Some novel necessary and sufficient conditions of exponential stability for discrete-time systems with multiple delays: A Lyapunov matrix approach |
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| Institution: | 1. School of Mathematical Science, Heilongjiang University, Harbin 150080, China;2. Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Heilongjiang University, Harbin 150080, China;1. Department of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, China;2. Key Laboratory for Electric Drive Control and Intelligent Equipment of Hunan Province, Zhuzhou 412007, China |
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| Abstract: | This paper aims at establishing necessary and sufficient conditions of exponential stability for linear discrete-time systems with multiple delays. Firstly, we introduce a new concept—Lyapunov matrix, and investigate its properties, existence and uniqueness by: (i) characterizing the solution of a boundary value problem of matrix difference equations; and (ii) constructing complete type Lyapunov–Krasovskii functionals with pre-specified forward difference. Secondly, a new constructive analysis methodology, named Lyapunov matrix approach, is proposed to establish necessary and sufficient exponential stability conditions for discrete-time systems with multiple delays. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results. It is worth emphasizing that, from a view of computation, the Lyapunov matrix approach proposed here is concerned with three key steps: (i) solve a systems of linear equations; (ii) check whether a constant matrix is of full-column-rank, and (iii) judge whether a constant matrix is positive definite. All of these can be easily realized by using the tool software—MATLAB. |
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