Brief paperPeriodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities☆
Introduction
Given an arbitrary finite set of complex matrices where and are two fixed integers, it generates a discrete-time linear inclusion Here the initial state is thought of as an -dimensional column vector. The solutions of this inclusion may be described by the discrete-time linear switched dynamics where is the switching sequence/signal.
For , one interesting and important question is how to determine the convergence of random infinite products of the matrices in . 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 of the matrices in is the “joint spectral radius” of , due to Rota and Strang (1960), defined as or equivalently, where we denote The quantity 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 converge to zero if and only if (f.g. see Barabanov, 1988, Shih et al., 1997). Moreover, according to Berger and Wang (1992) and Elsner (1995), is equal to the “generalized spectral radius” of , firstly introduced by Daubechies and Lagarias (1992a), given by or equivalently, Here, the spectral radius of a single matrix is defined by
Since the generalized spectral radius 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 , which leads to the following important problem:
Problem 1.1 The Spectral Finiteness Property For every finite set of matrices, there exists some word of finite length such that
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 . 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 and an infinite switching sequence , where for each , it defines a switched dynamical system : We denote the set of all such switching sequences by . It is easy to see that that consists of all the sequences , where , is a compact topological space in terms of the product topology.
Recall that is said to be “absolutely asymptotically stable” if or equivalently, If the above convergence is exponentially fast, i.e., for any switching sequence , then is said to be “absolutely exponentially stable”, where the quantity is called the “Lyapunov exponent” of at the switching sequence (cf. Barabanov, 1988). By Fenichel’s theorem Fenichel (1971), the absolute asymptotic stability is equivalent to the absolute exponential stability for .
For an arbitrary switching sequence in , if converges to 0 as , then or is called “asymptotically stable”; if is smaller than 0, then or is said to be “exponentially stable”. In general, it is quite challenging to determine the asymptotic or exponential stability of for an arbitrary switching sequence . However, if is “periodically switched” with period , i.e., there is some word, in , of length such that for and any , then is exponentially stable if and only if . Thus, determining the stability of is straightforward for a periodically switched sequence . 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 ? That is to say, if is asymptotically stable for each periodically switched sequence in then, is absolutely asymptotically stable?
For this problem, in Gurvits (1995), it is proved that the periodically switched stability implies the absolute asymptotic stability when has the extra restriction that all for an extremal “polytope” norm on . Its proof relays the problem to formal languages generated by finite automata there. However, the extremal “polytope” norm on is a special type of norms and may not exist; for example, see Jungers and Protasov (2009). Moreover, the condition is quite restrictive since it requires the multiplicative semigroup generated by 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 remains unknown. In this paper we will show that, if is periodically switched stable, then is exponentially stable for “almost all” switching sequences in . Here “almost all” means that the set of such switching sequences is of measure 1 for any canonical Markovian probability supported on . Furthermore, we will show that the set of such switching sequences has the same Hausdorff dimension as the entire switching sequence space . 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 be a given set of complex matrices, where and 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 be a set of square matrices, where and . Let We will show that under the hypothesis of the Main theorem, has the same Hausdorff dimension as that of .
Since Hausdorff dimension depends upon the choice of the metric of , without loss of generality, let the metric be chosen as in Section 2.1 with a preassigned constant . 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.
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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.