[Paper Review] Controller synthesis for bisimulation equivalence

Paulo Tabuada|ArXiv.org|Jun 6, 2007

Petri Nets in System Modeling20 references3 citations

TL;DR

This paper presents a categorical framework for solving the controller synthesis problem under bisimulation equivalence across diverse system classes—including discrete-event, nonlinear control, behavioral, and hybrid systems. By leveraging category theory and the open maps formalism, it establishes that a controller exists if and only if the plant simulates the specification, unifying and generalizing existing results with a single, abstract criterion applicable to multiple domains.

ABSTRACT

The objective of this paper is to solve the controller synthesis problem for bisimulation equivalence in a wide variety of scenarios including discrete-event systems, nonlinear control systems, behavioral systems, hybrid systems and many others. This will be accomplished by showing that the arguments underlying proofs of existence and methods for the construction of controllers are extraneous to the particular class of systems being considered and thus can be presented in greater generality.

Motivation & Objective

• To solve the controller synthesis problem for bisimulation equivalence in a wide range of system classes, including discrete-event, nonlinear control, behavioral, and hybrid systems.
• To demonstrate that the existence and construction of controllers are independent of the specific system class, by abstracting the core requirements using category theory.
• To unify and generalize known results in controller synthesis by identifying a single, abstract condition—simulation of the specification by the plant—that guarantees controller existence.
• To provide a computationally insightful framework that points toward efficient algorithms for verifying simulation relations in complex systems.
• To extend the applicability of existing results, such as those in [Tab04], to broader classes of systems through a minimal categorical foundation.

Proposed method

• Formalize systems as objects in categories, using transition systems and nonlinear control systems as illustrative examples.
• Apply the open maps framework of Joyal et al. to characterize bisimulation equivalence in a categorical setting.
• Use binary pullbacks and morphisms between systems to model system composition and controller interaction.
• Define a controller as a morphism from the plant to the specification such that the resulting composition satisfies a simulation relation.
• Establish that the existence of a simulation relation from the specification to the plant is both necessary and sufficient for controller synthesis.
• Leverage the existence of binary products and pullbacks in relevant categories (e.g., behavioral and hybrid systems) to ensure the framework’s applicability.

Experimental results

Research questions

• RQ1 Under what general conditions does a controller exist such that the closed-loop system is bisimilar to a given specification?
• RQ2 Can the controller synthesis problem be solved uniformly across disparate system classes like discrete-event, nonlinear control, and hybrid systems?
• RQ3 What categorical structures are essential for ensuring the existence and constructibility of controllers under bisimulation equivalence?
• RQ4 How can simulation relations be used as a unifying criterion for controller existence across different system models?
• RQ5 What are the minimal structural requirements on system categories that allow for effective controller synthesis via simulation?

Key findings

• A controller exists for bisimulation equivalence if and only if the plant simulates the specification, providing a single, unifying criterion across diverse system classes.
• The proposed framework generalizes prior results, such as those in [Tab04], to nonlinear control systems, behavioral systems, and hybrid systems by abstracting away system-specific details.
• The open maps formalism enables a uniform treatment of bisimulation across categories of systems, ensuring consistency in reasoning and construction.
• The existence of binary pullbacks in categories of behavioral and hybrid systems ensures that the controller synthesis problem is well-posed and solvable under the proposed conditions.
• The framework reveals that the core requirement for controller synthesis is not system-specific complexity but the existence of a simulation relation from specification to plant.
• The results suggest that efficient algorithms for computing simulation relations could significantly improve the scalability of controller synthesis in practice.