Transcript
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Lecture 8 - Model Identification
What is system identification?
Direct pulse response identification Linear regression
Regularization
Parametric model ID, nonlinear LS
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What is System Identification?
White-box identification
estimate parameters of a physical model from data
Example: aircraft flight model
Gray-box identification
given generic model structure estimate parameters from data
Example: neural network model of an engine
Black-box identification determine model structure and estimate parameters from data
Example: security pricing models for stock market
Data Identification ModelExperiment
Plant
Rarelyusedin
real-lifecontr
ol
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Industrial Use of System ID Process control - most developed ID approaches
all plants and processes are different
need to do identification, cannot spend too much time on each
industrial identification tools
Aerospace
white-box identification, specially designed programs of tests Automotive
white-box, significant effort on model development and calibration
Disk drives
used to do thorough identification, shorter cycle time
Embedded systems
simplified models, short cycle time
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Impulse response identification Simplest approach: apply control impulse and collect the
data
Difficult to apply a short impulse big enough such that the
response is much larger than the noise
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0
0.2
0.4
0.6
0.8
1IMPULSE RESPONSE
TIME
0 1 2 3 4 5 6 70. 5
0
0. 5
1NOISY IMPULSE RESPONSE
TIME Can be used for building simplified
control design models from complex sims
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Step response identification Step (bump) control input and collect the data
used in process control
Impulse estimate still noisy: impulse(t) = step(t)-step(t-1)
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0
0.5
1
1.5STEP RESPONSE OF P APER WEIGHT
TIME (SEC)
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0
0.1
0.2
0.3
IMPULSE RESPONSE OF PAPER WEIGHT
TIME (SEC)
Actuator bumped
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Noise reductionNoise can be reduced by statistical averaging:
Collect data for mutiple steps and do more averaging toestimate the step/pulse response
Use a parametric model of the system and estimate a few
model parameters describing the response: dead time, rise
time, gain
Do both in a sequence
done in real process control ID packages
Pre-filter data
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Linear regression Mathematical aside
linear regression is one of the main System ID tools
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Data Regression weights Regressor Error
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Linear regression Makes sense only when matrix is
tall, N >K, more data available thanthe number of unknown parameters.
Statistical averaging
Least square solution: ||e||2min
Matlab pinv or left matrix division \
Correlation interpretation:
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Example: linear first-order model
Linear regression representation
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This approach is considered in most of the technical
literature on identification
Matlab Identification Toolbox
Industrial use in aerospace mostly Not really used much in industrial process control
Main issue:
small error in a might mean large change in response
Lennart Ljung, System Identification: Theory for the User, 2nd Ed, 1999
( ) yTT = 1
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Regularization Linear regression, where is ill-conditioned
Instead of ||e||2
min solve a regularized problem
r is a small regularization parameter
Regularized solution
Cut off the singular values of that are smaller than r
ey +=
T
min22+ re
( ) yrI TT += 1
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Regularization Analysis through SVD (singular value decomposition)
Regularized solution
Cut off the singular values of that are smaller than r
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,, }diag{;;;=
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0 1 2 3 4 50
0.5
1
1.5
2
s
REGULARIZED INVERSE
1.02 +ss
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Linear regression for FIR model
Linear regression representation
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Identifying impulse response by
applying multiple steps
PRBS excitation signal
FIR (impulse response) model 0 10 20 30 40 50-1
-0.5
0
0.5
1
PRBS EXCITATION SIGNAL
PRBS =
Pseudo-Random Binary Sequence,see IDINPUTin Matlab
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Example: FIR model ID PRBS excitation
input
Simulated system
output: 4000
samples, random
noise of theamplitude 0.5
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-1
-0.5
0
0.5
1
PRBS excitation
0 200 400 600 800 1000
-1
-0.5
0
0.5
1
SYSTEM RESPONSE
TIME
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Example: FIR model ID
Linear regression
estimate of the FIR
model
Impulse response
for the simulated
system:T=tf([1 .5],[1 1.1 1]);
P=c2d(T,0.25);
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
0.2
FIR es timate
Impulse Res pons e
Time (sec)
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
0.2
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Nonlinear parametric model ID Prediction model depending on
the unknown parameter vector
Loss index
Iterative numerical optimization.
Computation of Vas a subroutine sim
Model including the
parameters
Optimizer
Loss Index V
)(tu
! =
2)|()( tytyV
! =2
)|()( tytyJ
)|()MODEL()( tytu
)(ty
Lennart Ljung, Identification for Control: Simple Process Models,
IEEE Conf. on Decision and Control, Las Vegas, NV, 2002
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Parametric ID of step response First order process with deadtime
Most common industrial process model
Response to a control step applied at tB
Example:
( )
,
-.
>+=
DB
DB
Ttt
Ttt
Tttegty
DB
for,0
for,1)|(
/)(
Papermachine
process
DT
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Gain estimation For given , the modeled step response can be
presented in the form
This is a linear regression
Parameter estimate and prediction for given
),|()|( 1 DTtygty +=
!==
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)()|(k
kk twty
DT,
( ) yTw TTD = 1
),( ),|(),|( 1 DD TtygTty +=
1)(),|()(
2
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2
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=
=
=
=
tTtyt
wgw D
DT,
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Examples: Step response ID Identification results for real industrial process data
This algorithm works in an industrial tool used in 500+
industrial plants, many processes each
0 10 20 30 40 50 60 70 80-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 100 200 300 400 500 600 700 800-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6Proce ss parameters: Gain = 0.134; Tdel = 0.00; Trise = 119.8969
time in se c.; MD res ponse - solid; es timated res ponse - das hed
Linear
Regression ID
of the first-ordermodel
Nonlinear
Regression ID
Nonlinear
Regression ID
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Linear filtering
Lis a linear filtering operator, usually LPF
A trick that helps: pre-filter data
Consider data model
euhy += *
$$
)(**)()*(
)*(
LuhuLhuhL
LeuhLLyf
fey
==
+=
Can estimate hfrom filteredyand filtered u
Or can estimate filtered h from filteredyand raw u
Pre-filter bandwidth will limit the estimation bandwidth
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Multivariable ID Apply SISO ID to various input/output pairs
Need ntests - excite each input in turn
Step/pulse response identification is a key part of the
industrial Multivariable Predictive Control packages.
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