Abstract
Co-Simulation of complex subsystems can lead to stiff problems, where, especially for applied explicit numerical schemes, small communication step sizes are mandatory to obtain stable and accurate results. Due to the utilization of subsystem information, i.e. interface-jacobians or partial derivatives, it is possible to increase the stability and accuracy of the overall co-simulation. The herein proposed co-simulation coupling method therefore performs three major steps: a global approximation of the monolithic solution utilizing an introduced Error Differential Equation; a global model-based extrapolation over the next communication step and, finally, a local pre-step input optimization is carried out for all subsystems. This method is validated along a two degree-of-freedom oscillator benchmark example. As for realistic use cases it is difficult to access interface-jacobians, system identification methods are applied for approximation. Associated interface-jacobian approximation errors are investigated with respect to the performance and especially the stability of the overall co-simulation.
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Notes
- 1.
A necessary assumption for this subsystem description is that the in- and outputs of a subsystem are continuous differentiable, which means e.g. that hybrid systems are not considered.
- 2.
This assumption is not a restrictive one because, this subsystem description will only be used in the context of co-simulation, where the subsystems are typically black boxes, which implies, that the number of states is unknown and therefore one can always choose the number of states as the number of outputs.
- 3.
Equilibrium point of an ordinary differential equation means that the right-hand side is zero.
- 4.
The size of the differential equation is based on the overall number of outputs of all subsystems, for this derivation, due to the assumptions, it is fixed to two.
- 5.
The terms \(\epsilon _1\) and \(\epsilon _2\) are denoted as coupling errors because they describe the deviation of the coupled signals, \(u_1=y_2\) and \(u_2=y_1\) at the communication points, over a communication step, see Fig. 1.
- 6.
\((I-\hat{B})\) is regular because of the zero-stability assumption, analogous to \((I-\tilde{C})\) before.
- 7.
If one wants to utilize equations (7) the number of states has to be chosen equal to the number of outputs.
- 8.
The choice of this threshold depend among others on the excitation, the dynamics of the subsystem and the accuracy requirement of the overall co-simulation.
- 9.
It should be mentioned that the name Error Differential Equation is kept although the equation is now an algebraic equation instead of an differential equation.
- 10.
Like in the continuous-time case this computation can be carried independently for every subsystem and therefore the subindex i is omitted.
- 11.
The force-displacement coupling has been chosen because, so no apperance of an algebraic loop is ensured and the zero-stability is guaranteed, see therefore [9].
- 12.
Due to the fact that, there is only one output but two states in \(S_2\) the Moore-Penrose inverse is utilized, instead of the classical inverse, to compute \(C_2^{-1}\).
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Acknowledgements
The publication was written at VIRTUAL VEHICLE Research Center in Graz, Austria. The authors would like to acknowledge the financial support of the COMET K2 – Competence Centers for Excellent Technologies Programme of the Federal Ministry for Transport, Innovation and Technology (bmvit), the Federal Ministry for Digital, Business and Enterprise (bmdw), the Austrian Research Promotion Agency (FFG), the Province of Styria and the Styrian Business Promotion Agency (SFG).
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Genser, S., Benedikt, M. (2020). A Pre-step Stabilization Method for Non-iterative Co-Simulation and Effects of Interface-Jacobians Identification. In: Obaidat, M., Ören, T., Rango, F. (eds) Simulation and Modeling Methodologies, Technologies and Applications. SIMULTECH 2018. Advances in Intelligent Systems and Computing, vol 947. Springer, Cham. https://doi.org/10.1007/978-3-030-35944-7_6
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