Stability Analysis Of Switched Delay Systems With All Unstable Subsystems

Switched delay systems are a class of models which are affected simultaneously by switching behavior and delay phenomenon. This class of systems plays an important role in control theory and control engineering and much attention has been paid to performance analysis and controller design for such systems, where one of the basic problems is the stability analysis for it. As we all known, both switching behavior and delays can make a system unstable. Therefore, the analysis for switching delay systems is more complicated under the interaction of switching behavior and delays. Many problems related to such a system remain unsolved.Affected by switching behavior and delays, the switched delay systems may contain unstable subsystems. In the extreme situation, all subsystems may become unstable, such as, the networked control systems, multi-agent systems, etc. Since switched delay systems with all unstable subsystems are often encountered in engineering and practical applications, stability analysis of such systems is of great significance both in theory development and practical applications.In this paper, stability analysis is presented for the switched delay systems with all subsystems unstable for the first time by dwell time approaches. Firstly, by adopting the techniques of multiple Lyapunov functional, the exponentially stable conditions are developed for the nonlinear switched delay systems with all unstable subsystems under the constraint of dwell time. Then, the obtained results are applied to linear switched delay systems with all unstable subsystems. Then by utilizing the linear interpolation technique, a novel discrete Lyapunov functional is constructed, based on which the exponentially stable conditions for such a linear system is proposed in the form of linear matrix inequalities (LMIs). Finally, a numerical example is given to show the effectiveness of the proposed results.