Covariance Steering of Discrete-Time Markov Jump Linear Systems with Multiplicative Noise
This work addresses control design for stochastic systems with jumps and noise, which is incremental as it extends existing covariance steering methods to handle multiplicative noise and chance constraints.
The paper tackles the covariance steering problem for discrete-time Markov jump linear systems with multiplicative noise, aiming to minimize a quadratic cost while steering from initial to terminal conditions, and introduces a lifted-state formulation that leads to a lossless relaxation and SDP reformulation, with numerical experiments on a finance application illustrating the results.
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.