Global asymptotic stability of limit cycle and [formula omitted]/[formula omitted] performance of discrete-time switched affine systems

Abstract

This paper treats the control design of a state-dependent switching function for discrete-time switched affine systems, assuring global asymptotic stability of a limit cycle. This limit cycle is determined according to criteria of interest related to the steady-state behaviour of the system trajectories, as for instance, its mean value, oscillation amplitude and frequency. More specifically, based on a time-varying convex Lyapunov function, a min-type switching rule is determined in order to orchestrate the state trajectories, evolving from any initial condition, towards the desired limit cycle. Most of the available methodologies are able to guide the state trajectories to a set of attraction containing the equilibrium point, assuring its practical stability. However, although important, the behaviour of the trajectories inside this set is not considered. Differently, the proposed technique assures an appropriate steady-state behaviour by adequately designing the limit cycle. Moreover, the proposed conditions take into account H2 and H∞ guaranteed costs, assuring also a suitable transient response. All the conditions are expressed in terms of linear matrix inequalities being simple to solve by readily available tools. Illustrative examples, one of them concerning the voltage regulation of a DC–DC multicellular converter, are used for validation and comparison.