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Koopman Meets Discrete-Time Control Barrier Functions: A Linear Model Predictive Control Framework

Shuo Liu, Liang Wu, Dawei Zhang, Jan Drgona, Calin. A. Belta

Abstract

This paper proposes a Koopman-based linear model predictive control (LMPC) framework for safety-critical control of nonlinear discrete-time systems. Existing MPC formulations based on discrete-time control barrier functions (DCBFs) enforce safety through barrier constraints but typically result in computationally demanding nonlinear programming. To address this challenge, we construct a DCBF-augmented dynamical system and employ Koopman operator theory to lift the nonlinear dynamics into a higher-dimensional space where both the system dynamics and the barrier function admit a linear predictor representation. This enables the transformation of the nonlinear safety-constrained MPC problem into a quadratic program (QP). To improve feasibility while preserving safety, a relaxation mechanism with slack variables is introduced for the barrier constraints. The resulting approach combines the modeling capability of Koopman operators with the computational efficiency of QP. Numerical simulations on a navigation task for a robot with nonlinear dynamics demonstrate that the proposed framework achieves safe trajectory generation and efficient real-time control.

Koopman Meets Discrete-Time Control Barrier Functions: A Linear Model Predictive Control Framework

Abstract

This paper proposes a Koopman-based linear model predictive control (LMPC) framework for safety-critical control of nonlinear discrete-time systems. Existing MPC formulations based on discrete-time control barrier functions (DCBFs) enforce safety through barrier constraints but typically result in computationally demanding nonlinear programming. To address this challenge, we construct a DCBF-augmented dynamical system and employ Koopman operator theory to lift the nonlinear dynamics into a higher-dimensional space where both the system dynamics and the barrier function admit a linear predictor representation. This enables the transformation of the nonlinear safety-constrained MPC problem into a quadratic program (QP). To improve feasibility while preserving safety, a relaxation mechanism with slack variables is introduced for the barrier constraints. The resulting approach combines the modeling capability of Koopman operators with the computational efficiency of QP. Numerical simulations on a navigation task for a robot with nonlinear dynamics demonstrate that the proposed framework achieves safe trajectory generation and efficient real-time control.
Paper Structure (16 sections, 2 theorems, 35 equations, 4 figures)

This paper contains 16 sections, 2 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman Operator
  4. Extended Dynamic Mode Decomposition
  5. Discrete-Time Control Barrier Functions (DCBFs)
  6. PROBLEM FORMULATION AND APPROACH
  7. Koopman-based MPC-DCBF framework
  8. DCBF-augmented Dynamic System
  9. Koopman Identification via Least Squares
  10. Koopman-based Linear MPC Formulation
  11. Complexity Analysis
  12. Numerical Examples
  13. System Dynamics and DCBF
  14. Koopman Identification
  15. MPC Design
  16. ...and 1 more sections

Key Result

Theorem 1

Given a DCBF $h(\mathbf{x})$ from Def. def:DCBF with the corresponding set $\mathcal{C}$ defined by eq:safe-set, if $\mathbf{x}_{0} \in \mathcal{C}$ then any Lipschitz controller $\mathbf{u}_{t}$ that satisfies the constraint in eq:dh_condition, $\forall t\ge 0$ renders $\mathcal{C}$ forward invaria

Figures (4)

  • Figure 1: Comparison between the true nonlinear dynamics and the identified Koopman predictor. Left: $x$--$y$ trajectories generated by the true system and the Koopman model. Middle: evolution of the DCBFs. Right: prediction error $e_t=\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|_2$.
  • Figure 2: Trajectory comparison in the presence of a circular obstacle. The robot moves from the start (blue diamond) to the goal (red diamond). The black arrow indicates the initial velocity direction of the robot. Trajectories generated by the proposed K-LMPC-DCBF with different prediction horizons $N$ and hyperparameters $\gamma$ are compared with iMPC-DCBF and NMPC-DCBF. The legend also reports the average computation time per step $t_s$ (ms).
  • Figure 3: Evolution of the DCBFs under different control strategies. The red dashed line indicates the safety boundary.
  • Figure 4: Control inputs $u_1$ and $u_2$ over time for K-LMPC-DCBF under different hyperparameter settings.

Theorems & Definitions (8)

  • Remark 1
  • Definition 1: DCBF xiong2022discrete
  • Theorem 1: Safety Guarantee xiong2022discrete
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4