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Recursive Identification of Switched ARX Hybrid Models: Exponential Convergence and Persistence of Excitation. Presentation Outline. Motivation Introduction to observation and identification problems Problem description Theory behind recursive algorithm Experimental Results Future work.

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Presentation Outline

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  1. Recursive Identification of Switched ARX Hybrid Models: Exponential Convergence and Persistence of Excitation

  2. Presentation Outline • Motivation • Introduction to observation and identification problems • Problem description • Theory behind recursive algorithm • Experimental Results • Future work

  3. Motivation • Previous work on hybrid systems • Modeling, analysis, stability • Control: reachability analysis, optimal control • Verification: safety • In applications, one also needs to worry about observability and identifiability

  4. Linear Systems state input output Hybrid System

  5. System Observation and Identification

  6. Given input/output data, identify Number of discrete states Model parameters of linear systems Hybrid state (continuous & discrete) Switching parameters (partition of state space) Problem Description and Challenges “Chicken-and-egg” problem

  7. Approach • Recursive identification algorithm for Switched Auto Regressive Exogenous systems is proposed • Algebraic approach • Hybrid decoupling polynomial • Persistance of Excitation conditions • Model parameter estimation from homogeneous polynomial

  8. Problem statement • Assume that each linear systems is in ARX form • input/output • discrete state • order of the ARX models • model parameters • Input/output data lives in a hyperplane • I/O data • Model params

  9. Number of regressors Number of models Hybrid Decoupling Polynomial and Model Parameters • The hybrid decoupling constraint • Independent of the value of the discrete state • Independent of the switching mechanism • Satisfied by all data points: no minimum dwell time • The hybrid model parameters • Veronese map

  10. Recursive Identification of Hybrid Model Parameters • Recursive equation error identifier for • single minimal ARX model • Persistence of Excitation for input/output data

  11. Recursive Identification of Hybrid Model Parameters • Hybridequation error identifier • Persistence of Excitation for SARX models

  12. Restrictions on Mode Sequences Choice of modes Times when mode i is visited Persistently exciting mode sequence

  13. Convergence of model hybrid parameters (h) Generalizing Persistance of Excitation to SARX models Is it true that Persistently exciting mode sequences + Persistently exciting input/output data + Bounded output X

  14. Identifying Parameters of Individual ARX models Normalization factor Normalization factor

  15. Experimental Results – noiseless data Results: top to bottom, λtperiods of 2, 30 and 200 s, no noise h a, c

  16. Experimental Results – noisy data h a, c Results: top to bottom, λtperiod 30 s, σ = 0.02, 0.05

  17. Future Work and Open Problems • Produce a recursive algorithm for identifying the parameters of SARX models of unknown and different orders • Determine persistence of excitation conditions on the input and mode sequences only • Extend the model to multivariate SARX models

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