Finite time stability of nonlinear strongly dissipative feedback systems

Abstract

In this paper, we introduce the notion of strongly dissipative dynamical systems. In particular, we construct a stronger version of the dissipation inequality that implies system dissipativity and generalises the notion of strict dissipativity but unlike strict dissipativity, which for a closed dynamical system implies asymptotic stability, the closed dynamical system possesses the property that system trajectories converge to a Lyapunov stable equilibrium state in finite time. The results are then used to derive Kalman-Yakubovich-Popov conditions for characterising necessary and sufficient conditions for strong dissipativity in terms of the system functions of the dynamical system using continuously differentiable storage functions and quadratic supply rates. Furthermore, using strong dissipativity concepts we present serval stability results for nonlinear feedback systems that guarantee finite time stability. For specific supply rates, these results provide generalisations of the feedback passivity and nonexpansivity theorems that additionally guarantee finite time stability.