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1. 51st IEEE Conference on Decision and Control A Convex Optimization Approach to Model (In)validation of Switched ARX Systems with Unknown Switches Yongfang Cheng1, Yin Wang1, Mario Sznaier1, Necmiye Ozay2, Constantino M. Lagoa3 1Department of Electrical and Computer Engineering Northeastern University, Boston, MA, USA 2Department of Computing and Mathematical Sciences Caltech, Pasadena, CA, USA 3Department of Electrical Engineering Penn State University, University Park, PA, USA Northeastern University

2. Outlines • Motivation • Problem Statement • Convex (In)validation Certificates • Sparsification Based Approach • Moments Based Approach • Extension to Systems with Structural Constraints • Examples • Conclusions Northeastern University

3. Cooling Air Conditioner Motivation --1 Powersaving Traffic Heating Can be modeled by hybrid systems with both continuous dynamics and discrete dynamics Northeastern University

4. Switched ARX System Motivation --2 are mode signal which can switch arbitrarily amongst subsystems, . Northeastern University

5. Motivation --3 NP-hard Switched ARX System Identification Given: Order of Submodels ; Number of submodels ; A priori bound on noise ; Experimental data Find: A piecewise linear affine model such that Solutions based on: Heuristics, Optimization, Probabilistic Priors, Convex Relaxation… Model Invalidation Problem Is the model identified valid for other data? Northeastern University

6. Outlines • Motivation • Problem Statement • Convex (In)validation Certificates • Sparsification Based Approach • Moments Based Approach • Extension to Systems with Structural Constraints • Examples • Conclusions Northeastern University

7. Model Invalidation of Switched ARX System Problem Statement --1 Given Parameters of submodels of a hybrid system: A priori bound on noise Experimental data Determine Whether the consistency set is nonempty, where Northeastern University

8. Problem Statement --2 not invalidated at t Hybrid decoupling constraint is feasible. Invalidated Not Invalidated NONCONVEX PROBLEM, HOW TO SOLVE? Northeastern University

9. Outlines • Motivation • Problem Statement • Convex (In)validation Certificates • Sparsification Based Approach • Moments Based Approach • Extension to Systems with Structural Constraints • Examples • Conclusions Northeastern University

10. is feasible. is feasible. Convex (In)validation Certificates 1Sparsification Based Approach --1 is feasible. Relaxation of cardinality Re-weighted heuristic Sparsification Based (In)validation Certificates Northeastern University

11. Sparsification Based (In)validation Certificates Solve iteratively Convex (In)validation Certificates 1Sparsification Based Approach --2 Update the weight by Northeastern University

12. Convex (In)validation Certificates 1Sparsification Based Approach --3 Northeastern University

13. Outlines • Motivation • Problem Statement • Convex (In)validation Certificates • Sparsification Based Approach • Moments Based Approach • Extension to Systems with Structural Constraints • Examples • Conclusions Northeastern University

14. The model is not invalidated Convex (In)validation Certificates 2Moments Based Approach --1 is feasible. • The model is invalidated Possible to use Positivstellensatz to get invalidation certificates. However, easier to utilize problem structure via moment-based polynomial optimization. Northeastern University

15. A Moments-based Relaxation Convex (In)validation Certificates 2Moments-Based Approach --2 • The model is invalidated if and only if There exists an N such that • The set is nonempty if and only if There exists an N such that and (the moment matrix hits the flat extension) Specifically, in our problem, T+1 is a choice for N. Northeastern University

16. A Moments-based Relaxation Problem has a sparse structure (running intersection property holds) Convex (In)validation Certificates 2Moments-Based Approach --3 Northeastern University

17. Outlines • Motivation • Problem Statement • Convex (In)validation Certificates • Sparsification Based Approach • Moments Based Approach • Extension to Systems with Structural Constraints • Examples • Conclusions Northeastern University

18. Time instant Extension to Systems with Structural Constraints Northeastern University

19. Outlines • Motivation • Problem Statement • Convex (In)validation Certificates • Sparsification Based Approach • Moments Based Approach • Extension to Systems with Structural Constraints • Examples • Conclusions Northeastern University

20. Given submodels without structural constraints and the measurement equation Examples --1 Northeastern University

21. Given submodels without structural constraints Examples --1 Northeastern University

22. Examples --2 run walk Northeastern University

23. A Priori information Experimental Data time Examples --2 Northeastern University

24. A Priori information Experimental Data time Examples --2 Northeastern University

25. Outlines • Motivation • Problem Statement • Convex (In)validation Certificates • Sparsification Based Approach • Moments Based Approach • Extension to Systems with Structural Constraints • Examples • Conclusions Northeastern University

26. The model (in)validation problem for switched ARX systems with unknown switches: Given Is there any consistency set ? Conclusions • Sparsification Based (In)validation Certificates • Sufficient Conditions • Moments Based (In)validation Certificates • Necessary and Sufficient Conditions • A finite order of moment matrix, N=T+1, is found to • hit the flat extension • Extension to systems with constraints on switch sequences Northeastern University