Exponential consensus of discrete-time systems based on a novel Krasovskii-LaSalle theorem under directed switching networks

This paper proposes a novel Krasovskii-LaSalle theorem for discrete-time switched linear time-invariant systems. The result is utilized to guarantee exponential convergence of both leaderless and leader-following consensus of discrete-time multi-agent systems under jointly connected switching network topology. Different from the continuous-time multi-agent systems under switching networks, where the magnitude of the dwell time of switching signals is irrelevant, in the discrete-time case, this value is critical for consensus reaching. A special index, called the constrained controllability index, is defined. This index can be easily calculated and provide a tighter lower bound of the dwell time when compared to those given in the literature. With our novel method, the controllability condition on the system matrix is relaxed to the stabilizability condition, and the proposed results not only expand the existing discussions to balanced switching networks but also cover some unbalanced networks. Finally, numerical examples and simulations are provided to verify the theoretical results.