgebraic systems with unknown inputs. IEEE Trans. Autom. Control 60(7), 1957–1962 (2015) 4. Boutat, D., Zheng, G., Boutat-Baddas, L., Darouach, M.: Observers design for a class of nonlinear singular systems. In: Proceedings of the 51st IEEE Conference on Decision and Control, pp. 7407–7412 (2012) 5. Campbell, S.L.V.: Singular Systems of Differential Equations. Pitman, London (1980) 6. Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, vol. 118. Springer, New York (1989) 7. Darouach, M., Boutat-Baddas, L.: Observers for a class of nonlinear singular systems. IEEE Trans. Autom. Control 53(11), 2627–2633 (2008) 8. Darouach, M., Zasadzinski, M., Hayar, M.: Reduced-order observer design for descriptor systems with unknown inputs. IEEE Trans. Autom. Control 41(7), 1068–1072 (1996) 9. Paraskevopoulos, P., Koumboulis, F.: Observers for singular systems. IEEE Trans. Autom. Control 37(8), 1211–1215 (1992) 10. Yip, E., Sincovec, R.: Solvability, controllability, and observability of continuous descriptor systems. IEEE Trans. Autom. Control 26(3), 702–707 (1981) 11. Zheng, G., Boutat, D., Wang, H.: Observer design for a class of nonlinear singular systems. In: Proceedings of the 55th IEEE Conference on Decision and Control, pp. 3910–3914 (2016) 12. Zheng, G., Boutat, D., Wang, H.: A nonlinear Luenberger-like observer for nonlinear singular systems. Automatica 86, 11–17 (2017)

Index

A Adaptive observer, 4, 23, 24, 103, 105 Auxiliary dynamics, 26, 91–93, 98, 102, 103, 105, 109, 124, 125

H High-gain observer, 4, 23, 69, 85, 87, 103, 104 Homogeneous observer, 141, 142

B Backstepping observer, 22 Brunovsky form, 12, 26, 56, 63, 65, 104, 107, 144, 156, 175

I Immersion technique, 26, 27, 82, 93 Implicit function theorem, 14 Indistinguishability, 7, 15, 16 Integrability, 35, 36, 44, 47, 187 Involutivity, 44, 132

C Cayley–Hamilton Theorem, 8, 17 Contractibility, 40

D Detectability, 2, 4, 5, 9 Differential algebraic method, 13 Differential forms, 31, 38 Differential geometric method, 13, 16, 27, 127, 173, 175 Differentiator, 3 Distinguishability, 5

E Exact differential, 57 Extended Kalman Filter, 20, 21

F Finite-time observer, 4, 127, 141

K Kalman–Bucy Filter, 11, 20, 104

L Lie bracket, 13, 31, 35–37, 39, 42, 45–48, 50, 55, 57, 70, 75, 78, 82, 96–98, 102, 107, 109–112, 114, 115, 127, 131, 134, 138, 146, 154, 160, 162, 183 Lie derivative, 13, 16, 17, 32, 35, 39, 56, 58, 75, 112–115, 131, 153, 159 Linear time-invariant system, 7, 187 Luenberger-like observer, 21, 55, 56, 66, 67, 86, 173, 175, 187, 188 Luenberger observer, 1, 4, 11, 12

M Modulo, 127, 131, 133, 135, 139, 140, 162, 164–166, 171

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Boutat and G. Zheng, Observer Design for Nonlinear Dynamical Systems, Lecture Notes in Control and Information Sciences 487, https://doi.org/10.1007/978-3-030-73742-9

191

192 N Non-holonomic constraints, 44, 45 Nonlinear Luenberger-like observer, 186– 188 Nonlinear Luenberger observer, 19–21 O Observability analysis, 1, 5, 13 Observability Gramian, 5, 8–10, 16, 18 Observability indices, 143–145, 151, 152, 154, 155, 162, 167, 171 Observability matrix, 8, 9, 138 Observability rank condition, 12, 16–18, 20, 22, 56, 60, 61, 70, 72, 94, 110, 111, 124, 127, 129, 144, 150, 157, 183 Observation error dynamics, 11, 12, 19, 22, 86 Observer canonical form, 3, 12 Observer design, 1, 4, 5, 10, 18, 25, 27, 66, 85, 103, 121, 141, 169, 175, 186 Observer normal forms, 24–27, 31, 56–59, 61, 65, 68–71, 82, 88, 89, 91, 93, 94, 99, 102, 106, 107, 109, 110, 117, 120, 124, 125, 127, 128, 141, 143, 144, 149, 155, 156, 160, 173–175, 186, 188 Observer normal form with output injection, 25, 26, 55, 56, 65, 69, 91, 107, 174 Optimization, 7, 18, 19 Output-depending observer normal form, 26, 27, 107, 109, 110, 124

Index Output derivative injection, 173–175, 181– 183, 186 Output diffeomorphism, 25, 27, 69, 71 Output injection, 3, 4, 25, 26, 55, 56, 62, 63, 65, 67, 69, 91, 107, 141, 174

P Partial observability, 129, 135, 136, 146, 157 Poincaré’s Lemma, 31, 40, 41, 43, 151, 165 Popov–Belevitch–Hautus test, 5, 9

R Reconstructibility, 6, 14 Reduced-order Luenberger-like observer, 143, 169, 171

S Singular, 16, 22, 173–176, 178, 179, 182, 184–186, 188 State transition matrix, 6 Step-by-step sliding mode observer, 107, 121–123

T Tangent fiber bundle, 31, 38, 41, 42, 132, 146, 163 Triangular form, 23, 103