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Unifying Sequential Quadratic Programming and Linear-Parameter-Varying Algorithms for Real-Time Model Predictive Control

Kristóf Floch, Amon Lahr, Roland Tóth, Melanie N. Zeilinger

TL;DR

The paper tackles real-time nonlinear MPC by unifying two prevalent solution strategies: SQP and iterative LPV-MPC. It develops a differential, FTC-based LPV embedding that, with a suitable anchor-point choice, recovers SQP steps and thus the same local optimality guarantees, while also enabling a zero-order approximation that reduces computational load. A common quadratic framework is derived, connecting the two methods and enabling analysis of convergence and complexity; the framework is validated through cart-pendulum simulations and autonomous racing experiments, including GP-MPC for learning residual dynamics. The results demonstrate that the unified approach can achieve comparable or improved real-time performance, with the added benefit of accommodating learning-based augmentations and real hardware deployment, supported by open-source tooling.

Abstract

This paper presents a unified framework that connects sequential quadratic programming (SQP) and the iterative linear-parameter-varying model predictive control (LPV-MPC) technique. Using the differential formulation of the LPV-MPC, we demonstrate how SQP and LPV-MPC can be unified through a specific choice of scheduling variable and the 2nd Fundamental Theorem of Calculus (FTC) embedding technique and compare their convergence properties. This enables the unification of the zero-order approach of SQP with the LPV-MPC scheduling technique to enhance the computational efficiency of stochastic and robust MPC problems. To demonstrate our findings, we compare the two schemes in a simulation example. Finally, we present real-time feasibility and performance of the zeroorder LPV-MPC approach by applying it to Gaussian process (GP)-based MPC for autonomous racing with real-world experiments.

Unifying Sequential Quadratic Programming and Linear-Parameter-Varying Algorithms for Real-Time Model Predictive Control

TL;DR

The paper tackles real-time nonlinear MPC by unifying two prevalent solution strategies: SQP and iterative LPV-MPC. It develops a differential, FTC-based LPV embedding that, with a suitable anchor-point choice, recovers SQP steps and thus the same local optimality guarantees, while also enabling a zero-order approximation that reduces computational load. A common quadratic framework is derived, connecting the two methods and enabling analysis of convergence and complexity; the framework is validated through cart-pendulum simulations and autonomous racing experiments, including GP-MPC for learning residual dynamics. The results demonstrate that the unified approach can achieve comparable or improved real-time performance, with the added benefit of accommodating learning-based augmentations and real hardware deployment, supported by open-source tooling.

Abstract

This paper presents a unified framework that connects sequential quadratic programming (SQP) and the iterative linear-parameter-varying model predictive control (LPV-MPC) technique. Using the differential formulation of the LPV-MPC, we demonstrate how SQP and LPV-MPC can be unified through a specific choice of scheduling variable and the 2nd Fundamental Theorem of Calculus (FTC) embedding technique and compare their convergence properties. This enables the unification of the zero-order approach of SQP with the LPV-MPC scheduling technique to enhance the computational efficiency of stochastic and robust MPC problems. To demonstrate our findings, we compare the two schemes in a simulation example. Finally, we present real-time feasibility and performance of the zeroorder LPV-MPC approach by applying it to Gaussian process (GP)-based MPC for autonomous racing with real-world experiments.

Paper Structure

This paper contains 19 sections, 4 theorems, 25 equations, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Nonlinear MPC
  3. General NMPC Problem
  4. SQP Solution
  5. Iterative LPV-MPC
  6. LPV systems
  7. LPV-MPC
  8. Unifying SQP and LPV-MPC
  9. Equivalence Condition
  10. Zero-order Approximation
  11. Simulation Study
  12. Convergence Properties and Computational Complexity
  13. Autonomous Racing
  14. Vehicle Model
  15. LPV-MPCC formulation
  16. ...and 4 more sections

Key Result

Lemma II.1

Consider the NMPC problem eqn:NLP, solved by SQP using Alg. alg:SQP-MPC. Suppose that standard SQP assumptions hold Boggs_Tolle_1995 and the algorithm has converged, i.e., $\Delta X^\star=0,\;\Delta U^\star=0$. Then, ${X}^\star, {U}^\star$ together with the corresponding Lagrange multipliers $(\lamb

Figures (3)

  • Figure 1: KKT residuals of the SQP and LPV-MPC iterations for a single OCP for the cart-pendulum system.
  • Figure 2: Number of iterations required to converge at each closed-loop step for the cart-pendulum system. Note that the red line overlays the black.
  • Figure 3: Miniature car racing hardware experiments with the MPCC implementations. A video of the experiments is available at https://youtu.be/-zPCEP9Ahok.

Theorems & Definitions (6)

  • Lemma II.1: Optimality of SQP Nocedal_Wright_2006
  • Lemma II.2: Suboptimality of LPV--MPC Hespe_Werner_2021
  • Proposition III.1: Equivalence of SQP and LPV-MPC
  • proof
  • Corollary III.2
  • proof