Switched systems that are periodically stable may be unstable (original) (raw)

The set of stable switching sequences for discrete-time linear switched systems

2011

In this paper we study the characterization of the asymptotical stability for discretetime switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.

Stability of a Switched Linear System

2012

Hybrid systems are dynamic systems that arise out of the interaction of continuous state dynamics and discrete state dynamics. Switched systems, which are a type of hybrid system, have been given much attention by control systems research over the past decade. Problems with the controllability, observability, converseability and stabilizability of switched systems have always been discussed. In this paper, the trend in research regarding the stability of switched systems will be investigated. Then the variety of methods that have been discovered by researchers for stabilizing switched linear systems with arbitrary switching will be discussed in detail.

Almost Sure Stability of Discrete-Time Switched Linear Systems: A Topological Point of View

SIAM Journal on Control and Optimization, 2008

In this paper, we study the stability of discrete-time switched linear systems via symbolic topology formulation and the multiplicative ergodic theorem. A sufficient and necessary condition for μ A -almost sure stability is derived, where μ A is the Parry measure of the topological Markov chain with a prescribed transition (0,1)-matrix A. The obtained μ A -almost sure stability is invariant under small perturbations of the system. The topological description of stable processes of switched linear systems in terms of Hausdorff dimension is given, and it is shown that our approach captures the maximal set of stable processes for linear switched systems. The obtained results cover the stochastic Markov jump linear systems, where the measure is the natural Markov measure defined by the transition probability matrix. Two examples are provided to illustrate the theoretical outcomes of the paper.

Semistability of switched dynamical systems, Part I: Linear system theory

2009

This paper develops semistability and uniform semistability analysis results for switched linear systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system's initial conditions. Since solutions to switched systems are a function of the system's initial conditions as well as the switching signals, uniformity here refers to the convergence rate of the multiple solutions as the switching signal evolves over a given switching set. The main results of the paper involve sufficient conditions for semistability and uniform semistability using multiple Lyapunov functions and sufficient regularity assumptions on the class of switching signals considered.

Stability of switched systems: a Lie-algebraic condition

Systems & Control Letters, 1999

We present a su cient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.

Conditions for stabilizability of linear switched systems

International Journal of Control Automation and Systems, 2011

This paper investigates some conditions that can provide stabilizability for linear switched systems with polytopic uncertainties via their closed loop linear quadratic state feedback regulator. The closed loop switched systems can stabilize unstable open loop systems or stable open loop systems but in which there is no solution for a common Lyapunov matrix. For continuous time switched linear systems, we show that if there exists solution in an associated Riccati equation for the closed loop systems sharing one common Lyapunov matrix, the switched linear systems are stable. For the discrete time switched systems, we derive a Linear Matrix Inequality (LMI) to calculate a common Lyapunov matrix and solution for the stable closed loop feedback systems. These closed loop linear quadratic state feedback regulators guarantee the global asymptotical stability for any switched linear systems with any switching signal sequence.

Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time Linear Switched Systems

2011

Let S = {S 1 , S 2 } ⊂ R d×d have a common, but not necessarily strict, Lyapunov matrix (i.e. there exists a symmetric positive-definite matrix P such that P − S T k PS k ≥ 0 for k = 1, 2). Based on a splitting theorem of the state space R d (Dai, Huang and Xiao, arXiv:1107.0132v1[math.PR]), we establish several stability criteria for the discrete-time linear switched dynamics x n = S σ n · · · S σ 1 (x 0 ), x 0 ∈ R d and n ≥ 1 governed by the switching signal σ : N → {1, 2}. More specifically, let ρ(A) stand for the spectral radius of a matrix A ∈ R d×d , then the outline of results obtained in this paper are: (1) For the case d = 2, S is absolutely stable (i.e., S σ n · · · S σ 1 → 0 driven by all switching signals σ) if and only if ρ(S 1 ), ρ(S 2 ) and ρ(S 1 S 2 ) all are less than 1; (2) For the case d = 3, S is absolutely stable if and only if ρ(A) < 1 ∀A ∈ {S 1 , S 2 } ℓ for ℓ = 1, 2, 3, 4, 5, 6, and 8. This further implies that for any S = {S 1 , S 2 } ⊂ R d×d with the generalized spectral radius ρ(S) = 1 where d = 2 or 3, if S has a common, but not strict in general, Lyapunov matrix, then S possesses the spectral finiteness property.