On Duality and Exponential Stability for Two Classes of Linear Switched Systems with Applications to Multiagent Systems
Tao Liu, Jie Huang · SIAM Journal on Control and Optimization · 2025
Abstract. Two classes of linear switched systems play an important role in studying the cooperative control of leader-follower multiagent systems. The first class arises from the design of an output-based distributed observer for a linear leader system over jointly connected switching networks, while the second class arises from the study of a leader-following consensus problem for linear multiagent systems over jointly connected switching networks. The exponential stability for both classes of linear switched systems has been established in the literature by using the generalized Krasovskii–LaSalle theorem. In this paper, by using the notion of uniform complete observability for linear time-varying systems, we first present an alternative proof for the exponential stability for the first class of linear switched systems. Compared with the existing results in the literature, this alternative approach only calls for elementary linear control concepts and is self-contained. In addition, we offer two new ingredients. First, we relax the symmetric assumption on a key switching matrix contained in the switched systems. Second, we give an explicit characterization of a lower bound for the exponential convergence rate. Then, by applying a simple duality argument, we further obtain the exponential stability for the second class of linear switched systems. As applications of these two stability results, we show that the first exponential stability result leads to an exponentially convergent output-based distributed observer for a linear leader system over jointly connected switching networks, while the second exponential stability result solves the leader-following exponential consensus problem for linear multiagent systems over jointly connected switching networks.