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Time-Certified and Efficient NMPC via Koopman Operator

Liang Wu, Yunhong Che, Bo Yang, Kangyu Lin, Ján Drgoňa

TL;DR

This work addresses certifying worst-case execution time for nonlinear MPC under state–input constraints by marrying a data-driven Koopman predictor with a dynamics-relaxed BoxQP formulation. The approach converts NMPC into a structured BoxQP via a Koopman lift and a penalty that enforces model dynamics within the objective, then solves it with a time-certified, feasible path-following IPM that exploits problem structure to drastically reduce the Newton system dimension. Key contributions include a) learning a high-dimensional Koopman model, b) a dynamics-relaxed construction that yields a structured BoxQP, c) a worst-case iteration bound with a cost-free initialization, and d) efficient linear-system computations leveraging Schur complements, enabling real-time, certificate-backed NMPC for large-scale PDE control. Practically, the method achieves substantial speedups over generic QP solvers while providing explicit execution-time certificates, making it well-suited for PDE-like control applications where tight timing guarantees are essential.

Abstract

Certifying and accelerating execution times of nonlinear model predictive control (NMPC) implementations are two core requirements. Execution-time certificate guarantees that the NMPC controller returns a solution before the next sampling time, and achieving faster worst-case and average execution times further enables its use in a wider set of applications. However, NMPC produces a nonlinear program (NLP) for which it is challenging to derive its execution time certificates. Our previous works, \citep{wu2025direct,wu2025time} provide data-independent execution time certificates (certified number of iterations) for box-constrained quadratic programs (BoxQP). To apply the time-certified BoxQP algorithm \citep{wu2025time} for state-input constrained NMPC, this paper i) learns a linear model via Koopman operator; ii) proposes a dynamic-relaxation construction approach yields a structured BoxQP rather than a general QP; iii) exploits the structure of BoxQP, where the dimension of the linear system solved in each iteration is reduced from $5N(n_u+n_x)$ to $Nn_u$ (where $n_u, n_x, N$ denote the number of inputs, states, and length of prediction horizon), yielding substantial speedups (when $n_x \gg n_u$, as in PDE control).

Time-Certified and Efficient NMPC via Koopman Operator

TL;DR

This work addresses certifying worst-case execution time for nonlinear MPC under state–input constraints by marrying a data-driven Koopman predictor with a dynamics-relaxed BoxQP formulation. The approach converts NMPC into a structured BoxQP via a Koopman lift and a penalty that enforces model dynamics within the objective, then solves it with a time-certified, feasible path-following IPM that exploits problem structure to drastically reduce the Newton system dimension. Key contributions include a) learning a high-dimensional Koopman model, b) a dynamics-relaxed construction that yields a structured BoxQP, c) a worst-case iteration bound with a cost-free initialization, and d) efficient linear-system computations leveraging Schur complements, enabling real-time, certificate-backed NMPC for large-scale PDE control. Practically, the method achieves substantial speedups over generic QP solvers while providing explicit execution-time certificates, making it well-suited for PDE-like control applications where tight timing guarantees are essential.

Abstract

Certifying and accelerating execution times of nonlinear model predictive control (NMPC) implementations are two core requirements. Execution-time certificate guarantees that the NMPC controller returns a solution before the next sampling time, and achieving faster worst-case and average execution times further enables its use in a wider set of applications. However, NMPC produces a nonlinear program (NLP) for which it is challenging to derive its execution time certificates. Our previous works, \citep{wu2025direct,wu2025time} provide data-independent execution time certificates (certified number of iterations) for box-constrained quadratic programs (BoxQP). To apply the time-certified BoxQP algorithm \citep{wu2025time} for state-input constrained NMPC, this paper i) learns a linear model via Koopman operator; ii) proposes a dynamic-relaxation construction approach yields a structured BoxQP rather than a general QP; iii) exploits the structure of BoxQP, where the dimension of the linear system solved in each iteration is reduced from to (where denote the number of inputs, states, and length of prediction horizon), yielding substantial speedups (when , as in PDE control).
Paper Structure (11 sections, 1 theorem, 31 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 11 sections, 1 theorem, 31 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Problem formulations
  4. Preliminary: Koopman transforms NMPC into general QP
  5. Methodology (1): Koopman-BoxQP via Dynamics-relaxed Construction
  6. Methodology (2): Time-certified and Efficient IPM Algorithm for BoxQP
  7. Cost-free initialization for Feasible IPMs
  8. Algorithm and worst-case iteration complexity
  9. Efficient computation of Newton systems
  10. Numerical Examples
  11. Conclusion

Key Result

Lemma 1

(see wu2025time) Let $\{(z^k,v^k,s^k) \}$ be generated by Algorithm alg_PC_IPM. Then Furthermore, Algorithm alg_PC_IPM requires at most

Figures (1)

  • Figure 1: Closed-loop simulation of the nonlinear KdV system with the dynamics-relaxed Koopman-BoxQP controller tracking a time-varying spatial profile reference. Left: time evolution of the spatial profile $y(t,x)$ and the state constraints $[-1,1]$. Middle: spatial mean of the $y(t,x)$ and the state constraints $[-1,1]$. Right: the four control inputs and the control input constraints $[-1,1]$.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Remark 4