Switching Laws Design for Stability of Finite and Infinite Delayed Switched Systems With Stable and Unstable Modes

This paper studies the switching laws designed to maintain the stability of delayed switched nonlinear systems with both stable and unstable modes. The addressed time delays include finite and infinite delays. First, we consider the finitely delayed nonlinear switched systems and establish some delay differential inequalities that play an important role in the design of average dwell time (ADT)-based switching laws. Then, by employing multiple Lyapunov functions and the ADT approach, some delay-dependent switching laws for globally uniform exponential stability are derived. This approach establishes a relationship between time delay, ADT constant, and the ratio of total dwell time between stable and unstable modes. This method can guarantee the stability of finitely delayed switched systems with stable and unstable modes if the divergence rate and total dwell times of unstable modes can be effectively controlled and balanced by an ADT-based switching control with stable modes. Furthermore, based on multiple Lyapunov functions coupled with the Razumikhin technique, we study the infinitely delayed nonlinear switched systems and present some delay-independent switching laws for uniform stability and globally uniformly exponential stability. These can be applied to the cases in which the time delay in stable or unstable modes cannot be exactly observed, and the bound of the time delay may be unknown or infinite. The proposed results in this paper are more general than several recent works. Finally, some numerical examples and their computer simulations are given to demonstrate the effectiveness and advantages of the designed switching laws.