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Koopman-BoxQP: Solving Large-Scale NMPC at kHz Rates

Liang Wu, Wallace Gian Yion Tan, Richard D. Braatz, Ján Drgoňa

TL;DR

A Koopman-BoxQP framework that learns a linear Koopman high-dimensional model, eliminates the high-dimensional observables to construct a multi-step prediction model of the states and control inputs, and penalizes the multi-step prediction model into the objective is proposed.

Abstract

Solving large-scale nonlinear model predictive control (NMPC) problems at kilohertz (kHz) rates on standard processors remains a formidable challenge. This paper proposes a Koopman-BoxQP framework that i) learns a linear Koopman high-dimensional model, ii) eliminates the high-dimensional observables to construct a multi-step prediction model of the states and control inputs, iii) penalizes the multi-step prediction model into the objective, which results in a structured box-constrained quadratic program (BoxQP) whose decision variables include both the system states and control inputs, iv) develops a structure-exploited and warm-starting-supported variant of the feasible Mehrotra's interior-point algorithm for BoxQP. Numerical results demonstrate that Koopman-BoxQP can solve a large-scale NMPC problem with $1040$ variables and $2080$ inequalities at a kHz rate.

Koopman-BoxQP: Solving Large-Scale NMPC at kHz Rates

TL;DR

A Koopman-BoxQP framework that learns a linear Koopman high-dimensional model, eliminates the high-dimensional observables to construct a multi-step prediction model of the states and control inputs, and penalizes the multi-step prediction model into the objective is proposed.

Abstract

Solving large-scale nonlinear model predictive control (NMPC) problems at kilohertz (kHz) rates on standard processors remains a formidable challenge. This paper proposes a Koopman-BoxQP framework that i) learns a linear Koopman high-dimensional model, ii) eliminates the high-dimensional observables to construct a multi-step prediction model of the states and control inputs, iii) penalizes the multi-step prediction model into the objective, which results in a structured box-constrained quadratic program (BoxQP) whose decision variables include both the system states and control inputs, iv) develops a structure-exploited and warm-starting-supported variant of the feasible Mehrotra's interior-point algorithm for BoxQP. Numerical results demonstrate that Koopman-BoxQP can solve a large-scale NMPC problem with variables and inequalities at a kHz rate.
Paper Structure (14 sections, 1 theorem, 25 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 1 theorem, 25 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Problem Formulation
  4. Preliminary: Koopman transforms NMPC into general QP
  5. Methodology
  6. Koopman-BoxQP: Dynamics-relaxed Construction
  7. Feasible Mehrotra's IPM Algorithm for BoxQP
  8. Cost-free strictly feasible cold- and warm-starting initializations
  9. Algorithm description and efficient computation of Newton systems
  10. Numerical Examples
  11. Practical behavior of cold-starting Mehrotra's IPM algorithm on random BoxQPs
  12. Performance validation of Koopman-BoxQP on PDE NMPC problems
  13. Conclusion
  14. Numerical experiments on general QPs

Key Result

lemma 1

Assume that the lifted mapping $\psi(\cdot)$ is Lipschitz continuous with constant $L_{\psi}$. The feedback policy of Koopman-BoxQP eqn_KoopmanMPC_BoxQP, where $C_{\text{policy}}=[I_{n_u},0]$), is Lipschitz continuous with constant $\frac{ \rho\sqrt{\lambda_{\max}(\mathbf{E}^\top(\mathbf{F}\mathbf{F}^\top+I)\mathbf{E})}L_{\psi}}{\lambda_{\min}(H)}$.

Figures (5)

  • Figure 1: Iteration counts of feasible and infeasible Mehrotra's IPM algorithms on ill-conditioned random BoxQP problems with dimensions ranging from $100$ to $2000$.
  • Figure 2: Iterations comparison of Algorithm \ref{['alg_IPM']} with Cold-start and Warm-start.
  • Figure 3: 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 $[-1,1]$.
  • Figure 4: Execution time comparison between Algorithm \ref{['alg_IPM']}, OSQP, and SCS solvers on random QPs (and their soft-constrained QPs with $\rho=10^6$) with different dimensions. Left: $n_y=20 n_x$. Right: $n_y=40 n_x$.
  • Figure : Feasible Mehrotra's predictor-corrector IPM for BoxQP \ref{['eqn_BoxQP']}

Theorems & Definitions (6)

  • remark 1: Feasibility-guaranteed
  • lemma 1: Lipschitz-guaranteed
  • proof
  • remark 2
  • remark 3
  • remark 4