Convergence of adaptive control schemes using least-squares parameter estimates

The stability, convergence, asymptotic optimality, and self-tuning properties of stochastic adaptive control schemes based on least-squares estimates of the unknown parameters are examined. It is assumed that the additive noise is i.i.d. and Gaussian, and that the true system is of minimum phase. The Bayesian embedding technique is used to show that the recursive least-squares parameter estimates converge in general. The normal equations of least squares are used 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 are given to characterize the limiting behavior precisely. 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