State covariance assignment problem

Covariance control theory provides a parameterisation of all controllers which assign a specified state covariance matrix to the closed-loop system according to different requirements on system performance. In this study, an innovative scheme for covariance system description based on the original linear stochastic system is proposed. The main idea of the proposed scheme is based on a special rearrangement of the corresponding covariance matrix Riccati equation to a new linear deterministic state space system. The variances of the states and cross-covariance between each two states of the original system would be the states of the new covariance system. By some mathematical manipulations, the covariance assignment problem is reformulated as a standard disturbance rejection problem. Since the new covariance system is linear and deterministic, all conventional and well-defined control strategies can be applied on it. Results are presented by a covariance feedback control law based on an integral control action applied to the new covariance system. The control law causes a stable closed-loop system and assigns a pre-specified state covariance matrix to the states of the closed-loop system.