Stability of Time-Delay Reset Control Systems

Abstract

This chapter presents results on the stability of time-delay systems under reset control. Since reset control is able to overcome fundamental limitations, and time-delay is one source of such limitations, it is of great interest to study the problem of delayed reset systems. The stability is addressed by choosing an appropriate Lyapunov–Krasovskii functional and by imposing that the functional should decrease in the continuous and reset modes. The resulting conditions take the form of linear matrix inequalities, and, depending on the chosen functional, these LMIs can be delay-dependent or delay-independent. In both cases, those LMIs, derived from time-domain stability conditions, are translated into equivalent frequency-domain conditions by means of appropriate tools, like the Kalman–Yakubovich–Popov lemma, or passivity techniques. From the latter frequency-domain conditions, useful interpretations are exhibited regarding the achieved robustness, in terms of scaled small-gain or positive realness of certain subsystems. Finally, several examples illustrate the application of the stability conditions, showing the potential of reset controls when applied to time-delay systems. This chapter is based on the publications (Baños and Barreiro in 32nd IECON Conference, Paris, France, 2006; Baños and Barreiro in Proceedings of the American Control Conference, ACC, New York, 2007; Baños and Barreiro in IEEE Trans. Autom. Control 54(2):341–346, 2009; Barreiro and Baños in Automatica 46:216–221, 2010).