Stabilization of NCSs via Discontinuous Lyapunov Functionals

Abstract

More general constructions of the time-dependent Lyapunov functionals addressed in Chap. 1 can be found in [46] for the analysis of sampled-data systems in the framework of time-delay approach. The introduced time-dependent terms of Lyapunov functionals lead to qualitatively new results, allowing a superior performance under the sawtooth delay, than the one under the constant delay. In this chapter, we extend the discontinuous Lyapunov functional method (in the framework of time-delay approach) for sampled-data systems to network-based $H_{\infty }$ control, where data packet dropouts and variable network-induced delays are taken into account. The proposed Lyapunov functional depends on the time and on the upper bound of the network-induced delay, and it does not grow along the input update times. We apply our analysis results to the state-feedback and to a novel static output-feedback $H_{\infty }$ control. It is noted that the observer-based control via network is usually encountered with some waiting strategy and buffers [204]. The implementation of the network-based static output-feedback controller is simple, provided that the system is stabilizable by such a controller. The sufficient conditions for the stabilization via the continuous static output-feedback can be found in the survey [219]. Following the sampled-data $H_{\infty }$ control [217], we consider an $H_{\infty }$ performance index that takes into account the updating rates of the measurement. This index is related to the energy of the measurement noise. Numerical examples show that the novel discontinuous terms in the Lyapunov functional essentially reduce the conservatism.