Exact SDP Formulation for Discrete-Time Covariance Steering with Wasserstein Terminal Cost

Abstract

In this paper, we present new results on the covariance steering problem with Wasserstein distance terminal cost. We show that the state history feedback control policy parametrization, which has been used before to solve this class of problems, requires an unnecessarily large number of variables and can be replaced by a randomized state feedback policy which leads to more tractable problem formulations without any performance loss. In particular, we show that under the latter policy, the problem can be equivalently formulated as a semidefinite program (SDP) which is in sharp contrast with our previous results that could only guarantee that the stochastic optimal control problem can be reduced to a difference of convex functions program. Then, we show that the optimal policy that is found by solving the associated SDP corresponds to a deterministic state feedback policy. Finally, we present nontrivial numerical simulations which show the benefits of our proposed randomized state feedback policy derived from the SDP formulation of the problem over existing approaches in the field in terms of computational efficacy and controller performance.

Optimal Covariance Steering for Discrete-Time Linear Stochastic Systems

“…One of the reasons for the above approach is that with chance constraints present on the sample paths of the input and the state, the state-mean and state-covariance constraints are coupled, which makes it difficult to find the optimal control analytically [7], [8], [10]. When there are no chance constraints, the desired terminal covariance can be replaced with a soft constraint on the Wasserstein distance between the desired and the actual terminal Gaussian distributions, which can be solved using a randomized state feedback control in terms of a (convex) semi-definite programming (SDP) [12]. Finite-horizon covariance control has also been applied to a model-predictive-control setting [13], [14], in which at each time step k, an optimal covariance steering problem is solved in a receding-horizon fashion.…”

Discrete-time Optimal Covariance Steering via Semidefinite Programming

“…Remark. A different approach that results in the same formulation is that of a randomized feedback control policy presented in [21]. Therein, the injected randomness on the control policy can be interpreted as a slack variable converting (8b) to equality.…”

Covariance Steering of Discrete-Time Linear Systems with Mixed Multiplicative and Additive Noise

“…Although the solution of the problem instance in Example 1 does not correspond to an affine state feedback policy since M k = L k Σ −1 k L T k is not satisfied, the mean and the covariance of the state and the control processes which can be found by solving (16) can still be realized by considering randomized affine state feedback policies as in [9].…”

“…Despite the fact that deterministic affine state feedback policies are sufficient for CS problems for systems excited by additive noise [2], [3], [9], Example 1 shows that the optimal policy for exact CS problem 1 may require randomized policies for systems excited by multiplicative noise.…”

“…Unconstrained CS problem formulations with continuous-time linear systems were first addressed in [2], [6] whereas the constrained CS problems for discrete-time linear systems are considered in [1], [7]. Soft constrained versions of the CS problems, in which the terminal covariance assignment constraint is replaced by a terminal cost term which corresponds to the (squared) Wasserstein distance between the terminal state distribution and the goal (Gaussian) distribution, are studied in [8], [9]. Furthermore, CS problems for partially observable systems are studied in [10] in which control policies based on histories of "purified outputs" are utilized.…”