Discussion on: “Suboptimal switching control consistency analysis for discrete-time switched linear systems”

Abstract

The past 20 years has seen a huge number of papers produced in the area of switched and hybrid systems. The principal focus of these papers has been to produce conditions under which stability can be guaranteed, either in the case of arbitrary switching, or in the case of constrained switching, where one constrains the switching through state constraints, or through limits on the rate of switching. Some important topics have, however, rather disappointingly, been the subject of far less study. These include the discretization of switched systems, the design of systems where constraints are fixed rather than a design variable, and the design of switched systems with performance guarantees (to name but a few). A major contribution of the paper under discussion is that it falls into the latter category, and provides results that are likely to be of interest, not only to the academic community, but also to practicing engineers. Following similar work in the context of continuous-time switched linear systems [1], the authors introduce the concept of consistency for the discrete-time counterpart. More specifically, a given switching strategy is consistent if it improves the performance of the closed loop system when compared with any of the isolated subsystems. The performance of the controlled system is evaluated in terms of input–output H2 and H∞ indexes, which are defined in such a way that whenever the switching policy is constant, then the indices coincide with the conventional definition for the single subsystem. Section 3 constitutes the main part of the paper. The authors present two theorems that guarantee the existence of a consistent switching function; the first based on conditions on the Hi matrices, and the second based on even other conditions on the class of switching systems considered. In both cases the use of Lyapunov–Metzler inequalities is heavily employed. Note in the discussion following both of these theorems, the investigation of strict consistency involves the formulation of a non-convex search that would almost certainly be