NonLyapunovUltimateBoundEstimation forPerturbed Systems

In the presence of nonvanishing perturbations, it is often the case that asymptotic stability is not possible and hence the trajectories of a system can only be guaranteed to be uniformly ultimately bounded. This paper presents a method for estimating ultimate bounds in both continuousand discrete-time systems. The method is introduced for linear systems and is then applied to nonlinear systems by treating the perturbed nonlinear system as a linear system with a perturbation bounded by a nonlinear function of the state. The methodology exploits the perturbation structure as well as the system geometry, without employing Lyapunov functions in the analysis. The suggested method is simpler than classical Lyapunov analysis and provides tighter bounds in many cases. Application of the method is illustrated by means of different examples.