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

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

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

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

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.

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

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)

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

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

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

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

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

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)

Identifying Parameters of Individual ARX models Normalization factor Normalization factor

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

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

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