H_∞ control and estimation of discrete-time linear systems with stochastic parameter uncertainties

Linear discrete-time systems with stochastic uncertainties in their state-space matrices are considered. The problems of finite-horizon state-feedback control and filtering are solved, taking into account possible cross-correlations between the uncertain parameters. In both problems, a cost function is defined which is the expected value of the standard H_∞ performance index with respect to the uncertain parameters. In the state-feedback case, we derive a necessary and sufficient condition for the cost to be non-positive, for all possible energy bounded exogenous signals. A solution to the filtering problem is obtained by applying the adjoint system. It guarantees a prescribed estimation level of accuracy while minimizing an upper-bound on the covariance of the estimation error. The filtering problem is also analyzed in the stationary case. The results for the state-feedback and the filtering problems are then used to solve the corresponding output-feedback problem. An example is given which demonstrates the use of the theory developed.