⚡ electrical engineering

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

This paper presents a computational framework based on tropical polyhedra to efficiently calculate one-step backward reachable sets for uncertain max-plus linear systems with additive disturbances, demonstrating that the backward operators preserve the tropical-polyhedral structure to enable constructive safety analysis.

Original authors: Yuda Li, Xiang Yin

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuda Li, Xiang Yin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a very special ship navigating a foggy, unpredictable ocean. This isn't just any ocean; it's a world where time and distance work differently, governed by a set of rules called Max-Plus Algebra. In this world, "adding" two numbers means taking the maximum of them, and "multiplying" means adding them. It sounds strange, but it's perfect for modeling things like traffic jams, factory assembly lines, or train schedules, where the whole system is only as fast as its slowest part.

The Problem: The Foggy Target

Your goal is to reach a specific "Safe Harbor" (a Target Set). However, your ship is being buffeted by Disturbances—unpredictable waves, wind, or errors in your map. You can steer the ship (apply Control), but you don't know exactly how strong the waves will be.

The big question is: From which starting points on the map can you guarantee that, no matter how hard the waves hit, you can still steer the ship into the Safe Harbor in one step?

This is called the Backward Reachability problem. Instead of asking "Where can I go from here?", you are asking "Where must I have started to ensure I can get there safely?"

The Old Way: A Messy Calculation

Previously, trying to solve this was like trying to calculate the path of a ship through a storm using a giant, messy spreadsheet. As the number of variables (waves, ship speed, wind direction) grew, the math became so complex that computers would crash or take forever to give an answer. It was like trying to count every grain of sand on a beach to find a single shell.

The New Solution: Tropical Polyhedra (The "Shape-Shifting" Map)

The authors of this paper, Yuda Li and Xiang Yin, have developed a new, elegant way to solve this. They use a concept called Tropical Polyhedra.

Think of a Tropical Polyhedron not as a rigid geometric shape, but as a flexible, stretchy net.

In normal math, if you push a square, it might turn into a weird, jagged mess.
In this "Max-Plus" world, these special nets have a superpower: They keep their shape.

No matter how you stretch them, twist them, or push them through the "disturbance waves," they remain recognizable as these special nets. This is the paper's biggest breakthrough.

How It Works: The Three-Step Dance

To find the safe starting points, the authors break the problem down into three simple moves, like a dance routine:

The Wave Filter (Universal Quantification): First, they look at the Safe Harbor and ask, "If a wave hits us from any direction, where could we have been to still land safely?" They use the "stretchy net" to filter out all the dangerous starting spots. Because the net keeps its shape, this step is easy to calculate.
The Steering Wheel (Existential Quantification): Next, they ask, "From the remaining spots, can we choose a steering angle to get to the harbor?" They expand the net to include all spots where a good steering choice exists. Again, the shape stays intact.
The Time Machine (Inverse Dynamics): Finally, they reverse the ship's engine. They ask, "Where was the ship one second ago to be at these new spots now?" They pull the net backward in time.

The Magic Result

Because the "stretchy nets" (Tropical Polyhedra) never lose their shape during these three steps, the authors can compute the final answer using simple, fast geometric rules. They don't need to check every single point in the ocean; they just need to track the "corners" of the net.

Why This Matters

This isn't just a math trick. It's a safety tool for the real world.

Traffic Control: It can tell traffic lights exactly when to turn green to ensure no car gets stuck in a gridlock, even if a driver is late.
Factory Safety: It can help a robot arm know exactly where to start moving so it never crashes into a human, even if the human moves unexpectedly.
Train Scheduling: It ensures that even if a train is delayed, the schedule can be adjusted so the whole system doesn't collapse.

The Bottom Line

The authors took a problem that was previously too hard and messy to solve (finding safe paths in a chaotic system) and turned it into a clean, geometric puzzle. By realizing that the "shapes" of the problem stay consistent, they gave engineers a powerful new tool to design systems that are safe, even when the world is unpredictable.

In short: They found a way to draw a "safety bubble" around a chaotic system, proving that as long as you start inside the bubble, you can steer your way to safety no matter what the storm throws at you.

1. Problem Statement

The paper addresses the backward reachability problem for Max-Plus Linear Systems (MPLS) subject to additive disturbances.

System Model: The system evolves according to $x_k = A \otimes x_{k-1} \oplus B \otimes u_k \oplus C \otimes w_k$ , where $x$ is the state, $u$ is the control input, and $w$ is the disturbance. All operations are defined over the max-plus algebra ( $\mathbb{R}_{\max} = \mathbb{R} \cup \{-\infty\}$ , where $\oplus = \max$ and $\otimes = +$ ).
Objective: Given a target set $E$ (representing a safe state region), compute the one-step backward reachable set $\Upsilon(E)$ . This set consists of all states $x$ from which there exists an admissible control input $u$ such that, for all admissible disturbances $w$ , the successor state remains within $E$ .
Challenge: Unlike disturbance-free systems, the presence of disturbances introduces a universal quantification ( $\forall w$ ) over the uncertainty set. Existing methods (e.g., Difference-Bound Matrices or DBMs) often suffer from high computational complexity as system dimensions and uncertainty structures grow. Furthermore, it is not immediately obvious whether the backward reachable set preserves the geometric structure of the initial sets under max-plus operations.

2. Methodology

The authors propose a computational framework based on Tropical Convexity and Tropical Polyhedra.

A. Mathematical Framework

Representation: The target set $E$ , control set $U$ , and disturbance set $W$ are all modeled as tropical polyhedra. A tropical polyhedron is defined as the intersection of finitely many tropical half-spaces (Outer/M-form) or the max-plus span of a finite set of vectors (Inner/V-form).
Operator Decomposition: The backward reachability operator $\Upsilon$ $Υ$ is decomposed into three elementary set-valued operators:
1. Inverse Operator ( $A^{-1}$ ): Handles the autonomous system dynamics.
2. Universal Inverse Operator ( $\phi_W$ ): Handles the universal quantification over disturbances (the most challenging part).
3. Existential Inverse Operator ( $\gamma_U$ ): Handles the existential quantification over control inputs.
- The composition is $\Upsilon = A^{-1} \circ \gamma_U \circ \phi_W$ .

B. Key Computational Steps

Preservation of Structure: The paper proves that if the input sets are tropical polyhedra, the images under $A^{-1}$ and $\gamma_U$ remain tropical polyhedra. This allows the use of a unified geometric framework.
Handling Disturbances ( $\phi_W$ ):
- Decomposition: The disturbance set $W$ is decomposed into a recession cone $R$ and a bounded polyhedron $P$ ( $W = R \oplus P$ ). The authors show that if $R$ is contained within the target set's recession cone, the problem reduces to computing $\phi_P(Z)$ .
- Pseudo Half-Spaces: The set $\phi_P(Z)$ is characterized as an intersection of "pseudo half-spaces." A pseudo half-space is defined by a universal quantification over a generating set of the disturbance cone.
- Recursive Generation: The authors develop an algorithm to compute the generating set (V-form) of the resulting set. They utilize a theorem analogous to the Minkowski-Carathéodory theorem, which states that the intersection of a tropical cone and a pseudo half-space can be generated by:
  - Vectors in the original set that satisfy the constraint.
  - New vectors formed by combining a "satisfying" vector and a "violating" vector at a specific threshold $\rho$ (the maximum scalar $\lambda$ such that the combination remains in the set).
Optimization: An extremal points filtering procedure is suggested to reduce computational complexity by removing non-extremal points from the generating set at each iteration.

3. Key Contributions

Structural Preservation: The paper establishes that the backward reachable set for uncertain MPLSs, when defined over tropical polyhedra, is itself a tropical polyhedron. This is a crucial theoretical result enabling constructive computation.
Novel Algorithm for Universal Quantification: The authors provide a constructive method to handle the universal quantification over disturbances ( $\phi_W$ ), which is significantly more complex than existential quantification. This involves defining pseudo half-spaces and deriving a recursive algorithm to compute their generating sets.
Unified Geometric Framework: The approach unifies the treatment of system dynamics, control, and disturbances within the tropical polyhedral framework, avoiding the need to switch between different representations (e.g., from interval arithmetic to piecewise affine).
Computational Procedure: The paper details a step-by-step algorithm to compute the generating vectors of the backward reachable set, including specific formulas for calculating the threshold $\rho$ required to generate new boundary vectors.

4. Results

Theoretical Validation: The paper provides rigorous proofs (Lemmas 1-3, Theorem 1) demonstrating that the proposed operators preserve the tropical polyhedral structure.
Case Study: A numerical example (Section VII) is presented involving a 2-dimensional MPLS with a specific disturbance set and safety target.
- The authors manually trace the computation of $\phi_W(S)$ , $\gamma_U(\cdot)$ , and $A^{-1}(\cdot)$ .
- The example demonstrates the calculation of generating vectors for the intersection of pseudo half-spaces and the final backward reachable set.
- Geometric visualizations (Figures 1 and 2) illustrate the transformation of the sets, confirming the method's ability to handle complex uncertainty structures.

5. Significance

Safety Verification: The method provides a rigorous tool for safety analysis and controlled invariance in discrete-event systems (e.g., manufacturing, transportation, scheduling) where timing uncertainties are critical.
Beyond Probabilistic Models: Unlike stochastic approaches that rely on probability distributions, this deterministic approach is suitable for scenarios where uncertainties are bounded but their distribution is unknown or non-Gaussian.
Scalability Potential: By leveraging the algebraic properties of tropical polyhedra, the proposed method offers a potential alternative to DBM-based approaches, which can become computationally expensive as dimensionality increases. The use of extremal point filtering further suggests a path toward handling larger systems.
Foundational Work: This work bridges tropical convexity theory with control theory, opening new avenues for the analysis of uncertain max-plus systems.

In summary, Li and Yin present a robust algebraic and geometric framework for solving backward reachability in uncertain max-plus systems, proving that the problem remains tractable within the tropical polyhedral domain and providing explicit algorithms for its computation.