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

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 Observer design Identification
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 persistence of excitation condition in lifted space
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
We see that the last equation does not hold if the model only predicts 1 state and there happen to be more than 1- what kind of restrictions does the condition place on the mode sequences? If the persistance of excitation for SARX models holds, then mode sequences are persistently exciting. A mode sequence is called persistently exciting if there is an S such that for all j >=max(na, nc), the equation holds.

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. Temporal video segmentation
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
dynamics
appearance
images x t + 1 = A v y C w

22. Temporal video segmentation
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
dynamics
appearance
images x t + 1 = A v y C w

23. Segmentation of moving dynamic textures
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