[Paper Review] Co-simulation of Continuous Systems: A Tutorial

Cláudio Gomes, Casper Thule+3 more|arXiv (Cornell University)|Sep 22, 2018

Simulation Techniques and Applications16 references6 citations

TL;DR

This tutorial presents a comprehensive introduction to co-simulation of continuous systems, explaining how heterogeneous models from different domains—such as mechanical, electrical, and control systems—can be integrated and simulated together using orchestrator algorithms. It details numerical methods, communication strategies, and stability analysis, offering practitioners a foundation to debug and optimize co-simulation setups with key insights into convergence, consistency, and numerical accuracy across diverse simulation units.

ABSTRACT

Co-simulation consists of the theory and techniques to enable global simulation of a coupled system via the composition of simulators. Despite the large number of applications and growing interest in the challenges, the field remains fragmented into multiple application domains, with limited sharing of knowledge. This tutorial aims at introducing co-simulation of continuous systems, targeted at researchers new to the field.

Motivation & Objective

• To provide researchers and practitioners with a foundational understanding of co-simulation for continuous systems, especially in the context of complex, multi-domain engineering systems.
• To explain the challenges in integrating independently developed models due to tool heterogeneity, intellectual property constraints, and differing numerical characteristics.
• To demonstrate how co-simulation enables early and frequent integration of partial solutions, reducing development costs and improving system validation.
• To equip readers with the ability to analyze and improve co-simulation performance by understanding tradeoffs between different orchestration strategies.
• To bridge the gap between modeling tools and holistic system simulation by formalizing co-simulation as a composition of black-box simulation units with defined input/output behaviors.

Proposed method

• Modeling each partial system as a black-box simulation unit with its own model, solver, input approximation, and input/output reactivity.
• Using an orchestrator to coordinate simulation units by setting inputs, retrieving outputs, and advancing time in discrete communication steps.
• Employing input approximation functions (e.g., constant, linear, extrapolation) to predict inputs between communication steps, enabling asynchronous simulation.
• Applying the Jacobi and Gauss-Seidel orchestration algorithms to iteratively correct interface values and improve consistency in co-simulation.
• Formalizing co-simulation steps using matrix equations to represent state transitions, incorporating system matrices and coupling terms (e.g., $C_1, C_2, D_1, D_2$).
• Analyzing stability and convergence through iterative fixed-point iterations and matrix-based error propagation models, with theoretical grounding in [17, 14, 13, 21, 3].

Experimental results

Research questions

• RQ1 How can heterogeneous models from different domains be co-simulated when they use different tools, solvers, and numerical methods?
• RQ2 What are the key tradeoffs between different co-simulation orchestration algorithms, such as Jacobi and Gauss-Seidel, in terms of convergence and stability?
• RQ3 How do input approximation functions and communication step size affect the accuracy and consistency of co-simulation results?
• RQ4 What mathematical framework enables the analysis of co-simulation convergence and stability across multiple simulation units?
• RQ5 How can co-simulation be used to integrate physical prototypes, software components, and human operators in a single simulation environment?

Key findings

• The Jacobi and Gauss-Seidel orchestration algorithms can be formalized using matrix equations that describe state transitions and iterative corrections across simulation units.
• The use of input approximation functions introduces errors that accumulate over time, and these must be accounted for in the co-simulation's consistency and accuracy analysis.
• Co-simulation convergence is influenced by the interplay between internal solver steps, communication step size, and the choice of input approximation, making it a complex but analyzable system.
• Theoretical stability and convergence of co-simulation methods are grounded in fixed-point iteration and matrix-based error propagation, as formalized in [17, 14, 13, 21, 3].
• The framework enables the analysis of co-simulation as a dynamic system, allowing practitioners to assess numerical behavior and improve performance through parameter tuning.