Convergence of Least-Squares Parameter Estimate Based Adaptive Control Schemes

Abstract

We consider adaptive controllers for linear stochastic systems which use least-squares parameter estimates. We develop a convergence theory which has a "broad-spectrum" application to various such schemes, including the self-tuning regulator with and without fixing bo' self-tuning pole-zero placement schemes with noise shaping, etc.By a process of "Bayesian embedding", we show that the parameter estimates always converge, a "universal" convergence result, whenever the noise is white and Gaussian, and except for a set of true parameters of Lebesgue measure zero. Next we analyze the normal cquations of least squares to exhib it that all certainty-equivalent based schemes which employ an underlying stabilizing "design" methodology, are stable. Then we obtain various results which indicate the precise role played by "delay", persistence of excitation, etc., in determining the limiting values of the parameters estimates.These results allows us to prove specific self-tuning results, and stability properties, of a variety of adaptive control schemes using least-squares estimates of the parameters.