When is a parameterized controller suitable for adaptive control?

In this paper we investigate when a parameterized controller, designed for a plant depending on unknown parameters, admits a realization which is independent of the parameters. It is argued that adaptation is unnecessary for this class of parameterized controllers. We prove that standard model reference controllers (state and output-feedback) for linear time invariant systems with a filter at the plant input admit a parameter independent realization. Although the addition of such a filter is of questionable interest, our result formally, and unquestionably, establishes the deleterious effect of such a modification, which has been widely publicized in the control literature under the name ℒ₁-adaptive control.