Observability, Observer Design and Identification of Hybrid Systems

1. Observability, Observer Design and Identification of Hybrid Systems Rene Vidal Aleksandar Juloski Yi Ma

2. Observability, Observer Design and Identification of Hybrid Systems • 09:50-10:10, Existence of Discrete State Estimators for Hybrid Systems on a Lattice (I)Del Vecchio, Domitilla, Murray, Richard M. California Inst. of Tech. • 10:10-10:30, Observability of Hybrid Systems and Turing Machines. Collins, Pieter, van Schuppen, Jan H. Cent. voor Wiskunde en Informatica • 10:30-10:50, A Bayesian Approach to Identification of Hybrid Systems. Juloski, Aleksandar, Weiland, Siep, Heemels, Maurice Eindhoven Univ. of Tech. • 10:50-11:10, Data Classification and Parameter Estimation for the Identification of Piecewise Affine Models. Bemporad, Alberto, Garulli, Andrea, Paoletti, Simone, Vicino, Antonio, Univ. of Siena • 11:10-11:30, On the Observability of Piecewise Linear Systems Babaali, Mohamed Univ. of Pennsylvania, Egerstedt, Magnus Georgia Inst. of Tech. • 11:30-11:50, Recursive Identification of Switched ARX Hybrid Models: Exponential Convergence and Persistence of Excitation Vidal, Rene Johns Hopkins Univ. Anderson, Brian D.O. Australian National Univ.

3. Recursive Identification of Switched ARX Hybrid Models:Exponential Convergence and Persistence of Excitation René Vidal Brian D.O. Anderson Center for Imaging Science Dept. of Inf. Engineering Johns Hopkins University Australian National Univ. & National ICT Australia

4. Given input/output data generated by a hybrid system, identify Number of discrete states Model parameters Hybrid state (continuous & discrete) Motivation: dynamic vision & hybrid ID Video segmentation Dynamic textures Gait recognition

5. Main challenges • Challenging “chicken-and-egg” problem • Given switching times, estimate model parameters • Given the model parameters, estimate hybrid state • Given all above, estimate switching parameters • Iterate via Expectation Maximization (EM) • Very sensitive to initialization • Prior work on hybrid system identification • Mixed-integer programming: (Bemporad et al. ’01) • K-means Clustering + Regression+ Classification + Iterative Refinement: (Ferrari-Trecate et al. ’03) • Greedy/iterative approach: (Bemporad et al. ‘03) • We proposed an algebraic geometric approach to the identification of switched ARX models (CDC’03) • Number of models =degree of a polynomial • Model parameters =roots (factors) of a polynomial

6. Related work • Observability • Mixed-integer linear program for PWAS (Bemporad et al. ’00) • Rank tests for JLS (Vidal et al. ’02 ’03, Babaali and Egerstedt ’04, Collins and Van Schuppen et al. ’04) • Observer design • Luenberger observers (Alessandri and Colleta ’01) • Location + Luenberger observers (Balluchi et al. ’02) • Moving horizon estimator via mixed-integer quadratic programming (Ferrari-Trecate et al. ’01) • Observers on lattices (Del Vecchio and Murray ’03 ’04) • Identification • Mixed-integer programming: (Bemporad et al. ’01) • K-means Clustering + Regression+ Classification + Iterative Refinement: (Ferrari-Trecate et al. ’03) • Greedy/iterative approach: (Bemporad et al. ‘03) • Algebraic geometric: (Vidal et al. ’03 ’04)

7. Our approach to hybrid system ID • Most existing methods are batch • Collect all input/output data • Identify model parameters using all data • Not suitable for online/real time operation • Contributions • Recursive identification algorithm for Switched ARX • No restriction on switching mechanism • Does not depend on value of the discrete state • Based on algebraic geometry and linear system ID • Key idea: identification of multiple ARX models is equivalent to identification of a single ARX model in a lifted space • Persistence of excitation conditions that guarantee exponential convergence of the identified parameters

8. Our approach: recursive hybrid ID • Most existing methods are batch • Collect all input/output data • Identify model parameters using all data • Not suitable for online/real time operation • Contributions of this paper: recursive identification algorithm for Switched ARX • No restriction on switching mechanism • Based on algebraic geometry and linear system ID • Provides persistence of excitation conditions that guarantee exponential convergence of the identifier

9. Problem statement • The dynamics of each mode are 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

10. Recursive identification of ARX models • True model parameters • Equation error identifier • Persistence of Excitation:

11. Recursive identification of SARX models • Identification of a SARX model is equivalent to identification of a single lifted ARX model • Can apply equation error identifier and derive persistenceofexcitationconditioninliftedspace Embedding Lifting Embedding

12. Decoupling identification from mode estimation • The hybrid decoupling polynomial • 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 Number of regressors Number of models

13. Recursive identification of hybrid model params • Recall equation error identifier for ARX models • Equation error identifier for SARX models

14. Recursive identification of ARX model params ARX Models by Polynomial Differentiation

15. Exponential convergence of the hybrid identifier • Recall persistence of excitation for ARX models • Persistence of excitation for SARX models • Convergence of individual ARX model parameters

16. Sufficiently exciting mode sequences • Choice of models • Number of times mode i is visited • Persistently exciting mode sequences • Condition is sufficient but not necessary

17. Experimental Results – noiseless data

18. Experimental Results – noiseless data

19. Experimental Results – noisy data

20. Conclusions and open issues • Contributions • A recursive identification algorithm for hybrid ARX models of equal and known orders • A persistence of excitation condition on the input/output data that guarantees exponential convergence • Open issues • Persistence of excitation condition on the mode and input sequences only • Recursive algorithm for identifying the parameters of SARX models of unknown and different orders • Extend the model to multivariate SARX models

21. Segmenting N=30 frames of a sequence containing n=3 scenes Host Guest Both Image intensities are output of linear system Apply GPCA to fit n=3 observability subspaces A + x x v = 1 + t t t C + dynamics y x w = t t t images appearance Temporal video segmentation

22. Segmenting N=60 frames of a sequence containing n=3 scenes Burning wheel Burnt car with people Burning car Image intensities are output of linear system Apply GPCA to fit n=3 observability subspaces A + x x v = 1 + t t t C + dynamics y x w = t t t images appearance Temporal video segmentation

23. Time varying model Segmentation of multiple moving subspaces Apply PCA to intensities in a moving time window Apply GPCA to projected data dynamics images appearance Segmentation of moving dynamic textures