Analysis of Nonlinear-Nonquadratic Discrete-Time Systems with t2 and tm Disturbances

In this paper we develop analysis results for nonlinearnonquadratic discrete-time systems with bounded exoge nous disturbances. 1. Introduction In a recent paper [l] an optimality-based disturbance rejection nonlinear control framework for nonlinear continuous-time systems with bounded exogenous disturbances was developed. In this paper, analogous analysis results are developed for the discrete-time case. Specifically, to address the optimality-based discrete-time disturbance rejection nonlinear control analysis problem we extend the nonlinear-nonquadratic controller analysis framework developed in [2]. The basic underlying ideas of the results in [2] are based on the fact that the steady-state solution to the Bellman equation is a control Lyaponov function for the nonlinear controlled system thus guaranteeing both optimality and stability. In this paper we extend the framework developed in [2] to address the problem of nonlinear-nonquadratic discrete-time systems with disturbance rejection guarantees to bounded input disturbances. Our framework guarantees that the nonlinear dynamical system input-output map is dissipative with respect to general supply rates. Specializing to quadratic supply rates involving net system energy flow and weighted input and output energy our results guarantee passive and nonexpansive (gain bounded) closed-loop input-output maps, respectively. In the special case where the system is linear our results, with appropriate quadratic supply rates, specialize to the mixed-norm H2/H, framework developed in [4] and the discrete-time analog of the continuous-time mixed Hz/positivity framework developed in [5]. The contents of the paper are as follows. In Section 2 we establish notation and mathematical preliminaries. In Section 3 we consider a nonlinear discrete-time system with bounded input disturbances and a nonlinearnonquadratic performance functional evaluated over the infinite horizon. The performance functional is then evaluated in terms of a Lyapunov function that guarantees stability and dissipativity with respect to general supply rates. This result is then specialized to linear systems