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Reach-Avoid Model Predictive Control with Guaranteed Recursive Feasibility via Input Constrained Backstepping

Jianqiang Ding, Nishant Jayesh Bhave, Shankar A. Deka

Abstract

This letter proposes a novel sampled-data model predictive control framework for continuous control-affine nonlinear systems that provides rigorous reach-avoid and recursive feasibility guarantees under physical constraints. By propagating both input and output constraints through backstepping process, we present a constructive approach to synthesize a reach-avoid invariant set that complies with control input limits. Using this reach-avoid set as a terminal set, we prove that the proposed sampled-data MPC framework recursively admits feasible control inputs that safely steer the continuous system into the target set under fast sampling conditions. Numerical results demonstrate the efficacy of the proposed approach.

Reach-Avoid Model Predictive Control with Guaranteed Recursive Feasibility via Input Constrained Backstepping

Abstract

This letter proposes a novel sampled-data model predictive control framework for continuous control-affine nonlinear systems that provides rigorous reach-avoid and recursive feasibility guarantees under physical constraints. By propagating both input and output constraints through backstepping process, we present a constructive approach to synthesize a reach-avoid invariant set that complies with control input limits. Using this reach-avoid set as a terminal set, we prove that the proposed sampled-data MPC framework recursively admits feasible control inputs that safely steer the continuous system into the target set under fast sampling conditions. Numerical results demonstrate the efficacy of the proposed approach.

Paper Structure

This paper contains 8 sections, 4 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Recursive Feasible Reach-avoid MPC
  4. Construction of Input-Constrained Reach-avoid Set
  5. Reach-avoid MPC with Recursive Feasibility Guarantees
  6. Experiments
  7. Conclusion
  8. Additional details for the proof of Lemma \ref{['lemma:sampled-data system reach-avoid guarantees']}

Key Result

Lemma 1

(Theorem 2 in ding2025backstepping) Given the system eq: dynamic system with safe and target sets satisfying the Assumption assume:safe_target_sets, suppose there exists locally Lipschitz continuous function $\bm{k}_1(\bm{y}(\bm{x})) = ^\top \in \mathbb{R}^m$ for some $\lambda>0$ such that, Let $\{\gamma_1, \cdots, \gamma_m \}$ be the vector relative degree for system eq: dynamic system with the

Figures (2)

  • Figure C1: (\ref{['fig:vanilla mpc']}), (\ref{['fig:unconstrained reach-avoid']}) & (\ref{['fig:constrained reach-avoid mpc']}) show controller success rates for $300$ random initial states. Blue dots indicate successful states achieving the reach-avoid objective, while red crosses represent failed ones. A closed-loop trajectory (dashed line from orange dot to green cross) is included for visualization. Initial angles are omitted for clarity. (\ref{['fig:control inputs comparison']}) represents control inputs comparison between unconstrained reach-avoid controller (blue) and the proposed MPC controller (purple).
  • Figure D1: (\ref{['fig:vanilla mpc manipulator']}), (\ref{['fig:unconstrained reach-avoid manipulator']}) & (\ref{['fig:constrained reach-avoid mpc manipulator']}) show controller success rates for $100$ random initial states. Trajectories starting from blue dots achieved the reach-avoid objective while the ones starting at the red crosses led to infeasibility. A feasible closed-loop trajectory (dashed line from orange dot to green cross) is included for visualization. (\ref{['fig:control inputs comparison manipulator']}) represents control inputs comparison between unconstrained reach-avoid controller (blue) and the proposed MPC controller (purple).

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Theorem 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 2
  • ...and 3 more