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Generalizing Linear Graphs and Bond Graph Models with Hetero-functional Graphs for System-of-Systems Engineering Applications

Ehsanoddin Ghorbanichemazkati, Amro M. Farid

TL;DR

This paper proves mathematically that hetero-functional graphs are a formal generalization of both linear graphs and bond graphs.

Abstract

In the 20th century, individual technology products like the generator, telephone, and automobile were connected to form many of the large-scale, complex, infrastructure networks we know today: the power grid, the communication infrastructure, and the transportation system. Progressively, these networked systems began interacting, forming what is now known as systems-of-systems. Because the component systems in the system-of-systems differ, modeling and analysis techniques with primitives applicable across multiple domains or disciplines are needed. For example, linear graphs and bond graphs have been used extensively in the electrical engineering, mechanical engineering, and mechatronic fields to design and analyze a wide variety of engineering systems. In contrast, hetero-functional graph theory (HFGT) has emerged to study many complex engineering systems and systems-of-systems (e.g. electric power, potable water, wastewater, natural gas, oil, coal, multi-modal transportation, mass-customized production, and personalized healthcare delivery systems). This paper seeks to relate hetero-functional graphs to linear graphs and bond graphs and demonstrate that the former is a generalization of the latter two. The contribution is relayed in three stages. First, the three modeling techniques are compared conceptually. Next, these techniques are contrasted on six example systems: (a) an electrical system, (b) a translational mechanical system, (c) a rotational mechanical system, (d) a fluidic system, (e) a thermal system, and (f) a multi-energy (electro-mechanical) system. Finally, this paper proves mathematically that hetero-functional graphs are a formal generalization of both linear graphs and bond graphs.

Generalizing Linear Graphs and Bond Graph Models with Hetero-functional Graphs for System-of-Systems Engineering Applications

TL;DR

This paper proves mathematically that hetero-functional graphs are a formal generalization of both linear graphs and bond graphs.

Abstract

In the 20th century, individual technology products like the generator, telephone, and automobile were connected to form many of the large-scale, complex, infrastructure networks we know today: the power grid, the communication infrastructure, and the transportation system. Progressively, these networked systems began interacting, forming what is now known as systems-of-systems. Because the component systems in the system-of-systems differ, modeling and analysis techniques with primitives applicable across multiple domains or disciplines are needed. For example, linear graphs and bond graphs have been used extensively in the electrical engineering, mechanical engineering, and mechatronic fields to design and analyze a wide variety of engineering systems. In contrast, hetero-functional graph theory (HFGT) has emerged to study many complex engineering systems and systems-of-systems (e.g. electric power, potable water, wastewater, natural gas, oil, coal, multi-modal transportation, mass-customized production, and personalized healthcare delivery systems). This paper seeks to relate hetero-functional graphs to linear graphs and bond graphs and demonstrate that the former is a generalization of the latter two. The contribution is relayed in three stages. First, the three modeling techniques are compared conceptually. Next, these techniques are contrasted on six example systems: (a) an electrical system, (b) a translational mechanical system, (c) a rotational mechanical system, (d) a fluidic system, (e) a thermal system, and (f) a multi-energy (electro-mechanical) system. Finally, this paper proves mathematically that hetero-functional graphs are a formal generalization of both linear graphs and bond graphs.
Paper Structure (35 sections, 2 theorems, 85 equations, 30 figures)

This paper contains 35 sections, 2 theorems, 85 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Original Contribution
  3. Paper Outline
  4. Overview of Linear Graph, Bond Graph, and Hetero-functional Graph Techniques
  5. Linear Graphs
  6. Bond Graphs
  7. Hetero-functional Graphs
  8. Objective Function
  9. Equality Constraints
  10. Inequality Constraints
  11. Device Model Constraints
  12. Illustrative Examples
  13. Linear Graphs by Example
  14. Electrical System
  15. Translational Mechanical System
  16. ...and 20 more sections

Key Result

Theorem 1

Given an arbitrary linear graph composed of organized in an arbitrary topology, and a minimal set of initial conditions on the associated state variables, the solution of its associated state space ODE model is equivalent to the solution of a specialized instance of the HFNMCF problem.

Figures (30)

  • Figure 1: Graphical and mathematical models of Bond Graph, Linear Graph, and Hetero-functional Graph.
  • Figure 2: Comparison of physical domains with respective Lagrangian and Eulerian views, illustrating the across and through, flow and effort variables associated with each domain.
  • Figure 3: Physical elements used in bond graphs and linear graphs
  • Figure 4: A SysML Block Definition Diagram of the System Form of the Engineering System Meta-Architecture
  • Figure 5: Different system domains studied in this paper: (a) Electrical system; (b) Translational mechanical system; (c) Rotational mechanical system; (d) Fluidic system; (e) Thermal system; (f) Electronic motor generator
  • ...and 25 more figures

Theorems & Definitions (15)

  • Definition 1: System Operand SE-Handbook-Working-Group:2015:00
  • Definition 2: System ProcessHoyle:1998:00SE-Handbook-Working-Group:2015:00
  • Definition 3: System Resource SE-Handbook-Working-Group:2015:00
  • Definition 4: BufferSchoonenberg:2019:ISC-BK04Farid:2022:ISC-J51
  • Definition 5: CapabilitySchoonenberg:2019:ISC-BK04Farid:2022:ISC-J51Farid:2016:ISC-BC06
  • Definition 6: The Negative 3$^{rd}$ Order Hetero-functional Incidence Tensor (HFIT) $\widetilde{\cal M}_\rho^-$Farid:2022:ISC-J51
  • Definition 7: The Positive 3$^{rd}$ Order Hetero-functional Incidence Tensor (HFIT)$\widetilde{\cal M}_\rho^+$Farid:2022:ISC-J51
  • Definition 8: Engineering System NetSchoonenberg:2022:ISC-J50
  • Definition 9: Engineering System Net State Transition FunctionSchoonenberg:2022:ISC-J50
  • Definition 10: Operand NetFarid:2008:IEM-J04Schoonenberg:2019:ISC-BK04Khayal:2017:ISC-J35Schoonenberg:2017:IEM-J34
  • ...and 5 more