Static H₂ and H_∞ output-feedback of discrete-time LTI systems with state multiplicative noise

A parameter dependent approach for designing static output-feedback controller for linear time-invariant systems with state-multiplicative noise is introduced which achieves a minimum bound on either the stochastic H2 or the H∞ performance levels. A solution is obtained also for the case where, in addition to the stochastic parameters, the system matrices reside in a given polytope. In this case, a parameter dependent Lyapunov function is described which enables the derivation of the required constant feedback gain via a solution of a set of linear matrix inequalities that correspond to the vertices of the uncertainty polytope. The stochastic parameters appear in both the dynamics and the input matrices of the state space model of the system. The problems are solved using the expected value of the standard performance indices over the stochastic parameters. The theory developed is demonstrated by a simple example.