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On the Computation of Backward Reachable Sets for Max-Plus Linear Systems with Disturbances

Yuda Li, Xiang Yin

Abstract

This paper investigates one-step backward reachability for uncertain max-plus linear systems with additive disturbances. Given a target set, the problem is to compute the set of states from which there exists an admissible control input such that, for all admissible disturbances, the successor state remains in the target set. This problem is closely related to safety analysis and is challenging due to the high computational complexity of existing approaches. To address this issue, we develop a computational framework based on tropical polyhedra. We assume that the target set, the control set, and the disturbance set are all represented as tropical polyhedra, and study the structural properties of the associated backward operators. In particular, we show that these operators preserve the tropical-polyhedral structure, which enables the constructive computation of reachable sets within the same framework. The proposed approach provides an effective geometric and algebraic tool for reachability analysis of uncertain max-plus linear systems. Illustrative examples are included to demonstrate the proposed method.

On the Computation of Backward Reachable Sets for Max-Plus Linear Systems with Disturbances

Abstract

This paper investigates one-step backward reachability for uncertain max-plus linear systems with additive disturbances. Given a target set, the problem is to compute the set of states from which there exists an admissible control input such that, for all admissible disturbances, the successor state remains in the target set. This problem is closely related to safety analysis and is challenging due to the high computational complexity of existing approaches. To address this issue, we develop a computational framework based on tropical polyhedra. We assume that the target set, the control set, and the disturbance set are all represented as tropical polyhedra, and study the structural properties of the associated backward operators. In particular, we show that these operators preserve the tropical-polyhedral structure, which enables the constructive computation of reachable sets within the same framework. The proposed approach provides an effective geometric and algebraic tool for reachability analysis of uncertain max-plus linear systems. Illustrative examples are included to demonstrate the proposed method.

Paper Structure

This paper contains 16 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Max-Plus Algebra $\mathbb{R}_{\max}$
  4. Semimodules, Sub-Semimodules, and Convex Cones
  5. Tropical Cones, Half-Spaces and Polyhedra
  6. Problem Formulation
  7. Computation for the operators $A^{-1}$ and $\gamma_\square$
  8. Computation for the image of $A^{-1}$
  9. Computation for $\gamma_\square$
  10. Computation for $\phi_\mathcal{W}$
  11. Simplification Step
  12. Reform $\phi_\mathcal{P}(Z)$ as An Intersection of Pseudo Half-Spaces
  13. Induction Step for Calculating Generating Set for $\phi_\mathcal{P}(Z)$
  14. Computation methods for $\phi_\square$
  15. Case study
  16. ...and 1 more sections

Figures (2)

  • Figure 2: The overview of example \ref{['exm: v1v2 in HcdU']}. The region in skin color represents the set $\langle c,d\rangle\cap \mathbb{R}_{\max}^2$, the region in blue represents the set $\mathcal{U}\cap \mathbb{R}_{\max}^2$, and the region on the right-hand side of the dot line represent the set $\mathscr{H}_{c,d}^\mathcal{U}$. The two red points represent respectively $v_1$ and $v_2$. It is easy to check that $v_2 \notin \mathscr{H}_{c,d}^\mathcal{U}$ and $v_1 \in \mathscr{H}_{c,d}^\mathcal{U}$.
  • Figure 3: The overview of the three regions: $S$, $\mathcal{W}$ and $\phi_{\mathcal{W}}(S)$. The skin color region represents $S$, $\mathcal{W}$ corresponds to the blue region, and $\phi_{\mathcal{W}}(S)$ is surrounded by the line of dots.

Theorems & Definitions (6)

  • proof
  • proof
  • proof
  • proof
  • proof
  • Remark 1: Extremal Points Filtering Procedure