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Data-driven MPC with stability guarantees using extended dynamic mode decomposition

Lea Bold, Lars Grüne, Manuel Schaller, Karl Worthmann

TL;DR

This work introduces a data-driven model predictive control (MPC) framework based on extended dynamic mode decomposition (EDMD) within the Koopman framework. By deriving a proportional error bound that scales with the state and input norms, the authors show that cost controllability is preserved when transitioning from the true dynamics to the EDMD surrogate, enabling semi-global practical asymptotic stability of the closed-loop when using the surrogate in MPC. The main theoretical contribution is a rigorous stability guarantee (PAS) for the surrogate-based MPC under the proposed error bound, complemented by a thorough sample-based analysis. Numerical simulations on a van der Pol oscillator and a chemical reactor example illustrate the practical stabilization behavior and the impact of surrogate accuracy on convergence. Overall, the paper provides a principled pathway to stability guarantees for data-driven MPC using EDMD, with explicit bounds tied to data quality and dictionary design.

Abstract

For nonlinear (control) systems, extended dynamic mode decomposition (EDMD) is a popular method to obtain data-driven surrogate models. Its theoretical foundation is the Koopman framework, in which one propagates observable functions of the state to obtain a linear representation in an infinite-dimensional space. In this work, we prove practical asymptotic stability of a (controlled) equilibrium for EDMD-based model predictive control, in which the optimization step is conducted using the data-based surrogate model. To this end, we derive novel bounds on the estimation error that are proportional to the norm of state and control. This enables us to show that, if the underlying system is cost controllable, this stabilizablility property is preserved. We conduct numerical simulations illustrating the proven practical asymptotic stability.

Data-driven MPC with stability guarantees using extended dynamic mode decomposition

TL;DR

This work introduces a data-driven model predictive control (MPC) framework based on extended dynamic mode decomposition (EDMD) within the Koopman framework. By deriving a proportional error bound that scales with the state and input norms, the authors show that cost controllability is preserved when transitioning from the true dynamics to the EDMD surrogate, enabling semi-global practical asymptotic stability of the closed-loop when using the surrogate in MPC. The main theoretical contribution is a rigorous stability guarantee (PAS) for the surrogate-based MPC under the proposed error bound, complemented by a thorough sample-based analysis. Numerical simulations on a van der Pol oscillator and a chemical reactor example illustrate the practical stabilization behavior and the impact of surrogate accuracy on convergence. Overall, the paper provides a principled pathway to stability guarantees for data-driven MPC using EDMD, with explicit bounds tied to data quality and dictionary design.

Abstract

For nonlinear (control) systems, extended dynamic mode decomposition (EDMD) is a popular method to obtain data-driven surrogate models. Its theoretical foundation is the Koopman framework, in which one propagates observable functions of the state to obtain a linear representation in an infinite-dimensional space. In this work, we prove practical asymptotic stability of a (controlled) equilibrium for EDMD-based model predictive control, in which the optimization step is conducted using the data-based surrogate model. To this end, we derive novel bounds on the estimation error that are proportional to the norm of state and control. This enables us to show that, if the underlying system is cost controllable, this stabilizablility property is preserved. We conduct numerical simulations illustrating the proven practical asymptotic stability.
Paper Structure (8 sections, 6 theorems, 45 equations, 3 figures, 1 algorithm)

This paper contains 8 sections, 6 theorems, 45 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Koopman-based prediction and control
  3. Proportional error bound for EDMD-based MPC and problem formulation
  4. Proportional error bound for sampled-data systems
  5. Problem statement
  6. Practical asymptotic stability of surrogate-based MPC
  7. Numerical simulations
  8. Conclusions

Key Result

Proposition 4

Suppose that Assumption ass:invariance holds. For every error bound $\varepsilon>0$ and probabilistic tolerance $\delta \in (0,1)$, there is an amount of data $d_0\in \mathbb{N}$ such that with probability $1-\delta$, the error bound holds for all $d \geq d_0$ and all $u\in \mathbb U$ for the Koopman operator $\mathcal{K}^{\Delta t}_u = e^{\Delta t \mathcal{L}^u}$.

Figures (3)

  • Figure 1: Averaged error of the EDMD-based solution for different number of data points for fixed random control sequence.
  • Figure 2: Optimal value functions along the closed-loop of system \ref{['eq:vanderPol']} for nominal MPC (black) and EDMD-based MPC (gray) for horizons $N = 30$ (solid) and $N = 50$ (dashed) for $\lambda = 0.25$.
  • Figure 3: MPC closed loop $\| x_{\mu_N}(\cdot;x_0) \|$ (left) and $\| x_{\mu_N}^\varepsilon(\cdot;x_0) \|$ (right, EDMD: $d = 1000$) for system dynamics \ref{['eq:chemical']} for different horizons $N$.

Theorems & Definitions (13)

  • Definition 1
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Definition 6: Practical asymptotic stability
  • Proposition 7
  • Lemma 8
  • proof
  • Proposition 9
  • ...and 3 more