Revisiting Model Reference Adaptive Control: Linear-Like Closed-Loop Behavior

Abstract

In this article, we examine the model reference adaptive control problem when the commonly used projection algorithm is utilized, subject to several common assumptions on the set of admissible parameters, in particular a compactness constraint as well as knowledge of the sign of the high-frequency gain. It is proven in the literature that for this setup, the closed-loop system is bounded-input bounded-state; since the closed-loop system is not linear time-invariant, this does not imply a bounded gain. Here, we prove a much crisper and detailed bound on the closed-loop behavior consisting of three terms: a decaying exponential on the initial condition, a linear-like convolution bound on the exogenous inputs, and a constant scaled by the square root of the constant in the denominator of the estimator update law; we also provide an upper bound on the two-norm of the tracking error. We then demonstrate that the same kind of bounds hold in the presence of a degree of unmodeled dynamics and plant parameter time-variation.