Information inequalities, rate of adaptation and nonlinear identification in adaptive stochastic control

We state the analytical expression for the maximum possible adaptation rates for the class of adaptive control strategies corresponding to the indirect adaptation approach which uses parallel identification procedures. Linear stationary multidimensional objects, belonging to some classes which may include moving average control and noise terms, are considered. It turns out that for any adaptive control scheme, the corresponding state space trajectories do not differ less than some lower bounds from those corresponding to an asymptotically optimal control scheme when the complete information on the parameters of the model is available. These information bounds present the generalization of the Cramer-Rao inequality to adaptive stochastic control field. The tracking and regulations problems are considered in detail. Some strengthening is attained in "white noise" case. To obtain the best adaptation rate we have to use a recursive version of maximum likelihood estimators with nonlinear residual transformation.