Stable Certainty Equivalence Adaptive Control using Normalized Parameter Adjustment Laws

Parameter estimation problem in dynamical systems can be addressed using the static adaptive observers to generate parameter estimates on line. However, if such estimates are used in a Certainty Equivalence Adaptive Control (CEAC) law, the stability problem arises. In this paper it is shown that the CEAC law, in which parameter estimates are generated using static observers and several adaptive laws with normalization, results in a stable system in which the tracking control objective is achieved. The adaptive laws include gradient algorithms with normalization and projection, and least squares with covariance resetting and exponential forgetting. This is followed by an analysis of the case when dynamic observers and adaptive laws with normalization are used to generate parameter estimates. It is shown that with such adaptive laws overall system stability can be guaranteed provided that the observer is sufficiently fast and that its gain is chosen to be larger than a calculable worst-case bound.