Stability and L1L1-gain controller design for positive switched systems with mixed time-varying delays

Keywords: Positive systems Switched systems Exponential stability L 1-gain performance Controller design Time-varying delays Average dwell time a b s t r a c t This paper investigates the problems of stability and L 1-gain controller design for positive switched systems with mixed time-varying delays. The mixed time-varying delays are presented in the forms of discrete delay and distributed delay. The purpose of this paper is to design a class of switching signals and a state feedback controller for the considered system such that the resulting closed-loop system is exponentially stable with L 1-gain performance. By constructing an appropriate co-positive type Lyapunov–Krasovskii functional and using the average dwell time approach, we propose a sufficient condition to ensure the exponential stability with weighted L 1-gain performance for the system. On the basis of the obtained results, an effective method for designing such controller is established. Finally, a numerical example is presented to demonstrate the feasibility of the results obtained. A switched system is a classical type of hybrid dynamical systems that organically combine the discrete event dynamic systems and the continuous variable dynamic systems. It consists of a family of subsystems and a collection of logical rules deciding the current state of system. The logical rules are usually referred to as the switching strategies or the switching laws, which coordinate the operation of the various subsystems to decide which one to be effective at one point. Many dynamical systems can be modeled as switched systems [1–3]. Based on the background of broad applications, the study of switched systems has become a hot pot [4–7]. A positive system means that its state and output are nonnegative whenever the initial condition and input are nonneg-ative [8,9]. A positive switched system is a switched system, each subsystem in which is itself a positive system. Recently, due to the broad applications in communication systems [10,11], formation flying [12], and systems theories [13–19], the importance of positive switched systems has been highlighted and investigated by many researchers. Notice that most of the existing achievements of positive switched systems are concerned with the issues of stability and stabilization [20– 23]. When considering the stability of positive systems, it is natural to apply a linear co-positive Lyapunov functional instead of the traditional quadratic Lyapunov functional, due to the fact that the former will yield a less conservative stability condition for positive systems [24]. A switched linear co-positive Lyapunov …