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Offset-free Nonlinear MPC with Koopman-based Surrogate Models

Irene Schimperna, Lea Bold, Karl Worthmann

TL;DR

This paper addresses offset-free tracking for data-driven MPC models learned via Extended Dynamic Mode Decomposition (EDMD). It proposes a nonlinear offset-free MPC using a disturbance observer for bilinear EDMD surrogates, with a reference calculator to handle unknown equilibria, and two planning modes: known-equilibrium (ke) and unknown-equilibrium (ue). The method augments the EDMD model with a disturbance state $\\hat{d}$ and solves a horizon-$N$ optimal control problem, achieving asymptotically zero steady-state error under asymptotically-constant modeling error. The approach is demonstrated on a van-der-Pol oscillator and a four-tanks process, showing faster convergence and reduced tracking error compared with standard EDMD-based MPC and even matching SafEDMD performance in some cases.

Abstract

In this paper, we design offset-free nonlinear Model Predictive Control (MPC) for surrogate models based on Extended Dynamic Mode Decomposition (EDMD). The model used for prediction in MPC is augmented with a disturbance term, that is estimated by an observer. If the full information about the equilibrium of the real system is not available, a reference calculator is introduced in the algorithm to compute the MPC state and input references. The control algorithm guarantees offset-free tracking of the controlled output under the assumption that the modeling errors are asymptotically constant. The effectiveness of the proposed approach is showcased with numerical simulations for two popular benchmark systems: the van-der-Pol oscillator and the four-tanks process.

Offset-free Nonlinear MPC with Koopman-based Surrogate Models

TL;DR

This paper addresses offset-free tracking for data-driven MPC models learned via Extended Dynamic Mode Decomposition (EDMD). It proposes a nonlinear offset-free MPC using a disturbance observer for bilinear EDMD surrogates, with a reference calculator to handle unknown equilibria, and two planning modes: known-equilibrium (ke) and unknown-equilibrium (ue). The method augments the EDMD model with a disturbance state and solves a horizon- optimal control problem, achieving asymptotically zero steady-state error under asymptotically-constant modeling error. The approach is demonstrated on a van-der-Pol oscillator and a four-tanks process, showing faster convergence and reduced tracking error compared with standard EDMD-based MPC and even matching SafEDMD performance in some cases.

Abstract

In this paper, we design offset-free nonlinear Model Predictive Control (MPC) for surrogate models based on Extended Dynamic Mode Decomposition (EDMD). The model used for prediction in MPC is augmented with a disturbance term, that is estimated by an observer. If the full information about the equilibrium of the real system is not available, a reference calculator is introduced in the algorithm to compute the MPC state and input references. The control algorithm guarantees offset-free tracking of the controlled output under the assumption that the modeling errors are asymptotically constant. The effectiveness of the proposed approach is showcased with numerical simulations for two popular benchmark systems: the van-der-Pol oscillator and the four-tanks process.

Paper Structure

This paper contains 12 sections, 1 theorem, 23 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Extended Dynamic Mode Decomposition
  4. EDMD for autonomous systems
  5. EDMD for control-affine systems
  6. Offset-free MPC
  7. Case of known system equilibrium (ke)
  8. Case of unknown system equilibrium
  9. Simulation results
  10. Van-der-Pol oscillator
  11. Four tanks system
  12. Conclusion

Key Result

Theorem 1

Assume that $w$ is bounded and asymptotically constant, i.e. there exists $\bar{w}$ such that Then $\lim_{k \to \infty} x(k) - \tilde{x}(k) = 0$ and $\lim_{k \to \infty} \hat{d}(k) = \bar{w}$, where $\tilde{x}$ is defined in eq:observer. Moreover, if the closed-loop system reaches an equilibrium with input $u_\infty$ and state $x_\infty$, then $r(x_\infty) = \bar{y}_c$.

Figures (3)

  • Figure 1: Norm $\|x(k)\|$ of the closed-loop solution of the van-der-Pol oscillator \ref{['eq:vdP:discretized']} for bilinear EDMD-based and EDMDc-based MPC and offset-free MPC, and for SafEDMD-based MPC.
  • Figure 2: Output error $\|r(x) - \bar{y}_\mathrm{c} \|$ of System \ref{['eq:fourTanks']} for EDMD-based MPC and offset-free MPC comparing EDMDc and the bilinear approach for a known equilibrium.
  • Figure 3: $\|r(x) - \bar{y}_\mathrm{c} \|$ of System \ref{['eq:fourTanks']} for EDMD-based MPC and offset-free MPC, in the cases with unknown equilibrium (ue) and known equilibrium (ke).

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4