Finite-horizon covariance control for discrete-time stochastic linear systems subject to input constraints

This work deals with a finite-horizon covariance control problem for discrete-time, stochastic linear systems with complete state information subject to input constraints. First, we present the main steps for the transcription of the covariance control problem, which is originally formulated as a stochastic optimal control problem, into a deterministic nonlinear program (NLP) with a convex performance index and with both convex and non-convex constraints. In particular, the convex constraints in this nonlinear program are induced by the input constraints of the stochastic optimal control problem, whereas the non-convex constraints are induced by the requirement that the terminal state covariance be equal to a prescribed positive definite matrix. Subsequently, we associate this nonlinear program, via a simple convex relaxation technique, with a (convex) semi-definite program, which can be solved numerically by means of modern computational tools of convex optimization. Although, in general, the endpoints of a representative sample of closed-loop trajectories generated by the control policy that corresponds to the solution of the relaxed convex program are not expected to follow exactly the goal terminal Gaussian distribution, they are more likely to be concentrated near the mean of this distribution than if they were drawn from the latter, which is a desirable feature in practice. Numerical simulations that illustrate the key ideas of this work are also presented.