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

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

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

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

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… Motivation --3 Northeastern University NP-hard Is the model identified valid for other data? Model Invalidation Problem

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

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

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

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

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

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

Northeastern University Convex (In)validation Certificates 1 Sparsification Based Approach --3 Results of the Sparsification AlgorithmInterpretation InfeasibleInvalidated Feasible,Not invalidated Feasible, ?

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

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

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

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

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

Extension to Systems with Structural Constraints Northeastern University Time instant

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

Examples --1 Northeastern University Given submodels without structural constraints and the measurement equation Actual A Priori Information Results using sparsificationfeasible, InterpretationNot invalidatedno decision Time (sec.) Results using Moments e e-07 Interpretationnot invalidated Times (sec.)

Examples --1 Northeastern University Actual A Priori Information Results using sparsificationinfeasible Interpretationinvalidated Time (sec.) Results using Moments Interpretationinvalidated Times (sec.) Given submodels without structural constraints

Northeastern University Examples --2 run walk

Examples --2 Northeastern University Algorithmsresults Results using Sparsification with ConstraintsInfeasible InterpretationInvalidated Time (sec.) Results using Moments with Constraints InterpretationInvalidated Times (sec.)1.2968e03 A Priori informationExperimental Data time

Algorithmsresults Results using Sparsification with ConstraintsFeasible, InterpretationNot Invalidated Time (sec.) Results using Moments with Constraints e-08 InterpretationNot Invalidated Times (sec.)0.8407e03 Examples --2 Northeastern University A Priori informationExperimental Data time

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

Conclusions Northeastern University The model (in)validation problem for switched ARX systems with unknown switches: Given Is there any consistency set ?  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