Fractional Order Systems―Control Theory and Applications: Fundamentals and Applications (Studies in Systems, Decision and Control, 364) 3030714454, 9783030714451

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Studies in Systems, Decision and Control 364

Omar Naifar Abdellatif Ben Makhlouf   Editors

Fractional Order Systems— Control Theory and Applications Fundamentals and Applications

Studies in Systems, Decision and Control Volume 364

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

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Omar Naifar · Abdellatif Ben Makhlouf Editors

Fractional Order Systems—Control Theory and Applications Fundamentals and Applications

Editors Omar Naifar CEM Laboratory Department of Electrical Engineering National School of Engineers of Sfax University of Sfax Sfax, Tunisia

Abdellatif Ben Makhlouf Department of Mathematics College of Science Jouf University Sakaka, Saudi Arabia

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-71445-1 ISBN 978-3-030-71446-8 (eBook) https://doi.org/10.1007/978-3-030-71446-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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

Preface

Since the middle of the last century, the control theory has been subject to a revolution and a very huge amount of research works in the literature. Rationally, the great majority of works, established until now, have focused on the classical integerorder systems, modeled with differential equations where an integer-order derivative is used. Meanwhile, with the development of science and applied mathematics, it has been discovered that several physical systems are really described with differential fractional-order equations, where a fractional derivative order is used. Consequently, such systems cannot be effectively modeled using the classical differential integer-order equations. As a result to this fact, a growing interest is being given by researchers in the last few decades, to investigate fractional-order systems; and various problems inside the control theory, such as state estimation, control and fault diagnosis, are being tackled. Note that, compared to the integer-order case, the fractional-order framework represents a fertile field of research, since it has been “recently” addressed by researchers and several specific questions are still to investigate for fractional-order systems. Fractional-order systems (FOS) are dynamical systems that can be modeled by a fractional differential equation carried with a non-integer derivative. Such systems are said to have fractional dynamics. Integrals and derivatives of fractional orders are used to illustrate objects that can be described by power-law nonlocality, power-law long-range dependence, or fractal properties. FOS are advantageous in studying the behavior of dynamical systems in electrochemistry, physics, viscoelasticity, biology, and chaotic systems. In the last few decades, the growth of science and engineering systems has considerably stimulated the employment of fractional calculus in many subjects of the control theory, for example, in stability, stabilization, controllability, observability, observer design, and fault estimation. The application of control theory in fractional-order systems is an important issue in many engineering applications. It is necessary to note that several physical systems are not truly modeled with integer-order differential equations. The reason is that their actual dynamics contain non-integer derivatives. So, to accurately describe these systems, the fractional-order differential equations have been introduced. Such systems are conventionally called fractional-order systems. Fractional-order systems with a fractional derivative between 0 and 1 correspond to an extension of the classical v

vi

Preface

integer-order ones, so that a broader set of real systems could be covered. As examples, image processing, electromagnetic systems, and dielectric polarization have been modeled using the fractional-order calculus. Indeed, this subject covers also important applications in engineering areas such as bioengineering, viscoelasticity, electronics, robotics, control theory, and signal processing. The aim of this book is to bring together knowledge, analysis, and synthesis of fractional control problem of nonlinear systems as well as some applications. Topics of interest include state estimation for fractional-order systems including, for example, works on new results on the state estimation problem for fractional-order systems and robust observer scheme for a class of linear fractional systems. The problem of stabilization can be presented also, namely the feedback control scheme for fractional systems or the model-reference control problem. Furthermore, fault estimation for fractional nonlinear systems is one of the most important subjects, for example, the adaptive estimation strategy of component faults and actuator faults for fractional-order nonlinear systems, etc. In regard to applications, some examples are introduced. This book is composed of 10 chapters: • On the Stability of Caputo Fractional-Order Systems: A Survey In this chapter, an overview on the stability of Caputo fractional-order systems is presented. To begin with, a general introduction is introduced then, and some preliminaries are given. After that, the concept of stability of Caputo fractional differential equations is described, including the stability of nonlinear fractional differential equations and some special cases. • Observers and Observability—Theory and Literature Overview In this chapter, an overview of observers and observability is given. To begin with, a general introduction is introduced, and then, observers and observability for continuous-time linear systems are given. Finally, some observers design for certain fractional-order systems is presented. • A Brief Overview on Fractional Order Systems in Control Theory In this chapter, an overview of the different parts that intervene in this thesis is given. Fractional-order systems are introduced, and a state of the art in relation with these systems in the control theory is given. Certain classes of nonlinear systems tackled in this book are presented, with a literature survey on the observer design problem for these nonlinear systems. The fault diagnosis problem in engineering is introduced, with the focus on the fault estimation task. The observer-based control problem for dynamic systems is presented, with introducing the two main kinds of control, which are the stabilization and the model-reference control. • State Estimation for Fractional-Order Systems In this chapter, some results on the state estimation problem for fractional-order systems are presented. First, a robust observer scheme for a class of linear fractional systems with matched uncertainties is proposed. Then, a novel observer design for nonlinear fractional-order systems is given. Finally, the specific problem of robust state estimation for fractional one-sided Lipschitz systems,

Preface

•

•

•

•

vii

in the presence of unknown inputs, is tackled. A full-order observer version, as well as a reduced-order observer version, is detailed and analyzed. Observer-Based Control for Fractional-Order Systems In this chapter, certain results on the observer-based control problem for fractionalorder systems are presented. The two control philosophies, which are the “stabilization” and the “model-reference control,” are investigated. Dealing with the stabilization problem, two contributions are given. First, a feedback control scheme for fractional Lipschitz systems is proposed. Second, an adaptive stabilization scheme for a class of uncertain fractional systems is detailed. In the final part, the model-reference control problem is investigated. A novel and advantageous model-reference control design for linear fractional-order systems is presented. Fault Estimation for Nonlinear One-Sided Lipschitz Systems In this chapter, some results on the fault estimation problem for nonlinear systems are presented. Both integer-order systems and fractional-order systems are investigated. First, using an original and advantageous fault modeling, an adaptive estimation strategy of component faults and actuator faults for integer-order onesided Lipschitz is given. Second, a sensor fault estimation scheme for fractionalorder descriptor one-sided Lipschitz is provided. Finally, a robust sensor fault estimation approach for fractional-order systems with monotone nonlinearities and unknown inputs is presented and detailed. Fractional Order CRONE and PID Controllers Design for Nonlinear Systems Based on Multimodel Approach This chapter deals with the output regulation of nonlinear control systems in order to guarantee desired performances in the presence of plant parameters variations. The proposed control law structures are based on the fractional-order PI (FOPI) and the CRONE control schemes. By introducing the multimodel approach in the closed-loop system, the presented design methodology of fractional PID control and the CRONE control guarantees desired transients. Then, the multimodel approach is used to analyze the closed-loop system properties and to get explicit expressions for evaluation of the controller parameters. The tuning of the controller parameters is based on a constrained optimization algorithm. Simulation examples are presented to show the effectiveness of the proposed method. Design of Robust Fractional Predictive Control for a Class of Uncertain Fractional Order Systems In this work, the robust fractional predictive control (RFPC) approach is designed for a class of fractional-order system with real uncertain parameters. The control law is obtained by the resolution of a min-max optimization problem that takes into account the uncertainties of the fractional-order model parameters. Subsequently, the resolution of this problem using standard approach can give local solutions. Hence, we propose the use of the global optimization methods with a view to solving min-max non-convex problem for the uncertain fractional-order system, which eventually helps to find the global optimum. The performances and the efficiency of the proposed RFPC controller are illustrated with practical applications to a thermal system.

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• Constant Phase Based Design of Robust Fractional PI Controller for Uncertain First Order Plus Dead Time Systems In this chapter, a new design method of a robust fractional-order PI controller for uncertain first-order plus dead time systems is proposed in this paper. The proposed design method uses a numerical optimization algorithm to determine the unknown controller parameters. The main objective of the proposed design method is improving the robustness in degree of stability to gain variations and the stability robustness to the other parameters variations that affect the phase by imposing a constant phase margin to the corrected open-loop system in a prespecified frequency band. Several simulation examples are presented to design the robust fractional PI controller and to test the robustness for different forms of uncertainty. • Identification of Continuous-Time Fractional Models from Noisy Input and Output Signals It is well known that, in some practical system identification situations, measuring both input and output signals can commonly be affected by additive noises. In this chapter, we consider the problem of identifying continuous-time fractional systems from noisy input and output measurements. The bias correction scheme, which aims at eliminating the bias introduced by the fractional-order ordinary least squares method, is presented, based on the estimation of variances of the input and output measured noises. The compensation method for the input and output noises is also studied by introducing an augmented high-order fractional-order system in the identification algorithm. The presented algorithm is established to perform unbiased coefficients and fractional-order estimation. The promising performances of the proposed method are assessed via the identification of a fractional model and a fractional real electronic system. Sfax, Tunisia Sakaka, Saudi Arabia April 2021

Omar Naifar Abdellatif Ben Makhlouf

Contents

On the Stability of Caputo Fractional-Order Systems: A Survey . . . . . . . Abdellatif Ben Makhlouf

1

Observers and Observability—Theory and Literature Overview . . . . . . . Assaad Jmal, Omar Naifar, Abdellatif Ben Makhlouf, Nabil Derbel, and Mohamed Ali Hammami

9

A Brief Overview on Fractional Order Systems in Control Theory . . . . . Assaad Jmal, Omar Naifar, Abdellatif Ben Makhlouf, Nabil Derbel, and Mohamed Ali Hammami

29

State Estimation for Fractional-Order Systems . . . . . . . . . . . . . . . . . . . . . . . Omar Naifar, Assaad Jmal, Abdellatif Ben Makhlouf, Nabil Derbel, and Mohamed Ali Hammami

49

Observer-Based Control for Fractional-Order Systems . . . . . . . . . . . . . . . . Omar Naifar, Assaad Jmal, Abdellatif Ben Makhlouf, Nabil Derbel, and Mohamed Ali Hammami

75

Fault Estimation for Nonlinear One-Sided Lipschitz Systems . . . . . . . . . . Omar Naifar, Assaad Jmal, Abdellatif Ben Makhlouf, Nabil Derbel, and Mohamed Ali Hammami

95

Fractional Order CRONE and PID Controllers Design for Nonlinear Systems Based on Multimodel Approach . . . . . . . . . . . . . . . . 123 Mohamed Lazhar Wardi, Rihab Abdelkrim, and Mohamed Naceur Abdelkrim Design of Robust Fractional Predictive Control for a Class of Uncertain Fractional Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Aymen Rhouma and Sami Hafsi Constant Phase Based Design of Robust Fractional PI Controller for Uncertain First Order Plus Dead Time Systems . . . . . . . . . . . . . . . . . . . 159 B. Saidi, Z. Yacoub, M. Amairi, and M. Aoun

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Contents

Identification of Continuous-Time Fractional Models from Noisy Input and Output Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Z. Yakoub, M. Aoun, M. Amairi, and M. Chetoui

On the Stability of Caputo Fractional-Order Systems: A Survey Abdellatif Ben Makhlouf

Abstract In this chapter, an overview on the stability of Caputo fractional-order systems is presented. To begin with, a general introduction is introduced then, some preliminaries are given. After that, the concept of stability of Caputo fractional differential equations is described, including the stability of nonlinear fractional differential equations and some special cases. Keywords Stability · Fractional order systems · Caputo derivative

1 Introduction Many dynamic systems are better characterized by a dynamic model of fractional order, usually based on the concept of differentiation or integration of fractional order. The study of the stability of fractional order systems is more delicate than their counterparts, the entire order systems. Indeed, fractional systems are, first, considered memory systems, in particular to take into account the initial conditions, and, secondly, they have a much more complex dynamic. In fact, stability for nonlinear systems derived from fractional order remains in automatic open problems due to the fractional nature and the nonlinearity of the systems. It is usually that several physical systems are characterized by fractional-order state equations [1], such as, fractional Langevin equation [2], fractional LotkaVolterra equation [3] in biological systems, fractional-order oscillator equation [4] in damping vibration, in anomalous diffusion and so on. Particularly, stability analysis is one of the most fundamental issues for systems. In this few years, there are many results about the stability of fractional order systems [5–13]. A. Ben Makhlouf (B) Mathematics Department, College of Science, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia Department of Mathematics, Faculty of Sciences of Sfax, Route Soukra, BP 1171, 3000 Sfax, Tunisia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Naifar and A. Ben Makhlouf (eds.), Fractional Order Systems—Control Theory and Applications, Studies in Systems, Decision and Control 364, https://doi.org/10.1007/978-3-030-71446-8_1

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The brief outline of this chapter is as follows. In Sect. 2, the preliminaries are studied. Section 3 describes some stability conditions of linear and nonlinear fractional differential equations. A general conclusion is given in the final section of this chapter.

2 Preliminaries In this section, some definitions, lemmas and theorems related to the fractional calculus are given. Definition 1 The Riemann-Liouville fractional integral of order α > 0 is defined as, t 1 α It0 x(t) = (t − τ )α−1 x(τ )dτ. (α) t0



+∞

(α) =

e−t t α−1 dt where  is the Gamma function generalizing factorial for

0

non-integer arguments. Definition 2 Given 0 < α < 1 and an absolutely continuous function x. The Caputo fractional derivative of x is defined as, C

Dtα0 ,t x(t)

1 = (1 − α)

t

(t − s)−α x  (s)ds.

(1)

t0

On the other hand, there exists a frequently used function in the solution of fractional order systems named the Mittag leffler function. Indeed, the proposed function is a generalization of the exponential function. In this context, the following definitions and Lemmas are presented. Definition 3 The Mittag-Leffler function with two parameters is defined as E α,β (z) =

+∞  k=0

zk , (kα + β)

where α > 0, β > 0, z ∈ C. When β = 1, one has E α (z) = E α,1 (z), furthermore, E 1,1 (z) = e z . Lemma 1 [7] If 0 < α < 2 and μ is a real number such that, πα < μ < min{π, π α}, 2

On the Stability of Caputo Fractional …

3

then, for p ∈ N∗ \ {1}, the following asymptotic expansions are valid:  1   1  1 1 1 1−β + O p+1 , z α exp z α − k α (β − αk) z z k=1 p

E α,β (z) =

with |z| −→ ∞, |arg(z)| ≤ μ; and E α,β (z) = −

p  k=1

 1  1 1 + O p+1 , (β − αk) z k z

with |z| −→ ∞, μ ≤ |arg(z)| ≤ π. For the proof of stability, particularly to justify some inequalities, the following Theorem is used Theorem 1 (Gronwall inequality) [14] Suppose x(t), a(t) are nonnegative and locally integrable on 0 ≤ t < T , some T ≤ +∞, and g(t) is a nonnegative, nondecreasing continuous function defined on 0 ≤ t < T , g(t) ≤ M, α > 0 with t x(t) ≤ a(t) + g(t)

(t − τ )α−1 x(τ )dτ

0

on this interval. Then x(t) ≤ a(t) +

t   +∞ (g(t)(α))n 0

n=1

(nα)

 (t − τ )nα−1 a(τ ) dτ , 0 ≤ t < T.

Moreover, if a(t) is a nondecreasing function on [0, T ), then x(t) ≤ a(t)E α (g(t)(α)t α ).

3 Stability of Caputo Fractional Differential Equations In this section, some results related to the stability of fractional differential equations are given.

3.1 Stability of Nonlinear Fractional Differential equations In this part, some basic stability results will be presented.

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We consider the following general type of Caputo fractional differential equations. C

Dtα0 ,t x(t) = f (t, x),

(2)

where x(t) = [x1 (t), . . . , xn (t)] ∈ Rn , 0 < α < 1, f : [0, +∞[×Rn −→ Rn is smooth. Definition 4 [9] The constant vector xeq is an equilibrium point of fractional differential system (2), if and only if f (t, xeq ) = 0, ∀t ≥ 0. For the proof of stability, the following Lemma is used Lemma 2 [6] Let α ∈ (0, 1) and P ∈ R n×n a constant, square, symmetric and positive definite matrix. Then the following relationship holds 1 2

C

Dtα0 ,t (x T (t)P x(t)) ≤ x T (t)P C Dtα0 ,t x(t).

Remark 1 When P = 1 is a constant, the Lemma 2 was introduced by [15]. As the Theorem 1, Lemma 2 has been used several times [8, 11, 13]. Definition 5 (Mittag-Leffler stability) [9] The solution of (2) is said to be MittagLeffler stable if

x(t) ≤ {m(x(t0 ))E α (−λ(t − t0 )α )}b , where t0 is the initial time, α ∈ (0, 1), λ > 0, b > 0, m(0) = 0, m(x) ≥ 0, and m is locally Lipschitz on x ∈ B ⊂ Rn with Lipschitz constant m 0 . Remark 2 Mittag Leffler stability imply asymptotic stability. The Mittag Leffler stability result was described by Li et al. in [9]. It is presented as follow: Theorem 2 Let x = 0 be an equilibrium point for the system (2) with t0 = 0 and D ⊂ Rn be a domain containing the origin. Let V (t, x(t)) : [0, ∞) × D −→ R be a continuously differentiable function and locally Lipschitz with respect to x such that α1 x a ≤ V (t, x) ≤ α2 x ab , C

β

Dt0 ,t V (t, x(t)) ≤ −α3 x ab ,

where β ∈ (0, 1), α1 , α2 , α3 , a and b are arbitrary positive constants. Then x = 0 is Mittag-Leffler stable. If the assumptions hold globally on Rn , then x = 0 is globally Mittag-Leffler stable. The result of asymptotic stability in the sense of Lyapunov of fractional order nonlinear systems using the comparison functions was firstly introduced by Yan Li et al. in [9] as follow:

On the Stability of Caputo Fractional …

5

Theorem 3 Let x = 0 be an equilibrium point for the system (2). Assume that there exists a Lyupunov function V (t, x(t)) and class K functions αi (i = 1, 2, 3) satisfying: α1 ( x ) ≤ V (t, x) ≤ α2 ( x ), C

β

Dt0 ,t V (t, x(t)) ≤ −α3 ( x(t) ),

(3) (4)

where β ∈ (0, 1). Then, the equilibrium point of system (2) is asymptotically stable. But, such theorem is correct as stated is not clear at this point. In fact, the arguments used in its proof don’t guarantee the asymptotic stability of the zero solution (see [16]).

3.2 Special Cases We consider the following linear system of fractional differential equations C

α D0,t x(t) = Ax(t),

(5)

where x(t) ∈ Rn , matrix A ∈ Rn×n and 0 < α < 1. Using algebraic approach combined with asymptotic results, Matignon was the first one who gave a will-known stability result. This result is as follows: Theorem 4 [10] The autonomous system (5) with Caputo derivative and initial value x0 = x(t0 ), is . 1. asymptotically stable if and only if |arg(spec(A))| > απ 2 2. stable if and only if either it is asymptotically stable, or those critical eigenvalues have geometric multiplicity one, spec(A) which satisfy |arg(spec(A))| = απ 2 denotes the eigenvalues of matrix A. We consider the following linear system of fractional differential equations C

  α D0,t x(t) = A + Q(t) x(t),

(6)

where x(t) ∈ Rn , matrix A ∈ Rn×n , Q : [0, +∞[−→ Rn×n is a continuous matrixvalued function and 0 < α < 1. Theorem 5 [17] Assume that the spectrum of the matrix A satisfies the condition |arg(spec(A))| > and, in addition, Q satisfies

απ 2

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t q := supt≥0

  (t − s)α−1 E α,α (t − s)α A Q(s) ds < 1.

0

Then the trivial solution of (6) is asymptotically stable. Theorem 6 [17] Assume that the spectrum of the matrix A satisfies the condition |arg(spec(A))| >

απ . 2

Then there exists a positive number ε > 0 such that if Q satisfies supt≥0 Q(t) < ε, Then the trivial solution of (6) is asymptotically stable. Theorem 7 [17] Assume that the spectrum of the matrix A satisfies the condition |arg(spec(A))| >

απ . 2

If the matrix Q is decaying to zero, i.e., lim Q(t) = 0.

t→+∞

Then the trivial solution of (6) is asymptotically stable. We consider the following linear system of fractional differential equations C

α D0,t x(t) = Ax(t) + f (t, x(t)),

(7)

with the initial condition x(0) = x0 , where x(t) ∈ Rn , matrix A ∈ Rn×n , 0 < α < 1 and f : [0, +∞[×Rn −→ Rn is a continuous function such that f (t, 0) = 0, ∀t ≥ 0, and there exists a continuous function K : [0, +∞[−→ R+ satisfying

f (t, x) − f (t, y) ≤ K (t) x − y , ∀t ≥ 0, x, y ∈ Rn . Theorem 8 [17] Assume that the spectrum of the matrix A satisfies the condition |arg(spec(A))| > and, in addition, K satisfies

απ , 2

On the Stability of Caputo Fractional …

t q := supt≥0

7

  (t − s)α−1 E α,α (t − s)α A K (s) ds < 1.

0

Then the trivial solution of (7) is asymptotically stable. Theorem 9 [17] Assume that the spectrum of the matrix A satisfies the condition |arg(spec(A))| >

απ . 2

Then there exists a positive number ε > 0 such that if K satisfies supt≥0 K (t) < ε, Then the trivial solution of (7) is asymptotically stable. Theorem 10 [17] Assume that the spectrum of the matrix A satisfies the condition |arg(spec(A))| >

απ . 2

If the matrix K is decaying to zero, i.e., lim K (t) = 0.

t→+∞

Then the trivial solution of (7) is asymptotically stable. In literature, many works have employed the Theorem 1. For example, in [5], authors have used theorem 1 to give sufficient conditions for the finite-time boundedness of the following fractional differential equation: C

α D0,t x(t) = Ax(t) + Dw(t), t ∈ [0, T ],

(8)

where x(t) ∈ Rn , 0 < α < 1 and A and D are a constant matrices. We have the following result. Theorem 11 [5] For the fractional order LTI system (8), suppose that there exist a scalar γ > 0 and two matrices P1 > 0, P1 ∈ Rn×n and P2 > 0, P1 ∈ Rm×m satisfying  A P + P A T − γ P D P2 < 0, P2 D T −γ P2 E α (γ T α )



 c1 c2 γ dT α + < , λmin (P2 )(α + 1) λmin (P1 ) λmax (P1 )

where P = R − 2 P1 R − 2 . Then the system (8) is FTB with respect to (c1 , c2 , T, R, d). 1

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4 Conclusion In this chapter, we have presented some results on the stability of fractional differential equations including nonlinear fractional differential equations and linear fractional differential equations. A fundamental result is presented in section 3 related to the asymptotic stability of the zero solution of fractional order nonlinear systems (Theorem 3). However, as it was mentioned, the proof of such result is incorrect. Thus, the statement of theorem 3 remains an open problem.

References 1. Hilfer, R.: Applications of fractional calculus in physics. World Scientific, Singapore (2000) 2. Burov, S., Barkai, E.: Fractional Langevin equation: overdamped, underdamped, and critical behaviors. Phys. Rev. (2008) 3. Das, S., Gupta, P.: A mathematical model on fractional Lotka-Volterra equations. J. Theor. Biol. 277, 1–6 (2011) 4. Ryabov, Y., Puzenko, A.: Damped oscillation in view of the fractional oscillator equation. Phys. Rev. 66, 184–201 (2002) 5. Ma, Y., Wu, B., Wang, Y.: Finite-time stability and finite-time boundedness of fractional order linear systems. Neurocomputing 173, 2076–2082 (2016) 6. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simulat. 22, 650–659 (2015) 7. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and application of fractional differential equations. Elsevier, New York (2006) 8. Leung, A., Li, X., Chu, Y., Rao, X.: Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling. Nonlinear Dynam. 82, 185– 199 (2015) 9. Li, Y., Chen, Y.Q., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 10. Matignon, D.: Stability result on fractional differential equations with applications to control processing, pp. 963–968. Proceedings of IMACS-SMC, Lille, France (1996) 11. Naifar, O., Ben Makhlouf, A., Hammami, M.A.: “Comments on” Lyapunov stability theorem about fractional system without and with delay. Commun. Nonlinear Sci. Numer. Simulat. 30, 360–361 (2016) 12. Qian, D., Li, C., Agarwal, R.P., Wong, P.: Stability analysis of fractional differential system with Riemann-Liouville derivative. Mathe. Comput. Modell. 52, 862–874 (2010) 13. Wei, Y., Chen, Y., Liang, S., Wang, Y.: A novel algorithm on adaptive backstepping control of fractional order systems. Neurocomputing 165, 395–402 (2015) 14. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007) 15. Alikhanov, A.A.: A priori estimates for solutions of boundary value problems for fractionalorder equations. Differ Equ. 46, 660–666 (2010) 16. Naifar, O., Ben Makhlouf, A., Hammami, M.A.: Comments on “ittag-Leffler stability of fractional order nonlinear dynamic systems. Automatical 75, 329 (2017) 17. Cong, N.D., Doan, T.S., Tuan, H.T.: Asymptotic stability of linear fractional systems with constant coefficients and small time dependent perturbations arXiv:1601.06538

Observers and Observability—Theory and Literature Overview Assaad Jmal, Omar Naifar, Abdellatif Ben Makhlouf, Nabil Derbel, and Mohamed Ali Hammami

Abstract In this chapter, an overview of Observers and Observability is given. To begin with, a general introduction is introduced then, observers and Observability for continuous-time linear systems is given. Finally, some observers design for certain fractional order systems is presented. Keywords Observers · Observability · Fractional order systems

1 Introduction An observer can be regarded as a technical structure that allows reconstructing the real states of dynamical systems. Designing observers is one of the prominent tasks in the control theory, as the knowledge of system states is necessary to solve many related problems. In most practical cases, not all the physical system states can be determined by direct measurement. Note that, if a system is observable, then it is possible to fully reconstruct its states from its accessible signals (inputs and outputs). Formally, a system is said to be observable if, for a possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs. Less formally, this means that from the system outputs, it is possible to determine the behavior of the entire system. In this chapter, the authors expose a state of the art, in relation to observability and observers, for linear and nonlinear systems, integer-order and fractional-order systems.

A. Jmal · O. Naifar (B) · N. Derbel Engineering National School, Electrical Engineering Department, Control and Energy Management Laboratory (CEM Lab), Sfax University, BP 1173, 3038 Sfax, Tunisia A. Ben Makhlouf Jouf University, Aljouf, Saudi Arabia A. Ben Makhlouf · M. A. Hammami Faculty of Sciences of Sfax, Department of Mathematics, Sfax University, BP 1171, 3000 Sfax, Tunisia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Naifar and A. Ben Makhlouf (eds.), Fractional Order Systems—Control Theory and Applications, Studies in Systems, Decision and Control 364, https://doi.org/10.1007/978-3-030-71446-8_2

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2 Observers and Observability for Continuous-Time Linear Systems 2.1 Observability To estimate the state of a linear process, the observability is a necessary condition regardless of the used technique. Consider a linear system, described by the following equation: 

x(t) ˙ = Ax(t) + Bu(t) y(t) = C x(t)

(1)

where x(t), u(t) et y(t) are vectors of dimensions n, m et p, that represent respectively the state, the input and the output vectors. The observability of such system corresponds to the fact that the state vector x(t) can be reconstructed in the time window [to ; t1], from the knowledge of the accessible signals u(t) ety(t). Definition 1 Observability [1, 2] System (1) is observable, if for the instant t0 , there exists a constant instant t1 such that the knowledge of y(t0 , t1 ) and u(t0 , t1 ) allows to determine, in a unique way, the state x(t1 ), and this ∀u ∈ U where U is the input set. The observability matrix O is given by:  T O = C C A C A2 . . . C A(n−1)

(2)

A necessary and sufficient condition for the observability of the system is the regularity of the observability matrix O. In other words, the observability is guaranteed when the rank of the observability matrix is maximum: rang(O) = n

(3)

If a linear system is completely observable, then it is globally observable, that is, all components of the system state vector are observable, and therefore can be reconstructed by an observer. If the system is nonlinear, we distinguish global observability from local observability.

2.2 Linear Observers In the literature, different techniques have been adopted to design linear observers, that is, observers for linear plants. Herein, three of the most used kinds of observers

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Fig. 1 Structural diagram of an observer

by researchers are exposed. The first is the full-order Luenberger observer, the second is the reduced-order Luenberger observer and the third is the Kalman filter.

2.2.1

The Luenberger Observer

In [3, 4], the authors have proposed a state reconstructor for linear systems. In the following, we present the Luenberger observer for linear systems.

The Full-Order Luenberger Observer From a deterministic model of the system defined in (1), a Luenberger observer can be designed. In the early 1970s, this theory was presented by David G. Luenberger [3, 4] for linear models (Fig. 1). The structure of the continuous-time Luenberger observer, for the system described by Eq. (1) is given by: 

.   xˆ = A x(t) ˆ + Bu(t) + L y(t) − yˆ (t) yˆ (t) = C x(t) ˆ

(4) 

In this equation, the corrective term clearly appears: L(y(t) − y (t)). This term uses the output reconstruction error (the difference between the measured outputs and those estimated). The correction gainL, called the observer gain, is to be determined. The state estimation error ε(t) is defined by: 

ε(t) = x(t) − x (t)

(5)

The state estimation error dynamics are then governed by ε˙ (t) = (A − LC)ε(t)

(6)

To guarantee the observer stability, one should compute L such that the eigenvalues of matrix (A − LC) would all have negative real parts. In this case, the estimation error asymptotically converges to 0 when times tends to infinity the eigenvalues of matrix (A − LC) can be set arbitrarily if and only if the pair (A, C) satisfies the

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Fig. 2 Structural diagram of the Luenberger observer

observability condition given by Eq. (3). Figure 2 shows the structural diagram of the Luenberger observer.

Reduced-Order Luenberger Observer A reduced-order observer is an observer that reconstructs a part of the state vector, while a full-order observer is an observer that completely reconstructs the state vector. These two types of observers come from the general form of Luenberger [4–6]. If a part of the state vector is already measured through sensors, then it is not necessary to use a full observer. In the following, is a summary of the reduced-order observer theory. Consider system (1), in which the output matrix is of the following form: C = ([0] I )

(7)

The state vector x(t) can be decomposed into two parts: the non-measured part v(t), and the measured oney(t). Equation (1) can then be rewritten as follows:     ⎧ v(t) B1 v˙ (t) A11 A12 ⎪ ⎪ + u(t) = ⎨ y(t) A21 A22 B2 y˙ (t)

 ⎪ v(t) ⎪ ⎩ y(t) = ([0] I) y(t)

(8)

Then: 

.

v˙ (t) = A11 v(t) + A12 y(t) + B1 u(t) y˙ (t) = A21 v(t) + A22 y(t) + B2 u(t)

(9)

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Fig. 3 Functional diagram of a linear system with reduced-order observer

Which gives the following equation: 

v˙ (t) = A11 v(t) + (A12 y(t) + B1 u(t)) y˙ (t) − A22 y(t) − B2 u(t) = A21 v(t)

(10)

System (10) is equivalent to (11), under: S(t) = ( y˙ (t) − A22 y(t) − B2 u(t)) and E(t) = (A12 y(t) + B1 u(t)). 

v˙ (t) = A11 v(t) + E(t) S(t) = A21 v(t)

(11)

Figure 3 represents the functional diagram of the reduced-order observer: The observer for system (11) is given by:   . ˆ vˆ (t) = A11 vˆ (t) + E + K o S(t) − S(t) ˆ = A21 vˆ (t) S(t)

(12)

Denote ε1 (t), as the reduced state estimation error: 

ε1 (t) = v(t) − v (t)

(13)

The estimation error dynamics are then given by: ε˙1 (t) = v˙ (t) − v˙ (t) = (A11 − K o A21 )ε1 (t) 

(14)

The observer (12) is stable if is only if the matrix ( A11 − K o A21 ) is stable. The observer (12) can be represented as (15):

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Fig. 4 Functional diagram of a linear system with a reduced-order observer

v˙ (t) = A11 v (t) + (A12 y(t) + B1 u(t)) + K o ( y˙ (t) − A22 y(t) − B2 u(t) − A21 v (t)) (15) 





Or v˙ (t) = (A11 − K o A21 )v (t) + (A12 y(t) + B1 u(t)) + K o ( y˙ (t) − A22 y(t) − B2 u(t)) (16)





K o is the gain of the reduced-order observer. One proposes to carry out the following change of variable: ·

Z (t) = vˆ (t) − K o y(t) < t ⇔ Z˙ (t) = vˆ (t) − K o y˙ (t)

(17)

Thus, one can obtain: Z˙ (t) + K o y˙ (t) = (A11 − K o A21 )(Z (t) + K o y(t)) + (A11 y(t) + B1 u(t)) + K o y˙ (t) + K o (−A22 − B2 u(t))

(18)

After simplification, one obtains the following Eq. (19) for the reduced-order observer. Figure 4 gives the detailed block diagram of the system and the reducedorder observer. ⎧ ⎨ Z˙ (t) = (A11 − K o A21 )Z (t) + ((A11 − K o A21 )K o + A12 − K o A22 ) y(t) + (B1 − K o B2 )u(t) (19) ⎩ vˆ (t) = Z (t) + K o y(t)

Observers and Observability—Theory and Literature Overview

2.2.2

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Kalman Filter

The Kalman filter is used for stochastic models, it makes it possible to take into account modeling errors and measurement noise. In [7], Kalman has introduced an approach based on a discrete state representation. Consider the stochastic system, described by the following equation: 

x(t) ˙ = Ax(t) + Bu(t) + Wc (t) y(t) = C x(t) + Vc (t)

(20)

where Vc (t) and Wc (t) are respectively the output noise and the state noise, with dimensions p and n To obtain an optimal estimate by the Kalman filter, the noises Vc (t) and Wc (t) must be white, Gaussian, centered and decorrelated from the statex( t ). Each noise is characterized by its covariance matrix denoted Q c for Wc (t) and Rc for Vc (t), such that: ⎧ E{Wc (t)} = 0,  E{Vc (t)} = 0 ⎪ ⎪ ⎨  E Wc (t1 )WcT (t2 ) = Q c δ(t2 − t1 ) (21) ⎪ E V (t )V T (t ) = Rc δ(t2 − t1 ) ⎪ ⎩  c 1 Tc 2  E Wc (t1 )Vc (t2 ) = 0 where E is the covariance function, δ is Dirac’s impulse. The Kalman filter is defined by: 

·   x(t) ˆ = A x(t) ˆ + Bu(t) + K (t) y(t) − yˆ (t) yˆ (t) = C x(t) ˆ

(22)

The filter optimal gain K(t) is defined by: K (t) = A P(t)C T Rc (t)−1

(23)

where P(t) = E{εε T } is the covariance of the estimation error, solution of the differential Eq. (24): ˙ P(t) = A P(t) + P(t)A T + Q c − K (t)Rc (t)K T (t)

(24)

The estimation error dynamics are given by: ·

ε˙ ( t ) = x(t) ˙ − x(t) ˆ = (A − K (t)C)ε( t ) + Wc − K (t)Vc The initial state in t0 is considered a random variable, one can then write:

(25)

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  E{x(t0 )} = m 0 et E ε(t0 )ε T (t0 ) = P0

(26)

The filter converges when matrices Q c and Rc are defined positive, bounded, the model is observable and the initial covariance matrix of the error P0 is definite positive.

2.2.3

Observers for Uncertain Linear Systems

The system uncertainties that appear in matrixA, can have many forms: • Unstructured parametric deterministic uncertainty: The state-space equation has then the following form: x(t) ˙ = (A + δ A)x(t) + Bu(t)

(27)

where δA represents the uncertainty in A, supposed to be bounded, such that δA < a

(28)

where a is a strictly positive scalar. • Unstructured parametric stochastic uncertainty: The plant is described by: x(t) ˙ = Ax(t) + Bu(t) + H (x)w(t)

(29)

where w(t) is a white noise. In this case, the influence of uncertainties is reflected in the elements of H . Uncertainties can be seen as a random disturbance, occurring over a wide frequency band, of the matrix A [8]. • Deterministic parametric structured uncertainty: In this case, every parameter of the system will be affected. The structure of the uncertainties is well known. The system has then the following form: x(t) ˙ = (A + δ A)x(t) + Bu(t)

(30)

where δA =

q  i=1

ki E i

(31)

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The ki are uncertain parameters which may vary. The structure of the matrices E i , which indicates the way in which the uncertainties act, is known. These uncertainties are due to variations in parameters or to the imprecision of parameter estimates. This model has been used for stability studies in [9, 10].

3 Observability and Observers for Nonlinear Systems 3.1 Observability A nonlinear system can be presented by the general Eq. (32), where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the input vector and y(t) ∈ R p is the output vector. 

x(t) ˙ = f (x(t), u(t)) y(t) = h(x(t))

(32)

Unlike the linear case, the observability problem for nonlinear systems is more complicated, as it can depend on the input applied. The observability of nonlinear models is characterized using the notion of indistinguishability [11]. Definition 2 Indistinguishability [11] Two initial states x(t0 ) = x1 and x(t0 ) = x2 are said to be indistinguishable for system (57), if ∀t ∈ [t0 ; t1 ], the outputs y1 (t) and y2 (t) are equivalent regardless of the adoptable input u(t). Definition 3 Observability The nonlinear system (32) is said to be observable if it does not admit an indistinguishable peer. In addition, a system is said to be observable if there are no distinct initial states which cannot be distinguished by examining the output of the system. Definition 4 Observability space [11] Consider system (32). The observability space, denotedO, is the smallest vector subspace of a function of Rn with values in the output space, enveloping the outputs h 1 ; h 2 ; . . . . ; h p and which is closed under the act of the Lie derivation with respect to the vector field f (x; u), The space of the elements differentials of O, is denoted d O Definition 5 [12] The space d O(x0 ) (evaluated inx0 ) characterizes the weak local observability in x0 of system (32). System (32) is said to satisfy the observability rank condition in x0 , if: dim d O( x0 ) = n

(33)

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System (32) satisfies the observability rank condition, if: ∀x ∈ R n ,

dim d O(x) = n

(34)

Definition 6 System (32) is generically observable if: dim O = n

(35)

This condition is the generic observability rank condition.

3.2 The Input Problem for Observability Definition 7 Universal input [12] An input u is universal in [0, t] if for every couple of distinct initial statesx1 = x2 , there exists τ ∈ [0, t] such that the corresponding outputs y(τ , x1 ) et y(τ, x 2 ) are different. A universal input is therefore an input which makes it possible to discern any pair of initial states by examining the output. Definition 8 Singular input [12] A non-universal input is called a singular input. There are particularities of inputs, "sufficiently universal", to allow the synthesis of observers. One of these inputs is the so-called “regularly persistent input”: Definition 9 Regularly persistent input [12, 13] An admissible input (measurable and bounded) u is said to be regularly persistent for the system (1.57) if it exists T > 0, α > 0 and such that ϒ(t, T, u) ≥ α for t ≥ t0 . Definition 10 Uniformly observable system In the case where the system has no singular inputs, it is called a uniformly observable system.

3.3 Nonlinear Observers For linear systems, Luenberger observers or Kalman filters represent a systematic solution. For nonlinear systems, the problem of reconstructing the states is naturally more challenging. Extensive works have been developed in the literature to reconstruct the nonlinear system states, for different classes of continuous-time systems, using various types of observers. In the following, some types of nonlinear observers are stated, together with stating some relevant literature works.

Observers and Observability—Theory and Literature Overview

3.3.1

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High-Gain Observers

The high-gain observer is a procedure for estimating state variables of nonlinear systems. Its convergence is theoretically proven for nonlinear systems, without requiring neither linearization nor approximation.

Canonical Form for Multi-Input, Single-Output, Affine Input and Uniformly Observable Systems The authors, in [14], have characterized a class of nonlinear multi-input, singleoutput, affine input and uniformly observable systems by a canonical form of uniform observability. They developed an exponentially converging observer whose speed of convergence can be adjusted. They treated class of systems, in [14], is: 

Z˙ = f˜(Z ) + g(Z ˜ )u y = h(Z )

(36)

where Z ∈ Rn , u ∈ Rl et y ∈ R are respectively the state, the input and the output vectors. f˜ : Rn → Rn , g˜ : Rn → Rn × Rl et h : Rn → R are linear functions of the state. Assume that system (36) is observable ∀ u(t) ∈ U where U is the set of inputs,  T and that x = φ(Z ) = h(Z ), L f¯ h(Z ), . . . . . . . . . ., L n−1 h(Z , where L f¯ h(Z ) is ) f¯ the Lie derivative along f¯ in point Z . With the change of variablex = φ(Z ), one can write: 

⎛ where:

A0

=

x˙ = A0 x + f (x) + g(x)u y = C0 x

⎞ 1 ... 0 .. . . .. ⎟ . . .⎟ ⎟,C0 0 ... 1⎠ 0 0 ... 0

0 ⎜ .. ⎜. ⎜ ⎝0

=

[1 0 . . . . 0] and

(37)

f (x)

=

   T 0, . . . , 0, L nf¯ h φ −1 (x) . Thanks to the uniform observability of system (37), the vector of nonlinear functions g has the following structure: g(x) = (g1 (x1 ), g2 (x1 , x2 ), . . . , gn (x1 , . . . , xn ))T

(38)

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This structure of the nonlinear function vector g is a necessary and sufficient condition for the uniform observability of the nonlinear system. System (36) constitutes the canonical form of uniform observability.

High-Gain Observer of Uniformly Observable Systems In [14], the authors have proposed a high gain observer for system (37), under the following two hypotheses: H1 : The nonlinear functions f and g are globally Lipschitz. H2 : The input u always remains inside a bounded set. The high-gain observer is defined through the following equation: 

·       xˆ = A0 xˆ + f xˆ + g xˆ u + S −1 (θ )C0T y − yˆ yˆ = C0 xˆ

(39)

where S(θ ) is the positive definite solution of the differential equation defined by: 0 = −θ S(θ ) − A0T S(θ ) − S(θ )A0 + C0T C0

(40)

where θ is a design parameter. Note that the observer (39), defined in [14], has been further discussed in a posterior research work [15], in which an improved version of [14] has been suggested.

3.3.2

Adaptive Observers

The purpose of the adaptive observer is to simultaneously estimate the state and the unknown parameters of the system. A convenient adaptation law is to be used together with the observer. This technique can be applied for either linear or nonlinear systems. Extensive works have been done, in the literature, to design adaptive observers for nonlinear systems, in different contexts. One of these contexts is the fault estimation one. Indeed, faults to be reconstructed are simply unknown parameters in the state-space system representation. One of the relevant related works is [16], where the authors have estimated both types of actuator and component faults for a nonlinear class of one-sided Lipschitz systems. In [16], the authors have considered the following class of systems:  x(t) ˙ =

A0 +

s  k=1

y(t) = C x(t),





θk Ak x(t) + B0 +

p 

 θk Bk u(t) + Dϕ(F x, u)

k=s+1

(41)

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where x ∈ Rn is the state, u ∈ Rm is a uniformly continuous input, y ∈ Rq is the output, ϕ(F x, u) is a nonlinear function that represents the system nonlinearity [16], A0 , Ak ∈ Rn×n , B0 , Bk ∈ Rn×m , C ∈ Rq×n , D and F two known matrices of appropriate dimensions. The parameter vector θc = [θ1 , .., θs ] appearing in the state distribution matrix, represents potential component faults, while the parameter vector  represents potential θa = θs+1 , .., θ p appearing in theinput distribution matrix,  actuator faults. One can writeθ = θ1 , .., θs , θs+1 , .., θ p . In such a situation, the system is described otherwise:  x(t) ˙ =

A0 +

p 





θk Ak x(t) + B0 +

k=1

p 

 θk Bk u(t) + Dϕ(F x, u)

k=1

y(t) = C x(t),

(42)

where Ak = 0 for k ∈ [s + 1, p] and Bk = 0 for k ∈ [1, s]. The parameter vector θ ∈ R p takes the well-known constant value θ = θh as long as the system behaves in the healthy mode. With some specific faulty mode, θ takes another constant value θ f , but unknown. In fact, the goal of this work is to estimate θ f under some conditions. The developed adaptive observer in [16] is a Luenberger-like observer, given by Eq. (43).  ˙ˆ = x(t)

A0 +

p 





θˆk Ak x(t) ˆ + B0 +

k=1

p 

 θˆk Bk u(t)

k=1

    + Dϕ F x, ˆ u + LC x(t) ˆ − x(t)

(43)

The state estimation error, the output estimation error and the fault estimation error are respectively e(t) = x(t) ˆ − x(t), e y = yˆ (t) − y(t) and eθ (t) = θˆ (t) − θ . Then the observer error dynamics are given by:  e(t) ˙ =

A0 +

p  k=1

 θk Ak + LC e(t) +

p 

  eθk Ak x (t) + Bk u(t) + D ϕ 

(44)

k=1

where  

ϕ = ϕ F x, ˆ u −ϕ(F x, u). The design goal is to find an observer gain matrix L such that estimates of x and θ f converge asymptotically. To do it and as mentioned above, an adaptation law is to be set. The authors, in [16], have defined the following one:   θ˙ˆk = −σk e Ty PC Ak xˆ + Bk u where σk is a positive scalar, and P is a symmetric definite positive matrix.

(45)

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3.3.3

Fractional-Order Nonlinear Observers

State-space representations, given above, (as for Eqs. (1), (8) and (20)) represent classical integer-order systems, where the state vector x(t) is derived exactly one time. Yet, the integer-order calculus is inconvenient for several real-world systems, whose actual dynamics contain fractional (non-integer) derivatives. In order to accurately model such systems, the fractional-order differential equations are to be used. For example, financial systems [17] and electromagnetic systems [18] have been successfully modelled using the fractional-order calculus. Note that, in the last years, the use of fractional-order equations in the stability theory has distinctly risen [19, 20]. A nonlinear fractional-order system can be described by the following general equation: C

Dtα0 ,t x(t) = Ax + Bu + f (x, u) y(t) = C x(t)

(46)

where x ∈ Rn is the state, u ∈ Rm is the input, y ∈ Rq is the output, A ∈ Rn×n ,B ∈ Rn×m , C ∈ Rq×n and f is a function representing the system nonlinearity. The operator C Dtα0 ,t represents the Caputo derivation concept, whose definition is the following: C

Dtα0 ,t x(t)

1 = (m − α)

t (t − τ )m−α−1

dm x(τ )dτ, (m − 1 < α < m) dτ m

t0

where α is the fractional derivation order. When 0 < α < 1, then the Caputo fractional derivative of order α of x(t) reduces to: C

Dtα0 ,t x(t)

1 = (1 − α)

t

(t − τ )−α

d x(τ )dτ dτ

(47)

t0

Further concepts, tools and definitions related to fractional-order calculus and fractional-order systems can be found in [21]. In the last decades, a particular interest has been given by researchers to investigate various control-theory queries, for fractional-order systems. This is the case for the observer design problem. In the rest of this sub-section, two interesting literature works, about concepting observers for nonlinear fractional-order systems, are briefly presented.

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High-Gain-Like Observer for Fractional-Order Nonlinear Systems In [22], the authors have developed a high-gain-like observer for different classes of nonlinear fractional-order systems: Lipschitz systems, One-Sided Lipschitz systems and Quasi-One-Sided Lipschitz systems. The class of systems, treated in [22], is the one given by (46). The authors have defined the following observer (48), under assumption (40). C

    Dtα0 ,t x(t) ˆ = A xˆ + Bu + f x, ˆ u − β S −1 C T C xˆ − y , t ≥ t0 , β ≥ 1

(48)

For each type of nonlinearity [22], the authors have given a theorem that ensures the stability of the state estimation error origin. It is noteworthy to mention that, within the fractional-order context investigated in [22], a stability notion dedicated to Caputo fractional calculus has been used in [22], which is the Mittag–Leffler stability. Theorem 1 [23] Let x = 0 be an equilibrium point for system (46). Let V : [0, ∞)× Rn → R be a continuously differentiable function and locally Lipschitz with respect to x such that. μ1 xc ≤ V (t, x) ≤ μ2 xcd C

Dtα0 ,t V (t, x(t)) ≤ −μ3 xcd

where t ≥ t0 ,x ∈ Rn , α ∈ (0, 1), μ1 , μ2 , μ3 , c and dare arbitrary positive constants. Then x = 0is globally Mittag–Leffler stable.

Luenberger-Like Observer for Fractional-Order Nonlinear Systems In [24], the authors have developed a Luenberger-like observer to reconstruct the states for fractional-order nonlinear systems, using the conformable fractional derivative concept [24]. In that study, the authors have investigated both a healthy operating case and a faulty operating case. The system under consideration is the one given by (49), with a conformable derivative order 0 < α ≤ 1: Ttα0 x(t) = Ax(t) + Bu(t) + ϕ(x(t)) + E f y(t) = C x(t)

(49)

where x ∈ Rn is the state, u ∈ Rm is the input, y ∈ R p is the output, f ∈ Rq represents possible constant actuator faults affecting the system. A ∈ Rn×n ,B ∈ Rn×m , E ∈ Rn×q and C ∈ R p×n are known constant matrices. The function ϕ(x(t)) represents the nonlinear part of the system, which has been supposed to be globally

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Lipschitz. The operator Ttα0 represents the Conformable derivation concept, whose definition is the following. Definition 11 [25] Given a function h defined on [a, + ∞), the conformable fractional derivative starting from a of h , with a derivative order α is defined as,   h t + ε(t − a)1−α − h(t) lim α Ta h(t) = ε→0 ε for all t > aand 0 < α ≤ 1. In the healthy operating case, the authors have defined the observer given by (50), while in the faulty operating case, they have defined the observer given by (51). 

  Ttα0 x (t) = Ax (t) + Bu(t) + ϕ x + L(y(t) − y (t)) y (t) = C x (t) ⎧   ⎪ ⎨ Ttα0 x (t) = Ax (t) + Bu(t) + ϕ x + E f (t) +L(y(t) − y (t)) ⎪ ⎩ α Tt0 f (t) = G(y(t) − y (t)) 

























(50)

(51)





where x (t) is the estimate of x(t), y(t) is the estimate of y(t) and L is the observer gain matrix to be designed, G is a matrix to be designed and f (t) is an internal reconstructed signal, needed in order to prove: x (t) − x(t) → 0 as t → ∞. 



3.3.4

Unknown Input Observers

In real practice situations, dynamic systems–either linear or nonlinear ones—are usually subject to some unknown inputs. One of the most efficient types of observers to such systems, is the so-called Unknown Input Observer. Indeed, Unknown Input Observers are well known for their ability to decouple the state estimation error dynamics from these unknown inputs, and consequently, the state estimation error can be proved to converge exactly to zero, even in the presence of such unknown inputs. Several research works have focused on designing unknown input observers for different classes of nonlinear systems. In the following, the authors present one interesting typical study [26], about designing unknown input observers. The interest of [26] is that it tackles an unknown input observer, which has the property of an adaptive observer (existence of an adaptation law), and that [26] tackles a fractionalorder system (a generalization of classical integer-order systems). In [26], the authors have considered the following nonlinear fractional-order system subject to unknown inputs, with a derivative order α ∈ (0, 1]: C

Dtα0 ,t x(t) = Ax(t) + Bu(t) + Ed(t) + (x)

Observers and Observability—Theory and Literature Overview

25

y(t) = C x(t) + D f

(52)

where C Dtα0 ,t is the Caputo fractional derivative operator, x(t) ∈ Rn is the state, u(t) ∈ Rm is the input, y(t) ∈ R p is the output, f ∈ Rq represents possible constant sensor faults and d(t) is an unknown input, assumed to be bounded. (x) is a continuous nonlinear function of x(t). A ∈ Rn×n , ∈ Rn×m , C ∈ R p×n , D ∈ R p×q and E are well known constant matrices. It is assumed that E and C T have full column rank. In order to design a nonlinear unknown input observer taking into consideration the structure of the nonlinearity, the term (x) is decomposed as follows: (x) = F1 1 (x) − F2 2 (H x),

(53)

where 1 (.) and 2 (.) : Rr → Rr are continuous functions and H ∈ Rr ×n . Let the −

−

−

matrix E and the vectors d and  1 (x) be, such that:

 − − − Ed(t) + F1 1 (x) = E d + 1 (x)

(54)

So, the model (52) can be rewritten as follows: C

  ¯ 1 (x) − F2 2 (H x) Dtα0 ,t x(t) = Ax(t) + Bu(t) + E¯ d¯ + 

y(t) = C x(t) + D f

(55)

−

Matrix E is determined by the choice of 1 (.) and 2 (.). The nonlinear part in system (52), treated in [26], is a monotone nonlinearity. This is expressed through function 2 (.), that satisfies the following multivariable slope property: ∂2 (s) + ∂s

∂2 (s) ∂s

T ≥ μIr , ∀s ∈ Rr

(56)

where μ is a constant scalar. The following unknown input observer is used: C

  Dtα0 ,t z(t) = N z(t) + L y(t) − T F2 2 H xˆ + T Bu(t)

x(t) ˆ = z(t) + M y(t)

(57)

where N = T A − K C,

L = K + N M,

T = In − MC,

ˆ e y (t) = K ∈ Rn× p and M ∈ Rn× p are design matrices. Let e(t) = x(t) − x(t), ∼



y(t) − yˆ (t) and f (t) = f − f (t). The adaptation law given by:

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Fig. 5 Observer structure with the adaptation law 

C

∼

Dtα0 ,t f (t) = Ge y = GCe + G D f

(58)

where G is a design matrix. Figure 5 [26] illustrates the structure of the unknown input observer, together with the adaptation law.

4 Conclusion In this chapter, the authors have presented a theoretical and literature overview on observability and observers for both classes of linear and nonlinear continuous-time systems. For linear systems, the authors have treated: the Luenberger type observers of reduced order, full order and the Kalman filter. For nonlinear systems, the authors have exposed: the adaptive observer, the high-gain observer and the unknown input observer. The authors have exposed some literature works about designing observers for nonlinear fractional-order systems, as well.

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References 1. Kailath, T.: In: Linear Systems. Prentice-Hall, Englewood, New Jersey (1980) 2. Sontag, E.D.: In: Mathematical Control Theory—Deterministic Finite Dimensional Systems. Springer (1990) 3. Luenberger, D.G.: Observing the state of linear system. IEEE Trans. Military Electron, 8, 74–80 (1964) 4. Luenberger, D.G.: An introduction to observers. IEEE Trans. Automatic Control 16(6), 596-602 (1971) 5. Muller, P., Diduch, C.: On the observer design for descriptor system. In: 30th Conference on Dexision and Control, pp 1960–1961. (1991) 6. Duan, G., Wu, A., Hou, W. et al.: Parametric approach for luenberger observers for descriptor linear systems. Bullet. Polish Acad. Sci. 55(1), 15–18 (2007) 7. Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. Series, 82D, 35–45 (1960) 8. Camozzi, P.: Systèmes a modélisation incertains : commande par placement de pole. thèse, Université Paul Sabatier de Toulouse, France (1995) 9. Gao, Z., Ho, D.:Explicit asymmetric bounds for robust stability of continuous and discret-time systems. IEEE Trans. Autom. Control. 38, 332–335 (1993) 10. Bien, Z., Kim, J. et al.: A robust stability bound of linear system with structured uncertainty. IEEE Trans Autom Control. 37, 1549–1551 (1992) 11. Hermann, R., Krener, A.J.: Nonlinear controllability and observability. IEEE Trans. Autom. Control 22, 728–740 (1977) 12. Besançon, G., Bornard, G., Hammouri, H. et al.: Observer synthesis for a class of nonlinear control systems. Europ. J. Control 2(3), 176–192 (1996) 13. Hammouri, H., Deleon, J.: Observer synthesis for state-afne systems. In: Proceedings 29th IEEE Conference on Decision and Control, Honolulu, Hawaii, pp. 784–785. (1990) 14. Gauthier, J.P., Hammouri, H., Othman, S.: A simple observer for nonlinear systems application to bioreactors. IEEE Trans. Autom. Control 37(6), 875–880 (1992) 15. Busawon, K.K., Leon-Morales, J.D.: An observer design for uniformly observable non-linear systems. Int. J. Control 73(15), 1375–1381 (2000) 16. Jmal, A., Naifar, O., Ben Makhlouf, A., Derbel, N., Hammami, M.A.: Adaptive estimation of component faults and actuator faults for nonlinear one-sided Lipschitz systems. Int. J. Robust Nonlinear Control 30(3), 1021–1034 (2020) 17. Laskin, N.: Fractional market dynamics. Phys. A 287(3), 482–492 (2000) 18. Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44(4), 554–566 (1996) 19. Naifar, O., Ben Makhlouf, A., Hammami, M.A.: Comments on ‘Lyapunov stability theorem about fractional system without and with delay’. Commun. Nonlinear Sci. Numerical Simul. 30(1–3), 360–361 (2016) 20. Naifar, O., Ben Makhlouf, A., Hammami, M.A.: Comments on “Mittag-Leffler stability of fractional order nonlinear dynamic systems [Automatica 45(8), (2009) 1965–1969]”. Automatica 75, 329 (2017) 21. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Diferential Equations. Elsevier, New York (2006) 22. Jmal, A., Naifar, O., Ben Makhlouf, A., Derbel, N., Hammami, M.A.: On observer design for nonlinear caputo fractional-order systems. Asian J. Control 20(4), 1533–1540 (2018) 23. Li, Y., Chen, Y.Q., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009) 24. Jmal, A., Elloumi, M., Naifar, O., Ben Makhlouf, A., Hammami, M.A.: State estimation for nonlinear conformable fractional-order systems: a healthy operating case and a faulty operating case. Asian J Control 22(5), 1870–1879 (2020)

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25. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014) 26. Jmal, A., Naifar, O., Ben Makhlouf, A., Derbel, N., Hammami, M.A.: Robust sensor fault estimation for fractional-order systems with monotone nonlinearities. Nonlinear Dynam. 90(4), 2673–2685 (2017)

A Brief Overview on Fractional Order Systems in Control Theory Assaad Jmal, Omar Naifar, Abdellatif Ben Makhlouf, Nabil Derbel, and Mohamed Ali Hammami

Abstract In this chapter, an overview of fractional-order systems, nonlinear systems and observer-related queries is given. To begin with, fractional-order systems are introduced, and a state of the art in relation with these systems in the control theory is given. Then, different classes of nonlinear systems are presented, with a literature survey on the observer design problem for these nonlinear systems. Next, we talk about the fault diagnosis problem in engineering, focusing on the fault estimation task. A state of the art of the fault estimation works in the literature is given. The final part of this chapter deals with the observer-based control of dynamic systems. The two main kinds of control, which are the stabilization and the model reference control, are introduced, and a literature survey is presented. Keywords Fractional-order systems · Nonlinear systems · Observers · Fault estimation · Observer-based control

1 Introduction Like the other sciences, the control theory has been subject, in the last decades, to a revolution in relation with the amount of research works done all over the world. Thanks to the advances in applied mathematics, several new practical problems are being solved, and new technological solutions and applications are being achieved. One of the mathematical fields of research that have attracted several researchers is the fractional-order calculus. A. Jmal · O. Naifar (B) · N. Derbel Engineering National School, Electrical Engineering Department, Control and Energy Management Laboratory (CEM Lab), Sfax University, BP 1173, 3038 Sfax, Tunisia A. Ben Makhlouf Jouf University, Aljouf, Saudi Arabia A. Ben Makhlouf · M. A. Hammami Faculty of Sciences of Sfax, Department of Mathematics, Sfax University, BP 1171, 3000 Sfax, Tunisia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Naifar and A. Ben Makhlouf (eds.), Fractional Order Systems—Control Theory and Applications, Studies in Systems, Decision and Control 364, https://doi.org/10.1007/978-3-030-71446-8_3

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Advances in the fractional-order calculus have lead to the development of researches related to a “new” class of systems, namely fractional-order systems. On the other hand, several pertinent questions, like state estimation fault diagnosis and control, are being notably investigated in the last few years. The scope of this chapter is to further present fractional-order systems, nonlinear systems, observers and their applications in estimating faults and control. This chapter includes, as well, a state of the art of various works done in the literature, in relation with: fractional-order systems, observers for different classes of nonlinear systems, fault estimation and control problems.

2 Stability Theory Consider a general nonlinear system, as follows: 

x(t) ˙ = g(x(t), t) x(t0 ) = x0

(1)

where x(t) ∈ Rn is the state vector and x0 is the initial condition. Definition 1 (Equilibrium point) Let xe ∈ Rn ·xe is an equilibrium point for system (1), if g(xe , t) = 0, ∀t ≥ t0 . Definition 1 means that, in the absence of external influence: the state x = xe does not change in all future time. Definition 2 (Stability) The equilibrium point xe is said to be stable, if for each δ > 0 and any t0 ≥ 0, there is ϑ = ϑ(t0 , δ) such that: x0 − xe ϑ = x(t) − xe  < δ, ∀t ≥ t0 Definition 3 (Attractiveness) Assume that the origin x = 0 is an equilibrium point. Then, this equilibrium point is said to be: (i)

attractive, if there is a neighborhood of the origin U (0) such that:

∀x0 ∈ U (0), x(t) → 0 as t → +∞ (ii)

globally attractive, if

∀x0 ∈ Rn , x(t) → 0 as t → +∞

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Definition 4 (Asymptotic stability) An equilibrium point is asymptotically stable, if it is both stable and attractive. Definition 5 (Uniform stability) The equilibrium point is uniformly stable, if for all δ > 0 there is ϑ = ϑ(δ) such that: x0 − xe ϑ = x(t) − xe < δ, ∀t ≥ t0 Definition 6 (Exponential stability) The equilibrium point x = 0 is exponentially stable, if it is stable and there is a neighborhood of the origin U(0) and λ1 , λ2 > 0 such that: x(t) ≤ λ1 x0 e−λ2 (t−t0 ) ,

∀x0 ∈ U (0), ∀t ≥ t0 ≥ 0

Definition 7 (Positive definite function) Let V : [0, ∞) × Rn → R be a continuously differentiable function. V is called a positive definite function in a region U ⊂ Rn , if V (t, 0) = 0, ∀t ≥ t 0 and V (t, x) > 0, ∀t ≥ t 0 , ∀x ∈ U\{0}. Definition 8 (Lyapunov function) Let U ⊂ Rn and V : [0, ∞) × U → R be a continuously differentiable function. Then: • V is called a large Lyapunov function, if: (i)

V is a positive definite function.

(ii) V˙ (t, x) ≤ 0, ∀x ∈ U • V is called a strict Lyapunov function, if: (ii)

V is a positive definite function.

(ii) V˙ (t, x) < 0, ∀x ∈ U \{0}

3 Fractional-Order Systems 3.1 Motivation Describing a physical system or process by the use of a mathematical model is an essential step before studying one of the control theory issues, like stability, control, state estimation or fault diagnosis. Most of the studied systems in the literature belong to the well known and traditional class of integer-order systems, where an integer-order derivative is used in the describing equations. However, several physical systems are not truly modeled with integer-order differential equations. The reason is that their actual dynamics contain non-integer derivatives. So, in order to accurately describe these systems, the fractional-order differential equations have

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been introduced. Such systems are conventionally called fractional-order systems. Fractional-order systems with a fractional derivative between 0 and 1 correspond to an extension of the classical integer-order ones, so that a broader set of real systems could be covered. In the last few years, a noteworthy growing interest is given by researchers to the fractional calculus, and several real phenomena have been modeled using fractional order equations. For instance, fractional differential equations have been used in modeling some electromagnetic systems [1]. As well, fractional equations have been exploited in describing financial systems [2]. In [3], a heat transfer system is modeled with fractional-order equations. As well, image processing [4] and dielectric polarization [5] have been modeled using the fractional-order calculus.

3.2 Fractional-Order Systems in the Control Theory: A State of the Art Thanks to the development of complex engineering systems and science, the use of fractional equations in different issues of the control theory, such as controllability [6], observability [7] and stability [8, 9] has significantly risen. Recently, the stability of fractional-order systems depending on one parameter has been tackled in [10]. Note that different concepts related to theory and applications of fractional-order systems have been presented in a special issue [11]. In the following of this paragraph, we present a survey on observer design for fractional-order systems. In [12], the authors have proposed a non-fragile observer design for a class of nonlinear fractional-order systems. A study about sliding mode observer design for a class of nonlinear fractional-order systems has been given in [13]. In the last decade, very few works related to the specific task of designing unknown input observers for fractional-order systems have been done. In [14], an unknown input observer has been proposed for fractional-order glucose-insulin system. In [15] and [16], unknown input observers for linear fractional-order systems have been developed. As well, very few attempts have been done to design reducedorder observers for fractional-order systems. Among these attempts, an interesting study about synchronization of nonlinear fractional order systems by means of a PI reduced order observer has been presented in [17]. In [18], the authors have developed an iterative LMI-based reduced-order observer for fractional-order chaos synchronization. A reduced-order observer scheme for linear fractional-order systems can be found in [19]. On another hand, two recent papers have been done in relation with the observer design problem for uncertain fractional-order systems: In [20], the authors have given a sliding mode observer method for a class of uncertain fractional-order nonlinear systems, while in [21], a robust functional observer design for uncertain fractional-order time-varying delay systems has been presented.

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3.3 Preliminaries In this sub-section, useful definitions and results related to the fractional-order calculus, are given. In the next two definitions, the Riemann–Liouville fractional integral and the Caputo fractional derivative are presented. In the literature, other definitions of the fractional-derivative can be found [22, 23]. Definition 9 The Riemann–Liouville fractional integral of order α > 0 is defined as, Itα0 x(t)

1 = (α)

t

(t − τ )α−1 x(τ )dτ

t0

 +∞ (α) = 0 e−t t α−1 dt, where  is the Gamma function generalizing factorial for non-integer arguments. Definition 10 Let a be the integer part of α + 1. The Caputo fractional derivative is defined as, C

Dtα0 ,t x(t) =

t da 1 ∫(t − τ )a−α−1 a x(τ )dτ, (a − α) t0 dτ

where a − 1 < α < a. For 0 < α < 1: the Caputo fractional derivative of order α of x(t) reduces to C

Dtα0 ,t x(t)

1 = (1 − α)

t

(t − τ )−α

d x(τ )dτ dτ

(2)

t0

On the other hand, there exists a frequently used function in the resolution of fractional order systems, named the Mittag–Leffler function. This function is a generalization of the exponential one. In this context, the following definition is presented. Definition 11 The Mittag–Leffler function with two parameters is defined as. E α,γ (z) =

+∞  k=0

zk (kα + γ )

where α > 0, γ > 0, z ∈ C. When γ = 1, one has E α (z) = E α,1 (z), furthermore, E 1,1 (z) = e z . In the following, we present the state-space equation of a general continuous-time nonlinear fractional-order system. This general equation will be used in different

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situations in the next chapters. C

Dtα0 ,t x(t) = Ax + Bu + f (F x, u), t ≥ t0

(3)

where x ∈ Rn is the state vector, u ∈ Rm is the input vector, A ∈ Rn×n , B ∈ Rn×m and F are known matrices and f (F x, u) represents the nonlinear part of the system. Some sufficient conditions for the existence and uniqueness of solutions for fractional differential equations are given in [24, 25]. Now, let as present the definition of global Mittag–Leffler stability: Definition 12 The system (3) is said to be globally uniformly Mittag–Leffler stable if there exist positive scalars b and λ such that the trajectory of (3) passing through any initial state x0 at any initial time t0 evaluated at time t satisfies:  b x(t) ≤ m(x0 )E α (−λ(t − t0 )α ) ,

∀t ≥ t0

(4)

With m(0) = 0, m(x) ≥ 0, m is locally Lipschitz and E α (.) is the Mittag–Leffler function presented in Definition 11. Theorem 1 [26] Let x = 0 be an equilibrium point for the system (3). Let V : [0, ∞) × Rn → R be a continuously differentiable function and locally Lipschitz with respect to x such that. μ1 xc ≤ V (t, x) ≤ μ2 xcd C

Dtα0 ,t V (t, x(t)) ≤ −μ3 xcd

where t ≥ t0 , x ∈ Rn , α ∈ (0, 1), μ1 , μ2 , μ3 , c and d are arbitrary positive constants. Then x = 0 is globally Mittag–Leffler stable. Lemma 1 [27] Let α ∈ (0, 1) and P ∈ Rn×n a constant square symmetric and positive definite matrix. Then the following relationship holds. 1C α  T Dt0 ,t x (t)P x(t) ≤ x T (t)P C Dtα0 ,t x(t) 2

4 Observers for Nonlinear Systems Observers are regarded as very important structures in the control theory, with a fundamental task of reconstructing the system states [28, 29]. In addition, these estimators have been widely used in other advanced tasks, such that fault diagnosis [30–32] and controlers [33, 34]. In the rest of this section, a detailed presentation

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and state of the art about observer design, for three types of nonlinearities is given. We first talk about the Lipschitz nonlinearity.

4.1 The Lipschitz Nonlinearity Up to now, the observer design problem for nonlinear systems has been investigated by several researchers, and various types of nonlinearities have been treated. One of the most famous and traditional nonlinearities, that researchers have extensively tackled, is the so-called Lipschitz condition. The system (3) is said to be globally Lipschitz with respect to x, if there exist a constant r > 0 satisfying:  f (F x1 , u) − f (F x2 , u) ≤ r F(x1 − x2 ),

∀x1 , x2 ∈ Rn ,

(5)

The observer design problem for Lipschitz nonlinear systems has been first considerd by Thau [35]. Then, several research works have been done to solve this problem. In the rest of this paragraph, we give a state of the art of some related works. In [36], the authors have proposed an approach to design adaptive observers for integer-order Lipschitz systems. More recently, Abbazadeh and Marquez [37] have developed a robust H∞ observer for integer-order sampled-data Lipschitz nonlinear systems with exact and Euler approximate models. Recently, two papers have been done to solve the problem of observer design for integer-order Lipschitz systems with time delay: in [29], the authors have tackled the Lipschitz continuous-time systems with unknown time delay, while in [38], the Lipschitz discrete-time systems have been treated. Dealing with Unknown Input Observers (which will be investigated in the next chapters) for integer-order Lipschitz systems, readers can refer to some research studies, for instance [39,40]. In the last decade, the conception of observers for fractional-order Lipschitz systems has been particularly investigated. In [3], a simple observer design scheme for fractional-order nonlinear Lipschitz systems has been presented. In [12], the authors have proposed a non-fragile observer design for Lipschitz fractional-order systems. A study about sliding mode observers for Lipschitz nonlinear fractional-order systems has been given in [13]. In [14], an unknown input observer has been proposed for the nonlinear fractional-order glucose-insulin systems, where the Lipschitz property has been used.

4.2 The One-Sided Lipschitz Nonlinearity We have mentioned in the previous sub-section that the observer design problem for nonlinear systems has been mostly investigated in the literature using the Lipschitz nonlinearity. However, Lipschitz observers present a major limitation: They can usually stabilize the error dynamics with a conveniently small Lipschitz constant. When this constant becomes large, the problem becomes unfeasible [41]. To relax

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this problem, the Lipschitz nonlinearity can be substituted by the one-sided Lipschitz nonlinearity. In fact, the one-sided Lipschitz constant is always smaller than its Lipschitz counterpart [41]. Another advantage of the one-sided Lipschitz property is the following: It has been proved in [41] that the one-sided Lipschitz class of systems is larger than the Lipschitz one, meaning that: using the one-sided Lipschitz nonlinearity, a broader family of nonlinear systems could be covered. Note that this nonlinear property has been first used in the mathematic literature [42, 43], then it has been for the first time introduced to the observer design problem by Hu [44, 45]. After Hu, an increasing interest is being shown by researchers to solving the observerrelated problems for one-sided Lipschitz nonlinear systesms. It is to be noted here, that one of the primal papers dealing with observers for one-sided Lipschitz systems (the paper of Marquez and Abbazadeh [41]) can be regarded as a key-paper, that have stimulated several subsequent research works. In [41], the authors have originally introduced a new nonlinear concept to the observer design problem for onesided Lipschitz systems, namely «the quadratic inner-boundedness». In that way: the considered class of nonlinear systems includes systems which satisfy the one-sided Lipschitz property and the quadratic inner-bounded property, simultaneously. With the following definitions, we present these two nonlinearities: Definition 13 In system (3), the nonlinear function f (F x, u) is said to be globally one-sided Lipschitz with respect to x, if there exist a constant ρ ∈ R, satisfying: f (F x1 , u) − f (F x2 , u), F(x1 − x2 ) ≤ ρF(x1 − x2 )2 , ∀x1 , x2 ∈ Rn

(6)

Definition 14 In system (3), the nonlinear function f (F x, u) is said to be globally quadratically inner bounded with respect to x, if there exist constants β, γ ∈ R, satisfying:  f (F x1 , u) − f (F x2 , u)2 ≤ βF(x1 − x2 )2 + γ  f (F x1 , u) − f (F x2 , u), F(x1 − x2 ), ∀x1 , x2 ∈ Rn

(7)

In the previous definitions: ρ is called the one-sided Lipschitz constant, β and γ are called the quadratic inner-boundedness constants. Note that the One Sided Lipschitz constant ρ can be positive, zero or even negative, unlike the Lipschitz constant which must be positive. On the other hand, any Lipschitz function is also one-sided Lipschitz and qaudratically inner bounded. The converse however is not true [41]. This leads to the following belonging-relation scheme (Fig. 1). In the last few years, various research works dealing with the observer design problem for integer-order one-sided Lipschitz systems have been done. In [46], the authors have proposed a Linear Matrix Inequality (LMI) method to design observers for integer-order one-sided Lipschitz systems. An adaptive observer scheme has been developed in [47] for the same class of systems. Another study about exponential observers for a class of integer-order one-sided Lipschitz stochastic nonlinear systems can be found in [48]. The observer design problem for integer-order descriptor onesided Lipschitz systems has been investigated in two very recent papers [49, 50].

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Fig. 1 Lipschitz, one-sided Lipschitz, and quadratically inner-bounded function sets

Dealing with the specific task of reduced-order observer design for integer-order one-sided Lipschitz systems in the presence of unknown inputs, two other papers can be found [51, 52]. On the other hand, it is prominent to note that designing observers for fractional-order one-sided Lipschitz systems is a fertile area of research. Indeed, only very few works have been done within this specific topic. Readers can refer to two existing papers in the literature: In [53], a non-fragile observer design for fractional-order one-sided Lipschitz systems has been presented, while in [54], the authors have developed a full-order observer and a reduced-order observer for fractional-order one-sided Lipschitz systems.

4.3 The Monotone Nonlinearity As mentioned in the previous sub-section, Lipschitz observer design schemes depend on the Lipschitz constant, and the problem becomes unfeasible when this constant is not sufficiently small. Furthermore, for systems which are only locally Lipschitz: only a local convergence of the estimation error can be obtained. These are two main limitations, when using the Lipschitz nonlinearity definition. In order to overcome these limitations, one can take advantage of another nonlinear condition, namely the monotone nonlinearity. Indeed, if one studies the nonlinear plant (3) using the monotone nonlinearity definition (15), instead of the Lipschitz definition (5), then the design approach will not depend on the Lipschitz constant of the system. Besides, using the monotone nonlinearity, the global convergence of the estimation error is guaranteed, even if the system is not gloabally Lipschitz. Definition 15 In system (3), let s = F x ∈ Rj . The nonlinear function f (F x, u) is said to be a monotone nonlinearity with respect to x, if there exist a constant μ ∈ R, satisfying: ∂ f (s, u) + ∂s



∂ f (s, u) ∂s

T ≥ μI j , ∀s ∈ Rj

(8)

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Observers for systems with monotone nonlinearities have been first introduced by Arack [55, 56], within the integer-order calculus framework. Then, other some works have been done dealing with this topic. For example, in [57], the authors have proposed an unknown input observer scheme for integer-order systems with monotone nonlinearities. In [58], an observer-based actuator fault estimation approach has been developed for the same class of systems. To the best of our knowledge, no research work has been done, dealing with observer design for fractional-order systems with monotone nonlinearities.

5 Fault Diagnosis in Engineering Problems 5.1 Introduction and Motivation Due to an incessant modernization in technology, several manufactured systems, which are becoming very crucial for modern human being, (such as industrial systems and transportation systems), are becoming more and more complex and sophisticated. Naturally, with this rising complexity, risks of malfunctioning increase, which may result in huge material losses and even human losses. It is within this context that improving reliability, safety and availability, is a major issue in the control area, nowadays [59–60]. This explains the fact that dynamic systems diagnosis is becoming a research axis of an increasing importance. Generally speaking, fault diagnosis systems can be divided into three classes [31]: • Fault Detection (FD) systems: a FD system enables to detect the fault in the process as well as its time of occurrence. • Fault Detection and Isolation (FDI) systems: together with detection, a FDI system determines the kind and location of the fault. • Fault Detection, Isolation and Estimation (FDIE) systems: a FDIE system determines, in addition, the size and time behavior of the fault. The final goal of Fault diagnosis is to synthesize a new control law that tolerates these faults, a task well-known in the control theory as « Fault Tolerant Control» (FTC). In the following figure, the different diagnosis techniques that can be used are presented: As it is mentioned in “Fig. 2”, the main two groups of analytical redundancy based diagnosis are model based approaches and computational-intelligence based approaches. Note that, since model based approaches rely on the mathematical model of the system, their reliability is directly related to the degree of model accuracy. It is for this reason that their major limitation is their performance degradation for complex and uncertain systems. A main advantage of model based methods, however, is that their online implementation is easier.

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Fig. 2 Diagnosis techniques

5.2 Fault Classification Faults that may happen in real systems include the following types:

5.2.1

Sensor Faults

Sensor faults denote every kind of malfunctioning that can affect one or several sensors in the process. They are the cause of bad image of the real system outputs. Different types of sensor faults can be distinguished. The most common ones are: bias, freezing, drift, loss of accuracy and calibration error (see Fig. 3).

5.2.2

Actuator Faults

Actuator faults are related to the operative part of the system. With these faults, we have a partial or total loss of one or more actuators. Different types of actuator faults can be distinguished, for instance: lock in place, float, hardover and loss of effectiveness (see Fig. 4).

5.2.3

Component Faults

Component faults are faults that appear in the system components. They include every malfunctioning that occurs in any part of the system, excepting sensors and actuators. These faults are the most difficult to detect, locate and estimate.

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Fig. 3 Types of sensor faults (adapted from [61])

5.3 Fault Estimation in the Literature Estimating faults that may appear in dynamic systems is a necessary task for some Diagnosis and Fault Tolerant Control (DFTC) schemes. Indeed, as mentioned in [61, 62], detecting and isolating faults is insufficient for several Fault Tolerant Controllers. These controllers need accurate estimates of the actual faults, in order to guarantee desired performances, in the fault-free mode as well as in the faulty mode. Up to now, a massive interest has been given by researchers to solve the fault estimation problem for integer-order systems, unlike the fractional-order systems case. In the following, we present a state of the art of some works treating this issue for integer-order systems. In [63, 64], the sensor fault estimation problem for integerorder linear systems has been tackled: in [63], the authors have investigated linear time-varying systems by means of an adaptive observer, while in [64], the authors have investigated linear discrete-time switched systems. A research work, treating robust sensor fault estimation and fault-tolerant control for integer-order uncertain Lipschitz nonlinear systems, can be found in [65]. In [66,67], simultaneous sensor and actuator fault estimation for the same class of systems has been done. For the same class of systems (integer-order uncertain Lipschitz nonlinear systems), readers can refer to [32] as an example of simultaneous component and actuator fault diagnosis (detection and estimation). Dealing with integer-order descriptor systems, readers can refer to [68–70]: in [68], the authors have estimated actuator faults, in [69],

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Fig. 4 Types of actuator faults (adapted from [61])

the authors have estimated sensor faults, while in [70] the authors have simultaneously estimated actuator faults and sensor faults. Concerning fault estimation for specific physical systems, we remind here two interesting papers: The first one [71] treats robust actuator fault estimation with unknown input decoupling for a wind turbine system. The second one [72] is about incipient sensor fault reconstruction and accommodation for inverter devices in electric railway traction systems. As indicated previously in this paragraph: only very few research works have been done to estimate faults for fractional-order systems. One of these works is given in [73], where the authors have used a second-order Step by Step sliding mode observer for fault estimation in a class of nonlinear fractional-order systems.

6 Control of Dynamic Systems It can be affirmed that the control of dynamic systems is the main and central problem in the control theory. Controlling a process means designing a suitable control law,

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so that the closed-loop system behaves in the desired way. There are two principal control philosophies: the « stabilization» and the « model reference control».

6.1 The Stabilization Problem The stabilization task consists in finding a convenient control law, so that the whole controled closed-loop system is stable (see the stability definition given in Sect. 2). In the last decades, several research works have been done, in relation with the stabilization of integer-order systems. Some of these works have used fuzzy techniques [74–77]. For example, in [77], the authors have tackled the stabilization problem for integer-order uncertain nonlinear systems using fuzzy models. Other works have used model observer-based techniques [78–81]. For instance, in [81], an observer-based controller has been designed for networked control systems with sensor quantisation and random communication delay. Fractional-order systems have been also subject to various research works in relation with the stabilization query. In [82], the authors have proposed a robust stabilization technique for uncertain linear descriptor fractional-order systems. In [83], a robust stabilization technique is designed for a class of three-dimensional uncertain fractional-order non-autonomous systems. More recently, a study [84] has used a fuzzy output feedback stabilization technique for uncertain fractional-order systems.

6.2 The Model Reference Control Problem The model reference control for dynamic systems consists in designing a control law, so that the plant outputs can follow accurately the outputs of a given model. Several research works applied to the classical integer-order systems have been reported in the literature [85–89]. In these cited works [85–89], it is assumed that all the plant states are available by measurements, meaning that there is no need to design a state observer. Concerning fractional-order systems, we present herein some new papers. In [90], the authors have proposed a model reference adaptive control strategy for fractional order systems using discrete-time approximation methods. In [91], a tracking differentiator based fractional order model reference adaptive control has been suggested. More recently, another study [92] has been done, tackling indirect model reference adaptive control for a class of fractional order systems.

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7 Conclusion In this chapter, a literature review about fractional-order systems, nonlinear systems and observer-related queries, has been given. In Sect. 1, a general introduction has been provided. In Sect. 2, basic notions of the stability theory have been reminded. In Sect. 3, the notion of fractional-order systems has been presented, with the motivation of treating such systems. A state of the art of previous works dealing with fractional-order systems in the control theory has been given. As well, this section contains useful definitions and results from the literature, in relation with fractionalorder systems. Section 4 has exclusively tackled the classes of nonlinear systems investigated, which are: Lipschitz systems, one-sided Lipschitz systems and systems with monotone nonlinearities. A special interest has been given to the observer design problem for these types of nonlinearities. In Sect. 5, we have introduced the fault diagnosis query, in general. Then, we have focused on the fault estimation task, which will be extensively tackled in Chap. 4. Finally, in Sect. 6, the system control problem has been presented, and a state of the art of previous works related to the two main philosophies of control (the stabilization and the model reference control) has been given.

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81. Liu, M., You, J.: Observer-based controller design for networked control systems with sensor quantisation and random communication delay. Int. J. Syst. Sci. 43(10), 1901–1912 (2012) 82. N’Doye, I., Zasadzinski, M., Darouach, M., Radhy, N.: Robust stabilization of uncertain descriptor fractional-order systems. Automatica 49(6), 1907–1913 (2013) 83. Haghighi, A.R., Pourmahmood Aghababa, M., Roohi, M.: Robust stabilization of a class of three-dimensional uncertain fractional-order non-autonomous systems. Int. J. Indus. Math. 6(2), 133–139 (2014) 84. Ji, Y., Su, L., Qiu, J.: Design of fuzzy output feedback stabilization for uncertain fractional-order systems. Neurocomputing 173(3), 1683–1693 (2016) 85. Oucheriah, S.: Robust tracking and model following of uncertain dynamic delay systems by memoryless linear controllers. IEEE Trans. Autom. Control 44(7), 1473–1477 (1999) 86. Ni, M.L., Er, M.J., Leithead, W.E., Leith, D.J.: New approach to the design of robust tracking and model following controllers for uncertain delay systems. IEEE Proceed. Control Theory and Appl> 148(6), 472–477 (2001) 87. Basher, H.A.: Linear model reference control for swing-free transport. In: Proceedings of the Southeastcon 2006, Memphis, Tennessee, pp. 34–39. (2006) 88. Wang, Q., Hou, Y., Dong, C.: Model reference robust adaptive control for a class of uncertain switched linear systems. Int. J. Robust Nonlinear Control 22(9), 1019–1035 (2012) 89. Bernardo, M.D., Montanaro, U., Olm, J.M., Santini, S.: Model reference adaptive control of discrete-time piecewise linear systems. Int. J. Robust Nonlinear Control 23(7), 709–730 (2013) 90. Abedini, M., Nojoumian, M.A., Salarieh, H., Meghdari, A.: Model reference adaptive control in fractional order systems using discrete-time approximation methods. Commun. Nonlinear Sci. Numer. Simul. 25(1), 27–40 (2015) 91. Wei, Y., Hu, Y., Song, L., Wang, Y.: Tracking differentiator based fractional order model reference adaptive control: the 1 0, γ > 0, z ∈ C. When γ = 1, one has E α (z) = E α,1 (z), furthermore, E 1,1 (z) = e z . Consider the general continuous-time nonlinear fractional-order system (1). C

Dtα0 ,t x(t) = Ax + Bu + f (F x, u), t ≥ t0

(1)

where x ∈ Rn is the state vector, u ∈ Rm is the input vector, A ∈ Rn×n , B ∈ Rn×m and F are known matrices and f (F x, u) represents the nonlinear part of the system. We present the definition of global Mittag–Leffler stability: Definition 2 The system (1) is said to be globally uniformly Mittag–Leffler stable if there exist positive scalars b and λ such that the trajectory of (1) passing through any initial state x0 at any initial time t0 evaluated at time t satisfies:  b x(t) ≤ m(x0 )E α (−λ(t − t0 )α ) , ∀t ≥ t0

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With m(0) = 0, m(x) ≥ 0, m is locally Lipschitz and E α (.) is the Mittag–Leffler function presented in Definition 1. Theorem 1 [1] Let x = 0 be an equilibrium point for the system (1). Let V : [0, ∞) × Rn → R be a continuously differentiable function and locally Lipschitz with respect to x such that

μ1 xc ≤ V (t, x) ≤ μ2 xcd C

Dtα0 ,t V (t, x(t)) ≤ −μ3 xcd

where t ≥ t0 , x ∈ Rn , α ∈ (0, 1), μ1 , μ2 , μ3 , c and d are arbitrary positive constants. Then x = 0 is globally Mittag–Leffler stable. Lemma 1 [2] Let α ∈ (0, 1) and P ∈ Rn×n a constant square symmetric and positive definite matrix. Then the following relationship holds.

 1C α  T Dt0 ,t x (t)P x(t) ≤ x T (t)P C Dtα0 ,t x(t) 2

2 Robust State Estimation for a Class of Linear Uncertain Fractional-Order Systems 2.1 Motivation and Problem Formulation In real practice situations, dynamical systems under consideration are subject to uncertainties. These uncertainties can be interpreted as modeling uncertainties, or uncertainties due to unexpected perturbations affecting the plant. Up to now, several works investigating the robust state estimation problem for integer-order uncertain systems, have been done. Readers can refer to [3–4], as examples of robust observers for integer-order uncertain linear systems. They can refer to [5–6], as examples where the robust observer design problem for integer-order uncertain nonlinear systems has been tackled. Dealing with fractional-order uncertain systems, the task of designing robust observers can be regarded as a fertile field of research. Indeed, only few works exist in the literature within this particular topic [7–9], and several existing results for integer-order systems are to be extended into the fractional-order framework. In this context, and based on [9], we present in this section a novel robust observer design for a class of linear fractional-order systems with matched uncertainty. In this section, inspired by [10, 11], the following linear fractional order system with matched uncertainties, is considered (1), for all t ≥ t0 .

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⎧C α ⎨ Dt0 ,t x(t) = A(q)x(t) + Bu(t) = (A + ϕ(q)C)x(t) + Bu(t) ⎩ y(t) = C x(t)

(2)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the input, y(t) ∈ R p is the output. A ∈ Rn×n , B ∈ Rn×m and C ∈ R p×n are known matrices. ϕ(q) ∈ Rn× p represents the uncertainty matrix of the system, that depends on the uncertainty real parameter q.

2.2 Robust Observer Design The goal is to design an observer for the uncertain linear fractional order system (2). This observer has to be insensitive to the parametric variations due to the uncertainty parameter q. In order to achieve this task, the following assumption should be introduced. Assumption 1 For any x, xˆ ∈ Rn and for any q ∈ ⊂ R, there exists a scalar λ > 0 such that the following inequality is satisfied:





ϕ(q)C x − ϕ(q0 )C xˆ ≤ λϕ(q0 ) C x − C xˆ

(3)

where xˆ is the estimated state vector and q0 is a nominal value of q. In the rest of this work, and for simplicity, we denote ϕ = ϕ(q) and ϕ0 = ϕ(q0 ). Consider the following Lumberger-like observer: C

  Dtα0 ,t xˆ = F xˆ + Bu + G C x − C xˆ yˆ = C xˆ

(4)

For convenience, we define the matrices F and G as follows: F = A + ϕ0 C

(5)

G = G1 + G2

(6)

where G 1 and G 2 are two design matrices to be found such that the observer (4) is a convergent observer in the Mittag–Leffler definition sense, given previously. Now, we are ready to present the main result of this research work, by stating the following theorem. Theorem 2 Consider the linear fractional-order system with matched uncertainties (2), under assumption 1, with the observer (4). The error origin xˆ − x = 0 is

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globally Mittag–Leffler stable, if there exist a symmetric definite positive matrix P and a matrix X with appropriate dimensions, and positive scalars r and ε such that:

P A + A T P + XC + C T X T + ε I P P −r I

0, one can have:



  2e T P ϕC x − ϕ0 C xˆ ≤ r −1 e T P Pe + r ϕC x − ϕ0 C xˆ 2 , Then: C

  Dtα0 ,t V (t) ≤ e T P(A − G 1 C) + (A − G 1 C)T P + r −1 P P e   − e T P G 2 C + (G 2 C)T P e + r (λϕ0 )2 Ce2

We have G 2 = r (λϕ0 )2 P −1 C T /2, then: P G2C =

1 r (λϕ0 )2 C T C 2

Then, it follows that: C

  Dtα0 ,t V (t) ≤ e T P(A − G 1 C) + (A − G 1 C)T P + r −1 P P e   ≤ e T P(A − G 1 C) + (A − G 1 C)T P + r −1 P P + ε I e − εe2

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Define X = −P G 1 . If one has: P A + A T P + XC + C T X T + r −1 P 2 + ε I < 0

(9)

then, we conclude that C Dtα0 ,t V (t) ≤ −εe2 . Consequently, according to Theorem 1, the error origin e = 0 is globally Mittag–Leffler stable. Now, according to the Schur complement lemma [12], the stability condition (9) is equivalent to (7). This ends the proof. Remark 1 For the extreme case: α = 1 (case of integer-order systems), Theorem 2 and its proof are still valid. Note that, in that case, the obtained result is the classical global exponential stability.

2.3 Simulation Results We propose in this sub-section a numerical study, in order to validate the suggested robust fractional order observer. The following numerical example, in the form of (2), is taken from [10–11]: ⎡

⎤ −2 + 0.5q 1 1 − q A(q) = ⎣ 1 + 0.5q −8 0 − q ⎦, 1 + 0.5q 0 −2 − q

⎡ ⎤ 1   B = ⎣ 0 ⎦, C = 0.5 0 −1 , 0

where the uncertainty parameter q ∈ [0, 50]. Matrices A and ϕ(q) are then given by: ⎡ ⎤ ⎤ q −2 1 1 A = ⎣ 1 −8 0 ⎦, ϕ(q) = ⎣ q ⎦, q 1 0 −2 ⎡

First of all, define a nominal parameter q0 = 4 and the constant λ = 14. ⎡ ⎤ 0 1 −3 Then, matrix F can be computed using Eq. (5): F = ⎣ 3 −8 −4 ⎦. According 3 0 −6 to [11], assumption 1 is well satisfied. Now, we can exploite Theorem 2. In order to use the observer (4), first, one should solve the LMI (7). The parameter r is assigned to r = 0.001. The LMI is found feasible with solutions ε = 4.51 10−4 , ⎡ ⎤ ⎡ ⎤ 0.0332 −0.001 −0.0642 −10.0347 P = ⎣ −0.001 0.0067 0.0003 ⎦, X = ⎣ 0.0167 ⎦. Then, we compute −0.0642 0.0003 0.1285 20.0696 the gain matrices G 1 and G 2 by using the expressions defined in Theorem 2:

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⎡

⎡ ⎤ ⎤ 13.5669 3.7114 G 1 = ⎣ 7.1718 ⎦ and G 2 = ⎣ 2.339 ⎦. For simulations, we use the input −149.4227 −34.7586  T  T u = 1(t), with the initial conditions x0 = 0 0 0 and xˆ0 = 0.5 0.5 0.5 . The fractional derivative order is set to α = 0.9. The actual states and their estimates for q = 0 are given in Fig. 1. The same signals for q = 50 are illustrated in Fig. 2. In the two cases, it is noticed that the Mittag–Leffler stability is obtained, as demonstrated in the last sub-section. This validates the robustness of our proposed observer.

3 New Observer Design for Nonlinear Fractional-Order Systems 3.1 Motivation and Problem Formulation Generally speaking, the state estimation problem for nonlinear systems can be regarded as one of the oldest problems ever investigated by researchers in the control area. One of the most famous and frequently used nonlinear properties in the scientific research community is the so-called Lipschitz nonlinearity. Designing observers for nonlinear Lipschitz systems has been tackled, for the first time, by Thau [13]. Obviously, the first done paper [13] has been developed for integer-order systems. After Thau, a massive amount of research works has been done to design observers for nonlinear integer-order systems. However, dealing with fractionalorder nonlinear systems, the problem is not yet treated in the same graetness, and many observer design schemes are still to be generalyzed into the fractional-order framework. In this context and inspired by [14–16], a particular form of observers, which is well tackled for the integer-order Lipschitz systems [16], is extended into the nonlinear fractional-order: Lipschitz, one-sided Lipschitz and quasi-one sided Lipschitz systems. In this section, the considered class of systems can be described by the following couple of state-space equation and output Eq. (2.III.1). Note that the mentioned state-space equation corresponds to a particular case of Eq. (1.3), where matrix F is supposed to be the identity matrix. C

Dtα0 ,t x(t) = Ax(t) + Bu(t) + f (x, u) y(t) = C x(t)

(10)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the input, y(t) ∈ Rq is the output. A ∈ Rn×n , B ∈ Rn×m and C ∈ Rq×n are known constant matrices. f (x, u) represents the nonlinear part of the plant, such that f (0, u) = 0. In the next sub-sections, a novel state estimation scheme for fractional-order Lipschitz, one-sided Lipschitz and quasi-one sided Lipschitz systems, is presented.

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Fig. 1 Actual states and their estimates using the robust observer (4) for q = 0

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Fig. 2 Actual states and their estimates using the robust observer (4) for q = 50

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3.2 Observer Design for Lipschitz Fractional-Order Systems The problem is to design a novel continuous-time observer scheme for Lipschitz fractional systems, such that the states estimates denoted by x(t) ˆ converge to x(t), in the Mittag–Leffler stability sense. For this purpose, we use the following assumption. Assumption 2 There exists a positive scalar θ such that.

−θ S − A T S − S A + C T C = 0

(11)

where S is a symmetric definite positive matrix. Here, the nonlinear function f (x, u) is supposed to satisfy the following Lipschitz property.  f (F x1 , u) − f (F x2 , u) ≤ r F(x1 − x2 ), ∀x1 , x2 ∈ Rn ,

(12)

where r > 0 is the Lipschitz constant. Under assumption 2, one proposes the following observer: C

    Dtα0 ,t x(t) ˆ = A xˆ + Bu + f x, ˆ u − β S −1 C T C xˆ − y ,

(13)

where β is a design scalar that satisfies β ≥ 1 [16]. Now we present the following theorem, that ensures the Mittag–Leffler stability of the proposed observer with the Lipschitz condition (12). Theorem 3 If assumption 2 and condition (12) hold, and if.   1 λmin θ S + (2β − 1)C T C >r 2 λmax (S)

(14)

where r is the Lipschitz constant defined in (12), then (13)is a global Mittag–Leffler stable observer for system (10). Proof Let e = xˆ − x, Then the state estimation error dynamic is given by.

C

    Dtα0 ,t e(t) = A xˆ + Bu + f x, ˆ u − β S −1 C T C xˆ − y − Ax − Bu − f (x, u)   = A − β S −1 C T C e + f

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  where f = f x, ˆ u − f (x, u). Now, consider the Lyapunov function V (e) = e T Se, Then using Lemma 1, we have C

Dtα0 ,t V (e) ≤ 2e T (t)S C Dtα0 ,t e(t), , ∀t ≥ t0    T    ≤ e T A − β S −1 C T C + f T Se + e T S A − β S −1 C T C e + f  T   ≤ e T A − β S −1 C T C S + S A − β S −1 C T C e + 2e T S f   ≤ e T A T S − 2βC T C + S A e + 2e T S f

Now, using (11), we have C

Dtα0 ,t V (e) ≤ −θ e T Se + (1 − 2β)e T C T Ce + 2e T S f

And so C

Dtα0 ,t V (e) ≤ −θ e T Se + (1 − 2β)e T C T Ce + 2eS f 

Then, using (12), one can have C

  Dtα0 ,t V (e) ≤ −λmin θ S + (2β − 1)C T C e2 + 2r λmax (S)e2

So, if (14) is satisfied, then one gets C Dtα0 ,t V (e) ≤ −le2 with l =   λmin θ S + (2β − 1)C T C − 2r λmax (S) > 0. Hence, the origin of the error e = 0 is globally Mittag Leffler stable.

3.3 Observer Design for One-Sided Lipschitz and Quasi One-Sided Lipschitz Fractional-Order Systems The problem is to design a novel continuous-time observer scheme for one-sided Lipschitz and quasi one-sided Lipschitz fractional systems, such that the states estimates denoted by x(t) ˆ converge to x(t), in the Mittag–Leffler stability sense. First, we consider one-sided Lipschitz nonlinear systems. In the following, the one-sided Lipschitz definition is given. Definition 3 In system (10), the nonlinear function f (x, u) is said to be globally one-sided Lipschitz with respect to x, if there exist a constant v ∈ R, satisfying:

S f (x1 , u) − S f (x2 , u), x1 − x2 ≤ vx1 − x2 2 , ∀x1 , x2 ∈ Rn

(15)

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By the following theorem, the Mittag–Leffler stability of the observer (13), under the one-sided Lipschitz condition (15), is ensured. Theorem 4 If assumption 2 and condition (15) hold, and if.   λmin θ S + (2β − 1)C T C − 2v > 0

(16)

then (13) is a global Mittag–Leffler stable observer for system (10). Proof The proof of Theorem 4 goes like the proof of Theorem 3. Let e = xˆ − x, then:

C

  Dtα0 ,t e(t) = A − β S −1 C T C e + f

Consider the Lyapunov function V (e) = e T Se. Then, using Lemma 1, the derivative of V along any trajectory of the error equation is given by C

Dtα0 ,t V (e) ≤ 2e T (t)S C Dtα0 ,t e(t), , ∀t ≥ t0        T ≤ e T A − β S −1 C T C + f T Se + e T S A − β S −1 C T C e + f  T   ≤ e T A − β S −1 C T C S + S A − β S −1 C T C e + 2e T S f   ≤ e T A T S − 2βC T C + S A e + 2e T S f

Now, using (11), we have C

Dtα0 ,t V (e) ≤ −θ e T Se + (1 − 2β)e T C T Ce + 2e T S f

Then, using (15), one can have C

  Dtα0 ,t V (e) ≤ −λmin θ S + (2β − 1)C T C e2 + 2ve2

So, if (16) is satisfied, then one gets C Dtα0 ,t V (e) ≤ −le2 with l =   λmin θ S + (2β − 1)C T C − 2v > 0. Hence, the origin of the error e = 0 is Mittag–Leffler stable. In the present sub-section, the second stage consists in estimating the states of system (10) by means of the observer (13), under the following quasi one-sided Lipschitz condition: Definition 4 In system (10), the nonlinear function f (x, u) is said to be globally quasi one-sided Lipschitz with respect to x, if there exist a constant real symmetric matrix M, satisfying:

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S f (x1 , u) − S f (x2 , u), x1 − x2 ≤ (x1 − x2 )T M(x1 − x2 ), ∀x1 , x2 ∈ Rn (17) Theorem 5 If assumption 2 and condition (17) hold, and if.   λmin θ S + (2β − 1)C T C − 2λmax (M) > 0

(18)

Then (13) is a global Mittag–Leffler stable observer for system (10). Proof Let e = xˆ − x and (e) = e T Se. One can find.

C

Dtα0 ,t V (e) ≤ −θ e T Se + (1 − 2β)e T C T Ce + 2e T S f

Then, using (17), one can have C

  Dtα0 ,t V (e) ≤ −λmin θ S + (2β − 1)C T C e2 + 2e T Me     ≤ − λmin θ S + (2β − 1)C T C − 2λmax (M) e2

Then, (18) ensures the Mittag–Leffler stability of the error origin e = 0.

4 Full-Order Unknown Input Observer Design for Fractional One-Sided Lipschitz Systems 4.1 Motivation and Problem Formulation In real practice situations, dynamic systems are usually subject to some unknown inputs. One of the most efficient types of observers to such systems, is the so-called Unknown Input Observer. Indeed, Unknown Input Observers are well known for their ability to decouple the state estimation error dynamics from these unknown inputs, and consequently, the state estimation error can be proved to converge exactly to zero, even in the presence of such unknown inputs. Up to now, several works investigating the robust full-order state estimation problem for integer-order systems with unknown inputs, have been done. For instance, readers can refer to [17–18]. Dealing with fractional-order systems with unknown inputs, only very few works have been done to design full-order Unknown Input Observers. To the best of our knowledge, no paper has been developed to solve the full-order state estimation problem for the general class of continuous-time fractional-order one-sided Lipschitz systems. In this context and inspired by Zhang et al. [19], we present in this section an original scheme in order to solve this problem for continuous-time fractional-order one-sided Lipschitz systems.

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In this section, the considered class of systems can be described by the following couple of state-space equation and output Eq. (19). C

Dtα0 ,t x(t) = Ax(t) + Bu(t) + D f f (F x, u) + Dv(t) y(t) = C x(t)

(19)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the input, y(t) ∈ Rq is the output, v(t) ∈ Rs is the disturbance (unknown input). A ∈ Rn×n , B ∈ Rn×m , C ∈ Rq×n , D ∈ Rn×s , D f and F are known matrices. Without loss of generality, we assume that the unknown input distribution matrix D is of full column rank. The nonlinear function f (F x, u) is supposed to satisfy the one-sided Lipschitz condition and the quadratic inner-bounded condition, given by the next two definitions. Definition 5 The nonlinear function f (F x, u) is said to be globally one-sided Lipschitz with respect to x, if there exist a constant ρ ∈ R, satisfying:

 f (F x1 , u) − f (F x2 , u), F(x1 − x2 ) ≤ ρF(x1 − x2 )2 , ∀x1 , x2 ∈ Rn (20) Definition 14 The nonlinear function f (F x, u) is said to be globally quadratically inner bounded with respect to x, if there exist constants β, γ ∈ R , satisfying:

 f (F x1 , u) − f (F x2 , u)2 ≤ βF(x1 − x2 )2 + γ  f (F x1 , u) − f (F x2 , u), F(x1 − x2 ) , ∀x1 , x2 ∈ Rn

(21)

4.2 Robust State Estimation in the Presence of Unknown Inputs Consider the following full-order Unknown Input Observer for system (19): C

  Dtα0 ,t ξ (t) = N ξ (t) + Gy(t) + T D f f F x, ˆ u + T Bu(t) x(t) ˆ = ξ (t) − E y(t)

(22)

where ξ (t) ∈ Rn is the state vector of the observer and x(t) ˆ ∈ Rn is the estimate of x(t). N , G and T are real matrices with appropriate dimensions, satisfying the following conditions: N = T A − KC

(23)

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G = K (I + C E) − T AE

(24)

T = I + EC

(25)

where I is the identity matrix and K and E are matrices to be designed later. Let the state estimation error be e(t) = x(t) ˆ − x(t) = ξ (t) − T x(t)   Then, defining f = f F x, ˆ u − f (F x, u), the error dynamics is given by C

Dtα0 ,t e(t) = C Dtα0 ,t ξ (t) − T C Dtα0 ,t x(t) = N e + (N T + GC − T A)x + T D f f − T Dv

From (23), (24) and (25), one can have N T + GC − T A = 0, which means that: C

Dtα0 ,t e(t) = N e + T D f f − T Dv

(26)

By the following theorem, a sufficient condition that guarantees the Mittag–Leffler stability for the observer (22) is given. Theorem 6 Consider the class of nonlinear fractional-order systems (19) satisfying (20) and (21), with the Unknown Input Observer (22). The error origin e = 0 is globally Mittag–Leffler stable, if there exist a symmetric definite positive matrix P and matrices E and K with appropriate dimensions, and positive scalars τ1 , τ2 and ε such that.

N T P + P N + 2ηF T F + ε I σ F T + P T D f ∗ −2τ2 I

0, condition (21) yields:   2τ2 βe T F T Fe + γ e T F T f − f T f ≥ 0

(32)

Then, adding the terms on the left-hand side of (31) and (32) to the right-hand side of (30) yields C

    Dtα0 ,t V (t) ≤ e T N T P + P N + 2ηF T F e − 2τ2 f T f + 2e T σ F T + P T D f f

T T



e N P + P N + 2ηF T F σ F T + P T D f e ≤

f ∗ −2τ2 I

f Then, for any positive scalar ε, one can have C

Dtα0 ,t V (t) ≤



e

f

T

e  − εe2

f

where: =

N T P + P N + 2ηF T F + ε I σ F T + P T D f ∗ −2τ2 I

Then, the condition (27) ensures that C Dtα0 ,t V (t) ≤ −εe2 . Then, according to Theorem 1, the origin e = 0 is globally Mittag–Leffler stable. This ends the proof. Remark 2 In the last given theorem, condition (27), which ensures the Mittag– Leffler stability of the origin, is a Nonlinear Matrix Inequality (NMI). In order to be able to solve this inequality using commercially available packages such that the Matlab solver, it should be transformed into a LMI. In a way similar to [19], this transformation is ensured through the following theorem. Theorem 7 Assume that conditions (20) and (21) hold and that C D is of full column rank. Then (22) is a Mittag–Leffler stable Unknown Input Observer for the fractional-order system (19), if there exist a symmetric definite positive matrix P and matrices X 1 and X 2 and positive scalars τ1 , τ2 and ε such that:

State Estimation for Fractional-Order Systems



 σ F T + P T + X 1 2 ∗ −2τ2 I

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0, it follows that:   2τ2 β z˜ 2T F2T F2 z˜ 2 + γ z˜ 2T F2T f − f T f ≥ 0

(51)

Then, adding the terms on the left-hand side of (50) and (51) to the right-hand side of (49) yields: C

  Dtα0 ,t V (t) ≤ z˜ 2T (A22 + L A12 )T P + P(A22 + L A12 ) + 2ηF2T F2   z˜ 2 − 2τ2 f T f + 2˜z 2T σ F2T + P D L f

Let Q = (A22 + L A12 )T P + P(A22 + L A12 ), then: C

Dtα0 ,t V (t)



z˜ 2 ≤

f

T

Q + 2ηF2T F2 σ F2T + P D L ∗ −2τ2 I

Then, for any positive scalar ε, one can have C

Dtα0 ,t V (t) ≤



z˜ 2

f

T

z˜  2 − ε˜z 2 2

f



z˜ 2

f

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Q + 2ηF2T F2 + ε I σ F2T + P D L . Then, the condition (45) ensures where  = ∗ −2τ2 I that C Dtα0 ,t V (t) ≤ −ε˜z 2 2 . Thus, according to Theorem 1, the origin z˜ 2 = 0 is globally Mittag–Leffler stable. This ends the proof.

Remark 4 In Theorem 8, condition (45) is a Nonlinear Matrix Inequality (NMI). This condition should be transformed into a LMI, so one can solve it using commercially available packages such that the Matlab solver. By using a suitable transformation technique, the following conclusion is obtained, then the proposed reduced-order observer can be effectively designed. Theorem 9 Assume that conditions  (20) and (21) hold and that rank (D1 ) = s.  Under the condition C = Iq 0 , system (42) is a Mittag–Leffler stable reducedorder Unknown Input Observer for the fractional-order system (19), if there exist a symmetric definite positive matrix P and a matrix S and positive scalars τ1 , τ2 and ε such that:

 σ F2T + P D + S F ∗ −2τ2 I

0, γ > 0, z ∈ C. When γ = 1, one has E α (z) = E α,1 (z), furthermore, E 1,1 (z) = e z . Consider a general continuous-time nonlinear fractional-order system, described by (1). C

Dtα0 ,t x(t) = Ax + Bu + f (F x, u), t ≥ t0

(1)

where x ∈ Rn is the state vector, u ∈ Rm is the input vector, A ∈ Rn×n , B ∈ Rn×m and F are known matrices and f (F x, u) represents the nonlinear part of the system. We present the definition of global Mittag-Leffler stability: Definition 2 The system (1) is said to be globally uniformly Mittag-Leffler stable if there exist positive scalars b and λ such that the trajectory of (1) passing through any initial state x0 at any initial time t0 evaluated at time t satisfies:  b x(t) ≤ m(x0 )E α (−λ(t − t0 )α ) , ∀t ≥ t0 With m(0) = 0, m(x) ≥ 0, m is locally Lipschitz and E α (.) is the Mittag-Leffler function presented in definition 1. Theorem 1 Li et al. [1] Let x = 0 be an equilibrium point for the system (1). Let V : [0, ∞)×Rn → R be a continuously differentiable function and locally Lipschitz with respect to x such that. μ1 xc ≤ V (t, x) ≤ μ2 xcd

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Dtα0 ,t V (t, x(t)) ≤ −μ3 xcd

where t ≥ t0 , x ∈ Rn , α ∈ (0, 1), μ1 , μ2 , μ3 , c and d are arbitrary positive constants. Then x = 0 is globally Mittag-Leffler stable. Lemma 1 Duarte-Mermoud et al. [2] Let α ∈ (0, 1) and P ∈ Rn×n a constant square symmetric and positive definite matrix. Then the following relationship holds.  1C α  T Dt0 ,t x (t)P x(t) ≤ x T (t)P C Dtα0 ,t x(t) 2 Definition 3 System (1)is said to be globally uniformly practically Mittag-Leffler stable, if there exist positive scalars b, λ and r such that the trajectory of (3.3.1) passing through any initial state x0 at any initial time t0 evaluated at time t satisfies:  b x(t) ≤ m(x0 )E α (−λ(t − t0 )α ) + r, ∀t ≥ t0 with m(0) = 0, m(x) ≥ 0 and m is locally Lipschitz. Lemma 2 Choi et al. [3] Suppose that: C

Dtα0 ,t m(t) ≤ λm(t) + d, m(t0 ) = m 0 , t ≥ t0 ≥ 0

where λ, d ∈ R. Then, one has: m(t) ≤ m(t0 )E α (λ(t − t0 )α ) + d(t − t0 )α E α,α+1 (λ(t − t0 )α ), t ≥ t0 ≥ 0 Morever, if λ < 0, then: m(t) ≤ m(t0 )E α (λ(t − t0 )α ) + d M, t ≥ t0 ≥ 0   where M = sups≥0 s α E α,α+1 (λs α ) .

3 Adaptive Stabilization for a Class of Uncertain Nonlinear Fractional-Order Systems 3.1 Motivation and Problem Formulation Up to now, the problem of designing output feedback controllers for uncertain nonlinear integer-order systems has been subject to several research works. Dealing with uncertain fractional-order systems, the stabilization task can be regarded as a

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fertile area of research. In fact, several existing results for integer-order systems are still to be extended to the fractional-order calculus. In this section, based on the works of Arefi [4], a novel and advantageous observer-based adaptive technique for the stabilization of a class of uncertain nonlinear fractional-order systems, is proposed. In the presented scheme, the unknown bound of uncertainties is estimated through a convenient adaptation law, then, an adaptive output feedback stabilizing control law is constructed. Since the state variables are not all measurable, an observer is firstly designed. It is prominent to note that, to the best of our knowledge, no similar methodology has been done in the literature for uncertain nonlinear fractional-order systems. The advantages of such a technique, compared to others already used in the literature, can be summarized in the following points: • In several existing works dealing with the stabilization problem for uncertain integer-order or fractional-order systems (such as [5] and [6]), a fuzzy system is used to solve the problem. However, such a methodology requires the knowledge of an expert. This limitation is avoided in the present work by the estimation of upper bound of uncertainties, thanks to an adaptive control strategy. • In some existing works, the unknown nonlinear function is assumed to be bounded by output injection terms. In other works, it is assumed to be globally Lipschitz. None of these two assumptions is required in this study. • A common assumption in the literature, is that the nonlinear uncertainties are only functions of the measurable signals. Such an assumption is not required in this work. In this section, the considered class of systems can be described by the following couple of state-space equation and output Eq. (2). C

Dtα0 ,t x(t) = Ax(t) + B[ f (t, x) + u(t)] y(t) = C x(t)

(2)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the input, y(t) ∈ Rm is the output. A ∈ Rn×n , B ∈ Rn×m and C ∈ Rm×n are known constant matrices. The term f (t, x) ∈ Rm is an unknown continuous function representing the uncertain part of the system.

3.2 Observer Design and Adaptive Feedback Controller Scheme A novel scheme is presented here, in order to stabilize the uncertain fractional-order plant (2). In other words, the goal is to design an output feedback controller, such that system (2) is globally uniformly practically Mittag-Leffler stable. For this purpose, the following assumptions are considered.

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Assumption 3.1 There exists a positive scalar β such that:

 f (t, x) ≤ β Assumption 3.2 (i) (ii)

There exists a matrix K such that A − B K is Hurwitz. There exists a matrix L such that A − LC is Hurwitz.

Remark 1 Generally speaking, if f (t, x) is known and the state vector x is entirely available, the controller could be chosen as u = − f (t, x) − K x. Since f (t, x) is unknown and only the system output y(t) is available by measurement, this controller can not be implemented in practice. An alternative is to estimate the upper bound β using the adaptive control strategy, and to design a suitable observer to reconstruct the unmeasurable state variables. Consider the following observer, where xˆ denotes the estimate of the state vector x. C

  Dtα0 ,t x(t) ˆ = (A − B K )x(t) ˆ + L y − C xˆ , yˆ (t) = C x(t) ˆ

(3)

Let the state estimation error be e = x − x. ˆ Then, it follows from (2 and 3) that: C

Dtα0 ,t e(t) = (A − LC)e(t) + B K x(t) ˆ + B f (t, x) + Bu

(4)

The output estimation error is given by e y = Ce. Assumption 3.3 There exist positive-definite matrices P and Q such that: 

(A − LC)T P + P(A − LC) = −Q P B = CT

(5)

Remark 2 Generally speaking, in order to solve (5), one can refer to the Kalman– Yakubovich–Popov lemma in [7]. This lemma elaborates conditions, under which there exist positive–definite matrices P and Q such that the problem (5) is feasible. Theorem 2 Consider the fractional-order system (2)and the observer (3), under assumptions 3.1-3.3. Let the adaptive controller and the adaptation law respectively be:

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e y βˆ 2 u = −K xˆ −   e y βˆ + ε C

  ˆ ˆ = ηe y  − ησ β(t) Dtα0 ,t β(t)

(6) (7)

where βˆ is the estimate of β, ε > 0, η > 0 and σ > 0 are design parameters. Then, the adaptive controller (6) globally uniformly practically Mittag-Leffler stabilizes the system (4, 5, 6 and 7). Proof Consider the following Lyapunov function.

1 T 1 2 ˜ V = e Pe + β 2 η

(8)

where β˜ = βˆ − β . Using Lemma 1, one can have: C

 1 T T e A L P + P A L e + e T P B K xˆ + e T P B f (t, x) + e T P Bu 2 1 ˜ + β˜ C Dtα0 ,t β(t) (9) η

Dtα0 ,t V (t) ≤

where A L = A − LC. By (3.III.5), we have e T P B = e Ty , then using (5, 6 and 9), yields: C

Dtα0 ,t V (t)

 2 2 e y  βˆ 1 1 T T ˜ + β˜ C Dtα0 ,t β(t) ≤ − e Qe + e y f (t, x) −   e y βˆ + ε η 2  2 2 e y  βˆ   1 1 2   ˆ ≤ − λmin (Q)e + e y β −   + β˜ C Dtα0 ,t β(t) e y βˆ + ε η 2

We have:  2 2 e y  βˆ   −  ≤ −e y βˆ + ε e y βˆ + ε Then: C

  1 1 ˆ Dtα0 ,t V (t) ≤ − λmin (Q)e2 − e y β˜ + ε + β˜ C Dtα0 ,t β(t) 2 η

Using the adaptation law (7), we obtain:

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1 Dtα0 ,t V (t) ≤ − λmin (Q)e2 + ε − σ β˜ βˆ 2   1   ≤ − λmin (Q)e2 + ε − σ β˜ 2 + σ β˜ β 2 2

Using −a 2 + ab < − a2 + C

b2 2

for any a, b we have:

β˜ 2 β2 1 +σ Dtα0 ,t V (t) ≤ − λmin (Q)e2 + ε − σ 2 2 2

(10)

From (8 and 10): ∃ λ > 0 such that: C

Dtα0 ,t V (t) ≤ −λV (t) + r

with = ε + σ β2 . Then, it follows from Lemma 3 that: 2

V (t) ≤ V (t0 )E α (−λ(t − t0 )α ) + r M

(11)

  where M = sups≥0 s α E α,α+1 (−λs α ) . 2 2 T Using (8) and the fact that λmin (P)e ≤ e Pe ≤ λmax (P)e , and defining 1 1 λ1 = min 2 λmin (P), 2η one can have: 

 2  ˜ V (t) ≥ λ1  e(t), β(t)  By the same way and defining λ2 = max



1 1 λ 2 max (P), 2η

(12)

, one can have:



  ˜ 0)  V (t0 ) ≤ λ2  e(t0 ), β(t 2

(13)

So, from (11, 12 and 13), it follows that:  



 2   ˜ ˜ 0)  λ1  e(t), β(t)  ≤ λ2  e(t0 ), β(t  2 E α (−λ(t − t0 )α ) + r M Then, using

√ √ √ a + b ≤ a + b for any positive scalars a, b we have:



1/2 



 √ λ2    2 α ˜ ˜ + rM  e(t), β(t)  ≤  e(t0 ), β(t0 )  E α (−λ(t − t0 ) ) λ1

˜ The last inequality is in the form of (2), so by definition 3, the system e(t), β(t) is globally uniformly practically Mittag-Leffler stable. Since A − B K is Hurwitz, we conclude from Eq. (3) that xˆ is bounded. Then, from e = x − x, ˆ x is also bounded. This ends the proof.

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3.3 Simulation Results Consider a fractional-order numerical system, in the form of (2), as follows: ⎡

⎤ ⎡ ⎤ −2 −1 0 0   A = ⎣ 1 0 0 ⎦, B = ⎣ 1 ⎦, C = 1 1 0 1 0 −1 0 f (t, x) = sin(x1 )

(14)

 T where x = x1 x2 x3 is the state vector. One can easily see that assumption III.1 is well satisfied, and the upper bound to be estimated is β = 1. The gain matrices are selected as:   K = 353 ,

 T L= 010

so that A − B K and A − LC are Hurwitz. Besides, by solving (5) for symmetric positive definite matrices P and Q, one finds: ⎡

⎤ 210 P = ⎣ 1 1 0 ⎦, 001

⎡

⎤ 8 5 −1 Q = ⎣ 5 4 0 ⎦, −1 0 2

meaning that assumption III.3 is fulfilled. The simulation is tackled with the initial  T  T conditions x0 = 1 −2 2 , e0 = 2 1 3 , βˆ0 = 0.1 and the derivative fractionalorder is set to α = 0.9. The design parameters are the following: ε = 0.05, η = 1 and σ = 0.001. Figure 1 shows the states trajectories when the system is under the control law (6). We can see that all the states are bounded in the presence of unknown nonlinear uncertainties, as found in the theoretical study. In Fig. 2, the state estimation errors are given. We can easily see that the theoretical practical Mittag-Leffler stability of the errors is obtained. The same note is concluded from Fig. 3, which presents the ˆ bound parameter estimate β(t).

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Fig. 1 States trajectories for system (14), under the control law (6)

Fig. 2 State estimation errors for system (14), under the observer (3) and the control law (6)

4 Model-Reference Control for Linear Fractional-Order Systems 4.1 Motivation and Problem Formulation The model-reference control task consists in designing a control law, so that the plant outputs could follow accurately the outputs of a given model. Up to now, several research works applied to the classical linear integer-order systems have been reported in the literature. Dealing with fractional-order systems, the problem has just been tackled by researchers. Note that several existing papers, treating integer-order systems, are still to be extended to the fractional-order framework. In this context and inspired by [8], we propose in the present section an original model-reference

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ˆ for system (14), under the control law (6) Fig. 3 Uncertainty bound estimate β(t)

controller for linear fractional-order systems. The scheme presented herein is unique and novel, since it associates the following features and advantages: • The designed model-reference control law is efficient for both integer-order systems (the case of [8]) and fractional-order systems, which is the major contribution of our work. • The state dimensions of the model and the plant could be unequal, which is not the case in most of the existing works in the literature. This aspect provides more freedom and less conservatism in the design. • An observer is incorporated, in order to guarantee the accessibility to all the plant states, even those unaccessible by measurement. • The feedback control law is linear and simple to implimente. In this section, a fractional-order linear plant is considered by the couple of statespace equation and output equation: (15). C

Dtα0 ,t x p (t) = Ax p (t) + Bu(t) y p (t) = C x p (t)

(15)

The model to be followed is described, for all t ≥ t0 , by: C

Dtα0 ,t xm (t) = Am xm (t) ym (t) = Cm xm (t)

(16)

In (15 and 16): x p (t) ∈ Rn is the plant state, u(t) ∈ Rm is the control to be designed, y p (t) ∈ Rq is the plant output, xm (t) ∈ Rn 1 is the model state and ym (t) ∈ Rq1 is the model output. A ∈ Rn×n , B ∈ Rn×m , C ∈ Rq×n , Am ∈ Rn 1 ×n 1 and Cm ∈ Rq1 ×n 1 are well known matrices.

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4.2 Model-Reference Controller Design In this section, we aim to design an observer-based model-reference control, such that the plant (15) could follow the model (16), with a tracking error converging to zero. In order to do so, the following assumption should be introduced. Assumption 4.1 There exist a matrix F ∈ Rn×n 1 and a matrix G ∈ Rm×n 1 , such that the following condition holds:

A B C 0



F G



=

F Am Cm

(17)

In order to reconstruct the plant states, including those unavailable by measurement, consider the following Luenberger-like observer: C

  Dtα0 ,t x z (t) = Ax z (t) + Bu(t) + LC x p (t) − x z (t)

(18)

where x z (t) ∈ Rn is the estimated state vector and L is a gain matrix to be conveniently assigned later. Define the error between the model states and the plant states as follows: e p (t) = F xm (t) − x p (t)

(19)

Then, using (15 and 16), its derivative is governed by: C

Dtα0 ,t e p (t) = Ae p (t) − Bu(t) + (F Am − AF)xm (t)

(20)

Let the state estimation error be: ez (t) = x z (t) − x p (t)

(21)

Then, its derivative is given by: C

Dtα0 ,t ez (t) = (A − LC)ez (t)

(22)

Define the following linear control law:   u(t) = γ B T P x z (t) − γ B T P F − G xm (t)

(23)

where γ is a design scalar and P is a design symmetric definite positive matrix. By the next theorem, it is announced that the suggested control law (23) is capable of stabilizing the tracking errors in the Mittag-Leffler sense, which guarantees the fact that the plant output follows accurately the model output.

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Theorem 3 Consider the linear fractional-order plant (15)and the linear fractionalorder model (16), under assumption IV.1. The observer-based control law  (23)guarantees the global Mittag-Leffler stability of the errors origin e p , ez = (0, 0) (consequently y p (t) → ym (t) as t → ∞), if there exists symmetric definite positive matrices P and Q, and a positive scalar ε such that:

−γ P B B T P P A + A T P + 2γ P B B T P + ε I T QX + XT Q + ε I −γ P B B P

0, γ > 0, z ∈ C. When γ = 1, one has E α (z) = E α,1 (z), furthermore, E 1,1 (z) = e z . Consider the general continuous-time nonlinear fractional-order system (1). C

Dtα0 ,t x(t) = Ax + Bu + ϕ(F x, u),

t ≥ t0

(1)

where x ∈ Rn is the state vector, u ∈ Rm is the input vector, A ∈ Rn×n , B ∈ Rn×m and F are known matrices and ϕ(F x, u) represents the nonlinear part of the system, which is supposed to satisfy the One-Sided Lipschitz and the Quadratic-Inner Boundedness properties, presented by the next two definitions. Definition 2 In system (1), the nonlinear function ϕ(F x, u) is said to be globally one-sided Lipschitz with respect to x, if there exist a constant ρ ∈ R, satisfying: ϕ(F x1 , u) − ϕ(F x2 , u), F(x1 − x2 ) ≤ ρF(x1 − x2 )2 , ∀x1 , x2 ∈ Rn

(2)

Definition 3 In system (1), the nonlinear function ϕ(F x, u) is said to be globally quadratically inner bounded with respect to x, if there exist constants β, γ ∈ R, satisfying: ϕ(F x1 , u) − ϕ(F x2 , u)2 ≤ βF(x1 − x2 )2 + γ ϕ(F x1 , u) − ϕ(F x2 , u), F(x1 − x2 ), ∀x1 , x2 ∈ Rn

(3)

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In the previous definitions: ρ is called the one-sided Lipschitz constant, β and γ are called the quadratic inner-boundedness constants. Note that the One Sided Lipschitz constant ρ can be positive, zero or even negative, unlike the Lipschitz constant which must be positive. Definition 4 The system (1) is said to be globally uniformly Mittag-Leffler stable if there exist positive scalars b and λ such that the trajectory of (1) passing through any initial state x0 at any initial time t0 evaluated at time t satisfies:  b x(t) ≤ m(x0 )E α (−λ(t − t0 )α ) , ∀t ≥ t0 With m(0) = 0, m(x) ≥ 0, m is locally Lipschitz and E α (.) is the Mittag-Leffler function presented in definition 1. Theorem 1 [3] Let x = 0be an equilibrium point for the system (1).Let V : [0, ∞)× Rn → R be a continuously differentiable function and locally Lipschitz with respect to x such that. μ1 xc ≤ V (t, x) ≤ μ2 xcd C

Dtα0 ,t V (t, x(t)) ≤ −μ3 xcd

where t ≥ t0 , x ∈ Rn , α ∈ (0, 1), μ1 , μ2 , μ3 , c and d are arbitrary positive constants. Then x = 0 is globally Mittag–Leffler stable. Lemma 1 [4] Let α ∈ (0, 1) and P ∈ Rn×n a constant square symmetric and positive definite matrix. Then the following relationship holds.  1C α  T Dt0 ,t x (t)P x(t) ≤ x T (t)P C Dtα0 ,t x(t) 2

2 Actuator Faults and Component Faults Estimation for Integer-Order One-Sided Lipschitz Systems 2.1 Motivation and Problem Formulation Until now, several fault estimation works have been done by researchers for nonlinear Lipschitz systems [5–7]. However, only very few results for nonlinear One-Sided Lipschitz systems exist in the literature. Through this section, a new component and actuator fault estimation result for integer-order One-Sided Lipschitz systems is presented and proved. Based on an adaptive observer technique and inspired by

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the works of Kahkeshi et al. [8], the estimated faulty parameters in the present study can correspond to some physical parameters of the system. Component faults, to be estimated here, appear in the state distribution matrix. Actuator faults are some parameters in the input distribution matrix. Thanks to this modeling strategy, actuator faults and component faults are distinguishable. This fact is not the case in the very few previous works dealing with component and/or actuator fault estimation for One-Sided Lipschitz systems. In these works, a lonely fault vector may contain both actuator faults and component faults, without giving a way to distinguish between them. The contributions of this work can be summarized as follows: • Compared to the great majority of works in the literature dealing with the fault estimation problem for nonlinear Lipschitz systems, this study tackles simultaneously component faults and actuator faults for a broader class of nonlinear systems. • Compared to the very few works dealing with the estimation of actuator and/or component faults for One-Sided Lipschitz systems, this work presents a novel and advantageous fault modeling, in which actuator faults and component faults can be distinguished. This is not the case for the already existing works, where actuator and/or component faults are modeled together in one vector, without being distinguished. • To the best of our knowledge: The fault modeling given in this section is original and the whole three-steps procedure (combining the Schur complement lemma, the two versions of Barbalat lemma and the persistency of excitation theory), by which the stability conditions are extracted, is unique. In this section, the considered class of systems can be described by the following couple of state-space equation and output Eq. (4). Note that, in the mentioned statespace equation, the derivative order is α = 1, matrices A and B are parametrized (the parameters are θk , k ∈ [1.. p]).



⎧ p s   ⎨ x(t) ˙ = A0 + θk Ak x(t) + B0 + θk Bk u(t) + Dϕ(F x, u) k=1 k=s+1 ⎩ y(t) = C x(t)

(4)

where x ∈ Rn is the state, u ∈ Rm is a uniformly continuous input, y ∈ Rq is the output, A0 , Ak ∈ Rn×n , B0 , Bk ∈ Rn×m , C ∈ Rq×n , D and F two known matrices of appropriate dimensions. The parameter vector θc = [θ1 , .., θs ] appearing in the state distribution matrix, represents potential component faults, while the parameter vector  represents potential θa = θs+1 , .., θ p appearing in the input distribution matrix,  actuator faults. One can write θ = θ1 , .., θs , θs+1 , .., θ p . In such a situation, the system can be described otherwise:

Fault Estimation for Nonlinear One-Sided Lipschitz Systems





⎧ p p   ⎨ x(t) θk Ak x(t) + B0 + θk Bk u(t) + Dϕ(F x, u) ˙ = A0 + k=1 k=1 ⎩ y(t) = C x(t)

99

(5)

where Ak = 0 for k ∈ [s + 1, p] and Bk = 0 for k ∈ [1, s]. The parameter vector θ ∈ R p takes the well-known constant value θ = θh as long as the system behaves in the healthy mode. With some specific faulty mode, θ takes another constant value θ f , but unknown until now. In fact, the goal of this work is to estimate θ f under some conditions. Without loss of generality, the nonlinear part ϕ(F x, u) is assumed to be a continuous vector function. Remark 1 The elements of the vector θ correspond to actual physical parameters of the system. Since θ1 , .., θs are in the state distribution matrix, then they refer to some properties of the process itself, and consequently, any change in their nominal values is considered as a component fault. Similarly, θs+1 , .., θ p are in the control distribution matrix, meaning that they are related to the actuator amplifier gains, and any fault with these parameters is regarded as an actuator fault. This principle is inspired from the simulation section of [8], where the authors have estimated a component fault appearing in the state distribution matrix, which was an abnormal friction in the motor, as well as an actuator amplifier gain fault, appearing in the control distribution matrix.

2.2 Component and Actuator Fault Estimation Strategy In the following, we address the problem of estimating the unknown fault vector θ f , under conditions (2) and (3). For simplicity, in the rest of this section, if θ is cited without any index, then it refers to θ f . We first employ the Luenberger-like observer:     ⎧ p p .     ⎪ ⎪ ⎪ ˆ = A0 + ˆ + B0 + ˆ u θˆ k Ak x(t) θˆ k Bk u(t) + Dϕ F x, ⎨ x(t) k=1  k=1  ⎪ + LC x(t) ˆ − x(t) ⎪ ⎪ ⎩ yˆ (t) = C x(t) ˆ

(6)

Let the state estimation error, the output estimation error and the fault estimation error respectively be: e(t) = x(t) ˆ − x(t), e y (t) = yˆ (t) − y(t), eθ (t) = θˆ (t) − θ Introduce the following adaptation law: .   θˆ k = −σk e Ty PC Ak xˆ + Bk u

(7)

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where σk is a positive scalar and P is a symmetric definite positive matrix. The observer error dynamics is given by:  e(t) ˙ =

A0 +

p 

 θk Ak + LC e(t)

k=1

+

p 

  eθk Ak x(t) ˆ + Bk u(t) + DΔϕ

(8)

k=1

  where Δϕ = ϕ F x, ˆ u −ϕ(F x, u) From Eq. (3), the following inequality holds: Δϕ T Δϕ ≤ βe T F T Fe + γ e T F T Δϕ

(9)

The design goal is to find an observer gain matrix L such that estimates of x and θ f converge asymptotically. For this purpose, the following assumption is introduced. Assumption 2.1 The states are bounded in the healthy mode as well as in the faulty mode. This is not a restrictive assumption, since, generally, systems to be diagnosed have naturally bounded states in the fault-free functioning. Here, we just assume that the possible faults do not affect the boundedness of states. Let η be a positive scalar and choose the following observer gain matrix: L=

η λmin (P) T −1 C P 2 λmax (P)

(10)

In the following, Theorem 2 ensures the asymptotical convergence of the states estimates as well as the component and actuator faults estimates, using (10). Theorem 2 Consider the class of continuous-time nonlinear systems (5),satisfying (2)and (3).If the inequalities (11)and (14)are feasible, then the adaptive observer (6),with the adaptation law (7),is asymptotically stable for any parameter θ ∈ N , where N is a specific neighborhood of the known vector θh . This yields: e → 0 as t → ∞ and eθ → 0 as t → ∞. A(θh )T C T PC + C T PC A(θh ) + (ε1 ρ + ε2 β)F T F + + ε3 I + η(C T C)2 < 0

1 H HT ε2 (11)

where ε1 , ε2 , and ε3 are all positive scalars, and: H = C T PC D +

(γ ε2 − ε1 )F T 2

(12)

Fault Estimation for Nonlinear One-Sided Lipschitz Systems

A(θ ) = A0 + 

p 

θk Ak

k=1 1 )F Λ Q D + (γ μ2 −μ 2 ∗ −μ2 I

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T

(13)

 0) is p . In other words, if μ3 > satisfied  p in the presence of the term μ0 k=1 m k Ak I p μ0 k=1 m k Ak , then μ3 takes the new value μ3 − μ0 k=1 m k Ak  > 0. Remark 3 The major limitation of Theorem 2 is the fact that it provides the accurate component and actuator fault estimates only for the faulty modes conveniently close to the healthy mode. In the following theorem, we present an interesting particular case related to only actuator faulty systems. The result obtained for such systems through that theorem is less conservative then theorem 2, since it allows estimating even severe faults. Theorem 3 Consider the class of nonlinear systems (5) , but with non-parameterized state distribution matrix , i.e.:

⎧ p  ⎨ x(t) ˙ = Ax(t) + B0 + θk Bk u(t) + Dϕ(F x, u) (31) k=1 ⎩ y(t) = C x(t) satisfying (2 and 3). If the inequality (32) has a symmetric definite positive solution P, using the adaptation law (33) and if the LMI (34) has a symmetric definite positive

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solution Q, then the adaptive observer (5) (with a non-parameterized well known matrix A) is asymptotically stable: e → 0 as t → ∞ and eθ → 0 as t → ∞. The observer gain matrix L has the same expression (10), used in theorem 2. A T C T PC + C T PC A + (ε1 ρ + ε2 β)F T F + ε3 I +

1 H H T + η(C T C)2 < 0 ε2 (32)

.

θˆ k = −σk e Ty PC Bk u 

1 )F Λ Q D + (γ μ2 −μ 2 ∗ −μ2 I

T

(33)

 0 such that: t 0 +T

α0 I ≤

f (u(τ )) f (u(τ ))T dτ ≤ α1 I t0

for all t0 , then eθ → 0. This ends the proof. 

2.3 Simulation Results In this sub-section, we apply the proposed fault estimation method to the following numerical system [13], described by (5) with the following healthy configuration:

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 A=

   T     1 −3 1 3 −x1 x12 + x22  , C = , D = F = I2 , ,B = , ϕ(F x, t) = −x2 x12 + x22 0 1 −6 0

Conditions (2) and (3) are well satisfied with ρ = 0, β = −200 and γ = −300 [13]. The treated scenario is the following: First, at t = 400s, a constant actuator fault happens. Then, after 100s, a constant component fault appears. Consider the  θ1 . This representation is in the form of (31) as case of the actuator fault B(θ ) = 0 follows:  x(t) ˙ = Ax(t) + (B0 + θ1 B1 )u(t) + Dϕ(F x, u) (37) y(t) = C x(t)     0 1 and B1 = . The actuator fault b = θ1 is presented by a decrease 0 0 of 30% of the nominal value: θh = 3. After solving Eq. (32) via the MATLAB LMI control toolbox, we obtain ε1 = 3625.3 ε2 = 375.4171, ε3 = 3641.5 and η = 7567. The matrix P, which is simply a scalar here,

is: P = 1208.4. Consequently, the 3.131 observer gain matrix is set to: L = . The resolution of the LMI (34) via the 0

4 6.5299 0.8854 MATLAB LMI control toolbox gives Q = 10 . After estimating 0.8854 1.1015   2.1 the actuator fault, matrix B takes the new value: B = . Now, the new fault 0 parameter to be treated is a component fault parameter: a = θ1, which is supposed θ1 1 − 10 to appear at t = 500s. This fault is represented by: A(θ ) = . Such a 1 −6 representation is in the form of (5): with B0 =





x(t) ˙ = (A0 + θ1 A1 )x(t) + Bu(t) + Dϕ(F x, u) y(t) = C x(t)

(38)

   0 1 −0.1 0 , A1 = . The component fault a = θ1 is presented 1 −6 0 0 by a decrease of 10% of the healthy value: θh = 30. Solving the inequality (11) for θ = θh is equivalent to solving (32). By fixing P = 1208.4 and η = 7567 (the found solutions for (32), corresponding to θ = θh = 30), and using the MATLAB LMI control toolbox, we find that (11) remains feasible for θ = 27. By following remark 2, the feasibility of the inequality (14) is also verified. To implement the proposed fault estimation method, the system is excited by u(t) = sin(t). The actual actuator fault and its estimate are given in Fig. 1. The actual component fault and its estimate are given in Fig. 2. From these figures, one with A0 =

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Fig. 1 Actuator fault and its estimate for system (37)

Fig. 2 Component fault and its estimate for system (4.2.38)

can clearly note the good performance of the proposed estimator. Indeed, the faults estimates reach perfectly the actual values, which validates our theoretical results.

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3 Sensor Fault Estimation for Fractional-Order Descriptor One-Sided Lipschitz Systems 3.1 Motivation Sensor faults, which are treated in the present section, denote every kind of malfunctioning that can affect one or several sensors in the process. To the best of our knowledge, only very few papers exist in the literature treating the sensor fault estimation problem for nonlinear One-Sided Lipschitz systems. On another hand, note that most of the observer design and/or fault diagnosis works in the literature use normal models, where no algebraic relations between the system variables exist. Though, various physical systems, such that power systems, robotics and electric systems [14] show in their models these algebraic equations, in addition to the ordinary differential equations. Such systems are called descriptor systems, singular systems or implicit systems. It is prominent to mention that, in the last decade, researchers are showing a particular and increasing interest to the observer design problem and the observer-related problems for descriptor systems. Recently, several papers treating fractional-order descriptor systems have been established, but without solving the fault estimation problem (see for instance [15 and 16]). On the other hand, observers for one-sided Lipschitz descriptor systems have been just tackled in the literature and only very few works have been very recently done within this specific topic [17, 18]. These very few works have treated the integer-order descriptor one-sided Lipschitz sytems without estimating any type of faults. Otherwise, note that only few works have been done in the literature, in relation with sensor fault estimation for descriptor integer-order systems. Some of these works can be found in [19 and 20]. Based on all the above discussions and inspired by the works of Gupta [21], a first solution is presented in this section to solve the problem of sensor fault estimation for one-sided Lipschitz descriptor integer-order and fractional-order systems. The advantages of the present work, compared to the previous ones can be summarized as follows: • In [22], the state estimation problem for normal fractional-order one-sided Lipschitz systems has been tackled. The present work is more general than [22], since it extends the problem to descriptor systems, with the estimation of sensor faults. • In [15 and 16], the stability analysis of linear fractional-order descriptor systems, without sensor faults, is achieved. The present paper is more general, since it treats nonlinear one-sided Lipschitz systems instead of linear systems, and the system model includes possible sensor faults. • In [17 and 18]: system states are estimated for integer-order descriptor one-sided Lipschitz systems. The present work is more general, since it extends the problem to fractional-order systems, with estimating sensor faults.

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3.2 Problem Formulation In this section, the considered class of systems can be described by the following couple of state-space equation and output Eq. (39). 

E ∗ C Dtα0 ,t x(t) = A∗ x(t) + B ∗ u(t) + D ∗ ϕ(H x, u) y(t) = C x(t) + F f

(39)

where x ∈ Rn is the state, u ∈ Rm is the input, y ∈ R p is the output, f ∈ Rq is a constant fault vector, E ∗ , A∗ ∈ Rn×n , B ∗ ∈ Rn×m , D ∗ ∈ Rn×n d , H ∈ Rn h ×n , F ∈ R p×q and C ∈ R p×n are known constant matrices. With the condition rank(E ∗ ) = r < n, the matrix E ∗ is singular and system (39) is a descriptor system, else it is a normal system. Without loss of generality, let rank(C) = p.

3.3 Sensor Fault Estimation Strategy Here, an overall scheme is given in order to reconstruct constant sensor faults. The organization of this sub-section is as follows. First, a system transformation step is presented. Then, the main part dealing with the observer design and the LMI formulation is detailed. Finally, a summarizing design procedure is given. But to begin, the following assumption should be cited. Assumption 3.1 The triple matrix (E ∗ , A∗ , C) satisfies the following conditions [23, 24]:  rank

E∗ C



 = n, rank

s E ∗ − A∗ C

 = n, ∀s ∈ C

where C denotes the complex plane.

3.3.1

System Transformation

Later, in order to guarantee the existence of the designed observer, system (39) has to be transformed into the following equivalent restricted system, by means of a non-singular matrix R ∈ Rn×n . 

E C Dtα0 ,t x(t) = Ax(t) + Bu(t) + Dϕ(H x, u) y(t) = C x(t) + F f

(40)

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where E = R E ∗ , A = R A∗ , B = R B ∗ and D = R D ∗ . Note that, according to [25]: when assumption III.1 holds, one has the following property: 

I−E rank C

 =p

(41)

 I − E∗ = p, then If (41) is satisfied from the beginning with E , i.e.: rank C there is no need to make the system transformation and to calculate the matrix R (one takes R = In ). The existence proof of matrix R can be found in [25]. Matrix R is constructed as follows [21]:   I − E∗ = p then take R = In and stop. If rank C Else (under assumption III.1), follow these instructions: 

∗

• Make the singular value decomposition of C: C = U1 [D1 0]V1T .  −1 T  D1 U 1 0 • Compute Q = V1 . 0 In− p   0 • Compute E = E ∗ Q . In− p   D2 • Make the singular value decomposition of E: E = U2 V2 . 0   0 Ip • Compute R0 = U2T . V2T D2−1 0 • Compute R = Q R0 . 3.3.2

Observer Design and LMI Formulation

Let system (42) be an observer for the fractional order descriptor system (39). Since (40) is the equivalent restricted system of (39), the proposed observer (42) gives the same results when applied to (40). C

  Dtα0 ,t z(t) = N z(t) + L y(t) + Bu(t) + Dϕ H x, ˆ u x(t) ˆ = z(t) + M y(t)

(42)

where z(t) is the state vector of the observer, x(t) ˆ is the estimate of x(t), N , L and M are design matrices with appropriate dimensions. These matrices satisfy the following conditions: E = I − MC

(43)

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N = A − KC

(44)

K = L − NM

(45)

where K is an introduced design matrix. Define the state estimation error as e = x − xˆ and the sensor fault estimation error as f˜ = f − fˆ. Then, one has: e = x − z − MC x − M F f = (I − MC)x − z − M F f e = Ex − z − M F f

(46)

  Taking into consideration: ϕ = ϕ(H x, u) − ϕ H x, ˆ u , the dynamics of the state estimation error are governed by: C

Dtα0 ,t e = E C Dtα0 ,t x −C Dtα0 ,t z

  = Ax + Bu + Dϕ(H x, u) − N z − Bu − LC x − L F f − Dϕ H x, ˆ u = (A − LC)x + Dϕ − N (E x − e − M F f ) − L F f = N e + (A − LC − N E)x + Dϕ + (N M − L)F f = N e + (A − LC − N + N MC)x + Dϕ − K F f

From (44) and (45), one has: A − LC − N + N MC = 0, then: C

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(47)

By introducing the following adaptation law, the possible sensor faults can be accurately estimated. The main Theorem of this section is then presented. C

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(48)

where fˆ(t) is the estimate of the fault vector, yˆ (t) is the estimate of the output vector and G is a design matrice. Theorem 4 Consider the descriptor fractional-order system (40), under assumption III.1, the observer (42) and the adaptation law (48).If there exist matrices K , G and η such that the LMI (49) is feasible P = P T > 0 and positive scalars τ1 , τ2 , εand  ˜ under condition (50), then the error origin e, f = (0, 0) is globally Mittag-Leffler stable. ⎤ ⎡ T − η1 C T G T P A + AT P − K C − C T K + X + ε I Y ⎥ ⎢ (49) ⎦ 100 the true system is model 2 The evolutions of the set point, output signal and input signal obtained by the robust predictive control are reproduced in Fig. 1. According to this simulation, we see that despite the change in the dynamics of the fractional system at the instant k = 100, the controller is able to develop a control signal which allows the output to follow the set point, but we see also that the control presents amplitude oscillations when changing the set point. In this work, the criterion to be optimized is non-convex which implies the use of a global optimization method in order to ensure good closed-loop performance.

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4 Global Optimization In this section, we present the Genetic Algorithms (GA) as a global solution to solve non-convex min–max optimization problem in order to obtain the robust control law for uncertain fractional order system.

4.1 Genetic Algorithm Method Genetic algorithms (GA) are numerical and stochastic optimization methods belonging to the family of evolutionary optimization techniques whose concept is based on Charles Darwin’s theory of evolution. The general principle of this type of optimization is to simulate the evolution of a population of individuals drawn

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randomly at the start. Each individual is assigned an index of quality by means of an adaptation function. A population succession is generated by applying different operators (recombination, mutation, selection, etc.) to the previous population. This process is repeated as many times until the satisfaction of one or more stopping criteria. Such an algorithm does not require any mathematical requirements regarding the optimization problem and it only needs evaluation of the objective function. This is why in many areas this methodology has provided solutions to previously unresolved problems and is more robust than deterministic methods. • Codification: The first step in implementing any genetic algorithm is coding. The basic mechanisms of an AG rely on binary coding. This representation has the advantage of easily coding all kinds of objects: real, integers, Boolean values, strings, etc. This simply requires the use of coding and decoding functions to switch from one representation to another. • Fitness function: An optimization algorithm requires the definition of a function called objective function which makes it possible to evaluate the potential solutions, from the magnitudes to be optimized. However, the GA uses a function called adaptation function which is a complement to the objective function whose role is to better measure the performance of an individual in relation to the entire population. The result provided by this function will be exploited by genetic operators to create a new population. As a result, the algorithm will converge towards an optimum of this function also called fitness function. • Generation of a new population: Generation of a new population: At each iteration called generation, a new population (offspring) is created from the current population using genetic operators: selection, crossing and mutation. The use of these three operators makes it possible to maintain a well diversified population and consequently to access all the research space. • Stopping conditions: Stopping the algorithm is conditioned by the validation of certain criteria which may be the maximum number of generations or when the individuals of a population do not evolve more quickly or the maximum execution time or even a combination of these criteria.

4.2 RFPC Based of Genetic Algorithm In this paragraph, the main ideas for implementing the GA strategy in the algorithm of the fractional robust predictive control method are explained. Equation (17) can be defined in the following way: min J ∗ ()

U ∈

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(21)

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(22)

The genetic algorithm has become widely used in solving complex and nonconvex optimization problems. Once the worst-case model is developed by solving the optimization problem (22), minimizing the performance criterion "J " with respect to future control increments becomes a quadratic optimization problem. The latter can then be solved by a conventional optimization method. Thus, the algorithm used is described as follows: • Step 1: Initialization: we specify the controller parameters, as well as the set point trajectory; the number of individuals in the population and the maximum number of generations are also determined. • Step 2: Finding the variables al and bm that optimize the problem (22) by genetic algorithm method. • Step 3: Calculating the future control sequence U by finding a solution to problem (21) using the identified values of al and bm . • Step 4: Implementing the control: u(k) = u(k − 1) + U (1) • Step 5: Moving to step (2), taking into consideration the new measured inputs/outputs.

5 Practical Application to a Fractional Thermal System The robust fractional predictive control technique developed in this work has been applied to a real fractional thermal system composed of a metal bar subjected to a heat flux. Indeed, the model, relating the density of heat flux through a metal bar to the measured temperature, is a fractional order model, because it follows from the resolution of the heat equation. This real system consists of a 41-cm-long aluminum bar and 1-cm-long radius, as shown in Fig. 2. The input signal for this system is a thermal flux (Q) produced through a heat resistor attached at the beginning of the bar. The output signal consists in the bar temperature which is measured with an LM35D temperature sensor. The objective of this application is to evaluate the robust predictive control of fractional systems based on the Grünwald-Letnikov definition with respect to the uncertain physical behavior of the system caused by the change in sensor position. The goal, in this case, is to maintain the temperature measured at the desired set point at two points P1 and P2 respectively at a distance d1 = 6 cm and d2 = 15 cm from the heat resistor [22].

5.1 Modeling and Identification This paragraph focuses, more specifically, on the modeling of the thermal system by an uncertain fractional order model. Thus, we applied to the heat resistor an input

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Fig. 2 Real schema of thermal process

sequence that can take two different values (4v and 0v) of varying durations and we saved the temperature values measured at the two positions of the sensors P1 and P2. Figure 3 shows the changes in the input signal and the actual and estimated temperatures at the two positions P1 and P2 with a sampling period equal to 20 s. For the sake of clarity of the figures, we have multiplied the input values by 10. 80 Measured Temperature (P1) Model H1 Measured Temperature (P1) Model H2 Input *10

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Using the SRIVCF fractional model identification method [23], we determined the two fractional model of our thermal system H1 (s) and H2 (s) for the two positions P1 and P2: 1.82 e−25s 37.14s 1.5 + 90.5s + 11.19s 0.5 + 1

(23)

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5.2 Controller Application The proposed RFPC control synthesis requires to take into account the parametric variations of the fractional order model. Using the two fractional models given by Eqs. (23) and (24), we determined this equation: G(s) = where:

p0 e−55s a2 s 1.5 +a1 s+a0 s 0.5 +1

(25)

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b0 ∈ [1.4, 1.82] a0 ∈ [3.08, 11.19] a1 ∈ [90.5, 188.42] a2 ∈ [37.14, 216.69] This fractional transfer function can be expressed by the Grünwald-Letnikov definition described by Eq. (8): ⎧ L = 3; M = 0; ⎪ ⎪ ⎪ ⎨ α = 0; α = 0; α = 0.5; α = 1; α = 1.5 b0 a0 a1 a2 a3 ⎪ a = 1; b ∈ 1.82]; a ∈ 11.19] [1.4, [3.08, 0 0 1 ⎪ ⎪ ⎩ a2 ∈ [90.5, 188.42]; a3 ∈ [37.14, 216.69] In this case, the worst case scenario is adopted for the determination of the robust predictive control law. That is, the proposed FRPC controller represents the best solution for the worst case of possible fractional models belonging to the set of uncertainties. Thus, the control law is obtained by solving a min–max optimization problem that was presented in this work in Sect. 3. The sampling period is equal to 20 s in all experiences and the control signal is topic to the limitations specified below: 0 ≤ u(k) ≤ 5v −0.5v ≤ u(k) ≤ 0.5v The robust fractional controller we have proposed is intended with the following parameters N1 = 1, N2 = 15 and λ = 1 Figure 5 reproduces the behavior of the thermal system closed loop obtained by the RFMPC controller with constraints. It can be seen from this figure, that despite the change in the dynamics of the system at iteration 300, the controller is able to develop a control signal which allows the output to follow the set point selected while respecting the constraints on the input signals. In this experiment, the solution found is not global, the control law would then be suboptimal. This can cause instability in the closed loop system. which involves the use of the Genetic Algorithm that was presented in Sect. 4 which eventually helps to find the global optimum. In this practical experience, we have used the ‘GA’ function defined in MATLAB. Figure 6 shows the measured temperature, the desired set point and the control signal obtained with the robust predictive control based on the genetic algorithm.

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From Fig. 6, we note that despite the change in system dynamics caused by the change in sensor position at the sampling period 300, the temperature measured follows the desired set point.

6 Conclusion This work has been devoted to the robust predictive control of uncertain fractional systems using the Grünwald-Letnikov definition. The synthesis of the control law is based on the min–max optimization method taking into account the constraints on the control signals and on the control increments. Consequently, the worst-case approach is then adopted for obtaining the predictive control law, i.e. the control represents the best solution for the worst case of possible fractional models belonging to the set of uncertainties. The robustness of the predictive control developed in this work has been tested on a fractional thermal system.

References 1. Fukushima, H., Kim, T., Sugie, T.: Adaptive model predictive control for a class of constrained linear systems based on comparison model. Automatica 43(2), 301–308 (2007) 2. Camacho, E.F., Bordons, C.: Model Predictive Control. Springer-Verlag, Berlin (2004) 3. Rhouma, A., Bouani, F.: Robust predictive controller based on an uncertain fractional order model. In: 12th International Multi-Conference on Systems, Signals and Devices, SSD (2015) 4. Rhouma, A., Bouzouita, B., Bouani, F.: Model predictive control of fractional systems using numerical approximation. 2014 World Symposium on Computer Applications and Research, WSCAR (2014) 5. Lijun, C., Shangfeng, D., Yaofeng, H., Meihui, L., Dan, X.: Robust model predictive control for greenhouse temperature based on particle swarm optimization. Inf. Process. Agric 5(3), 329–338 (2018) 6. Podlubny, I.: Fractional Differential Equations. Academie Press, New York (1999) 7. Bagley, R., Calico, R.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control. Dyn. 14, 304–311 (1991) 8. La, O.A.: commande CRONE (Commande Robuste d’Ordre Non Entier). Hermès, Paris (1991) 9. Baris, B.A., Abdullah, A., Celaleddin, Y.: Auto-tuning of PID controller according to fractionalorder reference model approximation for DC rotor control. Mechatronics 23(7), 789–797 (2013) 10. Dastjerdi, A.A., Vinagre, B.M., Chenc, Y., HosseinNiaa, S.H.: Linear fractional order controllers; A survey in the frequency domain. Annual Reviews in Control 47, 51–70 (2019) 11. Goyal, V., Mishra, P., Deolia, V.K.: A robust fractional order parallel control structure for flow control using a pneumatic control valve with nonlinear and uncertain dynamics. Arab. J. Sci. Eng. 14(3), 2597–2611 (2019) 12. Chen, K., Tang, R., Li, C., Lu, J.: Fractional order PIλ controller synthesis for steam turbine speed governing systems. ISA Trans. 77, 49–57 (2018) 13. Shabnam, P., Peyman, B.: Parallel cascade control of dead time processes via fractional order controllers based on Smith predictor. ISA Trans. (2019). https://doi.org/10.1016/j.isatra.2019. 08.047 14. Boudjehem, D., Boudjehem, B.: Robust Fractional Order Controller for Chaotic Systems. IFAC-PapersOnLine 49(9), 175–179 (2016)

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15. Sopasakisy, P., Ntouskas, S., Sarimveis, H.: Robust model predictive control for discrete-time fractional-order systems. 23rd Mediterranean Conference on Control and Automation (MED). Torremolinos, Spain (2015) 16. Tavazoei, M.S.: A note on fractional-order derivatives of periodic functions. Automatica 46, 945–948 (2010) 17. Hajiloo, A., Nariman-zadeh, N., Moeini, A.: Pareto optimal robust design of fractional-order PID controllers for systems with probabilistic uncertainties. Mechatronics, Elsevier 22, 788– 801 (2012) 18. Miller K.S., Ross B. An introduction to the fractional calculs and fractional differential equation. Wiley (1993) 19. Watanabet, K., Ikeda, K., Fukuda, T., zafestas, S. G.T.: Adaptive generalized predictive control using a state space approach. In: Proc. of International Workshop on Intelligent Robots and Systems, Osaka, Japan (1991) 20. Oustaloup, A., Olivier, C., Ludovic, L.: Representation et Identification Par Modèle Non Entier. Lavoisier, Paris (2005) 21. Rhouma, A., Bouani, F., Bouzouita, B., Ksouri, M.: Model predictive control of fractional order systems. J. Comput. Nonlinear Dyn. 9(3) (2014) 22. Rhouma, A., Bouani, F.: Robust model predictive control of uncertain fractional systems: a thermal application. IET Control Theory Appl. 8(17), 1986–1994 (2014) 23. Malti, R., Victor, S., Oustaloup, A., Garnier, H.: An optimal instrumental variable method for continuous-time fractional model identification. In: 17th IFAC World Congress, Seoul South Korea, pp. 14379–14384 (July 2008)

Constant Phase Based Design of Robust Fractional PI Controller for Uncertain First Order Plus Dead Time Systems B. Saidi, Z. Yacoub, M. Amairi, and M. Aoun

Abstract A new design method of a robust fractional order PI controller for uncertain First Order Plus Dead Time systems is proposed in this paper. The proposed design method uses a numerical optimization algorithm to determine the unknown controller parameters. The main objective of the proposed design method is improving the robustness in degree of stability to gain variations and the stability robustness to the other parameters variations that affect the phase by imposing a constant phase margin to the corrected open loop system in a pre-specified frequency band. Several simulation examples are presented to design the robust fractional PI controller and test the robustness for different forms of uncertainty. Keywords Numerical optimization · Fractional PI controller · Stability robustness · Uncertainty · Time delay system

1 Introduction For several decades and despite continuous advances in control theory, the PI and the PID controllers are widely used for many industrial systems. The popularity of these controllers refers not only to their simplicity of use and implementation or their effectiveness but also to the fact that they provide satisfactory performance in many applications. Although they guarantee sometimes the desired performance and in spite of their simplicity of use, they suffer in general from a lack of robustness and a slow transient response especially in presence of time delay. To improve those performances, the generalized form of the PID controller is now well used. The B. Saidi (B) · Z. Yacoub · M. Amairi · M. Aoun National Engineering School of Gabes (ENIG), Research laboratory Modeling, Analysis and Control of Systems (MACS)06/LR/11-12, University of Gabes, Omar Ibn el Khattab street, 6029 Gabes, Tunisia e-mail: [email protected] M. Aoun e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Naifar and A. Ben Makhlouf (eds.), Fractional Order Systems—Control Theory and Applications, Studies in Systems, Decision and Control 364, https://doi.org/10.1007/978-3-030-71446-8_9

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fractional (generalized) PID controller (PIα Dβ ) [15] has two extra-parameters (the integrator order α and the differentiator order β) which can lead to a more flexible structure. Recently, for improving results, many works focus on the design of the fractional PID. This controller takes advantage of the robustness characteristics of the fractional integrator and derivative. Different design methods are proposed since the development of the first control approach using the fractional PID controller [5, 14]. Most of these methods are used for controlling a First Order Plus Dead Time (FOPDT) system [1, 9, 12, 13, 20] or for fractional order system [3, 11, 16, 22]. Recently some works concern the non-minimum phase dead-time systems [17] or nonlinear chaotic systems [4] Concerning the Second Order Plus Dead Time system (SOPDT) they are few works in literature [6, 21]. In this paper, a design method based on numerical optimization is proposed. The work presented in this paper comes after a previous works that propose to control a SOPDT system with fractional PID controller [18] and to design of a reduced controller using multi-objective optimization method to control a non minimum phase second order system [19]. This paper presents a generalization of the two previous works for the design method of a robust fractional PI controller for especially an uncertain FOPDT. The paper is organized as follows: Section 2 details the problem statement, followed by a general formulation of the design method using numerical optimization. In Sect. 3, the design of the robust fractional PI controller for FOPDT system, followed by some particular cases. To show the effectiveness of the proposed design method, numerical examples are proposed for each case, followed by a comparative study with a robust classical PI controller proposed by Toscano in [23] in Sect. 4.

2 Problem Statement Systems with delays are frequently distributed in industry [8, 10] and they represent a class of systems where the dynamics not only depends on the value of the current time state value but also of the last state values taken on a certain temporal horizon. A First Order Plus Dead Time (FOPDT) is defined by the following transfer function: P(s) =

k e−δs τs + 1

(1)

where k is the process gain, τ is the time constant and δ is the time delay. Nowadays, in the face of the complexities surrounding industrial systems, most plants are affected by parametric uncertainties [2, 7]. In fact, an uncertain plant has at least one parametric uncertainty. An uncertain FOPDT system can be presented by the following transfer function P(s) =

[k k] e−[δ 1 + [τ τ ]s

δ]s

(2)

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where . and . are the lower and the upper bounds of each parameter. For controlling such a system, in this work the considered controller has the following form ki R(s) = k p + α (3) s where k p , ki are respectively the proportional and integrator gains. α is the fractional order of the integration. This controller has three tuning parameters (k p , ki , α). The fractional orders α leads to a more flexible controller that takes account of robustness to parameters uncertainties. That’s for the fractional PI controller will be designed using frequency specifications. The most used frequency specifications in design approaches are phase margin ϕm and unity-gain crossover frequency ωu . These specifications ensure robustness, and time specifications such as overshoot and settling time. To satisfy those design criterions, the fractional controller must achieve the robustness property and provides the desired damping which is related to the desired phase margin, the desired settling time which has a relation to unitygain crossover frequency ωu , and also the robustness to plant uncertainties which is related to a constant phase margin despite the parameters variations. The controller is then tuned by finding its three parameters satisfying the following constraints: • Minimise the following criterion: J 1 = |R( jωu )P( jωu )|d B

(4)

• Desired phase margin of the corrected open loop should verify: C1 = arg((R( jωu )P( jωu )) + π − ϕm = 0

(5)

• Robustness in stability degree to parameters variations: C2 =

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(6)

To determine the controller parameters “ Minimise the objective function J1 under constraints C1 and C2 . This famous design method can leads to a non robust controller in case of parameters uncertainties because the constant phase is in a limited frequency band. For this reason, the frequency band of constant phase will be more large.

3 Design Approach To ameliorate stability robustness and robustness in degree of stability, the proposed design approach is based on a large frequency band of constant phase and then, the constraints 5 and 6 will be changed as follows:

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i=1

To determine the controller parameters “ Minimize the objective function J1 under the constraint C3 . For simplifying the design and obtaining the stability robustness in case of parameters uncertainties, the constraint of the argument will be changed by considering the parameters uncertainties and also considering the maximum and the minimum values of phase in the pre-specified frequency band instead of all the frequency band and minimize the difference between those two values and the desired phase margin. Constraints are then changed as follows: • Unity-gain crossover frequency for the lower bound of the gain:     C1 = R( jωu )Pgain ( jωu )

dB

(8)

• Maximize the difference between the maximum phase value of the lower phase system arg + (R( jωi )Pphase ( jωi ))(respectively the minimum value of the lower phase system arg − (R( jωi )Pphase ( jωi ))) and the desired phase margin in the desired frequency band i.e. for all ωi ∈ [ωmin , ωmax ]: C2 = arg + (R( jωi )Pphase ( jωi )) + π − ϕm = 0

(9)

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To determine the controller parameters “ Minimize the objective function C1 under constraints C2 and C3 . To reduce the number of constraints and to simplify the deign procedure, only one function will be optimized. In fact, the objective function is a sum of all the desired objectives as follows:   3  w (C  )2 (11) J = 3

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3.1 Case of Gain Uncertainty If the uncertainty affects the gain, the system is then given by the following transfer function:   k, k −δn s (12) P (s) = e 1 + τn s For this particular case, the upper and the lower bounds of the argument are the same: arg(Pphase ( jω)) = arg(Pphase ( jω)) = atan(−τ ω) − δn ω

(13)

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However the gain of the system with uncertain time delay has the same value of the lower and the upper bounds that’s given as follows:        Pgain ( jω) =  Pgain ( jω) =

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3.3 Case of Constant Time Uncertainty In the case of uncertain constant time, the system in this case is given by the following transfer function: kn   e−δn s (20) P (s) = 1 + τ, τ s The lower bound of the system phase in this case is given by the following equation: arg(Pphase ( jω)) = atan(−τ ω) − δn ω

(21)

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(22)

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(25)

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(26)

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4 Numerical Example The problem statement is the design of a robust fractional PI controller for the uncertain FOPDT class system. The system is then given by the following uncertain transfer function: [0.5, 1.5] e−[0,1]s P (s) = (30) 1 + [0.5, 1.5] s The nominal case system is given as follows: Pn (s) =

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(31)

Table 1 presented the desired performances

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(33)

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From this figure, the controlled system guaranteed the desired performances. We remark that the unity gain crossover frequency is equal to 0.5rad/s which is almost equal to the desired one for the system with minimum value of gain Pgain . Also, we notice that the phase margin is constant in the pre-specified frequency band [ωmin , ωmax ] and equal to 50◦ . Then, the controller guaranteed desired performances of the unity gain crossover frequency and the phase margin. With a gain variation of ±50%, the step responses and the control signals of the closed loop corrected system with the robust fractional PI are shown in Fig. 2: The robustness of the P I α to the gain variation of the plant is then clearly shown. In fact, with a gain variation of ±50% of the nominal value kn = 1, the robustness is good enough and the overshoot is almost the same with the three different values of gain. Figure 2 shows also the control signal which is tolerant for all different cases.

4.2 Case of Time Delay Uncertainty The time delay uncertain system is given by the following uncertain transfer function: P (s) =

1 −[0,1]s e 1+s

(34)

Applying the criterion J3 with w1 = w2 = w3 , the proposed design method leads to the following controller transfer function:

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R(s) = 1.04 +

0.47 s 1.33

(35)

Bode diagrams of the controlled open loop system in case of time delay variations are shown in the following figure: From Bode diagrams of the controlled open loop system in case of time delay variations which are shown in Fig. 3, the frequency specifications are well guaranteed. In fact, in the desired frequency band [ωmin , ωmax ] = [0.05, 1] rad/s, the phase is equal to −130◦ for the system with the lower phase bound Pphase (system with delay δ = 1s). So the phase margin is greater then the desired one for the two other system (system with delay δ = 0s and system with delay δ = 0.5s). With a time delay variation of ±50% of the nominal case, the step responses of the closed loop corrected system with robust fractional PI controller are shown in the Fig. 4: We remark that if we modify the time delay δ ∈ {0, 0.5, 1}, the overshoot is steel constant which prove the robustness of the P I α to the time delay variation of the plant.

4.3 Case of Constant Time Uncertainty The variation of the constant time affects the variation of the phase and the gain. An uncertain constant time system is given by the following uncertain transfer function:

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P (s) =

1 e−0.5s 1 + [0.5, 1.5] s

(36)

Applying the criterion J3 , the proposed design method leads to the following controller transfer function: 0.55 R(s) = 0.76 + 1.23 (37) s By applying this controller to the uncertain constant time system, the obtained Bode diagrams of the controlled open loop are shown in the Fig. 5: From Bode Diagrams, the controlled uncertain constant time system with robust fractional PI controller guarantees well the frequency specifications. In fact the desired unity gain crossover frequency is obtained with the lower gain system (system with maximum value of constant time) and the desired phase margin in a pre-specified frequency band for the lower phase system (system with maximum value of constant time). The frequency results are well confirmed by the step responses presented in Fig. 6. From Fig. 6 which presents the step responses and the control signals are shown with a time constant variation of ±50% , we remark that for the fractional PI controller if we modify the time constant τ ∈ {0.5, 1, 1.5}s, the overshoot is almost constant. The robustness of the P I α to the time constant variation of the plant is again clearly demonstrated as it was for gain and time delay. Previous results of the corrected system show that the fractional PI is generally robust to the gain variation, time delay variation and to the variation of the time constant without lost of precision. What if the uncertainties affect all parameters at the same time?

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4.4 Case of Uncertainties of All Parameters Now the uncertainties affect all the parameters, the system is given by the following uncertain transfer function: P (s) =

[0.5, 1.5] e−[0,1]s 1 + [0.5, 1.5] s

(38)

The proposed design method leads to following fractional PI controller R (s) = 1.36 +

0.9286 s 1.30

(39)

In case of all parameters uncertainties, the obtained Bode diagrams of the controlled open loop are presented by the Fig 7: As all the previous cases, from the Bode diagrams of the controlled open loop, the frequency specifications are well proved for the specified systems. In fact, the unity gain crossover frequency is obtained by the lower gain system (system with maximum value of time constant τ and minimal value of gain k) and the desired phase margin in the pre-specified frequency band is obtained for the lower phase system (system with maximal value of time constant τ and maximal value of time delay δ). The step responses and control laws of the limited bound systems are presented as follows. Figure 8 shows that the proposed design approach ensures the robustness in stability and in degree of stability in case of extremum values of phase and gain at the same

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time (i.e. when uncertainties correspond to the lower bound value of the gain and the upper bound phase). The designed robust fractional PI controller is able to guarantee the stability robustness of the system even if the uncertainties have the maximum values, and can warranted that the parameters variation does not deteriorate the system stability.

4.5 Comparative Study For comparison, simulation results are presented by classical PI controller proposed by Toscano method and a fractional PI controlled by this proposed design method. Consider the following uncertain FOPDT: P (s) =

[0.5, 1.5] e−[0.1,0.5]s 1 + [0.5, 1.5] s

(40)

By Applying the criterion J3 with w1 = w2 = w3 , the proposed design method leads to the following controller transfer function: R (s) = 1.721 +

2.196 s 1.162

(41)

Applying the toscano method, the controller is The fractional PI controller is given by 2.66 R (s) = 0.86 + (42) s Figures 9, 10 and 11 show that with different uncertainties (i.e.: gain time delay or time constant) the proposed design method give more robust results than the design method of Toscano. This comparative study show the effectiveness of this deign method, especially in case time delay variations the proposed design method is stable but the other method give an oscillatory response with time delay δ = 0.5s. The robustness ensured is in stability and degree of stability.

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X: 4.589 Y: 0.98 Z: 5

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5 Conclusion In this paper a sample and efficient design approach of a robust fractional PI controller is considered for uncertain FOPDT process. This design approach uses numerical non linear optimization and includes the parameters uncertainties in the design method which is based on the optimization of a sum of desired specifications. It leads to the determination of the parameters of the fractional PI controller. The use of the controller have shown a good robustness of the closed loop regardless all the parameters variation i.e. the gain, the time constant and the time delay of the FOPDT system. The Numerical examples are presented to show the efficiency of the design and the obtained results shows the importance of considering the parameters uncertainties in the design procedure to maintain the robustness of the system to parameters variations. This approach ensures good stability robustness by ensuring a constant phase in a pre-specified frequency band. Acknowledgements This work was supported by the Ministry of the Higher Education and Scientific Research in Tunisia.

References 1. Amairi, M., Aoun, M., Saidi, B.: Design of robust fractional order pi for fopdt systems via set inversion. In: 2014 IEEE Conference on Control Applications (CCA), pp. 1166–1171. IEEE (2014) 2. Annal, A.W.P., Kanthalakshmi, S.: An adaptive pid control algorithm for nonlinear process with uncertain dynamics. Int. J. Automation Control 11(3), 262–273 (2017) 3. Ben Hmed, A., Amairi, M., Aoun, M.: Robust stabilization and control using fractional order integrator. Trans. Inst. Meas. Control 39(10), 1559–1576 (2017) 4. Bouyedda, H., Ladaci, S., Sedraoui, M., Lashab, M.: Identification and control design for a class of non-minimum phase dead-time systems based on fractional-order smith predictor and genetic algorithm technique. Int. J. Dyn. Control 7(3), 914–925 (2019) 5. Das, S., Saha, S., Das, S., Gupta, A.: On the selection of tuning methodology of FOPID controllers for the control of higher order processes. ISA Trans. 50(3), 376–388 (2011) 6. Feliu-Batlle, V., Rivas, R., Castillo, F.: Fractional order controller robust to time delay variations for water distribution in an irrigation main canal pool. Comput. Electronics Agric. 69, 185–197 (2009) 7. Guo, J., Dong, L.: Robust load frequency control for uncertain nonlinear interconnected power systems. Int. J. Automation Control 11(3), 239–261 (2017) 8. Kannan, G., Saravanakumar, G., Saraswathi, M.: Two-degree of freedom pid controller in time delay system using hybrid controller model. Int. J. Automation Control 12(3), 399–426 (2018) 9. Liang, T., Chen, J., Lei, C.: Algorithm of robust stability region for interval plant with time delay using fractional order P I λ D ν contoller. Commun. Nonlineair Sci. Numer. Simulat. 17, 979–991 (2011) 10. Lu, D., Tang, J.: Particle swarm optimisation algorithm for a time-delay system with piece-wise linearity. Int. J. Automation Control 11(3), 290–297 (2017) 11. Boudana, Marwa, Ladaci, S.J.J.L.: Fractional order pi λ and piμ d λ control design for a class of fractional order time-delay systems. Int. J. Cyber-Phys. Syst. (IJCPS) 1(2), 1–18 (2019) 12. Mercader, P., Banos, A., Vilanova, R.: Robust proportional-integral-derivative design for processes with interval parametric uncertainty. IET Control Theor. Appl. 11(7), 1016–1023 (2017)

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13. Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer (2010) 14. Neçaibia, A., Ladaci, S.: Self-tuning fractional order pi λ d μ controller based on extremum seeking approach. Int. J. Automation Control 8(2), 99–121 (2014) 15. Podlubny, I.: Fractional-order systems and PIλ Dν controller. IEEE Trans. Automatic Control 44, 208–214 (1999) 16. Pourhashemi, A., Ramezani, A., Siahi, M.: Designing dynamic fractional terminal sliding mode controller for a class of nonlinear system with uncertainties. Int. J. Automation Control 13(2), 197–225 (2019) 17. Rabah, K., Ladaci, S., Lashab, M.: Bifurcation-based fractional-order pi λ d μ controller design approach for nonlinear chaotic systems. Front. Inf. Technol. Electronic Eng. 19(2), 180–191 (2018) 18. Saidi, B., Amairi, M., Najar, S., Aoun, M.: Bode shaping-based design methods of a fractional order pid controller for uncertain systems. Nonlinear Dyn. 80(4), 1817–1838 (2014) 19. Saidi, B., Amairi, M., Najar, S., Aoun, M.: Multi-objective optimization based design of fractional PID controller. In: 12th International Multi-Conference on Systems, Signals & Devices (SSD), pp. 1–6. IEEE (2015) 20. Saidi, B., Amairi, M., Najjar, S., Aoun, M.: Fractional pid min-max optimisation-based design using dominant pole placement. Int. J. Syst. Control Commun. 9(4), 277–305 (2018) 21. Saidi, B., Najar, S., Amairi, M., Abdelkrim, M.N.: Design of a robust fractional PID controller for a second order plus dead time system. In: 10th International Multi-Conference on Systems, Signals & Devices (SSD), pp. 1–6. IEEE (2013) 22. Somasundaram, S., Benjanarasuth, T.: Cdm-based two degree of freedom pi controller tuning rules for stable and unstable foptd processes and pure integrating processes with time delay. Int. J. Automation Control 13(3), 263–281 (2019) 23. Toscano, R.: A simple robust PI/PID controller design via numerical optimization approach. J. Process Control 15, 81–89 (2005)

Identification of Continuous-Time Fractional Models from Noisy Input and Output Signals Z. Yakoub, M. Aoun, M. Amairi, and M. Chetoui

Abstract It is well known that, in some practical system identification situations, measuring both input and output signals can commonly be affected by additive noises. In this paper, we consider the problem of identifying continuous-time fractional systems from noisy input and output measurements. The bias correction scheme, which aims at eliminating the bias introduced by the fractional order ordinary least squares method, is presented, based on the estimation of variances of the input and output measured noises. The compensation method for the input and output noises is also studied by introducing an augmented high-order fractional-order system in the identification algorithm. The presented algorithm is established to perform unbiased coefficients and fractional orders estimation. The promising performances of the proposed method are assessed via the identification of a fractional model and a fractional real electronic system. Keywords Fractional calculus · Error in variables · Bias correction · Least squares · Nonlinear optimization

1 Notation See Table 1.

2 Introduction The fractional order calculus has been widely used in the field of engineering and applied sciences over the last two decades (see [1–3] and the references therein), where the stability analysis, [4–6], the controller design [7–10] and the system idenZ. Yakoub (B) · M. Aoun · M. Amairi · M. Chetoui Research Laboratory Modeling, Analysis and Control of Systems (MACS -LR16ES22), National Engineering School of Gabes, University of Gabes, Gabes, Tunisia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Naifar and A. Ben Makhlouf (eds.), Fractional Order Systems—Control Theory and Applications, Studies in Systems, Decision and Control 364, https://doi.org/10.1007/978-3-030-71446-8_10

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182 Table 1 Table of notations Symbol R R∗+ Rn Rn×m MT M −1 I x2 = xT x E d D = p = dt s h Ns

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Explanation The set of all real numbers The set of all strictly positive real numbers The set of real vectors of dimensions n The set of n × m matrices The transpose of the matrix M The inverse of the matrix M The identity matrix The squares of the 2-norm of x The mathematical expectation The differential operator The Laplace operator The sampled period The number of samples

tification [11–14] problems are studied. This deep interest aroused by the fact that several complex physical systems can be suitably depicted by fractional order model which provide a reliable modeling tool that allows the illustration of their properties. During the last decades, a considerable effort has been made in fractional system identification. As a result, several methods have been developed in the literature (the reader is referred to [14, 15] and the references therein). In a brief historical review, in 1998, the frequency domain and time domain open-loop system identification with fractional models was initiated by Le Lay [16]. In 2000, Battagalia et al. extended the least squares algorithm to estimate a fractional order model that produces a transient thermal behavior of a system [2]. An output error technique was developed to estimate the dynamics of a lead-acid battery by Lin et al [7]. Subsequently, an equation error method based on the least squares algorithm combined with the state variable filter approach was investigated by Cois et al. [11]. Later on, in 2005, Aoun et al. synthesized fractional orthogonal bases Laguerre functions [17]. The instrumental variable approaches were extended to the fractional case by Victor et al. [18]. An iterative simplified refined instrumental variables (SRIV) method was proposed for the identification of a fractional model [19] and every iteration, the instrumental variable method was applied in this paper. Besides, the signals are filtered using stable filters built by using the parameters estimation obtained in previous iteration. In [20], the modulating function method was generalized to the online identification of fractional order systems based on Riemann—Liouville derivative. In 2014, Volterra series were extended to identify a thermal diffusion with experimental data [21]. More recently, in 2017, the innovative parameter estimation for fractional order systems with the impulse noise was suggested by Cui et al. (see [22]). The principal contribution of this work was replacing the conventional least squares objective by a new approximate least absolute error function. This method improves the estimation accuracy.

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Moreover, the fractional closed-loop system identification was elaborated over the last three years. First, Yakoub et al. extended the SRIV method to the fractional closed-loop case in order to estimate both differentiations orders and linear coefficients using the prior knowledge of the controller (see [23]). In [24], the fractional order optimization closed-loop bias eliminated least squares method was developed to overcome the bias problem caused by the correlation between the excitation signal and the output noise signal. Mainly, the idea of this method is to subtract the bias estimation from the ordinary least squares estimation in order to obtain unbiased parameters. Despite their high efficiency, the above fractional system identification methods assumed that the system output is, typically, corrupted by an additive noise, whereas the system input is perfectly known and available for modelling procedures. While in fact, in such practical situations such as: control systems, signal processing, image processing . . ., the system input must be observed. Thus, the input signal can be, also, contaminated by an additive noise. Consequently, this assumption is violated. It is worth noting that the models which take into account the output noise as well as the input noise to identify the system are named errors-in-variables (EIV) models [25–27]. The problem of identifying the parameters of a rational system from a noisy input and output signal has been a well-studied as a subject of research and significant efforts have been spent for developing efficient approaches during the last years. For example, the Frish scheme [28], the instrumental variables approach [29], the joint input/output approach [30]. An overview and some comparison of different approaches have been presented in [29]. The main drawback of these methods is that they rely on assumptions of some noise variances. That is way, to overcome this issue, another approach called the bias eliminated least squares [27] has been frequently presented in the literature [27]. This approach can provide consistent parameters with less computational demand. More interesting, this approach does not requires any prior information about noise variances. Despite their efficiency, the extension of these approaches to the fractional EIV case is not yet well developed. Only a limited number of methods have been available in the literature for instance the method proposed in [31] which propose a recurrent identification algorithm for parameterizing autoregression. Within this context and motivated by the above analysis, the main contribution of this research is to investigate a parameter identification method for continuous-time fractional order systems, under the Grünwald-Letnikov definition. This method is a generalized version of the bias eliminated least squares method which was developed by Zheng et al. for error-in-variables with rational models identification [27]. This choice is justified by its acceptable accuracy using only a simple least squares based method. It is called fractional order bias eliminated least squares (fo − bels). It was started by the parameters vector estimation obtained by the fractional order ordinary least squares (fo − ols) algorithm combined with the state variable filter (SVF) approach. Then, the variances of the input and output measured noises are estimated using an appropriate algorithm. Finally, to achieve the estimation consistency, the bias produced by the fo − ols parameters estimates is eliminated. Both fractional orders differentiations and linear coefficients are estimated in this work.

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The main features of the proposed method are: (i) the consistent fractional system parameters estimation without any prior information about the input and output noise measurements. So, the calculation of variances estimation is the innovation of this paper. ; (ii) its robustness against the noises; (iii) its implementation simplicity; (iv) its faster convergence. The rest of the paper is organized as follows: Sect. 3 reviews, briefly, some useful definitions of the fractional calculus. Section 4 formulates the problem of the errorin-variables fractional order system identification. The development of the fractional order bias eliminated least squares method is established in Sect. 5. In Sect. 6, both fractional orders differentiations and coefficients are estimated using a nonlinear algorithm. The illustrative simulations using a fractional model and a real electronic system identification are presented in Sects. 7 and 8, respectively. Finally, concluding remarks and further research direction are discussed in Sect. 9.

3 Preliminaries In this section, we will recall some useful definitions of the fractional calculus that will be used for the fractional system identification. Fractional calculus extends the integral and differential operators to non-integer order. Three definitions of the fractional derivative are available can be presented, namely, Caputo derivative, Grünwald-Letnikov derivative and Riemann—Liouville derivative. In this paper, the fractional derivatives are evaluated using the GrünwaldLetnikov definition. Definition 1 The Grünwald-Letnikov derivative approximation of the order α is described as Dα g (t) 

  K 1  α k g (t − kh), ∀ t ∈ R∗+ , (−1) k hα

(1)

k=0

  α where t = Kh and is the Newton’s binomial generalized to fractional order k defined by the following equation    1 if k = 0, α = α(α−1)(α−2)...(α−k+1) k if k > 0. k!

(2)

The Laplace transform of the α−order fractional derivative of a function g(t), with zero initial conditions, is defined as follows L {Dα g(t)} = sα G(s), where G(s) = L (g (t)).

(3)

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Considering a SISO (Single Input Single Output) linear CT fractional order system whose u(t) and y(t) are related by the following differential equation N 

αn

an D y (t) =

n=0

M 

bm Dβm u (t),

(4)

m=0

where (an , bm ) ∈ R are the linear coefficients of the differential equation. The fractional orders αn and βm are non-integer positive scalars. To verify the identifiability condition, they are arranged as follows α0 < α1 < · · · < αN ;

β0 < β1 < · · · < βM .

Applying the Laplace transform to the fractional differential equation (4) provides the following transfer function M bm sβm B (s) = m=0 . G (s) = N αn A (s) n=0 an s

(5)

The transfer function G(s), given by Eq. (5), can be classified as a commensurate 1 transfer function and it can be rewritten as nb jϑ B (s) j=0 bj s = na G (s) = , (6) iϑ A (s) i=0 ai s where ϑ denotes the commensurate order.

4 Problem Formulation 4.1 Data Generating EIV Fractional System The main topic discussed in this paper is the identification problem. In this case, referring to Fig. 1, we consider the SISO fractional order system S0 , described by the following input-output model ⎧ ⎨ A (p) y0 (t) = B (p) u0 (t), u (t) = u0 (t) + v (t) , S0 : ⎩ y (t) = y0 (t) + e (t) ,

1

(7)

All differentiation orders are exactly divisible by the same number, an integral number of times.

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u0

y0

system

v

s

s

y

u

Fig. 1 Error-in-variable fractional model

where the polynomials A (p) and B (p) are expressed as: 

A (p) = 1 + a1 pϑ + a2 p2ϑ + . . . + ana pna ϑ , B (p) = b0 + b1 pϑ + b2 p2ϑ + . . . + bnb pnb ϑ ,

(8)

and where u0 (t) and y0 (t) are the system noise-free input and output signals, respectively. u(t) and y(t) are the available measurements of input and output signals. v (t) and e (t) are assumed to be zero-mean white noises with variances λv and λe corrupting the input and the output signals, respectively.

4.2 Model Structure for EIV Fractional System Identification Considering the following parameterized model M : y (t) = −

na  i=1

ai Diϑ y(t)+

nb 

bj Djϑ u(t) + ε(t),

(9)

j=1

where ε(t) is the equation error. For convenience, we introduce in this section, the following notations: The parameters vector θ is given by the below equation 



θ = aT bT = a1 , . . . , ana , b0 , . . . , bnb .

(10)

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The regression vector that includes fractional derivatives of the measurement input and output signals is written as follows 

ϕ T (t) = −Dϑ y (t) , . . . , −Dna ϑ y (t), Dϑ u (t) , . . . , Dnb ϑ u (t) .

(11)

It can be rewritten as a sum of the noise-free term and the noise term ϕ (t) = ϕ0 (t) + ϕn (t) ,

(12)

where 

ϕ0T (t) = −Dϑ y0 (t) , . . . , −Dna ϑ y0 (t), Dϑ u0 (t) , . . . , Dnb ϑ u0 (t) .

(13)



ϕnT (t) = −Dϑ e (t) , . . . , −Dna ϑ e (t), Dϑ v (t) , . . . , Dnb ϑ v (t) .

(14)

and

The main difficulty of the direct identification method for CT systems is related to the inability to measure the differentiations of all input and output signals. For this reason, a linear transformation can be applied to the input and the output signals which consists of extending the Poisson’s filter to the fractional case [32] pα Fα (p) =   , p NF 1+ ΩF

(15)

where α is the fractional order differentiation, the order NF is an integer chosen in a way that NF > na ϑ and ΩF denotes the filter cut-off frequency. Remark 1 This filter can estimate the initial conditions as well as the unknown parameters. But, if we treat these terms as an additional set of unknowns complicate the proposed algorithm. In addition, if the SVF approach is considered for a large time, the terms related to the initial conditions may be neglected after a the rise time of the filter. Thus, the terms associated with the initial conditions will not be considered in the present paper. The filtered input and output signals uf (t) and yf (t) are used to build up the filtered regression vector ϕf (t) which is rewritten as follows ϕf (t)T

=

= ϕf0 (t)T + ϕfn (t)T −Dϑ yf (t) , . . . , −Dna ϑ yf (t) ,  Dϑ uf (t) , . . . , Dnb ϑ uf (t) .

(16)

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where ϕf0 and ϕnf are defined below:

ϕf0 (t)T = −Dϑ y0 f (t) , . . . , −Dna ϑ y0f (t) , ϑ nb ϑ 

D uϑ0f (t) , . . . , D un0fϑ(t) , T ϕnf (t) = −D ef (t) , . . . , −D a ef (t) , Dϑ vf (t) , . . . , Dnb ϑ vf (t) . The input and output behavior of the fractional system can be expressed by the following linear regression form yf (t) = ϕfT (t) θ − ϕnf T θ + ef (t),

(17)

where Diϑ ef (t) = Fiϑ (p) e (t) ;

1 ≤ i ≤ na

(18)

The main purpose of this research work is to estimate the parameters vector using Ns samples of the measured input and output signals generated by the fractional system. It is admitted that, in presence of an additive noise, in either or both of the input and output measurements, the conventional least squares method fails to provide unbiased estimates. So, it is interesting to consistently estimate the parameters vector θ in the EIV framework. In this study we make the following assumptions for the model identification: • A(p) and B(p) are coprimes polynomials. • The noise-free input u0 (t) is sufficiently rich in frequency of a sufficient order [33]: the regression vector and the filtered regression vector are satisfied the persistent excitation (PE) condition i.e. there exists positive scalars σ and δ such that: t 

ϕˆ (tk ) ϕ T (tk ) > σ I

k=0 t 

ϕˆf (tk ) ϕˆfT (tk ) > δI .

k=0

• The noises v(t) and e(t) are independent of each other and of u (t).

5 Fractional System Parameters Estimation At this level, the fractional commensurate order ϑ is assumed to be known a priori and our aim to estimate the fractional differential equation coefficients. In Sect. 5.1, the fractional order ordinary least squares (fo − ols) identification method is discussed using the discretized difference equation. The compensation for the measurement noises existing in the input and the output is detailed in Sect. 5.2.

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5.1 Fractional Order Ordinary Least Squares Method For the purpose of identifying CT fractional order systems in EIV context, the fo − ols method is shown in this section. Consider the θˆfo−ols as the estimated vector of θ obtained by minimizing the 2norm of the following criterion function CNs

Ns −1



1  ˆθfo−ols = εf2 tk , θˆfo−ols , Ns

(19)

k=1

where εf (t) designates the filtered equation error given by the following expression εf (t) = yf (t) − ϕfT (t) θ.

(20)

Thus, the least squares estimation of θ is defined as θˆfo−ols = Φˆ N−1s

Ns −1 1  ϕf (tk )yf (tk ), Ns

(21)

k=0

where ΦNs is the auto-covariance matrix defined by Ns −1   1  ΦNs = E ϕf ϕfT = ϕf (tk )ϕfT (tk ). Ns

(22)

k=0

Note that the existence of the ΦN−1s is guaranteed by persistent excitation of the input signal vector (the first assumption). Theorem 1 Under the sufficiently exciting condition, the Gaussian noises v(t) and e(t) are independent of each other and of u (t), the difference between the estimated parameters vector θˆfo−ols and the true one is obtained by tending Ns to the infinity in Eq. 21 as follows lim θˆfo−ols (Ns ) − θ = −ΦN−1s M θ

Ns →∞

= Γfo−ols ,

(23)

where Γfo−ols denotes the bias introduced by the fo − ols method and defined by the following equation Γfo−ols = −ΦN−1s M θ.

(24)

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where the matrix M ∈ R(na +nb )×(na +nb ) as follows 

 M1 0 M = , 0 M2

(25)

where the matrices M1 and M2 are expressed, respectively, by M1 = λe Ina and M2 = λv Inb ,

(26)

Proof Replacing the filtered output signal yf (t) by its expression (17) in Eq. (21) results on the following equation θˆfo−ols =

 1 Ns

N s −1 k=0

−1  ϕf (tk )ϕfT



=θ+

1 Ns

(tk )

1 Ns

N s −1

ϕf (tk )

k=0 −1 



ϕfT

(tk ) θ −

ϕnTf

(tk ) θ + ef



 T ϕf (tk ) −ϕnf (tk ) θ + ef (tk ) k=0 k=0

  (27) = θ + ΦN−1s E ϕf (tk ) −ϕnTf (tk ) θ + ef .

N s −1

ϕf (tk )ϕfT

N s −1

1 Ns

Substituting Eq (16) in the following equation, yields to

 

 

= E ϕ0f (t) + ϕn (t) −ϕnT θ + ef E ϕf (t) −ϕnT θ + ef f f f   

T = −E ϕ0f (t) ϕn (t) θ + E ϕ0f (t) ef f     −E ϕn (t) ϕnT (t) θ + E ϕn (t) ef (t) . f

f

f

(28)

Since vf (t), ef (t) and u0f (t) are mutually independent, we obtain 

 E ϕf (t) −ϕnT θ + ef = −M θ. f

(29)

Replacing Eq. (29) in Eq. (27), we get the following equation lim θˆfo−ols (Ns ) = θ − ΦN−1s M θ

Ns →∞

= θ + Γfo−ols .

(30)

Consequently, subtracting θ from lim θˆfo−ols (Ns ) in Eq. (30) makes us obtain Ns →∞

Eq. (23)



It is obvious, from Eq. 23, that the presence of measurements error in the variables leads to biased parameters and the bias expression, given by Eq. (24) is related to the noises variances estimation. Therefore, to overcome this problem, it is necessary to

Identification of Continuous-Time Fractional Models …

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study a method to compensate the effect caused by input and output noises. So, the fractional order bias eliminated least squares (fo − bels) method will be explained in the next section.

5.2 Fractional Order Bias Eliminated Least Squares Method Considering that the input and output signals containing noises v(t) and e(t). The estimation of θ is biased using Eq. (23). Therefore, in the following section, the bias eliminated method is extended for the CT system identification with fractional models from noisy input and output measurements. Thus, the fo − bels estimator is given by the following expression θˆfo−bels = θˆfo−ols − Γˆfo−ols ˆ θˆfo−bels , = θˆfo−ols + Φˆ N−1s M

(31)

where Γˆfo−ols denotes the bias estimation and   ˆ1 0 M ˆ M = ˆ2 . 0 M

(32)

ˆ 1 and M ˆ 2 are expressed, respectively, by The matrices M ˆ 2 = λˆ v Inb ˆ 1 = λˆ e Ina and M M

(33)

where λˆ e and λˆ v are the estimates of the noises variances λe and λv , respectively. So, the main idea to practical application of the unbiased identification procedure lies in estimation of the noise variances (λˆ e and λˆ v ). However, the fo − ols method can provide one expression for computing the estimates of the noise variances. Hence, it is necessary to find another equation relating the noise variances to some other known variables. To attain this objective, it is required to use an augmented system in order to increase the denominator parameters by one dimension, where, the introduced parameter ana +1 = 0. As a result, the augmented transfer function is given by nb jϑ B (s) j=0 bj s = na +1 . G¯ (s) = A¯ (s) ai siϑ

(34)

i=0

˜ and the augmented polynomial A(p) is defined by A˜ (p) = A (p) = 1 + a1 pϑ + a2 p2ϑ + . . . + ana pna ϑ + ana +1 p(na +1)ϑ .

(35)

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Similarly to (17), the filtered output signal is given by yf (t) = ϕ˜fT (t) θ˜ − ϕ˜nf (t) θ˜ + ef (t) ,

(36)

where the augmented regression vector ϕ˜fT (t) is defined as follows ϕ˜ f (t) = ϕ˜ 0f (t) + ϕ˜nf (t) = −Dϑ yf (t) , . . . , −D(na +1)ϑ yf (t)  Dϑ uf (t) , . . . , Dnb ϑ uf (t)

(37)

and the augmented parameters vector is presented by



 θ˜ = a˜ T bT = a1 , . . . , ana +1 , b0 , . . . , bnb . The optimal augmented estimator using the fo − ols method is rewritten as Ns −1 1  ϕ˜f (tk ) yf (tk ) , θˆ˜fo−ols = Φˆ˜ N−1s Ns

(38)

k=0

where   Φ˜ Ns = E ϕ˜f ϕ˜fT =

1 Ns

N s −1 k=0

ϕ˜ f (tk )ϕ˜ fT (tk ),

(39)

and the matrix Φ˜ N−1s can be rewritten as follows   Φ11 Φ12 , Φ˜ N−1s = Φ21 Φ22

(40)

where ⎧ Φ11 ⎪ ⎪ ⎨ Φ22 Φ12 ⎪ ⎪ ⎩ Φ21

∈ R(na +1)×(na +1) ∈ Rnb ×nb ∈ Rnb ×(na +1) ∈ R(na +1)×nb

By analogy to Eq. (23) we obtain ˜ θ˜ lim θˆ˜fo−ols (Ns ) = θ˜ − Φˆ˜ N−1s M

Ns →∞

= θ˜ + Γ˜fo−ols ,

(41)

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193

˜ is presented as follows where M ˆ˜ = M



 ˆ˜ 0 M 1 , ˆ2 0 M

(42)

ˆ˜ is expressed by and the matrix M 1 ˆ˜ = λˆ I M 1 e na +1 .

(43)

Therefore, the optimal augmented estimator θ˜fo−bels is given by θˆ˜fo−bels = θˆ˜fo−ols − Γˆ˜fo−ols ˆ˜ θˆ˜ = θˆ˜fo−ols + Φˆ˜ N−1s M fo−bels .

(44)

Proposition 1 There exists a vector Q ∈ R(na +nb +1) defined as 

Q = qT 0nb , and verify the following equation λe qT Φ11 a˜ + λv qT Φ12 b = −qT aˆ˜ fo−ols .

(45)

Proof Consider the vector Q ∈ R(na +nb +1) defined by 

QT = qT 0nb ,

(46)

 where qT = 0 . . . 0 1 ∈ Rna +1 .

 ˜ Since the augmented polynomial A(p) = a1 . . . ana ana +1 has one known parameter an+1 = 0, so, it is evident that Qθ˜





= qT 0nb × a1 . . . ana ana +1 , b0 . . . . . . bnb

 

= 0 . . . 0 1, 0 . . . 0 × a1 . . . ana ana +1 , b0 . . . . . . bnb = 1 × ana +1 = 0

Pre-multiplying (41) with QT and using (47) yields to ˆ˜ θ˜ = −QT QT Φ˜ N−1s M



lim θˆ˜fo−ols (Ns )

Ns →∞

 (47)

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Replacing (46), (42) and (40) in (47) leads to   ˆ −1 ˆ˜ ˜ T ˜ ˜ lim θfo−ols (Ns ) Q ΦNs M θ = −Q Ns →∞    ˆ 

T  ˜1 0 Φ11 Φ12 M q 0nb × × × θ˜ = Φ21 Φ22 ˆ2 0 M 

− qT 0 × lim θˆ˜ (N ) T

nb

Ns →∞

fo−ols

(48)

s

ˆ˜ and M ˆ 2 by their expressions Replacing M 1      0 λI Φ11 Φ12 qT 0nb × × e na × θ˜ = Φ21 Φ22 0 λv Inb

 − qT 0nb × lim θˆ˜fo−ols (Ns )



(49)

Ns →∞

Finally, we obtain λe qT Φ11 a˜ + λv qT Φ12 b = −qT lim aˆ˜ fo−ols (Ns ) Ns →∞

(50)

Proposition 2 Ns −1



2 1  lim Cˆ˜ Ns θˆ˜fo−ols = lim εˆ˜ f tk , θˆ˜fo−ols Ns →∞ Ns →∞ Ns k=1  T   ˆ = λe 1 + lim a˜ fo−ols a˜

 + λv

Ns →∞

T  lim bˆ fo−ols b

Ns →∞

(51)

Proof Using the expression of the filtered output signal defined by Eq. (36), the filtered equation error is given by

εˆ˜ f t, θˆ˜fo−ols = yf (t) − ϕ˜fT (t)θˆ˜fo−ols = ϕ˜fT (t)θ˜ − ϕ˜nf (t)θ˜ + ef (t) − ϕ˜fT (t)θ˜ˆfo−ols

= ϕ˜fT (t) θ˜ − θˆ˜fo−ols − ϕ˜nT (t)θ˜ + ef (t) f

Using Eq. (52), the criterion Cˆ˜ Ns (θˆ˜fo−ols ) is written as follows

(52)

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195

Ns −1



1  Cˆ˜ Ns θˆ˜fo−ols = εˆ˜ f2 tk , θˆ˜fo−ols Ns k=1

N s −1 



1 ϕ˜fT (tk ) θ˜ − θˆ˜fo−ols − ϕ˜nTf (tk ) θ˜ + ef (tk ) Ns k=1

× εˆ˜ f tk , θˆ˜fo−ols   s −1

T 1 N

ϕ˜ fT (tk ) εˆ˜ f tk , θˆ˜fo−ols = θ˜ − θ˜ˆfo−ols Ns k=1    =

R

+

1 Ns 

N s −1 



ef (tk ) − ϕ˜nTf (tk ) θˆ˜fo−ols εˆ˜ f tk , θˆ˜fo−ols

k=1



(53)



Q

Using the definition of the error equation, we compute the term R in Eq. (53) by Ns −1 Ns −1

1  1  ϕ˜f (tk ) εˆ˜ f tk , θˆ˜fo−ols = ϕ˜f (tk ) yf (tk ) − ϕ˜fT (tk ) θˆ˜fo−ols Ns Ns k=1

k=1

N s −1 

1 ϕ˜f (tk ) yf (tk ) Ns k=1  N −1  s 1  − ϕ˜ f (tk ) ϕ˜fT (tk ) θˆ˜fo−ols Ns =

(54)

k=1

Replacing θˆ˜fo−ols by its expression (38) to Eq. (54) yields to Ns −1 Ns −1

1  1  ϕ˜f (tk ) εˆ˜ f tk , θˆ˜fo−ols = ϕ˜f (tk ) yf (tk ) Ns Ns k=1 k=1  N −1  s

1  − ϕ˜f tk , θˆ˜fo−ols ϕ˜fT (tk ) Ns k=1 ⎛ −1 N s −1  1 T ×⎝ ϕ˜f (tk )ϕ˜ f (tk ) Ns k=1  Ns −1 1  × ϕ˜f (tk )yf (tk ) Ns k=1

=0

(55)

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Using Eqs. (52) and (55) we obtain Ns −1

1  ˆ ˆ ˜ ˜ ef (tk ) − ϕ˜nTf (tk ) θˆ˜fo−ols CNs θfo−ols = 0 + Ns k=1

T × ϕ˜f (tk ) θ˜ − θˆ˜fo−ols

+ ef (tk ) − ϕ˜nTf (tk ) θˆ˜fo−ols   s −1

T 1 N

ϕ˜fT ef (tk ) − ϕ˜ nTf (tk ) θˆ˜fo−ols = θ˜ − θ˜ˆfo−ols Ns k=1

+

1 Ns

N s −1 

ef (tk ) − ϕ˜nTf (tk ) θˆ˜fo−ols

2 (56)

k=1

Tending Ns to the infinity in Eq. (56) provides T 



 ˆ ˆ ˜ ˜ ˜ lim CNs θfo−ols = θ − lim θfo−ols (Ns ) E ef − ϕ˜nTf θˆ˜fo−ols Ns →∞ Ns →∞ 

2  T ˆ˜ + E ef − ϕ˜nf θfo−ols

(57)

Similarly to Eq. (29), we have 

 ˜ θ˜ E ϕ˜fT ef − ϕ˜nTf θˆ˜fo−ols = −M

(58)

Using the noise independence assumptions on v(t) and e(t), we obtain   

2  

T ˆ˜ = E ef2 − 2θ˜ T E ϕ˜nf ef E ef − ϕ˜nf θfo−ols   + θ˜ T E ϕ˜nf ϕ˜nTf θ˜ ˜ θ˜ = λe − 0 + θ˜ T M Substituting Eqs. (58), (59) and (42) into Eq. (57) we find

(59)

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197





T ˜ θ˜ −M lim Cˆ˜ Ns θˆ˜fo−ols (Ns ) = θ˜ − θˆ˜fo−ols (Ns )

Ns →∞

˜ θ˜ + λe + θ˜ T M

T

˜ θ˜ = λe + θˆ˜fo−ols (Ns ) M Ns −1

2 1  εˆ˜ f tk , θ˜fo−ols Ns →∞ Ns k=1  T   ˆ = λe 1 + lim a˜ fo−ols a˜

= lim

 + λv

Ns →∞

T  lim bˆ fo−ols b

Ns →∞

(60) 

A detailed description of the proposed method is given in Algorithm 1. Theorem 2 The proposed fo − bels method for system identification with fractional models gives an unbiased estimation: The optimal augmented estimator θ˜fo−bels is given by ˆ˜ θˆ˜ i (64) θˆ˜ i+1 = θˆ˜fo−ols + Φˆ˜ N−1 M fo−bels . fo−bels

s

Proof We suppose that the noises variances λu and λu are known. Using the assumption that the noises v(t) and e(t) are independent of each other and of u (t) and using Eq. (36), the matrix Φ˜ Ns can be rewritten as follows 

Φ˜ Ns = E ϕ˜ f ϕ˜f T   T  = E ϕ˜0f (t) + ϕ˜nf (t) ϕ˜0f (t) + ϕ˜nf (t)     = E ϕ˜ 0f (t) ϕ˜0Tf (t) + E ϕ˜nf (t) ϕ˜ nTf (t)   ˜ = E ϕ˜ 0f (t) ϕ˜0T (t) + M

(65)

ˆ˜ = Φ˜ −1 M ˜ lim Φˆ˜ N−1s M Ns

(66)

f

since

Ns →∞

i obtained by the iterative scheme is convergent We assume that the θˆ˜fo−bels

lim θˆ˜ (i) i→∞ fo−bels

(Ns ) = θˆ˜fo−bels (Ns )

and using Eqs. (64) and (67) we obtain

(67)

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Algorithm 1 Fractional order -bias eliminated least squares algorithm !  Data uf (t) , yf (t) t = 1, . . . , Ns , a maximum number of iterations 1: Initialization Apply the fo − ols method to obtain the θˆ˜fo−ols θˆ˜fo−bels (0) = θˆ˜fo−ols

(61)

2: repeat • Solve the following system equation S0 to calculate the noises variances estimation λˆ e and λˆ v   ⎧

T (i) ⎪ Cˆ˜ N θˆ˜fo−ols = λˆ (i) ˆ ˆ ⎪ a ˜ 1 + a ˜ fo−ols e s ⎪ fo−bels ⎪ ⎪   ⎪

T ⎨ (i) (i) +λˆ v bˆ fo−ols bˆ fo−bels S0 : (62) ⎪ ⎪ (i) T ˆ ˆ (i) (i) T ˆ ˆ (i) ⎪ ˆ ˆ ⎪ λe q R11 a˜ fo−bels + λv q R12 bfo−bels = ⎪ ⎪ ⎩ −qT aˆ˜ fo−ols • Estimate the parameters vector using the following equation (i) ˆ˜ (i−1) θˆ˜ (i−1) hat θ˜fo−bels = θˆ˜fo−ols + Φˆ˜ N−1 M fo−bels s

where ˆ˜ (i−1) = M and

3: until

" " " ˆ (i+1) " (i) "θfo−bels −θˆfo−bels " " " " ˆ (i+1) " "θfo−bels "



ˆ˜ (i−1) M 1 0

(63)

 0

ˆ˜ (i−1) M 2

ˆ˜ (i−1) = λˆ (i−1) I M na +1 e 1 (i−1) (i−1) ˆ = λˆ v Inb M2 ≥  or a maximum number of iterations is reached. i denotes

the iteration number. 4: return θˆfo−bels .

ˆ˜ θˆ˜ θˆ˜fo−bels (Ns ) = θ˜ˆfo−ols (Ns ) + Φ˜ˆ N−1s M fo−bels (Ns ) .

(68)

The result is shown on the following equation

ˆ˜ = θˆ˜ θˆ˜fo−bels (Ns ) Ina + nb + 1 − Φˆ˜ N−1s M fo−ols (Ns )

(69)



ˆ˜ = lim θˆ˜ lim θˆ˜fo−bels (Ns ) Ina + nb + 1 − Φˆ˜ N−1s M fo−ols (Ns )

(70)

since Ns →∞

Using the following equation we find

Ns →∞

Identification of Continuous-Time Fractional Models …



˜ θ˜ lim θ˜ˆfo−ols (Ns ) = Ina + nb + 1 − Φ˜ N−1s M

Ns →∞

199

(71)

Finally, we replace Eq. (30) in Eq. (70) to obtain lim θˆ˜fo−bels (Ns ) = θ˜

Ns →∞

(72) 

6 Fractional Order Estimation Predominantly, the fractional orders differentiations are unknown. Therefore, it is valuable to develop an algorithm which entails the simultaneous estimation of both process coefficients and fractional order. This algorithm is named fractional order optimization-bias eliminated least squares (foo − bels). It resides in combining the fo − bels method for coefficients estimation with a nonlinear algorithm for differentiation order optimization. The Gauss-Newton algorithm is suggested as an alternative to solve the problem of optimization. Let us define the augmented process parameters vector as follows 

θ˜ = a1 , . . . , ana , ana +1 , b0 , . . . , bnb , ϑ .

(73)

This vector is composed by na + nb + 2 elements. Otherwise, the parameters identification problem is put as a functional minimization. So, the principal goal of this algorithm is to minimize the bias induced by the fo − ols estimates with respect to ϑ [24]. Furthermore, during the computational procedures, the commensurate order is updated, iteratively, using the Gauss-Newton algorithm. The quadratic criterion is defined as follows "2

1" " " V Γˆfo−ols = "Γˆ˜fo−ols " , 2

(74)

where the bias estimation Γˆ˜fo−ols is given by: ˆ˜ θˆ˜ Γˆ˜fo−ols = −Φˆ˜ N−1s M fo−ols . This iterative algorithm calculates the fractional differentiation order ϑ j+1 at the iteration j + 1   ∂V (75) ϑ j+1 = ϑ j − γ H−1 ∂ϑ ϑ j

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∂V is the gradient defined by the following where j denotes the iteration number and ∂ϑ equation T ∂ Γˆ˜fo−ols ∂V = Γˆ˜fo−ols (76) ∂ϑ ∂ϑ and H is the approximated hessian given by: H=

T ∂ Γˆ˜fo−ols ∂ Γˆ˜fo−ols

∂ϑ

∂ϑ

(77)

The description of the foo − bels method is given by the Algorithm 2. Algorithm 2 Fractional order optimization-bias eliminated least squares algorithm  ! Data uf (t) , yf (t) t = 1, . . . , Ns 1: Initialization

• Initialize ϑ 0 . • Apply the fo − ols method to obtain an initial estimation of the θˆ˜fo−ols . • Compute the bias introduced by the fo − ols method Γˆ˜ 0 . fo−ols

0 ). • Calculate V (Γˆ˜fo−ols • Initialize γ which is a positive scalar.

2: repeat • Update the commensurate-order by applying the Gauss–Newton algorithm as follows   ∂V ϑ j+1 = ϑ j − γ H−1 (78) ∂ϑ ϑ j • Estimate the fractional order parameters vector by the use of the fo − ols method. j+1 • Compute the bias introduced by this method Γfo−ols when ϑ = ϑ j+1 and compute j+1 . the fractional process parameters vector θˆ˜ fo−bels

j+1 • Calculate the quadratic criterion V (Γˆ˜fo−ols ). γ • γ = 2 " " " " Γˆ˜ j+1 −Γˆ˜ j " fo−ols fo−ols " 3: until " " <  or a maximum number of iterations is reached. j+1 " " Γ˜ˆfo−ols ˆ 4: return θfo−bels .

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7 Numerical Example To confirm the theoretical results derived in the previously sections, the proposed EIV identification with fractional models is illustrated. It is used for identifying a fractional model where the true system is given as follows   K −T pϑ + 1 G (p) =  ϑ 2ϑ   ϑ 2ϑ  , 1 + 2ξ1 wp1 + wp1 1 + 2ξ2 wp2 + wp2

(79)

where ϑ = 0.5, K = −1, T = 0.5, w1 = 0.2 rad/sec, ξ1 = −0.4, w2 = 1 rad/sec and ξ2 = −0.65. This system provides a suitable test environement for continuous time and discrete time estimation methods (the reader is referred to [32] and the references therein). It contains two resonant modes: • the first mode at w1 = 0.2 rad/s and ξ1 = −0.4, • the second mode at w2 = 1 rad/s and ξ2 = −0.65. The mathematical representation of the fractional model is given as follows G (p) =

b0 + b1 pϑ . 1 + a1 pϑ + a2 p2ϑ + a3 p3ϑ + a4 p4ϑ

(80)

7.1 Data Generating The input exciting the system is a pseudo random binary sequence (PRBS) with a uniform distribution between [0 10]. Its power spectral density is presented by Fig. 2. The input and the output signals are measured for a sampling period h = 0.02 s. The number of samples is Ns = 5000 data. The input and the output signals are contaminated by zero-mean white Gaussian noises with variances λv = 0.255 and λe = 0.0461, respectively. This corresponds to a signal-to-noise ratio (SNR) of 20 dB where the SNR on the input and the output are given, respectively, as  var (u (t)) , var (v (t))

(81)

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(82)

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where var(.) denotes the variance. So, it is clear that the simulated system satisfies the assumptions mentioned in Sect. 5.

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The normalized relative quadratic error (NRQE) is used as the evaluation criterion for the developed algorithms and is defined as follows # "2 " $ " "ˆ $ nmc θ − θ " " $ 1  % NRQE = , θ  nmc

(83)

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7.2 Choice of the SVF Parameters At this level, we will focus on the effect of the SVF parameters on the fo − ols and fo − bels methods. A Monte Carlo simulation of nmc = 100 runs is accomplished.

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A study on the effect of the SVF cut-off frequency on the fo − ols and the fo − bels methods is carried out in this paragraph. So, the SVF order NF is predetermined to 3 and the cut-off frequency ΩF is selected as:

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Table 2 Study on the effect of the SVF cut-off frequency Ωf on the fo − ols and fo − bels methods (SNRu = SNRy = 20 dB, nmc = 100 runs) NRQE ΩF (rad/s) fo − ols fo − bels

0.01 0.8680 0.2922

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30 0.9424 0.9316

Table 3 Study on the effect of the SVF order NF on the fo − ols and fo − bels methods (SNRu = SNRy = 20 dB, nmc = 100 runs) NRQE NF fo − ols fo − bels

2 1.9145 0.8826

3 1.1198 0.0175

4 0.1077 0.00601

 ! ΩF = 0.01, 1, 30 rad/s The related results are presented in Table 2. It is shown that the fo − bels performs good results, i.e. a small value of NRQE, when ΩF = 1 rad/s.

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The choice of the SVF order NF is studied in this section. So, the cut-off frequency is fixed to 1 rad/s and NF is selected as  ! NF = 1, 2, 3 . The variation of the NRQE values according to the SVF order is displayed in Table 3. The obtained results prove that the smallest value of NRQE corresponds to NF = 4. But, to guarantees the filter simplicity implementation, the best choice is NF = 3.

7.3 Comparative Study To provide meaningful results, a Monte Carlo simulation of nmc = 200 experiments is carried out. At each Monte Carlo run, a new noises realization on both input and output signals is generated. The unknown fractional model parameters are estimated through the fo − ols the fractional order-instrumental variable (fo − iv), the method developed in the paper [31] called the fractional order recurrent identification fo − RI

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Table 4 Parameters estimation of fractional system obtained by fo − ols, fo − iv, fo − RI and fo − bels methods (SNRu = SNRy = 20 dB, nmc = 200 runs)

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fo − iv

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a1 −5.3

a2 31.2

a3 −36.5

−5.4214 0.1430 0.0204 0.1212 −5.3918 0.1032 0.0107 0.1052 4.7610 0.2625 0.0689 0.0537 −5.2926 0.0472 0.0022 0.0037

31.7662 2.9343 8.6101

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−1.0040 0.0113 0.0001

0.5037 0.0102 0.0001

and the fo − bels method presented in Algorithm 1. The parameters vector of the fractional model is given by 

θ = a1 a2 a3 a4 b0 b1 .

(84)

The SVF parameters are chosen respectively as: ΩF = 1 rad/s and NF = 3. The obtained results are reported in Table 4 which shows the means, the standard deviation of the estimated model parameters which defined by & σX =

 2 E X − E[X ]2

(85)

their variances defined by Var (X ) = σX2

(86)

and the NRQE over 200 Monte Carlo simulations. We can, obviously, assert that the fo − ols method provides biased parameters estimates. The fo − iv method ensures unbiased estimates, but it gives parameters with non-minimal standard deviation and NRQE values. Besides, the fo − bels method produces unbiased parameters estimates in terms of comparatively low NRQE and standard deviation values. Table 5 presents the noises variances estimation using the fo − bels method.

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Table 5 Noises variances estimates using the fo − bels method (SNRu = SNRy = 20 dB, nmc = 200 runs) Method fo − bels True λv = 0.2551 λe = 0.0461

Estimated 0.2512 ± 0.0236 0.0446 ± 0.00125

It is shown from Table 5 that the fo − bels method presents consistent noises variances estimates. Figure 3 illustrates the histograms of fo − ols, fo − iv, fo − RI and fo − bels estimates. This figure depicts that the obtained estimates using the fo − bels method are around the true ones, however, the histograms obtained by the fo − RI method are far from the true values. The identification results validate the powerful capability of the fo − bels in dealing with EIV systems with fractional models.

7.4 Fractional Order Estimation In this section, the commensurate order ϑ is unknown and the foo − bels method, presented in Algorithm 2, is applied to estimate both commensurate order and fractional transfer function coefficients. The noise-free input signal is the same previously used signal. In order to show the effectiveness of this method a Monte Carlo study with nmc = 200 runs is realized. The same SVF parameters are utilized in this section. The Monte Carlo simulation results, obtained from the foo − bels method, are presented in Table 6. They show that the foo − bels method provides consistent estimates of the fractional system. This is justified by a low value of NRQE. Table 7 presents the noises variances estimates using the foo − bels method. It can be seen that the noises input-output variances are consistently estimated. Figure 4 plots the histograms of foo − bels estimates. The identification results show that the estimated parameters obtained by the use of this method converge towards the true values which confirm the theoretical analyses.

8 Application to a Real Fractional Order Electronic System This section proves the effectiveness of the theoretical analyses through its implementation with a real electronic system. It is divided into two paragraphs. In the first paragraph, a description of the real system is presented. The identification of the electronic system in the EIV context using the foo − bels method is presented in the second paragraph.

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8.1 Plant Description The fractional real electronic system shown in the Fig. 5 is a double integrator performed using resistors, capacitors and operational amplifiers. It is a combination of two integrators in cascade possessing a cut-off frequency wcg = 1 rad/s.

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The above system is described by the following equation H (p) = where k is an uncertain gain.

k , p2

(87)

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Table 6 Fractional system parameters estimates obtained by foo − bels method (SNRu = SNRy = 20 dB, nmc = 200 runs) foo − bels θˆ SNR True Variances NRQE 20 dB

a1 = −5.3 a2 = 31.2 a3 = −36.5 a4 = 25 b0 = −1 b1 = 0.5 υ = 0.5

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Estimated 0.2499 ± 0.0336 0.0402 ± 0.00425

In order to obtain the desired margin phase2 Φm = 45◦ , the electronic system is controlled by a fractional derivative controller of an order equal to 0.5. This controller is bounded on the frequency band [wb wh ] and described by the following expression Np



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(88)

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The difference between the open-loop transfer function phase and 180◦ .

(89)

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occurrence

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(90)

where α, η and Np are real parameters chosen as follows: αη = 8 and Np = 4. Each pair of pole and zero of the controller transfer function constitutes an electronic cell depicted in Fig. 6. It consists of a pair of resistances, a pair of capacitors and an operational amplifier. The fractional closed-loop system is illustrated in the Fig. 7 Remark 2 The circuits presented in Figs. 5, 6 and 7 are based on perfect operational amplifier (OA), capacitors and resistors. Thus analytical models describing system of Figs. 5, 6 and 7 are linear. The circuit equivalent to the controller C(p) (in Fig. 7), is an infinite series of electronic cells of Fig. 6 with recursive values of resistance and capacitor. In earlier work of [1], it is proven that this infinite series is modeled with a linear fractional derivative.

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Fig. 8 Photo of the real electronic system

Figure 8 shows a real photo of the fractional electronic system. One of the main issues of the real-time identification is the model structure selection. This way, it is required to have a mathematical model that can duly represent this system. For this reason, it was verified in [23] that the appropriate model representation of this system is given by: G (p) =

b0 1 + a1 pϑ

(91)

8.2 System Identification This section presents the identification of the fractional order real electronic system using the foo − bels method. The noise-free input is a PRBS. It is worthy to note that the measured signals are almost noise-free. So, in order to show the performances of the proposed method, we added two noise signals v(t) and e(t), with a SNRu =

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SNRy = 10 dB. The estimation data are presented by Fig. 9 where the sampling period is h = 0.1 s and the number of samples is fixed to Ns = 1000 data points. Table 8 contains the estimated coefficients, the estimated fractional order and the FIT , a scalar measure is used to evaluate the performances of the identification method, which is defined as follows ⎞ ⎛ # $ Ns  2 $ ˆ (tk ) ⎟ ⎜ k=1 y (tk ) − y (92) FIT = 100 × ⎝1 − %  ⎠, Ns ¯ )2 k=1 (y (tk ) − y where y¯ is the mean of the output signal.

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Table 8 Parameters estimation of the fractional electronic system obtained with the foo − bels method (SNRu = SNRy = 10 dB) Method aˆ 1 bˆ 0 υˆ FIT Foo − bels

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The estimates of noises variances, presented in Table 9, demonstrate the effectiveness of the proposed method for estimating the real electronic system in the EIV framework To validate the obtained model using the foo − bels method, we have generated another input/output data presented in Fig. 10. Figure 11 shows that the output error is small which confirms the consistency of the proposed identification method. In summary, the simulation results verify the effectiveness of the proposed method to identify both coefficients and fractional order. The advantage of this method is the faster convergence of the estimates to the true values of the process parameters. However, the main drawback of the proposed method is the knowledge of the structure parameters (na , nb ) of the process. But, in practice, this information may not be available. Hence, to overcome this issue, it must over-parameterize the unknown process and then apply the presented algorithm. On the other hand, if one of the

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parameters na and/or nb are replaced by another parameters less than na and/or nb , respectively, the fo − bels algorithm fails to work correctly. In addition, the proposed method is sensitive in the case of white noises.

9 Conclusion In this paper, the fractional system identification from noisy input and output signals has been developed. The fractional order bias eliminated least squares method has been proposed. Starting with the fractional order ordinary least squares algorithm

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and without any information about noises variances, the proposed method estimates, iteratively, the bias introduced by the least square method. Thereafter, this bias is eliminated to obtain unbiased parameters. Next, the proposed method has been extended to a more realistic case where the fractional orders are unknown and estimated along with the coefficients using a nonlinear algorithm. The numerical example and the real electronic system emphasize that the proposed method has proved its success in providing consistent estimates. The future works will include the fractional multi-input multi-output (MIMO) system identification in error in variables context.

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