Some quadratic functionals and self-tuning control

A quadratic functional of the state and the control is studied using a somewhat arbitrary linear feedback control law for a linear stochastic system. If the linear stochastic system is stable with this control law using the true parameter values, then the limiting average cost exists and can be easily described. Furthermore, the asymptotic distribution of the suitably normalized cost functional is normal. Likewise with a stability condition the average costs, using the same control law based on the family of estimates of the unknown parameters, converge to the same constant as the control law based on the true parameter values, and the asymptotic distribution of the suitably normalized cost functionals is again normal. It is shown that there are feedback control policies for a stable system for which the asymptotic distribution of the average costs for the control policy using the maximum-likelihood estimates of the unknown parameters has smaller variance than the asymptotic distribution of the average costs for the control policy using the true values of the parameters, so that the former has less risk than the latter