Some Results on Practical Stabilizability of Discrete-Time Switched Systems

In this paper, we report some recent development on practical stabilizability of discrete-time switched systems. We first introduce some practical stabilizability notions for discrete-time switched systems. Then we propose some sufficient conditions for the -practical asymptotic stabilizability of such systems. Furthermore, we focus on a class of discrete-time switched systems — namely, switched systems with constant increments, and present an approach to estimating the minimum bound for practical stabilizability. Since such class of systems are usually derived by discretizing continuous-time switched systems with integrator subsystems, we also explore the relationship between the minimum bound and the sampling period. Extended Abstract Recently, in our papers [7, 8, 9, 10, 11, 12], we have noted that, under appropriate switching laws, switched systems whose subsystems have different or no equilibria may still exhibit interesting behaviors similar to those of conventional stable or asymptotically stable systems near an equilibrium. Such behaviors are defined as practical stabilizability (local behavior) and practical asymptotic stabilizability (behavior in a larger region) in these papers. They are natural extensions of the traditional concepts of practical stability [2, 3], which are concerned with bringing the system trajectories to be within given bounds. The results reported in [7, 8, 9, 10, 11, 12] are mainly concerned with practical stabilizability of continuous-time switched systems. For a survey of practical stabilizability and its relationship to the conventional stabilizability of continuous-time switched systems, the reader is referred to our recent paper [11] for more information. Up to now, there have only been very few results reported in the literature which are related to the boundedness or practical stability of discrete-time switched systems (see, e.g., [1, 4, 5, 6]). In this paper, we will formally propose some notions of practical stability and stabilizability for discrete-time switched systems and present some sufficient conditions for the -practical asymptotic stabilizability of such systems. Based on the sufficient conditions, we will then focus on a special class of discrete-time system, i.e., switched systems with constant increments, and present an approach to estimating the minimum bound for practical stabilizability. Since such class of systems are usually derived by discretizing continuous-time switched systems with integrator subsystems, we also explore the relationship between the minimum bound and the sampling period. A longer version of this paper can be found in [13]. In the sequel, we use ‖ ·‖ to denote the 2-norm, B(x, r) to denote the open ball {y ∈ R : ‖y−x‖ < r} and B[x, r] the closed ball. We use Sr to denote the r-sphere around the origin, i.e., Sr = {x ∈ R : ‖x‖ = r}. By a domain D around the origin, we mean an open connected subset of R containing the origin. Int(A) denotes the interior of a set A ∈ R. A. Discrete-Time Switched Systems and Practical Stabilizability Notions In this paper, we consider discrete-time switched systems which consist of discrete-time subsystems x(k + 1) = fi ( x(k) ) , i ∈ I 4 = {1, 2, · · · ,M}. (1) ∗Corresponding author. Department of Electrical and Computer Engineering, Penn State Erie, Erie, PA 16563 USA. Tel: 1-814-8987169; E-mail: Xuping-Xu@psu.edu. †Department of Electrical and Computer Engineering, Penn State Erie, Erie, PA 16563 USA. Tel: 1-814-898-6390; E-mail: sxh63@psu.edu. ‡Department of Mechanical Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, JAPAN. Tel: 81-72-254-9218; E-mail: zhai@me.osakafu-u.ac.jp.