Elsevier

Automatica

Volume 47, Issue 7, July 2011, Pages 1512-1519
Automatica

Brief paper
Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities

https://doi.org/10.1016/j.automatica.2011.02.034Get rights and content

Abstract

The conjecture that periodically switched stability implies absolute asymptotic stability of random infinite products of a finite set of square matrices, has recently been disproved under the guise of the finiteness conjecture. In this paper, we show that this conjecture holds in terms of Markovian probabilities. More specifically, let SkCn×n,1kK, be arbitrarily given K matrices and ΣK+={(kj)j=1+1kjK for each j1}, where n,K2. Then we study the exponential stability of the following discrete-time switched dynamics S: xj=SkjSk1x0,j1 and x0Cn where (kj)j=1+ΣK+ can be an arbitrary switching sequence.

For a probability row-vector p=(p1,,pK)RK and an irreducible Markov transition matrix PRK×K with pP=p, we denote by μp,P the Markovian probability on ΣK+ corresponding to (p,P). By using symbolic dynamics and ergodic-theoretic approaches, we show that, if S possesses the periodically switched stability then, (i) it is exponentially stable μp,P-almost surely; (ii) the set of stable switching sequences (kj)j=1+ΣK+ has the same Hausdorff dimension as ΣK+. Thus, the periodically switched stability of a discrete-time linear switched dynamics implies that the system is exponentially stable for “almost” all switching sequences.

Introduction

Given an arbitrary finite set of complex n×n matrices S={S1,,SK}Cn×n, where K2 and n2 are two fixed integers, it generates a discrete-time linear inclusion x{S1,,SK}x1,1 and x0Cn. Here the initial state x0Cn is thought of as an n-dimensional column vector. The solutions of this inclusion may be described by the discrete-time linear switched dynamics x=SkSk1x0,1 and x0Cn, where (kj)j=1+ is the switching sequence/signal.

For S, one interesting and important question is how to determine the convergence of random infinite products SkjSk2Sk1 of the matrices Sk in S. These types of questions arise, for example, in the study of the stability of discrete-time linear switched dynamical systems, coding theory, compactly supported wavelets and solutions of two-scale dilation equations, stochastic process associated with probability transition matrices, combinatorics and numerical solution to ordinary differential equations, and it has been receiving much attention in the past two decades. For instance, see Barabanov (1988), Daubechies and Lagarias (1992a), Daubechies and Lagarias (1992b), Dumont, Sidorov, and Thomas (1999), Guglielmi and Zennaro (2001), Gurvits (1995), Kozyakin (1990), Lee and Dullerud (2006a), Lee and Dullerud (2006b), Maesumi (1998), Moision, Orlitsky, and Siegel (2001) and Xiao (2005), survey papers (Barabanov, 2005, Lin and Antsaklis, 2009, Shorten et al., 2007, Sun and Ge, 2005) and the references therein.

A critical characterization of the convergence of all infinite products SkjSk2Sk1 of the matrices in S is the “joint spectral radius” of S, due to Rota and Strang (1960), defined as ρˆ(S)=lim supj+max(k1,,kj)KjSkjSk11/j, or equivalently, ρˆ(S)=limj+max(k1,,kj)KjSkjSk11/j, where we denote K={1,,K}and Kj=K××Kj−timesj1. The quantity ρˆ(S) is well defined and it is independent of the matrix norm used here Elsner (1995) and Rota and Strang (1960). It is well known that all infinite products of the matrices in S converge to zero if and only if ρˆ(S)<1 (f.g. see Barabanov, 1988, Shih et al., 1997). Moreover, according to Berger and Wang (1992) and Elsner (1995), ρˆ(S) is equal to the “generalized spectral radius” ρ(S) of S, firstly introduced by Daubechies and Lagarias (1992a), given by ρ(S)=supj1max(k1,,kj)Kjρ(SkjSk1)1/j, or equivalently, ρ(S)=lim supj+max(k1,,kj)Kjρ(SkjSk1)1/j. Here, the spectral radius of a single matrix ACn×n is defined by ρ(A)=max{|λ|:λ is an eigenvalue of A}.

Since the generalized spectral radius ρ(S) is the quantity for the characterization of the growth rate of a discrete-time linear switched system, it is essential to determine whether there is an effectively computable procedure for the computation of ρ(S), which leads to the following important problem:

Problem 1.1 The Spectral Finiteness Property

For every finite set S={S1,,SK} of n×n matrices, there exists some word (k1,,kj) of finite length j1 such that ρ(S)=ρ(SkjSk1)1/j.

This spectral finiteness property was conjectured, respectively, by Daubechies and Lagarias (1992a) and by Lagarias and Wang (1995). However, this finiteness conjecture has been disproved, respectively, by Bousch and Mairesse (2002) using measure-theoretical ideas, Blondel, Theys, and Vladimirov (2003) exploiting combinatorial properties of permutations of products of positive matrices, and by Kozyakin, 2005, Kozyakin, 2007 employing the theory of dynamical systems, all offered the existence of counterexamples in the case of n=2. Moreover, an explicit counterexample for the finiteness property has recently been found by Hare, Morris, Sidorov, and Theys (2011).

Although the finiteness conjecture fails to exist, the idea is still to be attractive and important since the algorithms for the computation of the joint spectral radius must be implemented in finite arithmetic. In this paper, we will study this problem in a border framework under the guise of the stability of discrete-time linear randomly switched systems. Here “randomly” means that we will only care “almost every” switching sequence/signal instead of every switching sequence.

For given S={S1,,SK} and an infinite switching sequence σ=(kj)j=1+, where kjK for each j1, it defines a switched dynamical system Sσ:xj=SkjSk1x0,j1 and x0Cn. We denote the set of all such switching sequences (kj)j=1+ by ΣK+. It is easy to see that ΣK+=KN that consists of all the sequences σ:NK, where N={1,2,}, is a compact topological space in terms of the product topology.

Recall that S is said to be “absolutely asymptotically stable” if SkjSk10as j+(kj)j=1+ΣK+; or equivalently, SkjSk10n×nas j+(kj)j=1+ΣK+. If the above convergence is exponentially fast, i.e., for any switching sequence σ=(kj)j=1+ΣK+, χ(σ)lim supj+1jlogSkjSk1<0, then S is said to be “absolutely exponentially stable”, where the quantity χ(σ) is called the “Lyapunov exponent” of S at the switching sequence σ (cf. Barabanov, 1988). By Fenichel’s theorem Fenichel (1971), the absolute asymptotic stability is equivalent to the absolute exponential stability for S.

For an arbitrary switching sequence σ=(kj)j=1+ in ΣK+, if SkjSk1 converges to 0 as j+, then Sσ or σ is called “asymptotically stable”; if χ(σ) is smaller than 0, then Sσ or σ is said to be “exponentially stable”. In general, it is quite challenging to determine the asymptotic or exponential stability of Sσ for an arbitrary switching sequence σ. However, if σ is “periodically switched” with period π1, i.e., there is some word, (w1,,wπ) in Kπ, of length π such that ki+π=wi for 1iπ and any 0, then Sσ is exponentially stable if and only if ρ(SwπSw1)<1. Thus, determining the stability of Sσ is straightforward for a periodically switched sequence σ. This naturally leads to the following important question.

Problem 1.2 Cf. Pyatnitskii˘ and Rapoport, 1991, Shorten et al., 2007

Does the periodically switched stability imply the absolute asymptotic stability for S? That is to say, if Sσ is asymptotically stable for each periodically switched sequence σ=(kj)j=1+ in ΣK+ then, is S absolutely asymptotically stable?

For this problem, in Gurvits (1995), it is proved that the periodically switched stability implies the absolute asymptotic stability when S has the extra restriction that all Si1 for an extremal “polytope” norm on Rn. Its proof relays the problem to formal languages generated by finite automata there. However, the extremal “polytope” norm on Rn is a special type of norms and may not exist; for example, see Jungers and Protasov (2009). Moreover, the condition Sk1 is quite restrictive since it requires the multiplicative semigroup generated by S to be bounded.

According to Shih et al. (1997), this problem is closely related, in fact equivalent, to the spectral finiteness property discussed at the beginning. Thus, the periodically switched stability does not imply the absolute asymptotic stability in general. But what degree of the periodically switched stability can characterize the stability of S remains unknown. In this paper we will show that, if S is periodically switched stable, then Sσ is exponentially stable for “almost all” switching sequences σ in ΣK+. Here “almost all” means that the set of such switching sequences is of measure 1 for any canonical Markovian probability supported on ΣK+. Furthermore, we will show that the set of such switching sequences has the same Hausdorff dimension as the entire switching sequence space ΣK+. The obtained results provide important characteristics on the finiteness conjecture problem (as well as Problem 1.2) from all probabilistic and topological viewpoints.

The rest of this paper is organized as follows: in Section 2, after introducing some basic notations, we will present the main results of the paper and examples. Statement (1) in the Main theorem shown in Section 2 will be proved in Section 3, which is the core outcome of this paper. And statement (2) in the Main theorem will be shown in Section 4. In Section 5 an open question related to our obtained results will be proposed for a further study.

Section snippets

Preliminary notations, main results and examples

In this section, we will recall some basic concepts of ergodic measure and Markovian probability which will be used for our approach, and we then state the main results of this paper and provide an illustrate example.

Throughout the section, we let S={S1,,SK}Cn×n be a given set of complex matrices, where K2 and n2 are two preassigned integers.

Exponential stability in Markovian probability

In this section, we will prove statement (1) in the Main theorem stated in Section 2, based on a series of important lemmas.

Dimension of stable switching sequences

As in the previous sections, we let S={S1,,SK}Cn×n be a set of square matrices, where K2 and n2. Let ΣK,es+={(kj)j=1+ΣK+lim supj+1jlogSkjSk1<0}. We will show that under the hypothesis of the Main theorem, ΣK,es+ has the same Hausdorff dimension as that of ΣK+.

Since Hausdorff dimension depends upon the choice of the metric of ΣK+, without loss of generality, let the metric dϱ(,) be chosen as in Section 2.1 with a preassigned constant ϱ>1. See Dai et al. (2008, Section 3.2) for the

Concluding remarks and an open problem

In this paper, we studied the stability of a discrete-time linear switched dynamics that is periodically switched stable, using dynamics and ergodic-theoretic approaches. A switched system that is periodically switched stable implies the stability of “almost” all switching sequences. Although such a dynamics need not be absolutely asymptotically stable, this paper shows that the set of all stable switching sequences has the same “size” as the entire set of switching sequences from the

Xiongping Dai was born on 13 June 1968, in Chongqing (China). He received Ph.D. in Mathematics from Zhongshan (Sun Yat-Sen) University, China, in 1997. He has been a professor in Department of Mathematics, Nanjing University since 2007. His research interests include the stability theory of switched dynamics, ergodic theory, and differentiable dynamical systems.

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    Xiongping Dai was born on 13 June 1968, in Chongqing (China). He received Ph.D. in Mathematics from Zhongshan (Sun Yat-Sen) University, China, in 1997. He has been a professor in Department of Mathematics, Nanjing University since 2007. His research interests include the stability theory of switched dynamics, ergodic theory, and differentiable dynamical systems.

    Yu Huang received his B.S. and M.S. from Zhongshan (Sun Yat-Sen) University, Guangzhou, P.R.China, in 1983 and 1986, respectively, and Ph.D. from the Chinese University of Hong Kong in 1995. Since 1986, he has been a faculty member with the Department of Mathematics, Zhongshan (Sun Yat-Sen) University, and currently he is a professor. Dr. Huang is an associate editor of Journal of Mathematical Analysis and Applications. His research interests include dynamical system theory and control of hybrid systems.

    Mingqing Xiao got his Ph.D. in 1997 from the University of Illinois at Urbana-Champaign. He had been a visiting research assistant professor at University of California at Davis from 1997 to 1999. He currently is a professor at the Department of Mathematics, Southern Illinois University at Carbondale. Dr. Xiao is a senior Member of IEEE and his research interests include dynamical system theory, nonlinear observer design, and control of distributed parameter systems.

    The project was supported partly by National Natural Science Foundation of China (Grant Nos. 11071112 and 11071263) and in part by NSF 1021203 of the United States. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Fabrizio Dabbene, under the direction of Editor Roberto Tempo.

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