Characterization of Forced Vibration for Difference Inclusions: A Lyapunov Approach

A Lyapunov approach to the characterization of forced vibration for continuous-time systems was recently proposed in. The objective of this paper is to derive counterpart results for discrete-time systems. The class of oscillatory input signals to be considered include sinusoidal signals, multitone signals, and periodic signals which can be described as the output of an autonomous system. The Lyapunov approach is developed for linear systems, homogeneous systems (difference inclusions), and nonlinear systems (difference inclusions), respectively. It is established that the steady-state gain can be arbitrarily closely characterized with Lyapunov functions if the output response converges exponentially to the steady state. We also evaluate other output measures such as the peak of the transient response and the convergence rate. Applying the results to linear difference inclusions, optimization problems with linear matrix inequality constraints can be formulated for the purpose of estimating the output measures. This paper´s results can be readily applied to the evaluation of frequency responses of general nonlinear and uncertain systems by restricting the inputs to sinusoidal signals. Similar to continuous-time systems, it is observed that the peak of the frequency response can be strictly larger than the gain for a linear difference inclusion.