Active identification for discrete-time nonlinear control. I. Output-feedback systems

The problem of controlling nonlinear systems with unknown parameters has received a great deal of attention in the continuous-time case. In contrast, its discrete-time counterpart remains largely unexplored, primarily due to the difficulties associated with utilizing Lyapunov design techniques in a discrete-time framework. Existing results impose restrictive growth conditions on the nonlinearities to yield global stability. In this paper, we propose a novel approach, which removes this obstacle and yields global stability and tracking for systems that can be transformed into an output-feedback canonical form, in which the nonlinearities depend only on the measured output, but are otherwise arbitrary. The main novelties of our design are: (i) the temporal and algorithmic separation of the parameter estimation task from the control task, and (ii) the development of an active identification procedure, which uses the control input to actively drive the system state to points in the state space that allow the orthogonalized projection estimator to acquire all the necessary information about the unknown parameters. We prove that our algorithm guarantees complete (for control purposes) identification in a finite time interval, whose maximum length we compute. Thus, the traditional structure of concurrent online estimation and control is replaced by a two-phase control strategy: first use active identification, and then utilize the acquired parameter information to implement any control strategy as if the parameters were known.