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Data-driven Koopman MPC using Mixed Stochastic-Deterministic Tubes

Zhengang Zhong, Ehecatl Antonio del Rio-Chanona, Panagiotis Petsagkourakis

TL;DR

This work addresses controlling nonlinear systems with disturbances by embedding the dynamics in a lifted linear Koopman space and applying a mixed stochastic-deterministic tube SMPC. The method combines a data-driven lifted model $s^{+}=A s+B u+d+Dw$ with a deterministic tube for modeling error and a stochastic tube learned via distributionally robust optimization under a Wasserstein ambiguity set, enabling offline tightening that guarantees online constraint satisfaction. Finite-sample error bounds and recursive feasibility are established, and the approach is demonstrated on a mass-spring example where it is less conservative than purely robust methods while maintaining probabilistic guarantees. The resulting framework is computationally efficient (solved as a quadratic program) and offers practical robustness for data-driven control of nonlinear systems.

Abstract

This paper presents a novel data-driven stochastic MPC design for discrete-time nonlinear systems with additive disturbances by leveraging the Koopman operator and a distributionally robust optimization (DRO) framework. By lifting the dynamical system into a linear space, we achieve a finite-dimensional approximation of the Koopman operator. We explicitly account for the modeling approximation and additive disturbance error by a mixed stochastic-deterministic tube for the lifted linear model. This ensures the regulation of the original nonlinear system while complying with the prespecified constraints. Stochastic and deterministic tubes are constructed using a DRO and a hyper-cube hull, respectively. We provide finite sample error bounds for both types of tubes. The effectiveness of the proposed approach is demonstrated through numerical simulations.

Data-driven Koopman MPC using Mixed Stochastic-Deterministic Tubes

TL;DR

This work addresses controlling nonlinear systems with disturbances by embedding the dynamics in a lifted linear Koopman space and applying a mixed stochastic-deterministic tube SMPC. The method combines a data-driven lifted model with a deterministic tube for modeling error and a stochastic tube learned via distributionally robust optimization under a Wasserstein ambiguity set, enabling offline tightening that guarantees online constraint satisfaction. Finite-sample error bounds and recursive feasibility are established, and the approach is demonstrated on a mass-spring example where it is less conservative than purely robust methods while maintaining probabilistic guarantees. The resulting framework is computationally efficient (solved as a quadratic program) and offers practical robustness for data-driven control of nonlinear systems.

Abstract

This paper presents a novel data-driven stochastic MPC design for discrete-time nonlinear systems with additive disturbances by leveraging the Koopman operator and a distributionally robust optimization (DRO) framework. By lifting the dynamical system into a linear space, we achieve a finite-dimensional approximation of the Koopman operator. We explicitly account for the modeling approximation and additive disturbance error by a mixed stochastic-deterministic tube for the lifted linear model. This ensures the regulation of the original nonlinear system while complying with the prespecified constraints. Stochastic and deterministic tubes are constructed using a DRO and a hyper-cube hull, respectively. We provide finite sample error bounds for both types of tubes. The effectiveness of the proposed approach is demonstrated through numerical simulations.
Paper Structure (19 sections, 7 theorems, 39 equations, 5 figures, 1 table)

This paper contains 19 sections, 7 theorems, 39 equations, 5 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. System dynamics, constraints and objective
  4. Koopman-based model
  5. Mixed stochastic-deterministic tube Koopman MPC
  6. Methodology
  7. Estimating the modeling error and additive disturbance
  8. Additive disturbance estimation
  9. Modeling error estimation
  10. Chance constraints design
  11. CVaR constraints
  12. Distributionally robust optimization and Ambiguity Sets
  13. Recursively feasible tubes for mixed uncertainty
  14. Measure Concentration for shifted samples
  15. Recursive feasibility, Stability and Convergence
  16. ...and 4 more sections

Key Result

Lemma 1

Let Assumption 1 hold, then $\|\hat{w}_i - w_{i}\| \le L_x \|x_i\|$.

Figures (5)

  • Figure 1: Minkowski sum $\mathbb{A} \oplus \mathbb{B}$
  • Figure 2: Pontryagin difference $\mathbb{A} \ominus \mathbb{B}$
  • Figure 3: Red: 500 trajectories of different realizations. Green: Expected trajectory. Blue: 90-quantile trajectory of 500 realizations.
  • Figure 4: Solid light green: Expected trajectory using our method. Dashed dark green: Expected trajectory using method in paulson2019mixed. Solid light Blue: 90-quantile trajectory of 500 realizations using our method. Dashed dark blue: 90-quantile trajectory of 500 realizations using method in paulson2019mixed.
  • Figure 5: Comparison between the method proposed in our paper and in zhang2022robust. Green indicates expected trajectory of 100 realizations with the same initial point and blue indicates the 90-quantile trajectory of 500 realizations.

Theorems & Definitions (21)

  • Remark 1
  • Definition 1: Robust positively invariant set
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Definition 2: Hoeffding’s Inequality
  • Lemma 2
  • proof
  • Definition 3: Wasserstein Metric ambrosio2005gradient
  • ...and 11 more