Near-Optimal Performance of Stochastic Model Predictive Control

TL;DR

This work analyzes stochastic model predictive control (SMPC) for discrete-time linear systems with quadratic costs under additive and multiplicative uncertainties, under the finite-support assumption. By reformulating the problem as a multistage stochastic program and employing a horizon-truncated SMPC strategy, the authors prove a near-optimal performance guarantee: the dynamic regret decays exponentially with the prediction horizon length $W$, provided stabilizability and detectability hold. A key technical contribution is a perturbation analysis of the KKT system on scenario trees, using graph-structured optimization results and probability-scaled norms to establish exponential decay of both open- and closed-loop policies. The results rigorously justify the practical effectiveness of SMPC: with a moderate horizon, SMPC can achieve near-optimal performance while significantly reducing computational complexity. The work additionally relaxes stringent determinism assumptions, paving the way for extensions to nonlinear, constrained, or distributionally robust settings.

Abstract

This article presents a dynamic regret analysis for stochastic model predictive control (SMPC) in linear systems with quadratic performance index and additive and multiplicative uncertainties. Under a finite support assumption, the problem can be cast as a finite-dimensional quadratic program, but the problem becomes quickly intractable as the problem size grows exponentially in the horizon length. SMPC aims to compute approximate solutions by solving a sequence of problems with truncated prediction horizons and committing the solution in a receding-horizon fashion. While this approach is widely used in practice, its performance relative to the optimal solution is not well understood. This article reports for the first time a rigorous near-optimal performance guarantee of SMPC: Under stabilizability and detectability conditions, the dynamic regret of SMPC is exponentially small in the prediction horizon length, allowing SMPC to achieve near-optimal performance at a substantially reduced computational expense.

Figures (3)

• Figure 1: Illustration of settings
• Figure 2: Schematic of stochastic model predictive control
• Figure 3: Structure of a typical scenario tree.

Theorems & Definitions (52)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Definition 1
• Definition 2
• Proposition 1
• Remark 6
• Theorem 1: Perturbation Bound (Open-Loop)