Adaptive Control—A Departure from the Certainty-Equivalence Paradigm

Abstract

A fundamental study of indirect adaptive control addressed towards a class of uncertain mechanical and aerospace systems is the subject of this paper. We utilize and build upon certain recent techniques within adaptive control that permit nonlinear parameterization of uncertainty, thereby amounting to a significant shift from the classical certainty equivalence methodology. Instead of the more popular direct adaptive control formulations, our focus is restricted to the indirect methods because imposing prior information on properties of physically meaningful plant parameters is convenient within this framework. For two different types of uncertainty parameterizations, namely scalar parameters and proper orthogonal matrix parameters, we present smooth, non-overparameterized (lower order), and stable adaptive control algorithms that provide improved transient performance while at the same time utilize all the available prior information on the unknown parameters. In the particular case when there exists a single scalar uncertain parameter, the most novel attribute of the proposed controller is that adaptation gets automatically frozen once the parameter estimate reaches the corresponding true value, a highly desirable feature that is clearly missing within all existing certainty equivalence adaptive control schemes. Through both analysis and simulations, we have been able to confirm that this “nice” feature amounts to having a proportional term within the parameter update law; and is in fact enabled by a crucial role played by the introduction of certain design functions that are embedded within the new controller. All technical descriptions will be further elaborated by illustrative physical examples as well as numerical simulations of the resulting closed-loop dynamics to demonstrate the potential advantages of this new approach.