Explicit Lyapunov functions for stability and performance characterizations of FOREs connected to an integrator

In this paper we provide explicit Lyapunov functions that prove that a first order reset element (FORE) in negative feedback interconnection with an integrator is exponentially stable for any, positive or negative, value of the pole of the FORE. The Lyapunov functions also allow to establish finite gain Lscr₂ stability from a disturbance input acting at the input of the plant to the plant output. Lscr₂ stability is established by giving a bound on the corresponding Lscr₂ gains. The framework used for the characterization of the system dynamics and for the stability and performance analysis corresponds to the ideas first proposed in (Nesic et al., IFAC 2005) and (Zaccarian et al., ACC 2005)