Optimal Discrete-Time Control for Nonlinear Cascade Systems - Control Conference, 1997. Proceedings of the 1997 American

In this paper we develop an optimality-based framework for designing controllers for discrete-time nonlinear cascade systems. Specifically, using a nonlinear-nonquadratic optimal control framework we develop a family of globally stabilizing backstepping-type controllers parameterized by the cost functional that is minimized. Furthermore, it is shown that the control Lyapunov function guaranteeing closed-loop stability is a solution to the steady-state Bellman equation for the controlled system and thus guarantees both optimality and stability. 1. In t roduct ion Since most physical processes evolve naturally in continuous time, it is not surprising that the bulk of nonlinear control theory has been developed for continuoustime systems. Nevertheless, it is the overwhelming trend to implement controllers digitally. Despite this fact the development of nonlinear control theory for discrete-time systems has lagged its continuous-time counterpart. This is in part due to the fact that concepts such as zero dynamics, normal forms, and minimum phase are much more intricate for discrete-time systems. For example, in contrast to the continuous-time case, technicalities involving passivity analysis tools needed to prove global stability via smooth feedback controllers [l] as well as system relative degree requirements [2] are more involved in the discretetime case. Recent work involving differential geometric methods [3] employing concepts of zero dynamics and feedback linearization have been applied to discrete-time systems. In particular, these results parallel continuous-time results on linearization of nonlinear systems via state and output feedback. However, as in the continuous-time case, these techniques cancel out system nonlinearities and may therefore lead to inefficient designs since the resulting feedback linearizing controller may generate large control effort to cancel beneficial nonlinearities. Backstepping control for continuous-time systems has recently received a great deal of attention in the nonlinear control literature [4]. The popularity of this control methodology can be explained in a large part due to the fact that it provides a framework for designing stabilizing nonlinear controllers for a large class of nonlinear cascade systems. Even though discrete-time recursive backstepping techniques have not been developed, the closest discrete-time analog to backstepping is given in [2,5 . Specifically, in [2,5] discrete-time passivity analyguaranteeing global asymptotic stability for block cascade discrete-time systems. In this paper we develop an optimality-based control design theory for nonlinear discrete-time cascade systems. The key motivation for developing an optimal nonlinear control theory framework for discrete-time cascade syssis too 1 s are used to construct control Lyapunov functions