Discrete-time stochastic H ₂ optimal control

Abstract

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.