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

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 Hybrid System state input output
Consider the following system. This is a typical linear system representation, where the next state is a function of the current state and input, and the output is a function of the state and input. An example of a state: consider a chess game. The state at a particular time t reflects the location of all the chess pieces and the player who has the next turn.
Hybrid System

5. System Observation and Identification
Type of Problem
Known
Unknown
Observability
y(t), A(), B(), C(), D(), u(t)
x(t), λ(t)
Identification
y(t), u(t)
A(), B(), C(), D(), x(t), λ(t)
System Identification has numerous applications including process control, biomedicine, seismology, environmental systems, aircraft dynamics, signal processing, macroeconomy, etc.

6. Problem Description and Challenges
Given input/output data, identify
Number of discrete states
Model parameters of linear systems
Hybrid state (continuous & discrete)
Switching parameters (partition of state space)
Given switching times, can estimate model parameters
Given the model parameters, estimate hybrid state
Given all above, estimate switching parameters
Iterate
“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
A hybrid coupling polynomial is proposed- this is an algebraic representation of the system during all times (regardless of the state of the system).
After modeling the system in this fashion, persistence of excitation conditions are proposed; these guarantee the exponential convergence of the recursive identifier (these are natural generalizations of results from ARX models)

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. 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
By solving for the hybrid model parameters, we can solve for the ARX model parameters afterwards, using the same techniques as were described in CDC 2003
Number of regressors
Number of models

10. Recursive Identification of Hybrid Model Parameters
Recursive equation error identifier for
single minimal ARX model
Pi 1 = [I()K-1 0] and mu >0 produces an exponentially convergent estimate of the model parameters if the input output data is persistently exciting
It should be noted that the persistent excitement condition is satisfied if the input sequence is persistently exciting (see eqn 12)
Give an example for persistent excitement
This is nice, but how do we generalize for SARX models when we have a hybrid system? Create an analogous error identifier and condition for hybrid model parameters h
Persistence of Excitation for input/output data

11. Recursive Identification of Hybrid Model Parameters
Hybrid equation error identifier
In this case, we have convergence of the model parameters h
Persistence of Excitation for SARX models

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

13. Generalizing Persistance of Excitation to SARX models
Is it true that
Persistently exciting mode sequences +
Persistently exciting input/output data
Bounded output
Go through the example. Main difficulty in decoupling the persistence of excitation condition on I/O data generated by SARX model into n persistence of excitation conditions on mode sequence and I/O data generated by individual models is that eqn can potentially satisfy homogeneous polynomial ofdegree n other than that defining SARX model X
Convergence of model hybrid parameters (h)

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

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

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

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