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A New Hybrid Automaton Framework with Partial Differential Equation Dynamics

Tianshu Bao, Hengrong Du, Weiming Xiang, Taylor T. Johnson

TL;DR

This work addresses the gap between hybrid automata and PDE-driven dynamics by introducing Partial Differential Hybrid Automata (PDHA), which incorporate a spatial domain $X$ and domain partitions to model PDE-based CPS behavior that cannot be captured by ODE-only HA. To enable analysis, the authors propose Discrete Space PDHA (DSPDHA) via finite-difference discretization, establishing a discretization relation that links continuous PDHA to discrete ODE-like automata. They formalize the PDHA and DSPDHA frameworks, define flows, transitions, and executions, and demonstrate with heater and traffic-flow running examples how PDE dynamics can be captured and discretized for analysis. The work lays groundwork for reachability, invariants, and safety analysis in PDE-enabled hybrid systems, while acknowledging area for further study such as well-posedness, invariant design, and extensions to irregular domains.

Abstract

This paper presents the syntax and semantics of a novel type of hybrid automaton (HA) with partial differential equation (PDE) dynamic, partial differential hybrid automata (PDHA). In PDHA, we add a spatial domain $X$ and harness a mathematic conception, partition, to help us formally define the spatial relations. While classically the dynamics of HA are described by ordinary differential equations (ODEs) and differential inclusions, PDHA is capable of describing the behavior of cyber-physical systems (CPS) with continuous dynamics that cannot be modelled using the canonical hybrid systems' framework. For the purposes of analyzing PDHA, we propose another model called the discrete space partial differential hybrid automata (DSPDHA) which handles discrete spatial domains using finite difference methods (FDM) and this simple and intuitive approach reduces the PDHA into HA with ODE systems. We conclude with two illustrative examples in order to exhibit the nature of PDHA and DSPDHA.

A New Hybrid Automaton Framework with Partial Differential Equation Dynamics

TL;DR

This work addresses the gap between hybrid automata and PDE-driven dynamics by introducing Partial Differential Hybrid Automata (PDHA), which incorporate a spatial domain and domain partitions to model PDE-based CPS behavior that cannot be captured by ODE-only HA. To enable analysis, the authors propose Discrete Space PDHA (DSPDHA) via finite-difference discretization, establishing a discretization relation that links continuous PDHA to discrete ODE-like automata. They formalize the PDHA and DSPDHA frameworks, define flows, transitions, and executions, and demonstrate with heater and traffic-flow running examples how PDE dynamics can be captured and discretized for analysis. The work lays groundwork for reachability, invariants, and safety analysis in PDE-enabled hybrid systems, while acknowledging area for further study such as well-posedness, invariant design, and extensions to irregular domains.

Abstract

This paper presents the syntax and semantics of a novel type of hybrid automaton (HA) with partial differential equation (PDE) dynamic, partial differential hybrid automata (PDHA). In PDHA, we add a spatial domain and harness a mathematic conception, partition, to help us formally define the spatial relations. While classically the dynamics of HA are described by ordinary differential equations (ODEs) and differential inclusions, PDHA is capable of describing the behavior of cyber-physical systems (CPS) with continuous dynamics that cannot be modelled using the canonical hybrid systems' framework. For the purposes of analyzing PDHA, we propose another model called the discrete space partial differential hybrid automata (DSPDHA) which handles discrete spatial domains using finite difference methods (FDM) and this simple and intuitive approach reduces the PDHA into HA with ODE systems. We conclude with two illustrative examples in order to exhibit the nature of PDHA and DSPDHA.
Paper Structure (14 sections, 18 equations, 12 figures)

This paper contains 14 sections, 18 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Running Examples
  3. Heater Model
  4. Traffic Flow Model
  5. Partial Differential Hybrid Automata
  6. Transitions, Trajectories and Executions
  7. Partial Differential Hybrid Automaton (PDHA)
  8. Discrete Space Partial Differential Hybrid Automaton
  9. Discretization Scheme and Discretization Relation
  10. Discrete Space Partial Differential Hybrid Automaton
  11. Relation to Classical Hybrid Automaton
  12. Experiments on Running Examples
  13. Conclusion
  14. Acknowledgements

Figures (12)

  • Figure 1: Diagram for heater model.
  • Figure 2: Diagram for traffic model.
  • Figure 3: Switching cases for two scenarios, full domain switching on left and partial domain switching on right. BC denotes boundary conditions and IC denotes initial conditions. $x$ axis denotes the spatial domain and $y$ axis denotes time.
  • Figure 4: Heater PDHA corresponding to the system in Figure \ref{['fig:heater']}.
  • Figure 5: Traffic PDHA corresponding to the system in Figure \ref{['fig:car_highway']}.
  • ...and 7 more figures

Theorems & Definitions (28)

  • Definition 1: Spatial Control Mode
  • Definition 2: Partition halmos2017naive
  • Definition 3: State
  • Definition 4: Flow
  • Definition 5: Transition
  • Definition 6: Reset
  • Definition 7: Partial Differential Hybrid Automaton (PDHA)
  • Definition 8: Discretization Scheme
  • Example 4.1: Finite Difference Method Discretization Scheme
  • Definition 9: Discretization Relation
  • ...and 18 more