Elsevier

Automatica

Volume 46, Issue 2, February 2010, Pages 471-474
Automatica

Technical communique
Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

https://doi.org/10.1016/j.automatica.2009.11.007Get rights and content

Abstract

This paper investigates linear systems with polynomial dependence on time-invariant uncertainties constrained in the simplex via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs). It is shown that a sufficient condition for establishing whether the system is either stable or unstable can be obtained by solving a generalized eigenvalue problem. Moreover, this condition is also necessary by using a sufficiently large degree of the HPD-QLF.

Introduction

Various methods have been proposed for the stability of linear systems with time-invariant uncertainty constrained in a polytope. Generally, these methods exploit parameter-dependent Lyapunov functions and LMIs; see e.g. Leite and Peres (2003), which considers Lyapunov functions with linear dependence, Bliman (2004), which proposes Lyapunov functions with polynomial dependence, Chesi, Garulli, Tesi, and Vicino (2005), which introduces the class of HPD-QLFs, Scherer (2006), which proposes a general framework for LMI relaxations, Oliveira and Peres (2007), where homogeneous solutions are characterized, Lavaei and Aghdam (2008), which addresses the case of semi-algebraic sets, and Oishi (2009) and Peaucelle and Sato (2009), where matrix-dilation approaches are considered.

Some of these methods provide necessary and sufficient conditions for robust stability. However, the necessity is achieved for an unknown degree of the polynomials used. This implies that, if the system is unstable, no conclusion can be reached. This paper addresses this problem via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs) for the case of polynomial dependence on the uncertainty. It is shown that a sufficient condition for establishing either stability or instability can be obtained by solving a generalized eigenvalue problem, and that this condition is also necessary by using a sufficiently large degree of the HPD-QLF. The idea behind this condition is to exploit the LMI relaxation introduced in Chesi et al. (2005) via the square matrical representation (SMR)1 in order to characterize the instability via the presence of suitable vectors in certain eigenspaces.

Before proceeding, it is worth explaining that the proposed approach differs from Chesi (2005), which proposes a non-Lyapunov method for establishing stability and instability of uncertain systems, from Chesi (2007), which investigates robust H performance via eigenvalue problems, from Ebihara, Onishi, and Hagiwara (2009), which exploits D/G-scaling in the case of rational dependence on the uncertainty, from Masubuchi and Scherer (2009), which derives a recursive algorithm based on the linear fractional representation, and from Goncalves et al., 2006, Goncalves et al., 2007, which propose a branch-and-bound method in the case of linear dependence on the uncertainty.

Section snippets

Preliminaries

Notation. R,C: real and complex numbers; R0: R{0}; In: n×n identity matrix; A>0: symmetric positive definite matrix; AB: Kronecker’s product; A, tr(A), det(A): transpose, trace and determinant of A; vec(A): vector with the columns of A stacked below each other; spc(A)={λC:det(λIA)=0}; span(v1,,vk)={a1v1++akvk,a1,,akR}; sq(p)=(p12,,pq2) with pRq; CT, DT: continuous-time and discrete-time; s.t.: subject to.

Let us consider the uncertain system {(CT case)ẋ(t)=A(p)x(t)(DT case)x(t+1)=A(p

Stability and instability condition

Let us define T(β)=Δ(K,IdqS(β)) where K is the matrix satisfying ppd timesp{m}=Kp{m+d} (see also (18) for a key property of T(β)) and define η=supα,β,ηηs.t.{S(β)>0R(β)+U(α)ηT(β)>0tr(S(β))=1. Let us define V=R(β)+U(α) where α,β are optimal values of α,β in (12), and let c1,,cr be the eigenvectors of the non-positive eigenvalues of V, i.e. {cici=1Vci=λicifor some λiRλi0.

Theorem 2

The setAis stable if and only if there existsmsuch thatη>0. Moreover,Ais unstable if and only if there existm

Examples

Here we present some illustrative examples. The computational time on a standard computer with Matlab of the proposed condition for all these examples (and of Goncalves et al. (2007) for the examples in Section 4.1) is lesser than 5 s.

Conclusion

We have proposed a condition via HPD-QLFs for establishing the stability and instability of linear systems with polynomial dependence on uncertainties constrained in the simplex. This condition is necessary and sufficient.

Acknowledgements

The author would like to thank the Associate Editor and the Reviewers for their comments, and the authors of Goncalves et al., 2006, Goncalves et al., 2007 for providing the code of their method.

References (20)

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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Mikael Johansson, under the direction of Editor André L. Tits.

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