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Efficient Nonlinear MPC by Leveraging LPV Embedding and Sequential Quadratic Programming

Dimitrios S. Karachalios, Hossam S. Abbas

TL;DR

The paper addresses the real-time solvability of nonlinear MPC by embedding nonlinear dynamics into a linear parameter-varying (LPV) form and solving the NMPC as a sequence of quadratic programs. It then develops LPVMPC-SQP, an SQP-based approach that uses LPV-informed inexact Hessians and Jacobians to efficiently update QP subproblems in both noncondensed and condensed LPV-MPC formulations. Empirical results on forced Van der Pol, a dynamic unicycle, and autonomous driving tracking show substantial reductions in computation time and iterations with modest optimality loss in some cases, while achieving near-NMPC performance in others. The method clarifies the connection between NLP-based NMPC and LPV-MPC and demonstrates practical benefits for real-time control, with potential for further convergence guarantees and globalization enhancements.

Abstract

In this paper, we present efficient solutions for the nonlinear program (NLP) associated with nonlinear model predictive control (NMPC) by leveraging the linear parameter-varying (LPV) embedding of nonlinear models and sequential quadratic programming (SQP). The corresponding quadratic program (QP) subproblem is systematically constructed and efficiently updated using the scheduling parameter from the LPV embedding, enabling fast convergence while adapting to the behavior of the controlled system. Furthermore, the approach provides insight into the problem, its connection to SQP, and a clearer understanding of the differences between solving NMPC as an NLP and using the LPV-MPC approach, compared to similar methods in the literature. The efficiency of the proposed approach is demonstrated against state-of-the-art methods, including NLP algorithms, in control benchmarks and practical applications.

Efficient Nonlinear MPC by Leveraging LPV Embedding and Sequential Quadratic Programming

TL;DR

The paper addresses the real-time solvability of nonlinear MPC by embedding nonlinear dynamics into a linear parameter-varying (LPV) form and solving the NMPC as a sequence of quadratic programs. It then develops LPVMPC-SQP, an SQP-based approach that uses LPV-informed inexact Hessians and Jacobians to efficiently update QP subproblems in both noncondensed and condensed LPV-MPC formulations. Empirical results on forced Van der Pol, a dynamic unicycle, and autonomous driving tracking show substantial reductions in computation time and iterations with modest optimality loss in some cases, while achieving near-NMPC performance in others. The method clarifies the connection between NLP-based NMPC and LPV-MPC and demonstrates practical benefits for real-time control, with potential for further convergence guarantees and globalization enhancements.

Abstract

In this paper, we present efficient solutions for the nonlinear program (NLP) associated with nonlinear model predictive control (NMPC) by leveraging the linear parameter-varying (LPV) embedding of nonlinear models and sequential quadratic programming (SQP). The corresponding quadratic program (QP) subproblem is systematically constructed and efficiently updated using the scheduling parameter from the LPV embedding, enabling fast convergence while adapting to the behavior of the controlled system. Furthermore, the approach provides insight into the problem, its connection to SQP, and a clearer understanding of the differences between solving NMPC as an NLP and using the LPV-MPC approach, compared to similar methods in the literature. The efficiency of the proposed approach is demonstrated against state-of-the-art methods, including NLP algorithms, in control benchmarks and practical applications.
Paper Structure (12 sections, 29 equations, 6 figures, 2 tables)

This paper contains 12 sections, 29 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Setup
  3. LPV-based MPC and QP Formulations
  4. Noncondensed Form
  5. Condensed Form
  6. Basic SQP for solving NLP
  7. Proposed Approach
  8. NUMERICAL EXAMPLES
  9. Example 1 - The forced Van der Pol oscillator
  10. Example 2 - The dynamic unicycle
  11. Example 3 - Reference tracking in Autonomous Driving
  12. Conclusion

Figures (6)

  • Figure 1: Phase plot of the closed-loop state trajectories of the Van der Pol oscillator.
  • Figure 2: The closed-loop state and input trajectories of the Van der Pol oscillator.
  • Figure 3: Solver execution time and iterations per MPC step for the considered approaches applied to the Van der Pol oscillator averaged over 100 simulation runs.
  • Figure 4: Closed-loop trajectories of the regulation problem for the unicycle.
  • Figure 5: Phase plot of the reference tracking control problem.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Remark 1