Data-driven distributionally robust MPC for systems with multiplicative noise: A semi-infinite semi-definite programming approach

TL;DR

This paper develops a distributionally robust model predictive control method for systems with multiplicative noise, formulating it as a semi-infinite semi-definite program and providing a numerical solution approach.

Contribution

It introduces a novel DRMPC framework for multiplicative noise systems, recasting the control problem as a SI-SDP and proposing a solution method.

Findings

Effective algorithm demonstrated through a numerical example

Reformulation as semi-infinite semi-definite program

Applicable to systems in mathematical finance

Abstract

This article introduces a novel distributionally robust model predictive control (DRMPC) algorithm for a specific class of controlled dynamical systems where the disturbance multiplies the state and control variables. These classes of systems arise in mathematical finance, where the paradigm of distributionally robust optimization (DRO) fits perfectly, and this serves as the primary motivation for this work. We recast the optimal control problem (OCP) as a semi-definite program with an infinite number of constraints, making the ensuing optimization problem a \emph{semi-infinite semi-definite program} (SI-SDP). To numerically solve the SI-SDP, we advance an approach for solving convex semi-infinite programs (SIPs) to SI-SDPs and, subsequently, solve the DRMPC problem. A numerical example is provided to show the effectiveness of the algorithm.