Convergence of adaptive control schemes using least-squares estimates
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abstract
An examination is made of the stability, convergence, asymptotic optimality, and self-tuning properties of stochastic adaptive control schemes based on least-squares estimates of the unknown parameters, when the additive noise is i.i.d. and Gaussian, and the true system is of minimum phase. The author exploits the normal equations of the least-squares method to establish that all stable control law designs used in a certainty-equivalent (i.e., indirect) procedure generally yield a stable adaptive control system. Four results that characterize the limiting behavior precisely are obtained. 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 one of the polynomials are consistently estimated for systems with delay greater than one. These general results are specialized to establish the convergence, asymptotic optimality, and self-tuning properties of a variety of proposed adaptive control schemes.
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Proceedings of the 28th IEEE Conference on Decision and Control
