Stability of switched linear systems on non-uniform time domains

Abstract

Abstract-A recent development in Lyapunov stability theory allows for analysis of switched linear systems evolving on nonuniform, discrete time domains. The analysis makes use of an emerging mathematical framework termed dynamic equations on time scales. We will present stability conditions for a general, arbitrarily switched system and then for system with a "constrained" switching signal. The results take the form of a compute-able inequality, which imposes conditions on the time domain itself.

On common quadratic Lyapunov functions for dynamic normal switched systems

“…One particular method of proving the stability of dynamic switched systems is by finding a CQLF for the system, which has been studied extensively [2], [3], [7], [10], [11], [12], [13], [18], and implies global uniform exponential stability [17].…”

“…A main contribution of this paper is that we relax the pairwise commuting hypothesis used in [2], [17], generalize the results in [20] to arbitrary time scale domains, and expand these results to allow arbitrary eigenvalue bounds on the QLF. In particular, we investigate sets of normal matrices; that is, matrices that commute with their own transpose (A T A = AA T ).…”

“…Both switched systems and dynamic equations on time scales are also of particular interest due to their numerable applications, as shown in [2], [8], [11], [16], [17]. In particular, stability of switched systems under arbitrary switching can be determined by the identification of a single quadratic Lyapunov function applicable to all component systems [11].…”

“…In particular, stability of switched systems under arbitrary switching can be determined by the identification of a single quadratic Lyapunov function applicable to all component systems [11]. Common quadratic Lyapunov functions (CQLFs) have been used as a method for determining stability under arbitrary switching, and discussed in several papers, both for time scale, continuous, and uniform discrete domains [2], [3], [10], [11], [12], [17], [18]. In this paper, we extend the results in [7], [13], [14], [19], [20] to time scale domains and utilize these results to investigate the existence of a stabilizing switching pattern.…”

Stability of switched linear systems on time scale

“…When focusing on stability analysis of switched systems, there are three basic problems in stability and design of switched systems: (i) find conditions for stability under arbitrary switching; (ii) identify the limited but useful class of stabilizing switching laws; and (iii) construct a stabilizing switching law. There are many previous works on this problem in the cases where the switched systems are composed of continuous-time subsystems [5], [6], or where switched systems are composed of discrete-time subsystems [14], [15], [16], and there are many results was recently generalized to include switched systems on arbitrary time scales T, with additional constraints imposed upon the graininess of the time scales [8], [10], [17]. In this work we introduce the theory of time scale on the context of stability of special class of switched systems where the dynamical system will commutate between a continuous-time dynamic subsystem considering dynamic ofẋ = f 1 (x), and after that, it will switch off (in continuous sense), during a certain period µ(t), and it will retake its dynamic with an initial state which depends on the stop time t according to the discrete dynamic x(σ (t))−x(t)…”