Covariance Control of Discrete-Time Gaussian Linear Systems Using Affine Disturbance Feedback Control Policies

In this paper, we present a new control policy parametrization for the finite-horizon covariance steering problem for discrete-time Gaussian linear systems (DTGLS) which can reduce the latter stochastic optimal control problem to a tractable optimization problem. The covariance steering problem seeks for a feedback control policy that will steer the state covariance of a DTGLS to a desired positive definite matrix in finite time. We consider two different formulations of the covariance steering problem, one with hard terminal LMI constraints (hard-constrained covariance steering) and another one with soft terminal constraints in the form of a terminal cost which corresponds to the squared Wasserstein distance between the actual terminal state (Gaussian) distribution and the desired one (soft-constrained covariance steering). We propose a solution approach that relies on the affine disturbance feedback parametrization for both problem formulations. We show that this particular parametrization allows us to reduce the hard-constrained covariance steering problem into a semi-definite program (SDP) and the soft-constrained covariance steering problem into a difference of convex functions program(DCP). Finally, we show the advantages of our approach over other covariance steering algorithms in terms of computational complexity and computation time by means of theoretical analysis and numerical simulations.