Discrete-time stochastic H 2 optimal control

In this chapter the problem of H2 control of a discrete-time linear system subject to Markovian jumping and independent random perturbations is considered. Several kinds of H2-type performance criteria (often called H2 norms) are introduced and characterized via solutions of some suitable linear equations on the spaces of symmetric matrices. The purpose of such performance criteria is to provide a measure of the effect of additive white noise perturbation over an output of the controlled system. Different aspects specific to the discrete-time framework are emphasized. Firstly, the problem of optimization of H2 norms is solved under the assumption that a full state vector is available for measurements. One shows that among all stabilizing controllers of higher dimension, the best performance is achieved by a zero-order controller. The corresponding feedback gain of the optimal controller is constructed based on the stabilizing solution of a system of discrete-time generalized Riccati equations. The case of discrete-time linear stochastic systems with coefficients depending upon the states both at time t and at time t-1 of theMarkov chain is also considered. Secondly, the H2 optimization problem is solved under the assumption that only an output is available for measurements. The state space realization of the H2 optimal controller coincides with the stochastic version of the well-known Kalman-Bucy filter. In the construction of the optimal controller the stabilizing solutions of two systems of discrete-time coupled Riccati equations are involved. Because in the case of the systems affected by multiplicative white noise the optimal controller is hard to implement, a procedure for designing a suboptimal controller with the state space realization in a state estimator form is provided. Finally a problem of H2 filtering in the case of stochastic systems affected by multiplicative and additive white noise and Markovian switching is solved.