Impulsive stabilization of a class of singular systems with time-delays

This paper deals with the impulsive stabilization problem for a class of linear singular systems with time-delays. The stabilization is achieved by only exerting impulsive action on the slow state variables. Two novel Lyapunov methods are presented to determine exponential stability of the impulsively controlled systems. For the case where the time-delay is unknown and may be time-varying, a Lyapunov-Razumikhin method is developed, in which the Razumikhin condition is constructed by exploiting the relation among the fast state variables, the slow state variables, and their initial values. For the case where the delay derivative is strictly less than 1, a descriptor type of impulse-time-dependent Lyapunov functional is introduced, which is discontinuous at impulse times but does not grow along the state trajectories by construction. By using a convex technique, the stability criteria are expressed in terms of linear matrix inequalities (LMIs). Then, the impulsive controllers can be designed in the framework of LMIs. The effectiveness and advantages of the proposed methods are confirmed through simulation results.