Self-Tuning Property of a Class of Adaptive Controllers based on the Generalized Minimum Variance Control Strategy

Convergence analysis of a class of adaptive controllers based on the generalized minimum variance control strategy (GMV) is considered. Using the ordinary differential equation (ODE) method of Ljung, under a certain boundedness assumption, and positive realness of the noise transfer function, it is shown that this class of adaptive controllers have self-tuning property with probability one (w.p.1). That is, the controller parameters converge to values which yield the same performance as if the true system parameters were used. The convergence results are also shown to hold for a closely related prediction algorithm as a special case. Local convergence analysis reveals that if the positive realness condition is not satisfied, counter-examples to convergence of the algorithms can be constructed. The numerical solution of the associated ODE, are shown to give further insight into the convergence properties of the algorithm.