Stability conditions for linear discrete‐time switched systems in block companion form

Abstract

Switched models whose dynamic matrices are in block companion form arise in theoretical and applicative problems such as representing switched ARX models in state-space form for control purposes. Inspired by some insightful results on the delay-independent stability of discrete-time systems with time-varying delays, in this work, the authors study the arbitrary switching stability for some classes of block companion discrete-time switched systems. They start from the special case in which the first block-row is made of permutations of non-negative matrices, deriving a simple necessary and sufficient stability condition under arbitrary switching. The condition is computationally less demanding than the sufficient-only existence of a linear common Lyapunov function. Then, both non-negativity and combinatorial assumptions are dropped, at the expense of introducing conservatism. Some implications on the computation of the joint spectral radius for the aforementioned families of matrices are illustrated.

Delay‐independent conditions of exponential ISS for linear time‐varying delay differential systems

“…Remark 7. The discussed asymmetry between the right-and left-conditions ( 5) and ( 6) also appears in the case of discrete-time positive delay systems, investigated in De Iuliis and Manes, 11 where the left-and right-positive vector conditions are as follows:…”

“…This work can be seen as a continuous-time counterpart of a recent contribution 11 which dealt with time-varying discrete-time systems with time-varying delays, and proposed novel delay-dependent and delay-independent conditions of global exponential stability and input-to-state stability for both positive and general systems of that class. We previously explored with similar techniques the class of delay differential systems with time-varying delays, although these previous works 24,25 addressed the simpler case of constant matrices and did not study input-to-state stability.…”

“…Among these results, we may reference the recent contributions by Zhou and Egorov, 3 Mazenc and Malisoff, 4 and Li et al 5 Other recent approaches have leveraged trajectory-based methodologies 6 and Halanay-like inequalities 7 ; an interesting idea is to look for a continuous-time counterpart 8 of the popular discrete-time approach based on lifted switched representations. [9][10][11] This work follows a different approach, which is mainly based on state-bounding and the properties of positive systems. The latter have been popular since the 70's, and held in high esteem due to their "deep and elegant theory -yet pleasantly consistent with intuition" (in the words of David G. Luenberger 12 ).…”

“…Analyzing such systems is known to be very challenging, and conservative approaches have been proposed in the literature based on popular extensions of the classical Lyapunov theory established by Krasovskii and Razumikhin. Among these results, we may reference the recent contributions by Zhou and Egorov, 3 Mazenc and Malisoff, 4 and Li et al 5 Other recent approaches have leveraged trajectory‐based methodologies 6 and Halanay‐like inequalities 7 ; an interesting idea is to look for a continuous‐time counterpart 8 of the popular discrete‐time approach based on lifted switched representations 9–11 …”

Feedback stabilization of switched linear systems via a parametric approach

“…In the past decades, the study of the stability of switched systems has attracted increasing research activities; see the survey papers [7,8] and book [9]. Many stability criteria have been investigated, such as stability conditions for linear discrete-time switched systems in block companion form [10] and for continuous-time switched T-S fuzzy systems [11]. It is well-known that the uniform global exponential stability of a switched system under arbitrary switching is equivalent to the existence of a common Lyapunov function for the component subsystems [12].…”