Stability and Performance of SISO Control Systems With First-Order Reset Elements

We consider set-point regulation and L₂ robust stability properties of a class of reset control systems consisting of a minimum-phase relative degree-one linear SISO plant controlled by a novel first-order reset element (FORE). These results rely on necessary and sufficient conditions for exponential and L₂ finite gain stability of a class of planar reset systems consisting of a scalar linear plant controlled by the novel FORE. We show that the L₂ gain of the planar reset system decreases to zero as the pole and/or the gain of the FORE are increased to infinity. A number of stability results, including Lyapunov conditions for L_p and exponential stability, for a larger class of reset and hybrid systems are presented and used to prove our main results.