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Learning Economic Model Predictive Control via Clustering and Kernel-Based Lipschitz Regression

Weiliang Xiong, Defeng He, Haiping Du

TL;DR

This paper tackles learning-based EMPC for uncertain nonlinear systems with input/state constraints and unknown dynamics. It introduces CKLR, a clustering-enhanced kernel Lipschitz regression method, to learn a Lipschitz predictor that bounds prediction errors and supports fast online predictions. The LEMPC framework then integrates a constraint-tightening scheme and a Lyapunov-inspired contraction to guarantee recursive feasibility and input-to-state stability, even under learning errors, with probabilistic bounds. The approach is demonstrated on a numerical example and a CSTR model, showing improved prediction accuracy, scalable handling of big data, and real-time feasibility compared to several nonparametric learners. The work offers a practical, theoretically grounded pathway for deploying learning-enabled MPC in chemical processes and other constrained, uncertain nonlinear systems.

Abstract

This paper presents a novel learning economic model predictive control scheme for uncertain nonlinear systems subject to input and state constraints and unknown dynamics. We design a fast and accurate Lipschitz regression method using input and output data that combines clustering and kernel regression to learn the unknown dynamics. In each cluster, the parallel convex optimization problems are solved to estimate the kernel weights and reduce the Lipschitz constant of the predictor, hence limiting the error propagation in the prediction horizon. We derive the two different bounds of learning errors in deterministic and probabilistic forms and customize a new robust constraint-tightening strategy for the discontinuous predictor. Then, the learning economic model predictive control algorithm is formulated by introducing a stabilized optimization problem to construct a Lyapunov function. Sufficient conditions are derived to ensure the recursive feasibility and input-to-state stability of the closed-loop system. The effectiveness of the proposed algorithm is verified by simulations of a numerical example and a continuously stirred tank reactor.

Learning Economic Model Predictive Control via Clustering and Kernel-Based Lipschitz Regression

TL;DR

This paper tackles learning-based EMPC for uncertain nonlinear systems with input/state constraints and unknown dynamics. It introduces CKLR, a clustering-enhanced kernel Lipschitz regression method, to learn a Lipschitz predictor that bounds prediction errors and supports fast online predictions. The LEMPC framework then integrates a constraint-tightening scheme and a Lyapunov-inspired contraction to guarantee recursive feasibility and input-to-state stability, even under learning errors, with probabilistic bounds. The approach is demonstrated on a numerical example and a CSTR model, showing improved prediction accuracy, scalable handling of big data, and real-time feasibility compared to several nonparametric learners. The work offers a practical, theoretically grounded pathway for deploying learning-enabled MPC in chemical processes and other constrained, uncertain nonlinear systems.

Abstract

This paper presents a novel learning economic model predictive control scheme for uncertain nonlinear systems subject to input and state constraints and unknown dynamics. We design a fast and accurate Lipschitz regression method using input and output data that combines clustering and kernel regression to learn the unknown dynamics. In each cluster, the parallel convex optimization problems are solved to estimate the kernel weights and reduce the Lipschitz constant of the predictor, hence limiting the error propagation in the prediction horizon. We derive the two different bounds of learning errors in deterministic and probabilistic forms and customize a new robust constraint-tightening strategy for the discontinuous predictor. Then, the learning economic model predictive control algorithm is formulated by introducing a stabilized optimization problem to construct a Lyapunov function. Sufficient conditions are derived to ensure the recursive feasibility and input-to-state stability of the closed-loop system. The effectiveness of the proposed algorithm is verified by simulations of a numerical example and a continuously stirred tank reactor.
Paper Structure (15 sections, 80 equations, 8 figures, 3 tables)

This paper contains 15 sections, 80 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem description and preliminary
  3. CKLR-based model learning
  4. K-means clustering
  5. Kernel regression
  6. Computational complexity analysis
  7. Properties of predictors
  8. LEMPC ALGORITHM
  9. Constraint-tightening method
  10. LEMPC algorithm design
  11. Stability analysis
  12. Examples
  13. A numerical example
  14. A CSTR example
  15. Conslusion

Figures (8)

  • Figure 1: A schematic of the CKLR-based LEMPC scheme
  • Figure 2: K-means clustering, where red points are centers, blue points are data, and the orange lines are interfaces
  • Figure 3: Visualization of four distributions. (a) bounded continuous distribution; (b) discrete distribution with nonzero probability in maximum; (c) mixed distribution with zero probability in maximum; (d) unbounded distribution. The (a) and (b) satisfy Assumption 2 whilst (c) and (d) do not.
  • Figure 4: (a) The actual model represented by the gray grid and 200 random data points represented by blue crosses; (b) The CKLR predictor ($K=2$) and interface of clusters represented by a blue line; (c) SLR predictor; (d) KI (PKI) predictor; (e) SPKI predictor; (f) RBFNN predictor; (g) Boxplot of prediction time on 100 test points of five algorithms; (h) Boxplot of absolute prediction error on 100 test points of five algorithms.
  • Figure 5: Comparison of prediction time and error
  • ...and 3 more figures