Covariance Steering of Discrete-Time Markov Jump Linear Systems with Multiplicative Noise

TL;DR

This paper addresses the problem of steering the mean and covariance of discrete-time Markov jump linear systems with multiplicative noise, proposing a new control formulation and SDP-based solutions.

Contribution

It introduces a lifted-state formulation for covariance steering with multiplicative noise, enabling SDP reformulation and chance-constrained control with iterative refinement.

Findings

Lifted-state formulation embeds mean and covariance into a second-moment description.

SDP reformulation provides a computational approach for covariance steering.

Chance constraints are handled via convex surrogates with iterative updates.

Abstract

We study a finite-horizon covariance steering problem for discrete-time Markov jump linear systems (MJLS) with both state- and control-dependent multiplicative noise. The objective is to minimize a quadratic running cost while steering the system from given mode-conditioned initial means and covariances to a prescribed terminal mean and covariance. We first show that, without loss of generality, feasible controls may be represented by mode-dependent linear feedback together with feedforward and independent random components, and we highlight that, in contrast to the case without multiplicative noise, a purely affine state-feedback law does not in general suffice. To this end, we introduce a lifted-state formulation that embeds the mean and covariance information into a unified second-moment description, and we prove that the resulting lifted problem is equivalent to the original covariance steering problem formulation. This leads to a lossless relaxation in moment variables and an SDP reformulation for the unconstrained case. We further study chance-constrained covariance steering with ball and half-space constraints on the state and control, derive tractable sufficient convex surrogates, and establish an iterative reference-update scheme to reduce conservatism. Numerical experiments on a finance application illustrate our results.