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

Abstract

In this paper, we study the covariance steering (CS) problem for discrete-time linear systems subject to multiplicative and additive noise. Specifically, we consider two variants of the so-called CS problem. The goal of the first problem, which is called the exact CS problem, is to steer the mean and the covariance of the state process to their desired values in finite time. In the second one, which is called the "relaxed" CS problem, the covariance assignment constraint is relaxed into a positive semi-definite constraint. We show that the relaxed CS problem can be cast as an equivalent convex semi-definite program (SDP) after applying suitable variable transformations and constraint relaxations. Furthermore, we propose a two-step solution procedure for the exact CS problem based on the relaxed problem formulation which returns a feasible solution, if there exists one. Finally, results from numerical experiments are provided to show the efficacy of the proposed solution methods.

Optimal Covariance Steering for Discrete-Time Linear Stochastic Systems

“…Furthermore, if there exists a unique solution to the equations ( 6), ( 7), ( 8), then, the optimal control u * is unique. The corresponding optimal process is given by (11).…”

“…Later, the optimal control that assigns a prescribed stationary state covariance with minimum control energy was developed in [6]. More recent studies address the problem of optimally steering the state covariance of a linear stochastic system over a finite time horizon [7]- [11]. For discrete-time optimal covariance steering, the approach taken by most state-of-the-art current works involves three steps: 1) reformulate the problem as an optimization problem; 2) relax the non-convex constraints to convex constraints for a tractable convex optimization problem; 3) solve the convex optimization problem numerically to approximate the optimal control.…”

Discrete-time Optimal Covariance Steering via Semidefinite Programming

“…An exact convex relaxation was proposed to efficiently solve this problem using linear semidefinite programming (SDP). At the same time, but independently, the authors of [20] used the same relaxation for solving the optimal covariance steering problem with an inequality terminal boundary condition for a system with multiplicative noise.…”

Covariance Steering for Systems Subject to Unknown Parameters

“…The literature on multiplicative disturbances is much less developed. In particular, the authors of [11], [12] also investigated numerical solutions for the covariance steering problem with parametric uncertainties. However, the work of [11], [12] assumes that the disturbances are independently, identically distributed in time, whereas this work assumes that the disturbances are timeinvariant, which is a more realistic assumption for model uncertainty, as system parameters are typically unknown but constant.…”

“…In particular, the authors of [11], [12] also investigated numerical solutions for the covariance steering problem with parametric uncertainties. However, the work of [11], [12] assumes that the disturbances are independently, identically distributed in time, whereas this work assumes that the disturbances are timeinvariant, which is a more realistic assumption for model uncertainty, as system parameters are typically unknown but constant. Furthermore, the proposed formulation allows for the dependence between prior states and the disturbance realization, whereas an assumption of state-disturbance independence is a key assumption enabling the approach of [11], [12].…”