Co-Simulation

The system is decomposed into constituent subsystems (e.g., mechanical, electrical, hydraulic, thermal), each modeled in its optimal domain-specific tool. These models are not merged into a single monolithic solver. Instead, they are coupled through a master algorithm that orchestrates the exchange of variables at their interfaces. This preserves the high-fidelity physics of each domain and allows engineering teams to work concurrently with their preferred software, such as using Simulink for controls and Adams for multi-body dynamics.

Unlike a continuous-time monolithic solver, co-simulation advances by exchanging data at discrete communication points or macro time steps. Between these points, each solver integrates its internal states independently. The master algorithm manages this handshake, providing input values at the start of a step and collecting outputs at the end. This synchronization is critical; the choice between explicit (Jacobi) and implicit (Gauss-Seidel) coupling schemes directly impacts numerical stability and accuracy.

Co-simulation relies on standardized interfaces to encapsulate solver internals. The dominant standard is the Functional Mock-up Interface (FMI). A tool exports its model as a Functional Mock-up Unit (FMU), a self-contained package containing the dynamic equations and a solver. The FMI standard defines the API for the master algorithm to initialize, step, and reset the FMU without needing to know its internal mathematics. This enables true tool-independent integration.

A dedicated master algorithm is the central conductor of the co-simulation. It is responsible for:

• Synchronization: Managing the communication time grid.
• Input/Output Mapping: Routing signals between the correct output and input ports of different FMUs.
• Error Control: Implementing step-size control and detecting algebraic loops that require iterative correction.
• Data Logging: Capturing all exchanged variables for post-processing and verification. This algorithm can be a simple script or a sophisticated real-time controller.

The FMI standard defines two distinct modes of integration:

• Co-Simulation Mode: Each FMU contains its own internal solver. The master algorithm only controls the communication step size. This is the most common mode for coupling black-box models from different vendors.
• Model Exchange Mode: The FMU provides only the mathematical equations (ODEs/DAEs) without a solver. A single external solver in the master algorithm integrates the entire coupled system. This mode offers higher numerical stability for stiff systems but requires the master to have a capable solver.

Co-simulation can be executed in two distinct temporal domains:

• Non-Real-Time (Offline): The simulation runs as fast as the CPU allows, used for system design, parameter sweeping, and what-if analysis. A 10-second maneuver might simulate in milliseconds.
• Real-Time (Hardware-in-the-Loop): The simulation must advance one second of model time for every second of wall-clock time. This is mandatory when a physical controller or human operator is in the loop, requiring strict deterministic execution to avoid overruns.

The central solver responsible for coordinating the time progression and data exchange between coupled subsystem models. The master algorithm manages critical tasks:

• Step size negotiation: Determining the communication interval between subsystems
• Data interpolation: Extrapolating values when models request data at different time points
• Error control: Monitoring coupling errors introduced by sequential solving
• Jacobian computation: Calculating sensitivity derivatives for implicit coupling schemes Common approaches include Gauss-Seidel (sequential, fast) and Jacobi (parallel, more stable) iteration. The choice of master algorithm directly impacts simulation stability and computational performance.

A specialized co-simulation configuration where a physical controller (the hardware under test) interacts with a real-time virtual model of the plant. Unlike pure software co-simulation, HIL requires hard real-time execution—the simulation must compute its outputs within strict timing deadlines to avoid misleading the physical controller. This technique is essential for validating ECUs in automotive, flight controllers in aerospace, and PLCs in manufacturing before deployment. HIL catches integration faults that pure software simulation misses, such as signal latency, electrical noise, and timing jitter.

Coupling subsystems introduces numerical artifacts not present in monolithic simulation. Key challenges include:

• Aliasing: High-frequency dynamics in one subsystem can be misrepresented at the communication step rate
• Energy errors: Explicit coupling schemes can artificially inject or dissipate energy at the interface, causing non-physical behavior
• Algebraic loops: Circular dependencies where Subsystem A needs Subsystem B's output, but B simultaneously needs A's output Mitigation strategies include implicit coupling (iterating until convergence at each step), energy-preserving interfaces, and variable step-size control. Understanding these phenomena is critical for trusting co-simulation results.

The application of co-simulation to validate control code against a digital model of the physical machine before installation. The PLC program connects to a simulated I/O layer that mirrors the behavior of sensors and actuators. This enables:

• Offline debugging: Finding logic errors without risking physical equipment
• Cycle time validation: Verifying that sequences complete within required takt times
• Exception testing: Injecting fault conditions that would be dangerous or expensive to create physically Virtual commissioning can reduce on-site commissioning time by up to 80% and is a cornerstone of modern mechatronic system development.

FMI defines two fundamentally different coupling paradigms:

Model Exchange: The FMU provides the model equations (ODEs/DAEs) but no solver. The importing environment integrates the entire coupled system monolithically. This yields higher accuracy but requires all models to be compatible with the master solver's numerical methods.

Co-Simulation: Each FMU contains its own embedded solver. The master algorithm only coordinates data exchange at discrete communication points. This enables coupling of models with incompatible mathematics (e.g., continuous-time mechanical systems with discrete-event controllers) but introduces coupling errors at the interfaces.