Reduced Order and Observer-Based Reset Control Systems with Time Delays

This paper establishes a new mechanism to stabilize plants using reduced order reset controllers. The proposed method uses state feedback to change the dynamics of plants to guarantee oscillation behavior instead of stability, then the reset mechanism will lead to stability. We show that the base system could be unstable while the reset mechanism drives the states to the equilibrium point. The order of the reset controller equals the rank of the plant’s input matrix. We show that the controller dynamics force some states to converge to the equilibrium point within a finite time. The behavior of the rest of the plant’s states depends greatly on the selection of the state feedback gain which can be selected by any appropriate conventional method. Moreover, the stability of reset time-delay systems is addressed based on a similar theorem of the Lyapunov-Krasovskii theory. Sufficient conditions are given in terms of linear matrix inequalities to guarantee asymptotic stability of the overall dynamics. Simulation results are presented to demonstrate the effectiveness of the proposed reset approaches.