Self-convergence of weighted least-squares with applications to stochastic adaptive control

A recursive least-squares algorithm with slowly decreasing weights for linear stochastic systems is found to have self-convergence property, i.e., it converges to a certain random vector almost surely irrespective of the control law design, Such algorithms enjoy almost the same nice asymptotic properties as the standard least-squares. This ''universal convergence'' result combined with a method of random regularization then easily can be applied to construct a self-convergent and uniformly controllable estimated model and thus may enable us to form a general framework for adaptive control of possibly nonminimum phase autoregressive-moving average with exogenous input (ARMAX) systems, As an application, we give a simple solution to the well-known stochastic adaptive pole-placement and linear-quadratic-Gaussian (LQG) control problems in the paper.