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Near-optimal performance of stochastic economic MPC

Jonas Schießl, Ruchuan Ou, Timm Faulwasser, Michael H. Baumann, Lars Grüne

TL;DR

Addresses near-optimal performance in stochastic economic MPC by deriving performance guarantees in expectation. Builds a framework around a distributional turnpike concept and stochastic DP to prove finite-horizon near-optimality and infinite-horizon overtaking/average optimality. Introduces two MPC formulations—one theoretical on random-variable trajectories and a practically implementable path-based variant—and shows their expected performance coincides. Numerical example demonstrates turnpike behavior and confirms the derived bounds, underscoring practical relevance for stochastic MPC.

Abstract

This paper presents first results for near optimality in expectation of the closed-loop solutions for stochastic economic MPC. The approach relies on a recently developed turnpike property for stochastic optimal control problems at an optimal stationary process, combined with techniques for analyzing time-varying economic MPC schemes. We obtain near optimality in finite time as well as overtaking and average near optimality on infinite time horizons.

Near-optimal performance of stochastic economic MPC

TL;DR

Addresses near-optimal performance in stochastic economic MPC by deriving performance guarantees in expectation. Builds a framework around a distributional turnpike concept and stochastic DP to prove finite-horizon near-optimality and infinite-horizon overtaking/average optimality. Introduces two MPC formulations—one theoretical on random-variable trajectories and a practically implementable path-based variant—and shows their expected performance coincides. Numerical example demonstrates turnpike behavior and confirms the derived bounds, underscoring practical relevance for stochastic MPC.

Abstract

This paper presents first results for near optimality in expectation of the closed-loop solutions for stochastic economic MPC. The approach relies on a recently developed turnpike property for stochastic optimal control problems at an optimal stationary process, combined with techniques for analyzing time-varying economic MPC schemes. We obtain near optimality in finite time as well as overtaking and average near optimality on infinite time horizons.
Paper Structure (11 sections, 12 theorems, 44 equations, 2 figures, 2 algorithms)

This paper contains 11 sections, 12 theorems, 44 equations, 2 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. SETTING AND PRELIMINARIES
  3. Problem formulation
  4. Stochastic distributional turnpike
  5. Stochastic dynamic programming
  6. STOCHASTIC MPC-ALGORITHMS
  7. PERFORMANCE ESTIMATES
  8. Non-average performance
  9. Average performance
  10. NUMERICAL SIMULATIONS
  11. CONCLUSION

Key Result

Theorem 3

Consider the optimal control problem eq:stochOCP with $N \in \mathbb{N}$ and $X_0 \in \mathbb{X}$. Let $\mathbf{U}^*_N \in \mathcal{U}^N(X_0)$ be an optimal control sequence on horizon $N$ and define $V_0 \equiv 0$. Then for all $N \in \mathbb{N}$ and all $M=1,\ldots,N$ it holds that

Figures (2)

  • Figure 1: Evolution of all realization paths for the optimal trajectories (left) and controls (right) for $X_0=3$ and $N = 3,5,\ldots,15$.
  • Figure 2: Cumulative and averaged costs of MPC Algorithm \ref{['alg:stochMPC2']} for $N=3$ (red), $N=4$ (green), and $N=5$ (blue).

Theorems & Definitions (27)

  • Definition 1: Stationary pair
  • Definition 2: Stochastic distributional turnpike
  • Theorem 3: Finite horizon DPP
  • Theorem 4: Infinite horizon DPP
  • Theorem 5: Bertsekas1996b
  • Theorem 6
  • proof
  • Corollary 7
  • Remark 8
  • Definition 9: Shifted cost
  • ...and 17 more