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SLICOT system identification

software and applicationsVasile Sima

National Institute for Research & Development in Informatics,Bucharest (Romania)

[email protected]

Diana Maria Sima, Sabine Van HuffelDepartment of Electrical Engineering (ESAT), KU Leuven (Belgium)

diana.sima,sabine. [email protected] leuven.ac.be

AcknowledgmentsWork supported by the European Community BRITE-EURAM III

Thematic Networks Programme NICONET (project BRRT-CT97-5040).


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OVERVIEW1. Introduction2. SLICOT Identification Algorithms

3. Performance investigation of SLICOTsystem identification toolbox4. Conclusions


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System identification : Finding mathematical models of dynamic systems from measured data.

Increasing complexity of CACSD problems need forvery reliable and efficient algorithms and software for systemidentification.

SLIDENT : new, multivariable system identification toolbox,incorporated in the Fortran 77 S ubroutine Library in Co ntrolTheory ( SLICOT ).

1. Introduction

SLIDENT objectives:reliability,efficiency,ability to solve industrial

identification problems.

SLIDENT provides:drivers,computational routines,Matlab or Scilab

interfaces.


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SLIDENT Outline: Linear Systems

Wiener Systems (linear part + static nonlinearity)Linear systems: identified with subspace-based techniques:

state-space models are directly estimated;

no parameterizations are needed;

robust linear algebra tools (QR decomposition and SVD);

only one parameter has to be chosen.

Approaches:

MOESP (Multivariable Output Error state SPace)

N4SID (Numerical algorithm for Subspace State Space SystemIDentification)


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SLIDENT Outline: Linear Systems

Wiener Systems (linear part + static nonlinearity)

Wiener Systems:

linear part parameterized with output normal form (ONF);

nonlinearity modeled by a neural network with one hiddenlayer.

Parameters are estimated by a tailored Levenberg-Marquardt(LM) algorithm.

Implementations for LM:

widentc - using Cholesky factorization or CG algorithm;

wident - MINPACK-like, LAPACK-based, structureexploiting implementation, using QR factorization with

block-column pivoting.


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SLIDENT features

flexibility of usage: various options, e.g., MOESP and N4SIDapproaches + their combination;

standard or fast techniques for data compression (includingabilities to exploit the block-Hankel structure);

multiple (possibly connected) data batches processing;

availability of fully documented drivers and computationalroutines;

structure exploiting algorithms and dedicated linear algebratools;

optional assessment of the intermediate results byinspecting the associated condition numbers.


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2. SLICOT Identification Algorithms

Mathematical problem descriptionWiener system state space representation:

If f () is the identity function, the system reduces to a

standard linear model .Identification problem for the linear model:

Find order n and system matrices A , B , C , D using

s>n and I/O data sequences {u ( k)} and {y( k)}, k = 1:t .


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1. Build I/O data matrix H as two concatenated

block Hankel matrices;

2. Factorize H = QR for data compression (Q not needed) ;3. Use matrix R and an SVD to compute the system

order n and the system matrices;

4. Covariance matrices residuals LS problem;5. Kalman gain matrix solving a DARE.

Linear Subspace-based System Identification

Implemented steps:


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Linear Subspace-based System Identification

Implemented options:

Data processing: non-sequential or sequential;

Fast algorithms for getting the factor R :

structure-exploiting Cholesky factorization of H T H ;

fast QR algorithm, based on the generalized Schuralgorithm.

when fast algorithms fail, QR factorization isautomatically used.


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Wiener System IdentificationApproach

state space representation for linear part;

single layer neural network (NN) for nonlinear part.

Steps :

1. Estimating linear part : fast subspace identificationalgorithm based on available I/O data.

2. Estimating nonlinear part : Levenberg-Marquardt(LM) algorithm.

3. Estimating whole system : structure-exploitingLAPACK-based LM algorithm.


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Wiener System Identification

- Done as if there are no nonlinearities.

- Obtain n , ( A , B , C , D ) and x(1).

Step 1. Estimating linear part


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Step 2. Estimating nonlinear part

- Compute estimated linear output, .

- Compute initial estimate, nonlinear model,

Parameter vector nl built by stacking

Solve NLS, .))((

min2

1nl

N

k k k z f y

)()1,(),()(),,(),(

:)())((

))((

1 1

k sbisbk z jisis

k k z f k z f

si

l

j j

sss

).1,(and),(),,,(),,( sbisb jisis


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Step 3. Estimating whole system

1. Use nl for nonlinear part, and l for linear part,where l obtained from ONF parameterization of

( A , C ) , B , D and x (1). := [ nlT

lT]

T.

2. Estimate the whole model, solving NLS

.)(

min2

1

N

k k k y y

Options :CG;

structured Cholesky;

structured QR.


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3. Performance investigation of SLICOT system identification toolbox

I/O data sets from the free DAISY collectionwww.esat.kuleuven.ac.be/sis

22 applications from simulated or real industrialsettings

Results obtained on an IBM PC at 500 MHz, 128Mb memory, Compaq Visual Fortran V5.1, non-optimized BLAS, and Matlab 6.1 (R12)

Many data sets of DAISY collection are wellmodelled by linear systems, but using Wienermodels usually gave significant improvement.
http://www.esat.kuleuven.ac.be/sista/daisy


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Performance investigation of SLIDENT

Linear system identification results

Timing comparison : SLICOT slmoen4 with fast QR factorization

versus Matlab 6.1 n4sid with QR factorization (default options).

Speed-upfactors up

to 240!


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Wiener system identification results

MINPACK-like SLIDENT implementation;

Wiener model much better + smoothing effect.

Mean values of prediction errors for Application 1(Ethane-ethylene distillation column):

Linear model

Wiener model

n = 3

5 inputs

3 outputs

4 sets of 90 I/Osamples, all used,after detrending;


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Wiener system identification results (ctnd.)Error norm trajectory during the optimization

calculations for Application 1:

first three parts correspondto solution of NLS problemsfor the 3 outputs;

the last part correspondsto the optimization of wholeWiener system.


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Wiener system identification results (ctnd.)Mean values of prediction errors for

Application 2 (Glass furnace):

The second half of data set not very well modeled.

Reason: the size of estimation set smaller than the number of unknown parameters (670).

n = 7

3 inputs

6 outputs

1247 I/O samples;

first 623 samplesused for estimation;

all samples used

for validation.

Linear model

Wiener model


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Wiener system identification results (ctnd.)Mean values of prediction errors for

Application 2 (Glass furnace):

Execution times:

n = 7

3 inputs

6 outputs

1247 I/O samples;

all samples usedfor estimation;

Cholesky

factorization-basedimplementation

Linear model

Wiener model

1352.10 sec. (MINPACK-like);

2580.39 sec. (Cholesky-based);

19374.70 sec. (CG-based).


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4. Conclusions

Algorithmic and numerical issues concerningnumerical techniques for linear and Wienermultivariable system identification were described.

These techniques are implemented in the newsystem identification toolbox, SLIDENT, for SLICOTLibrary.

Numerical results show that SLIDENT is reliable,efficient, and powerful enough to solve largeidentification problems.

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