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This presentation outlines the motivation, theory, experimental results, and future work related to recursively identifying switched ARX hybrid models with exponential convergence and persistence of excitation conditions. The algorithm proposed involves hybrid decoupling polynomials and Veronese map techniques.

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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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