Stable Adaptive Co-simulation: A Switched Systems Approach. Cláudio Gomes, Benoît Legat, Raphaël M. Jungers, Hans Vangheluwe

Transcription

1 Stable Adaptive Co-simulation: A Switched Systems Approach Cláudio Gomes, Benoît Legat, Raphaël M. Jungers, Hans Vangheluwe

2 Agenda 1. Background 2. Adaptive Orchestration 3. Contribution 4. Conclusion 2

3 1. Background Motivation and definition of co-simulation 3

4 Why Co-simulation? Tool interoperability Veitl, A., & Arnold, M. (1999). Coupled simulation of multibody systems and elastic structures. Advances in Computational Multibody Dynamics, Multi-rate Parallelism Newton, A. R., & Sangiovanni-Vincentelli, A. L. (1983). Relaxation-Based Electrical Simulation. SIAM Journal on Scientific and Statistical Computing, 4(3), Gomes, C., Thule, C., Broman, D., Larsen, P. G., & Vangheluwe, H. (2017). Co-simulation: State of the art. Retrieved from 4

5 Running Example Original System Busch, M. (2016). Continuous approximation techniques for co-simulation methods: Analysis of numerical stability and local error. ZAMM - Journal of Applied Mathematics and Mechanics, 96(9),

6 Running Example Co-simulation Orchestrator 6

7 Orchestration Orchestrator getoutput( ) setinput( ) Orchestrator getoutput( ) setinput( ) simulateuntil(t+h, ) simulateuntil(t+h, ) t := t + H t t+h t t+h 7

8 Internal Behavior Orchestrator getoutput( ) setinput( ) Orchestrator getoutput( ) setinput( ) simulateuntil(t+h, ) simulateuntil(t+h, ) t t+h1 t+h t := t + H t t+h2 t+h 8

9 2. Adaptive Orchestration Why? What? And How? 9

10 Simulator Internals Model Solver Input Approximation Orchestrator 10

11 Orchestration Space Inputs Model Solver Input Approximation Busch, M., & Schweizer, B. (2011). Stability of Co-Simulation Methods Using Hermite and Lagrange Approximation Techniques. In ECCOMAS Thematic Conference on Multibody Dynamics (pp. 1 10). Brussels, Belgium. Orchestrator Stettinger, G., Horn, M., Benedikt, M., & Zehetner, J. (2014). Model-based coupling approach for non-iterative real-time co-simulation. In 2014 European Control Conference (ECC) (pp ). Input Approximations Polynomial 0 Extrapolation Interpolation Polynomial 1 Context-aware C 0 Continuous C 1 Continuous Model ID ed Ben Khaled-El Feki, A., Duval, L., Faure, C., Simon, D., & Ben Gaid, M. (2017). CHOPtrey: contextual online polynomial extrapolation for enhanced multicore co-simulation of complex systems. SIMULATION, Burden, R. L., & Faires, J. D. (2010). Numerical Analysis (9th ed.). Cengage Learning. 11

12 Orchestration Space Solvers Model Solver Input Approximation Orchestrator Numerical Solvers Parallel Sequential Order 0 Order 1 Implicit Semi-Explicit Step size Explicit Hairer, E., & Wanner, G. (1996). Solving ordinary differential equations II: Stiff and differentialalgebraic problems. 12

13 Orchestration Space Synchronization Arnold, M., Clauß, C., & Schierz, T. (2014). Error Analysis and Error Estimates for Co-simulation in FMI for Model Exchange and Co- Simulation v2.0. In S. Schöps, A. Bartel, M. Günther, W. E. J. ter Maten, & C. P. Müller (Eds.), Progress in Differential-Algebraic Equations (pp ). Berlin, Heidelberg: Springer Berlin Heidelberg. Model Solver Input Approximation Schweizer, B., & Lu, D. (2015). Predictor/corrector co-simulation approaches for solver coupling with algebraic constraints. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik Und Mechanik, 95(9), Orchestrator Orchestration Algorithms Parallel Sequential Jacobi Gauss-Seidel Implicit Semi-Explicit Step size Explicit Gomes, C., Thule, C., Broman, D., Larsen, P. G., & Vangheluwe, H. (2017). Co-simulation: State of the art. Retrieved from 13

14 Capability Interaction Model Solver Input Approximation Orchestrator Input Approximations Polynomial 0 Extrapolation Polynomial 1 Interpolation Orchestration Algorithms C 0 Continuous Parallel Sequential Jacobi Gauss-Seidel Implicit Semi-Explicit Step size Explicit 14

15 Adaptive Orchestration Model Forward Euler: h=0.04 Midpoint: h=0.01 Model Forward Euler: h=0.2 h=0.1 Constant Extrapolation Constant Extrapolation Jacobi: H=0.2 H=0.1 15

16 Adaptive Orchestration Model Forward Euler: h=0.04 Midpoint: h=0.01 Model Forward Euler: h=0.2 h=0.1 Constant Extrapolation Constant Extrapolation Performance Forward Euler: h=0.04 Forward Euler: h=0.2 Jacobi: H=0.2 H=0.1 H=0.2 Research problem: Is this policy stable? Midpoint: h=0.01 Forward Euler: h=0.1 H=0.1 Stability 16

17 3. Contribution Certification of adaptive orchestration algorithms 17

18 Non-adaptive Stability Analysis Model Forward Euler: h=0.04 Model Forward Euler: h=0.2 Constant Extrapolation Constant Extrapolation Jacobi: H=0.2 Busch, M., & Schweizer, B. (2010). Numerical stability and accuracy of different co-simulation techniques: analytical investigations based on a 2-DOF test model. In 1st Joint International Conference on Multibody System Dynamics (pp ). 18

19 Stability definition Non-adaptive: Adaptive: for any permutation 19

20 Non-adaptive Stability Analysis Theys, J. (2005). Joint Spectral Radius: theory and approximations. Hamilton Institute (Ireland) Upper bound: k upper bound

21 Adaptive Cosim Stability Analysis Stability: for any permutation Stable? Upper bound can be computed: Ahmadi, A. A., Jungers, R. M., Parrilo, P. A., & Roozbehani, M. (2014). Joint Spectral Radius and Path-Complete Graph Lyapunov Functions. SIAM Journal on Control and Optimization, 52(1), Parrilo, P. A., & Jadbabaie, A. (2008). Approximation of the joint spectral radius using sum of squares. Linear Algebra and Its Applications, 428(10), Jungers, R. (2009). The joint spectral radius: theory and applications (Vol. 385). Springer Science & Business Media. 21

22 Example Performance Forward Euler: h=0.04 Forward Euler: h=0.2 H=0.2 Midpoint: h=0.01 Forward Euler: h=0.1 H=0.1 Accuracy 22

23 Example Performance Forward Euler: h=0.04 Forward Euler: h=0.2 H=0.2 Legat, B., Jungers, R. M., & Parrilo, P. A. (2016). Generating Unstable Trajectories for Switched Systems via Dual Sum-Of-Squares Techniques. In Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control - HSCC 16 (pp ). New York, New York, USA: ACM Press. Midpoint: h=0.01 Forward Euler: h=0.1 H=0.1 Accuracy 23

24 Example Performance Forward Euler: h=0.04 Forward Euler: h=0.2 H=0.2 Midpoint: h=0.01 Forward Euler: h=0.1 H=0.1 Accuracy 24

25 3. Conclusion Summary, limitations, and future work 25

26 Summary of the Approach 1. Capture possible orchestration decisions IP can be protected if solver is embedded 26

27 Summary of the Approach 1. Capture possible orchestration decisions IP can be protected if solver is embedded 2. Check stability of unrestricted adaptive orchestration 27

28 Summary of the Approach 1. Capture possible orchestration decisions IP can be protected if solver is embedded 2. Check stability of unrestricted adaptive orchestration 3. Restrict if needed 28

29 Limitations & Future Work Scalability How to efficiently restrict the orchestration policy? Optimization of policies, subject to constraints E.g., real-time constraints 29

30 Thank you! Questions? 30

31 Appendix 31

32 Tackling Complexity Scale Heterogeneity Market Pressure Van der Auweraer, H., Anthonis, J., De Bruyne, S., & Leuridan, J. (2013). Virtual engineering at work: the challenges for designing mechatronic products. Engineering with 32 Computers, 29(3),

33 Simulation is not Enough Original System: Tool Specialization: Actuators Actuators Overture Plant Control Plant Control Sensors Sensors Co-simulation 33

34 Simulation is not Enough Tool Specialization: Refinement: Actuators Actuators Plant Control Plant Control Sensors Sensors Co-simulation34

35 Adaptive Cosim Stability Analysis At each co-simulation step: Stability: for any permutation 35