Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions

doi:10.1016/j.automatica.2006.08.024

Automatica

Volume 43, Issue 2, February 2007, Pages 309-316

https://doi.org/10.1016/j.automatica.2006.08.024 Get rights and content

Abstract

This paper deals with robust stability analysis of linear state space systems affected by time-varying uncertainties with bounded variation rate. A new class of parameter-dependent Lyapunov functions is introduced, whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms. This class of Lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time-invariant perturbations. The main result of the paper is a sufficient condition to determine the sought Lyapunov function, which amounts to solving an LMI feasibility problem, derived via a suitable parameterization of polynomial homogeneous forms. Moreover, lower bounds on the maximum variation rate for which robust stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems. Numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature.

Introduction

Robust analysis of linear systems subject to time-varying parametric uncertainties has been tackled via Lyapunov functions since long time (see e.g., S˘iljak, 1969; Narendra & Taylor, 1973). Within this context, a popular uncertainty model is the one assuming that the uncertain parameters affect affinely the state space matrix and belong to a given polytope. The simplest approach consists in looking for a common quadratic Lyapunov function that proves stability of the entire polytope of matrices (see e.g., Horisberger & Belanger, 1976). However, it is well known that this condition, although appealing from a computational point of view, can lead to conservative results. In order to reduce conservativeness, more general classes of Lyapunov functions have been considered, including polyhedral Lyapunov functions (see e.g. Brayton & Tong, 1979; Blanchini, 1995), piecewise quadratic Lyapunov functions (Almeida et al., 2001, Xie et al., 1997), and homogeneous polynomial Lyapunov functions (Chesi et al., 2003a, Zelentsovsky, 1994). In particular, several examples are reported in the last reference, showing the effectiveness of conditions based on Lyapunov functions whose dependence on the state vector is expressed as a polynomial homogeneous form.

Comparably less attention has been devoted to the interesting case when bounds on the variation rate of the uncertain parameters are given, except for the special case of null variation rate, corresponding to time-invariant parametric uncertainty. In the latter case, the class of parameter-dependent Lyapunov functions has been successfully employed (see e.g., Barmish, 1994, pp. 342–347; Dasgupta, Chockalingam, Anderson, & Fu, 1994), and several conditions in terms of linear matrix inequalities (LMIs) have been recently derived (Bliman, 2002, Chesi et al., 2003b, Leite and Peres, 2003, Peaucelle et al., 2000, Trofino, 1999). In Chesi et al. (2003b), a new class of parameter-dependent quadratic Lyapunov functions has been introduced, whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form.

On the other hand, few results are available for the case when the variation rate is bounded by a known quantity, and they mostly concern the use of affine parameter-dependent quadratic Lyapunov functions. A first contribution has been given in Gahinet, Apkarian, and Chilali (1996), where an LMI-based sufficient condition has been provided exploiting a multiconvexity property of the time derivative of the Lyapunov function. More recently, different conditions have been derived in Montagner and Peres (2003) and Geromel and Colaneri (2005), taking into account the dynamics of the uncertain parameter vector. Sufficient conditions based on quadratic Lyapunov functions with polynomial dependence on the parameters have been given in Bliman, 2004, Wu and Prajna, 2005.

In this paper, the class of Homogeneously Parameter-Dependent Homogeneous Lyapunov Functions (simply abbreviated as HPD-HLFs) is introduced to investigate robust stability in presence of time-varying uncertainties with a polytopic bound on the variation rate. Such a class, where the dependence on the uncertain parameter vector and the state vector are both expressed as polynomial homogeneous forms, can be seen as a generalization of the Lyapunov functions previously employed for unbounded variation rates (Chesi et al., 2003a) and null variation rates (Chesi et al., 2003b). It is first shown that the existence of a HPD-HLF to prove robust stability can be established through an LMI feasibility test. Then, lower bounds on the maximum variation rate for which the system remains stable are obtained by solving generalized eigenvalue problems (GEVPs) (see Boyd, El Ghaoui, Feron, & Balakrishnan, 1994, for details about LMIs and GEVPs). Finally, numerical examples are reported to illustrate the behavior of the proposed condition, in comparison with results available in the literature. In particular, it is shown that in presence of bounds on the parameter variation rate, one can benefit from increasing the degree of the HPD-HLF in both the state variables and the uncertain parameters. This turns out to be a remarkable difference with respect to the cases of time-invariant parameters (when one can restrict to quadratic Lyapunov functions) and unbounded variation rates (where parameter dependence in the Lyapunov function is useless).

Section snippets

Problem formulation and preliminaries

In the paper, the following standard notation is adopted: $0_{n}, 0_{m \times n}$ : origin of $R^{n}$ and of $R^{m \times n}$ ; $R_{0}^{n}$ : $R^{n} ⧹ {0_{n}}$ ; $I_{n}$ : identity matrix $n \times n$ ; $A^{'}$ : transpose of matrix A; $A > 0 (A ⩾ 0)$ : symmetric positive definite (semidefinite) matrix A; $A \otimes B$ : Kronecker's product of matrices A and B; $A^{[k]}$ : kth Kronecker power $A \otimes A \otimes \dots \otimes A$ (k times); $co {d^{1}, \dots, d^{h}}$ : convex hull of vectors $d^{1}, \dots, d^{h}$ . Moreover: $p = [p_{1}, \dots, p_{q}]^{'} \in R^{q}$ ; $p^{i} = p_{1}^{i_{1}} \dots p_{q}^{i_{q}}, i = [i_{1}, \dots, i_{q}]^{'} \in N^{q}$ ; $sq (p) = [p_{1}^{2}, \dots, p_{q}^{2}]^{'} \in R^{q}, p \in R^{q}$ ; $sqr (p) = [\sqrt{p_{1}}, \dots, \sqrt{p_{q}}]^{'} \in R^{q}, p \in R^{q}$ .

Consider the continuous-time

Stability via HPD-HLFs

Let us consider problem P1. A sufficient condition to establish whether (1) is asymptotically stable under the constraint (2), can be obtained by finding a HPD-HLF $v (x, p)$ such that $\{\begin{matrix} v (x, p) > 0 & \forall (x, p) \in R_{0}^{n} \times P, \\ \dot{v} (x, p) < 0 & \forall (x, p, \dot{p}) \in R_{0}^{n} \times P \times D . \end{matrix}$ We have that $\dot{v} (x, p) = \frac{\partial v (x, p)}{\partial x} A (p) x + \frac{\partial v (x, p)}{\partial p} \dot{p} = \frac{\partial v (x, p)}{\partial x} A (p) x + {(\sum_{i = 1}^{q} p_{i})}^{2} \frac{\partial v (x, p)}{\partial p} \dot{p}$ that is $\dot{v} (x, p)$ can be written as a homogeneous form of degree $2 m$ in x and $s + 1$ in p, because $\sum_{i = 1}^{q} p_{i} = 1$ for all $p \in P$ . This is important because it allows us to get rid of the set $P$ in (8)

Example 1

Let us consider the following system, which is quite popular in the literature (Almeida et al., 2001, Chesi et al., 2003a, Montagner and Peres, 2003, Xie et al., 1997, Zelentsovsky, 1994), $\dot{x} (t) = [\begin{matrix} 0 & 1 \\ - 2 - r (t) & - 1 \end{matrix}] x (t), 0 ⩽ r (t) ⩽ k .$ We want to compute the maximum variation rate $γ$ of $r (t)$ , such that (21) is asymptotically stable for any $| \dot{r} (t) | ⩽ γ$ . System (21) can be rewritten as (1)–(2) with $A_{1} = [\begin{matrix} 0 & 1 \\ - 2 & - 1 \end{matrix}], A_{2} = [\begin{matrix} 0 & 1 \\ - 2 - k & - 1 \end{matrix}], D = co \{{[\frac{γ}{k},- \frac{γ}{k}]}^{'}, {[- \frac{γ}{k}, \frac{γ}{k}]}^{'}\} .$ Lower bounds $\hat{γ}$ have been computed by using HPD-HLFs with $m = 1, 2, 3$ and $s = 0, 1, 2, 3$

Conclusions

Robust stability of linear systems affected by time-varying polytopic uncertainty has been considered for the case when a polytopic bound on the variation rate of the uncertain parameters is given. Sufficient conditions have been obtained in terms of LMIs, via the use of homogeneously parameter-dependent homogeneous Lyapunov functions (HPD-HLFs), which are a generalization of the affine parameter-dependent quadratic Lyapunov functions previously employed in the literature. Numerical examples

Graziano Chesi received the Laurea degree in Information Engineering from the University of Firenze in 1997 and the Ph.D. in Systems Engineering from the University of Bologna in 2001. He was a visiting scientist at the University of Cambridge (1999–2000) and University of Tokyo (2001–2004). He joined the University of Siena as Research Associate in 2000 and Assistant Professor in 2004. Since 2006 he his with the University of Hong Kong. Dr. Chesi was the recipient of the best student award of

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Andrea Garulli was born in Bologna, Italy, in 1968. He received the Laurea in Electronic Engineering from the Università di Firenze in 1993, and the Ph.D. in System Engineering from the Università di Bologna in 1997. In 1996 he joined the Dipartimento di Ingegneria dell’Informazione of the Università di Siena, where he is currently Professor of Automatic Control. He has been a member of the Conference Editorial Board of the IEEE Control Systems Society and Associate Editor of the IEEE Transactions on Automatic Control. He is author of more than 80 technical publications and co-editor of the books “Robustness in Identification and Control”, Springer 1999, and “Positive Polynomials in Control”, Springer 2005.

His present research interests include system identification, robust estimation and filtering, LMI-based optimization for robust control, mobile robotics and autonomous navigation.

Alberto Tesi received the Laurea degree in Electronics Engineering from the Università di Firenze, Italy, in 1984. In 1989 he obtained the Ph.D. degree from the Università di Bologna, Italy. In 1990 he joined the Dipartimento di Sistemi e Informatica of the Università di Firenze, where he is currently a Professor of Automatic Control. Since 2006 he is Dean of the Engineering Faculty.

He has served as Associate Editor for the IEEE Transactions on Circuits and Systems from 1994 to 1995 and the IEEE Transactions on Automatic Control from 1995 to 1998. Presently he serves as an Associate Editor for Systems and Control Letters. His research interests are mainly in robust control of linear systems and nonlinear systems analysis.

Antonio Vicino was born in 1954. He received the Laurea in Electrical Engineering from the Politecnico di Torino, Torino, Italy, in 1978. From 1979 to 1982 he held several Fellowships at the Dipartimento di Automatica e Informatica of the Politecnico di Torino. He was an assistant professor of Automatic Control from 1983 to 1987 at the same department. From 1987 to 1990 he was an Associate Professor of Control Systems at the Università di Firenze. In 1990 he joined the Dipartimento di Ingegneria Elettrica, Università di L’Aquila, as Professor of Control Systems. Since 1993 he is with the Università di Siena, where he founded the Dipartimento di Ingegneria dell’Informazione and covered the position of Head of the Department from 1996 to 1999. From 1999 to 2005 he was Dean of the Engineering Faculty. In 2000 he founded the Center for Complex Systems Studies (CSC) of the University of Siena, where he presently covers the position of Director. He has served as an Associate Editor for the IEEE Transactions on Automatic Control from 1992 to 1996. Presently he serves as an Associate Editor for Automatica, IEEE Transactions on Circuits and Systems II, and an Associate Editor at Large for IEEE Transactions on Automatic Control. He is Fellow of the IEEE.

He is the author of more than 200 technical publications, co-editor of two books on ‘Robustness in Identification and Control’, Guest Editor of the Special Issue ‘Robustness in Identification and Control’ of the International Journal of Robust and Nonlinear Control and of the Special Issue ‘System Identification’ of the IEEE Transactions on Automatic Control. He has worked on stability analysis of nonlinear systems and time series analysis and prediction. Presently, his main research interests include robust control of uncertain systems, robust identification and filtering, mobile robotics and applied system modelling.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publications in revised form by Associate Editor Carsten W. Scherer under the directions of Editor Roberto Tempo.

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Brief paperRobust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions☆

Abstract

Introduction

Section snippets

Problem formulation and preliminaries

Stability via HPD-HLFs

Example 1

Conclusions

Automatica

Automatica

Systems and Control Letters

Systems and Control Letters

A team algorithm for robust stability analysis and control design of uncertain time-varying linear systems using piecewise quadratics Lyapunov functions

International Journal of Robust and Nonlinear Control

New tools for robustness of linear systems

A universal class of smooth functions for robust control

IEEE Transactions on Automatic Control

A convex approach to robust stability for linear systems with uncertain scalar parameters

SIAM Journal on Control and Optimization

Linear matrix inequalities in system and control theory

Stability of dynamical systems: A constructive approach

IEEE Transactions on Circuits and Systems

Solving quadratic distance problems: An LMI-based approach

IEEE Transactions on Automatic Control

Robust analysis of LFR systems through homogeneous polynomial Lyapunov functions

IEEE Transactions on Automatic Control

Brief paper
Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions☆