Estimation of Time-Varying Parameters and Its Application to Extremum-Seeking Control

Abstract

This dissertation considers the adaptive estimation of time-varying parameters and its use in extremum-seeking control problems. The ability to estimate uncertain time-varying behaviour can have a significant impact on a control system's performance. Hence, the problem of time-varying parameter estimation has been of considerable interest over the last two decades.

The present work provides a formal scheme for time-varying parameter estimation in a class of nonlinear systems. The geometric concept of invariance is the key concept for the parameter estimation techniques developed in this thesis. The techniques use a number of high gain estimators and filters that generate an almost invariant manifold. The almost invariance property allows an implicit mapping and a parameter update law that guarantees exponentially convergence to a small region of the true values of the time-varying parameters. A generalization of the invariant manifold approach is considered to deal with the estimation of periodic parameters with unknown periodicity.

In another step, this thesis seeks to apply the proposed time-varying estimation technique to the solution of extremum-seeking control problems. In extremum-seeking control, a gradient descent algorithm is used to find the optimal value of a measured but unknown cost functions. The contribution of this aspect of the thesis is the formulation of the extremum-seeking control problem where the unknown gradient of the cost is estimated as a time-varying parameter using the proposed invariance based estimation technique. The proposed approach is extended for the solution of constrained steady-state optimization problems. We establish two methods for finding the optimal points for systems with unknown objective functions that are subject to unknown/uncertain dynamics. For systems with unknown dynamics, a nonlinear proportional-integral controller is designed to find the optimal solution. Then for a class of control affine systems with known high frequency gains, an inverse optimal control technique is used for the direct design of a gradient-based controller.