Robust ISS of uncertain discrete-time singularly perturbed systems with disturbances

This paper is concerned with robustly input-to-state stable (ISS) and Robust ISS by feedback of uncertain discrete-time singularly perturbed systems (SPSs) with disturbances. Meanwhile, robust stability and stabilisation of uncertain discrete-time SPSs are also obtained as the particular cases of robust ISS and robust ISS by feedback. We first find a sufficient condition by using the fixed-point principle in terms of linear matrix inequalities (LMIs) to guarantee that the considered system is always standard discrete-time SPSs subject to uncertainty and disturbances. Then, the full systems could decompose into the continuous-time uncertain slow subsystem with disturbance and discrete-time uncertain fast subsystems with disturbance, respectively. Based on the two-time-scale decomposition technique, sufficient condition in terms of LMIs is given such that the full systems are uniformly standard and robust ISS simultaneously. In addition, a state feedback controller is constructed by using the LMI approach such that the resulting closed-loop systems are robust ISS. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.