Supervisory Adaptive Control Revisited: Linear-like Convolution Bounds

Abstract

Classical feedback control for LTI systems enjoys many desirable properties including exponential stability, a bounded noise-gain, and tolerance to a degree of unmodeled dynamics. However, an accurate model for the system must be known. The field of adaptive control aims to allow one to control a system with a great deal of parametric uncertainty, but most such controllers do not exhibit those nice properties of an LTI system, and may not tolerate a time-varying plant. In this thesis, it is shown that an adaptive controller constructed via the machinery of Supervisory Control yields a closed-loop system which is exponentially stable, and where the effects of the exogenous inputs are bounded above by a linear convolution - this is a new result in the Supervisory Control literature. The consequences of this are that the system enjoys linear-like properties: it has a bounded noise-gain, is robust to a degree of unmodeled dynamics, and is tolerant of a degree of time-varying plant parameters.

This is demonstrated in two cases: the first is the typical application of Supervisory Control - an integral control law is used to achieve step tracking in the presence of a constant disturbance. It is shown that the tracking error exponentially goes to zero when the disturbance is constant, and is bounded above by a linear convolution when it is not. The second case is a new application of Supervisory Control: it is shown that for a minimum phase plant, the d-step-ahead control law may be used to achieve asymptotic tracking of an arbitrary bounded reference signal. In addition to the convolution bound, a crisp bound is found on the 1-norm of the tracking error when a disturbance is absent.