Convergence of adaptive control schemes using least-squares parameter estimates
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We examine the stability, convergence, asymptotic optimally, and self-tuning properties of stochastic adaptive control schemes based on leastsquares estimates of the unknown parameters, when the additive noise is i.i.d. and Gaussian, and the true system is of minimum phase. Our analysis starts with the technique of Bayesian embedding, introduced earlier by Sternby [1], which shows that the recursive leastsquares parameter estimates converge in general. Then exploiting the normal equations of leastsquares we establish that all stable control law designs used in a certainty-equivalent (i.e., indirect) procedure generally yield a stable adaptive control system. Next we obtain four results which characterize the limiting behavior precisely. The first determines the possible limits of the parameter estimates, and yields all self-tuning type results. The next shows that the square of a certain linear combination of outputs and inputs has average value zero, and yields all optimality results. The third is a similar result on exogenous inputs, and yields all results based on persistency of excitation. The final result shows that the first few coefficients of the A polynomial are consistently estimated for systems with delay greater than one. By specializing these general results we establish the convergence, asymptotic optimality, and self-tuning properties of a variety of proposed adaptive control schemes, including the original self-tuning regulator with unit delay, both with and without fixing bo, a certainty-equivalent minimum-variance self-tuning regulator for systems with general delay, self-tuning trackers, and self-tuning pole-zero placement schemes. As an illustrative example of the results obtained, a certainty equivalent self-tuning regulator is shown to yield strongly consistent parameter estimates when the delay is strictly greater than one, even without any excitation in the reference trajectory. 1990 IEEE
