Stochastic adaptive controllers with and without a positivity condition

The study of robust adaptive controllers has led us to introduce a new modified least squares algorithm. It incorporates a normalization signal, a covariance matrix regularization, and a parameter projection. In this paper we investigate properties of minimum variance controllers using this parameter adaptation. First, we show that for any mean square bounded driving noise, the input output signals are mean square bounded. Secondly, if the noise is a moving average and its noise model parameters satisfy a very strict passivity condition, then the controller is asymptotically optimal. The price paid to remove the passivity condition, in the first part, is the a priori knowledge of a compact set containing a stabilizing regulator and the sign and a lower bound on its leading coefficient.