Classical Discrete-Time Adaptive Control Revisited: Exponential Stabilization

Abstract

Classical discrete-time adaptive controllers provide asymptotic stabilization. While the original adaptive controllers did not handle noise or unmodelled dynamics well, redesigned versions were proven to have some tolerance; however, exponential stabilization and a bounded gain on the noise was rarely proven. Here we consider a classical pole placement adaptive controller using the original projection algorithm rather than the commonly modifed version; we impose the assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are projected onto that set at every step. We demonstrate that the closed-loop system exhibits very desireable closed-loop behaviour: there are linear-like convolution bounds on the closed loop behaviour, which implies exponential stability and a bounded noise gain, as well an easily proven tolerance to unmodelled dynamics and plant parameter variation. We emphasize that there is no persistent excitation requirement of any sort.

The first version appeared in revised form in the Proceedings of the 1st IEEE Conference on Control Technology and Applications. The second version is a significant extension, and includes several new theorems relating to time-variations and robustness