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EE392m - Spring 2005Gorinevsky
Control Engineering 10-1
Lecture 10 - Model Identification
• What is system identification? • Direct impulse response identification • Linear regression• Regularization • Parametric model ID, nonlinear LS
EE392m - Spring 2005Gorinevsky
Control Engineering 10-2
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 System Identification Model
ExperimentPlant
Rar
ely
used
in
real
-life
con
trol
EE392m - Spring 2005Gorinevsky
Control Engineering 10-3
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
EE392m - Spring 2005Gorinevsky
Control Engineering 10-4
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
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1IMPULSE RESPONSE
TIME
0 1 2 3 4 5 6 7-0.5
0
0.5
1NOISY IMPULSE RESPONSE
TIME
• FIR modeling can be used for building simplified control design models from complex sims
EE392m - Spring 2005Gorinevsky
Control Engineering 10-5
Step response identification
• Step (bump) control input and collect the data– used in process control
• Impulse estimate: impulse(t) = step(t)-step(t-1)• Still noisy
0 200 400 600 800 10000
0.5
1
1.5STEP RESPONSE OF PAPER WEIGHT
TIME (SEC)
0 100 200 300 400 500 600
0
0.1
0.2
0.3
IMPULSE RESPONSE OF PAPER WEIGHT
TIME (SEC)
Actuator bumped
EE392m - Spring 2005Gorinevsky
Control Engineering 10-6
Noise reduction
Noise can be reduced by statistical averaging:• Collect data for multiple step inputs and perform more
averaging to estimate 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
EE392m - Spring 2005Gorinevsky
Control Engineering 10-7
Linear Regression - univariate• Simple fitting problem:
– Given model step response y (t)
– And process step response ϕ (t)
– Find the gain factor θ
)()()( tetty += θϕ
0 200 400 600 800 10000
0.5
1
1.5STEP RESPONSE OF PAPER WEIGHT
TIME (SEC)
)(tϕ
y(t)
ey +Φ= θ
,)(
)1(
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
Ny
yy M
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=Φ
)(
)1(,
)(
)1(
Ne
ee
NMM
ϕ
ϕ
ΦΦΦ= T
T yθ
Solution assuming uncorrelated noise:
EE392m - Spring 2005Gorinevsky
Control Engineering 10-8
Linear Regression
• Linear regression is one of the main System ID tools
)()()(1
tetty j
K
jj +=∑
=
ϕθ
DataRegression weights Regressor Error of the fit
ey +Φ= θ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=Φ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)(
)1(,,
)()(
)1()1(,
)(
)1( 1
1
1
Ne
ee
NNNy
yy
KK
K
MM
K
MOM
K
M
θ
θθ
ϕϕ
ϕϕ
EE392m - Spring 2005Gorinevsky
Control Engineering 10-9
Linear regression - least squares• Makes sense only when matrix Φ is tall,
N > K, more data available than the number of unknown parameters. – Statistical averaging
• Least square solution: ||e||2 → min
( ) yTT ΦΦΦ= −1θ
( ) ( ) min→Φ−Φ−= θθ yyL T
( ) 02 =Φ−Φ−=∂∂ θθ
yL T
• Can be computed using Matlab pinv or left matrix division \
EE392m - Spring 2005Gorinevsky
Control Engineering 10-10
Linear regression - least squares• Correlation interpretation of the least squares solution
( ) yTT ΦΦΦ= −1θ
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
∑
∑
∑∑
∑∑
=
=
==
==
N
tK
N
t
N
tK
N
tK
N
tK
N
t
tyt
tyt
Nc
ttt
ttt
NR
1
11
1
2
11
11
1
2
)()(
)()(1,
)()()(
)()()(1
1
ϕ
ϕ
ϕϕϕ
ϕϕϕ
M
K
MOM
K
cR 1ˆ −=θ
Information matrix Correlation vector
ΦΦ= T
NR 1 y
Nc TΦ= 1
EE392m - Spring 2005Gorinevsky
Control Engineering 10-11
Example: First-order ARMA model
• Linear regression representation
)()1()1()( tetgutayty +−+−=
⎥⎦
⎤⎢⎣
⎡=−=−=
ga
tuttyt
θϕϕ
)1()()1()(
2
1
• This (type of) approach is considered in most of the technical literature on identification
• Matlab Identification Toolbox– Limited industrial use
• Fundamental issue: – Small error in a might mean large change in the system response
Lennart Ljung, System Identification: Theory for the User, 2nd Ed, 1999
( ) yTT ΦΦΦ= −1θ
)()()()( 2211 tettty ++= ϕθϕθ
EE392m - Spring 2005Gorinevsky
Control Engineering 10-12
Regularization• Linear regression, where is ill-conditioned• Instead of ||e||2 → min solve a regularized problem
where r is a small regularization parameter• A.N.Tikhonov (1963)
– see http://solon.cma.univie.ac.at/~neum/ms/regtutorial.pdf• Regularized solution
• Cut off the singular values of Φ that are smaller than r
ey +Φ= θ
ΦΦT
min22 →+ θre
( ) yrI TT Φ+ΦΦ= −1θ
EE392m - Spring 2005Gorinevsky
Control Engineering 10-13
Regularization• Analysis through SVD (singular value decomposition)
• Regularized solution
• Cut off the singular values of Φ that are smaller than r
Kjj
KNKK sSRURV 1,, }diag{;; ==∈∈
( ) yUrs
sVyrI T
K
jj
jTT
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+=Φ+ΦΦ=
=
−
12
1 diagθ
10-2
10-1
100
101
10210
-2
10-1
100
101
102
INV
ER
SE
SINGULAR VALUE
Inverse singular values 1/s
Regularized inverse values
1.02 +ss
s
IVVUU TT ==TUSV=Φ
EE392m - Spring 2005Gorinevsky
Control Engineering 10-14
Linear regression for FIR model
• Linear regression representation
)()()()(1
tektukhtyK
k
+−=∑=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−−−−−
−−−=Φ
−=
−=
)(
)1(,
)1()1()(
)()2()1(,
)()(
)1()(1
Kh
h
KNtuNtuNtu
Ktututu
Ktut
tut
K
M
K
MOMM
K
M θϕ
ϕ
( ) yrI TT Φ+ΦΦ= −1θ
• 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
1PRBS EXCITATION SIGNAL
PRBS =Pseudo-Random Binary Sequence See IDINPUT in Matlab
Regularized LS solution:
EE392m - Spring 2005Gorinevsky
Control Engineering 10-15
Example: FIR model ID
• PRBS excitation input
• Simulated system output: 4000 samples, random noise of the amplitude 0.5
0 200 400 600 800 1000-1
-0.5
0
0.5
1PRBS excitation
0 200 400 600 800 1000
-1
-0.5
0
0.5
1
SYSTEM RESPONSE
TIME
EE392m - Spring 2005Gorinevsky
Control Engineering 10-16
Example: FIR model ID
• Linear regression estimate of the FIR model
• Impulse response for the simulated system:
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
0.2
FIR estimate
Impulse Response
Time (sec)0 1 2 3 4 5 6 7
-0.05
0
0.05
0.1
0.15
0.2
H = tf([1 .5],[1 1.1 1]); P = c2d(H,0.25);
EE392m - Spring 2005Gorinevsky
Control Engineering 10-17
Nonlinear parametric model ID
• Prediction model depending on the unknown parameter vector
• Nonlinear regression: loss index
• Iterative numerical optimization. Computation of L as a subroutine sim
Model including the parameters
Optimizerθ Loss Index L
)(tu
∑ −= 2)|(ˆ)( θtytyL
θ
min)|(ˆ)( 2 →−= ∑ θtytyL
)|(ˆ)MODEL()( θθ tytu →→
)(tyθ
Lennart Ljung, “Identification for Control: Simple Process Models,”IEEE Conf. on Decision and Control, Las Vegas, NV, 2002
EE392m - Spring 2005Gorinevsky
Control Engineering 10-18
Parametric SysID of step response• First order process with deadtime• Most common industrial process model• Response to a control step applied at tB
Example:
( )⎩⎨⎧
−≤−>−
+=−−
DB
DBTtt
TttTtteg
tyDB
for ,0for ,1
)|(/)( τ
γθ
Paper machine process
DT
τ
g
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
DT
gτ
γ
θ
EE392m - Spring 2005Gorinevsky
Control Engineering 10-19
Step1: Gain and Offset EstimationTwo-step approach: linear regression + nonlinear regression • For given , the modeled step response can be presented
in the form
• This is a linear regression
• Parameter estimate and prediction for given
),|(),;|( 1 DD TtygTty τγτθ ⋅+=
∑=
=2
1
)(),;|(k
kkD tTty ϕθτθ
DT,τ
( ) yT TTD ΦΦΦ== −1),(ˆˆ τθθ ),|(ˆˆ),|(ˆ 1 DD TtygTty τγτ ⋅+=
1)(),|()(
2
11
2
1
==
==
tTtytg D
ϕτϕ
γθθ
DT,τ
EE392m - Spring 2005Gorinevsky
Control Engineering 10-20
Step 2: Rise Time & Dead Time Estimation
• For any given , the loss index is
• Grid and find the minimum of
DT,τ
∑=
−=N
tDTtytyL
1
2),|(ˆ)( τ
),( DTLL τ=DT,τ
EE392m - Spring 2005Gorinevsky
Control Engineering 10-21
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
1.6Process parameters: Gain = 0.134; Tdel = 0.00; Trise = 119.8969
time in sec.; MD response - solid; estimated response - dashed
Linear Regression ID
of the first-ordermodel
Nonlinear Regression ID
Nonlinear Regression ID
EE392m - Spring 2005Gorinevsky
Control Engineering 10-22
Linear Filtering in SysID
• F is a linear filtering operator, usually LPF
• A trick that helps: pre-filter data• Consider data model
euhy += *
{ {
)(**)()*(
)*(
FuhuFhuhF
FeuhFFyff
ey
==
+=
• Can estimate h from filtered y and filtered u• Or can estimate filtered h from filtered y and ‘raw’ u• Pre-filter bandwidth limits the estimation bandwidth
Plantu y
SysID
Plantu y
SysIDF
F
h
h
EE392m - Spring 2005Gorinevsky
Control Engineering 10-23
Multivariable Identification
• Apply SISO ID to various input/output pairs
• Need n tests: excite each input in turn and collect all outputs at that
• Step/impulse response identification is a key part of the industrial multivariable Model Predictive Control packages
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