Output covariance tracking as a disturbance rejection problem

Covariance control theory provides a parameterization of all controllers which assign a specified state covariance matrix to the closed loop system. Our approach deals with the problem of describing a covariance state space model for stochastic systems. We derived the closed-from of the state covariance equations based on the model of the original stochastic system. The state space model contains known disturbance terms, hence covariance assignment problem could be reformulated as the design of a compensator with disturbance rejection features. Basically the idea of covariance feedback is used to assign desirable covariance matrices to the system. Firstly we tune a PID controller for the covariance system. Then a more systematic approach based on the internal model principle is employed. The persistent disturbance is compensated by constructing a disturbance compensator. Then, the optimal tracker with an infinite-time horizon cost functional is designed for the covariance system by using classical optimal tracker theory. It is notable that the convergence of the covariance matrix of the system can be guaranteed.