Necessary and sufficient conditions for stabilisability of discrete‐time time‐varying switched systems

Abstract

Abstract This paper is concerned with necessary and sufficient conditions for stabilisability of time‐varying discrete‐time switched systems. Starting with an asymptotically stable function, an exponentially stable function and a uniformly exponentially stable function, we successively propose necessary and sufficient conditions for asymptotic stabilisability, exponential stabilisability and uniform exponential stabilisability of time‐varying switched linear systems. Further, considering the broad applications of finite‐time stability in practical systems, based on an additionally introduced concept of finite‐time stable functions, we derive a necessary and sufficient condition for finite‐time stabilisability of time‐varying switched linear systems. Afterwards, three illustrative examples are given to show the applicability of our theoretical results. In the end, we further discuss the necessary and sufficient conditions for global exponential stabilisability and global uniform exponential stabilisability of time‐varying switched non‐linear systems. Compared to traditional difference Lyapunov inequalities, we release the requirement on negative definiteness of the time‐difference of Lyapunov functions.

Keywords

• Linear control systems
• Stability in control theory
• Discrete control systems
• Time-varying control systems
• Nonlinear control systems