Stabilizing control design for state-dependent impulsive dynamical linear systems

The stabilizing control design problem for state-dependent impulsive dynamical linear systems (SD-IDLS) is tackled in this paper. This class of systems consists of a continuous-time, linear time-invariant system combined with discrete-time linear time-invariant dynamics in a prescribed region of the state space. The former regulates the evolution of the system between any two consecutive resetting events, while the latter governs the instantaneous state jumps occurring whenever the system trajectory intersects a resetting set, which is a portion of the state space assumed to be time-independent. Stabilization of SD-IDLS through a state feedback control design is specifically discussed in this paper, by making use of a candidate quadratic control Lyapunov function. By considering the conical hulls of the resetting set subregions and imposing some cone copositivity properties on the chosen control Lyapunov function, a sufficient and constructive condition for the global asymptotic stabilization of SD-IDLS is provided. Such a result, based on the solution of a feasibility problem that involves a set of Linear Matrix Inequalities (LMIs), is shown to be less conservative and more numerically amenable with respect to other results available in the literature. An example illustrates the effectiveness of the proposed approach.