Networked-based stabilization via discontinuous Lyapunov functionals

This paper presents a new stability and L ₂-gain analysis of linear Networked Control Systems (NCS). The new method is inspired by discontinuous Lyapunov functions that were introduced by Naghshtabrizitextitet al. (Syst. Control Lett. 2008; 57:378-385; Proceedings 26th American Control Conference, New York, U.S.A., July 2007) in the framework of impulsive system representation. Most of the existing works on the stability of NCS (in the framework of time delay approach) are reduced to some Lyapunov-based analysis of systems with uncertain and bounded time-varying delays. This analysis via time-independent Lyapunov functionals does not take advantage of the sawtooth evolution of the delays induced by sample-and-hold. The latter drawback was removed by Fridman (Automatica 2010; 46:421-427), where time-dependent Lyapunov functionals for sampled-data systems were introduced. This led to essentially less conservative results. The objective of the present paper is to extend the time-dependent Lyapunov functional approach to NCS, where variable sampling intervals, data packet dropouts, and variable network-induced delays are taken into account. The Lyapunov functionals in this paper depend on time and on the upper bound of the network-induced delay, and these functionals do not grow along the input update times. The new analysis is applied to the state-feedback and to a novel network-based static output-feedback H _∞ control problems. Numerical examples show that the novel discontinuous terms in Lyapunov functionals essentially improve the results.