A CONCEPTUAL-FRAMEWORK FOR PARAMETER ADAPTIVE-CONTROL

A conceptual framework is described in which a parameter adaptive control system is taken to be the feedback interconnection of a process SIGMA(P) and a parameterized controller SIGMA(C)(k) whose parameter vector k is adjusted by a tuner SIGMA(T). 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 SIGMA(C)'s outputs called a tuning error e(T), which is the main signal driving SIGMA(T). For the dosed-loop parameterized system SIGMA(k) consisting of SIGMA(P) and SIGMA(C)(k), definitions and characterizations are given of the concepts of weak tunability and tunability of SIGMA(k) on a subset epsilon of the parameter space P in which k takes values. It proves to be necessary to know a subset epsilon on which SIGMA(k) is weakly tunable in order to be able to construct a tuner SIGMA(T) which adaptively stabilizes SIGMA(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.