A conceptual framework for parameter adaptive control

Abstract

A conceptual framework is described in which a parameter adaptive control system is taken to be the feedback interconnection of a process ΣP and a parameterized controller ΣC(k) whose parameter vector k is adjusted by a tuner ΣT. The framework is general enough to encompass almost all parameterized controllers proposed in the literature for stabilizing linear process models. Emphasis is placed on the importance to adaptation of one of ΣC's outputs called a tuning error eT, which is the main signal driving ΣT. For the closed-loop parameterized system Σ(k) consisting of ΣP and ΣC(k), definitions and characterizations are given of the concepts of weak tunability and tunability of Σ(k) on a subset ɛ of the parameter space p in which k takes values. It proves to be necessary to know a subset ɛ on which Σ(k) is weakly tunable in order to be able to construct a tuner ΣT which adaptively stabilizes Σ(k). For a large class of linear multivariable process models, a properly designed certainty equivalence controller results in a tunable closed-loop parameterized system. The importance of this result to both the analysis and synthesis of parameter adaptive control systems is discussed. It is demonstrated by means of examples how the connection between certainty equivalence and tunability, together with the concept of tunability itself, can be used to markedly simplify the stability analysis of adaptive control systems. A new family of indirect parameterized controllers is described which have capabilities comparable to those of the well-known direct parameterized controllers which for a long time have served as basic building blocks for adaptive control systems of all types. The concept of implicit tuning is formalized and its potential importance to adaptive control is briefly discussed.