Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-certified Input-constrained NMPC via Koopman Operator

Liang Wu, Krystian Ganko, Richard D. Braatz

TL;DR

This paper addresses the lack of worst-case solving-time certificates for nonlinear, input-constrained MPC by marrying Koopman operator lifting with a condensing step to yield a small Box-QP. The online problem is solved with a time-certified, feasible-path-following interior-point method that provides an exact iteration bound and a detailed FLOP budget. The approach is validated on a high-dimensional nonlinear PDE (KdV) control problem, where the lifted predictor and condensed QP enable a rigorous certificate that the online solve time remains within the sampling interval, while achieving accurate tracking and adherence to input limits. The work advances practical real-time NMPC deployment by delivering provable time bounds and demonstrating scalability to complex, high-dimensional dynamics.

Abstract

Determining solving-time certificates of nonlinear model predictive control (NMPC) implementations is a pressing requirement when deploying NMPC in production environments. Such a certificate guarantees that the NMPC controller returns a solution before the next sampling time. However, NMPC formulations produce nonlinear programs (NLPs) for which it is very difficult to derive their solving-time certificates. Our previous work, Wu and Braatz (2023), challenged this limitation with a proposed input-constrained MPC algorithm having exact iteration complexity but was restricted to linear MPC formulations. This work extends the algorithm to solve input-constrained NMPC problems, by using the Koopman operator and a condensing MPC technique. We illustrate the algorithm performance on a high-dimensional, nonlinear partial differential equation (PDE) control case study, in which we theoretically and numerically certify the solving time to be less than the sampling time.

Time-certified Input-constrained NMPC via Koopman Operator

TL;DR

This paper addresses the lack of worst-case solving-time certificates for nonlinear, input-constrained MPC by marrying Koopman operator lifting with a condensing step to yield a small Box-QP. The online problem is solved with a time-certified, feasible-path-following interior-point method that provides an exact iteration bound and a detailed FLOP budget. The approach is validated on a high-dimensional nonlinear PDE (KdV) control problem, where the lifted predictor and condensed QP enable a rigorous certificate that the online solve time remains within the sampling interval, while achieving accurate tracking and adherence to input limits. The work advances practical real-time NMPC deployment by delivering provable time bounds and demonstrating scalability to complex, high-dimensional dynamics.

Abstract

Determining solving-time certificates of nonlinear model predictive control (NMPC) implementations is a pressing requirement when deploying NMPC in production environments. Such a certificate guarantees that the NMPC controller returns a solution before the next sampling time. However, NMPC formulations produce nonlinear programs (NLPs) for which it is very difficult to derive their solving-time certificates. Our previous work, Wu and Braatz (2023), challenged this limitation with a proposed input-constrained MPC algorithm having exact iteration complexity but was restricted to linear MPC formulations. This work extends the algorithm to solve input-constrained NMPC problems, by using the Koopman operator and a condensing MPC technique. We illustrate the algorithm performance on a high-dimensional, nonlinear partial differential equation (PDE) control case study, in which we theoretically and numerically certify the solving time to be less than the sampling time.
Paper Structure (10 sections, 6 theorems, 39 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 6 theorems, 39 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulations
  3. Koopman and Extended Dynamic Mode Decomposition
  4. Transforming NMPC to condensed MPC
  5. Time-certified IPM algorithm
  6. Strictly feasible initial point
  7. Newton direction
  8. Iteration complexity and algorithm implementation
  9. Nonlinear PDE control case study
  10. Conclusion

Key Result

Lemma 1

Let $\xi:=\xi(\beta,\tau) < 1$. Then the full Newton step is strictly feasible, i.e., $v_{+}>0$ and $s_{+}>0$.

Figures (1)

  • Figure 1: Closed-loop simulation of the nonlinear KdV system with MPC controller -- Tracking a piecewise constant spatial profile reference. Left: time evolution of the spatial profile $y(t,x)$. Middle: spatial mean of the $y(t,x)$. Right: the four control inputs.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2: Initialization strategy
  • Lemma 1: See wu2023direct
  • Lemma 2: See wu2023direct
  • Lemma 3: See wu2023direct
  • Lemma 4: See wu2023direct
  • Lemma 5: See wu2023direct
  • Theorem 1
  • Proof 1
  • Remark 3