Analytical verification of undesirable properties of direct model reference adaptive control algorithms

In the past few years many model reference adaptive control algorithms have been shown to be globally asymptotically stable. However, there have been no analytical results on the performance of such systems in the transient adaptive phase. Simulations have shown that there are three particular problem areas in which the performance of these algorithms may be unsatisfactory when viewed from a practical context. The problems are (a) the generation of high frequency control inputs, (b) high susceptibility to instability in the presence of unmodeled dynamics, and (c) poor performance in the presence of observation noise. This paper displays analytically how these problems arise for a number of algorithms. The analysis technique employs linearization of the nonlinear time varying dynamic equations that describe the closed-loop system; this analysis technique is referred to as "final approach analysis" because the linearization is valid when the system and reference model outputs are close to each other, a fact that occurs during the final phases of adaptation. By studying simple first order systems one can analytically examine different adaptive algorithms, and pinpoint their shortcomings. However, the analytical studies are constructive because they indicate now to modify the algorithms so as to improve their practical utility.