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

Souvik Das, Siddhartha Ganguly, Ashwin Aravind, Debasish Chatterjee

TL;DR

The paper addresses control of discrete-time systems with multiplicative noise by formulating a data-driven distributionally robust MPC problem as a semi-infinite SDP (SI-SDP). It develops a convex relaxation-based methodology that reduces the SI-SDP to a finite, solvable problem and introduces a specialized algorithm to solve the outer max over semi-infinite noise-covariance parameters, leveraging an ambiguity set $\mathcal{P}_M$ defined by $\mathsf{E}[w_t]=0$ and $\Sigma_w \preceq \gamma \hat{\Sigma}_w$. The DRMPC formulation uses a linear feedback structure and translates constraints into LMIs via the Schur complement, yielding a tractable SDP-based framework. A numerical example demonstrates stabilizing behavior and tractability, while the authors note that explicit stability guarantees remain for future work. Overall, the work provides a data-driven, SDP-based approach to DRMPC for systems with multiplicative noise and lays groundwork for real-time or explicit MPC extensions in finance-like settings.

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.

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

TL;DR

The paper addresses control of discrete-time systems with multiplicative noise by formulating a data-driven distributionally robust MPC problem as a semi-infinite SDP (SI-SDP). It develops a convex relaxation-based methodology that reduces the SI-SDP to a finite, solvable problem and introduces a specialized algorithm to solve the outer max over semi-infinite noise-covariance parameters, leveraging an ambiguity set defined by and . The DRMPC formulation uses a linear feedback structure and translates constraints into LMIs via the Schur complement, yielding a tractable SDP-based framework. A numerical example demonstrates stabilizing behavior and tractability, while the authors note that explicit stability guarantees remain for future work. Overall, the work provides a data-driven, SDP-based approach to DRMPC for systems with multiplicative noise and lays groundwork for real-time or explicit MPC extensions in finance-like settings.

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.
Paper Structure (10 sections, 3 theorems, 36 equations, 1 figure, 2 algorithms)

This paper contains 10 sections, 3 theorems, 36 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Preliminary results
  4. Algorithm to solve $\text{SI-SDP}$
  5. $\text{SI-SDP}\text{-}$based algorithm to solve distributionally robust MPC
  6. DRMPC formulation
  7. $\text{SI-SDP}$ formulation of \ref{['eq:DRMPC']}
  8. Algorithm to solve \ref{['eq:g_func']}
  9. Numerical experiment
  10. Concluding remarks

Key Result

Proposition 3

Consider the optimization problem eq:mtns:CSIP_original along with its associated data and let Assumption assum:slater's hold. Then $\mathcal{G}(\cdot)$ admits the following properties:

Figures (1)

  • Figure 1: Phase portrait of the trajectories using our DRMPC Algorithm \ref{['alg:mtns:sabaro']}. It can be seen that the state trajectories are getting attracted towards the origin.

Theorems & Definitions (8)

  • Remark 1
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5.1
  • proof
  • Remark 6