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Frequency Domain Identification
Johan Schoukens
Vrije Universiteit Brussel
2/67
Basic goal
Built a parametric model for a linear dynamic system from sampled data
Initial questions
- sampled data: what’s in betweenthe samples?
- plant and noise model?
- cost function?
Gu t( ) y t( )
Noisy data
Model
Cost
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Outline
Introduction
Data: what is going on between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions
Noisy data
Model
Cost
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Sampled dataWhat is going on in between the samples?
Two popular assumptions
ZOH zero order hold: signal piece wise constant
BL band limited assumption: no power above f fs 2¤>
BL-spectrum
ZOH-spectrum
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Relation signal assumption / experimental setup
Choice driven by the application
- ZOH: discrete control design
- Band Limited: other applications
Actuator y1
+
+
+
y kTs( )
+Generator ZOH
u kTs( )
Actuator y1
+
y kTs( )
+Generator ZOH G G
ZOH Band limited
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ZOH: discrete control design
- input exactly known- high frequency components in input ( )- absolute calibration- no anti-alias filter allowed- model: from generator to output (ZOH, actuator, plant, acquisition)
Actuator y1 t( )
+
y kTs( )
my t( )
yAA t( )
u t( )+
np t( )
Generatorud k( )
ZOH
uzoh t( )
Gc jw( )
GZOH e j– wTs( )
G jw( )
Gy
Uzoh w( )
ud k( )f fs>
Gy 1=
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BL: other applications
- input and output measured- band limited data: no power above --> anti alias filters- relative calibration- only plant modelled
U w( )
Actuator Planty1 t( )u1 t( )
+
+
+
y kTs( )
ng t( )
mu t( ) my t( )uAA t( ) yAA t( )
ug t( )+
np t( )
Generatorud k( )
ZOH
uzoh t( )
Gu Gy
u kTs( )
f fs 2¤> Gu Gy,Gy Gu¤ 1=
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Conclusions signal assumption
ZOH-assumption- imposes experimental condition on the excitation signal- discrete time model from generator to measured plant output- possible to transform DT --> CT model (perfect ZOH)
BL-assumption- imposes condition on the observation of the signals- no constraints on the applied excitation
(e.g. BL-observations of ZOH-signals can be made).- continuous-time model of the plant in the observed frequency band
Violating the signal assumption- still possible to get a behaviour model- model is no longer independent of the measurement environment- the inter sample behaviour becomes an intrinsic part of the model
BL-Assumption --> approximate DT-models for simulationImperfect ZOH --> model linked to the generator
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Choice of the model.
Possible combinations of continuous/discrete-time data and models.
DT-model(Assuming ZOH-setup)
CT-model(Assuming BL-setup)
ZOH-setup
exact DT-model
‘standard conditions DT modelling’
Not studied
BL-setup
approximate DT model
,
‘digital signal processing field’
exact CT-model
‘standard conditions CT modelling’
G z( ) 1 z 1––( )Z G s( )s----------
î þí ýì ü
=
G z( )
G z = ejwTs( ) G s = jw( )» wws2------
<
G s( )
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Cascading models in simulations
Conclusion: BL-assumption is needed
L jw( ) G jw( )
L jw( )
G jw( )
Cascade of ZOH models
ZOH of cascaded models
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Outline
Introduction
Data: what is going on between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions
Noisy data
Model
Cost
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Parametric models of LTI systems
Continuous time
Diffusion
Discrete time
General model
with
G s q,( ) B s q,( )A s q,( )----------------
brsrr 0=nbå
arsrr 0=naå
----------------------------= =
G s q,( ) B s q,( )A s q,( )--------------------
bmsm 2/m 0=nbå
ansn 2/n 0=naå
--------------------------------------= =
G z 1– q,( ) B z 1– q,( )A z 1– q,( )---------------------
brz r–r 0=nbå
arz r–r 0=naå
------------------------------= =
G W q,( ) B W q,( )A W q,( )------------------
brWrr 0=nbå
arWrr 0=naå
------------------------------= = W
s continuous-time
s diffusion
z 1– discrete-timeîïíïì
=
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Parametric models of LTI systemsRelation between input/output DFT spectra
Input/output DFT spectra
,
with
Remark
are an for random excitations
U k( ) 1N
-------- u tTs( )zkt–
t 0=N 1–å= Y k( ) 1
N-------- y tTs( )zk
t–t 0=N 1–å=
zk ej2pk N¤=
U k( ) Y k( ), O N0( )
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Parametric models of LTI systemsRelation between input/output DFT spectra
Periodic signals
if- steady state response- integer number of periods are observed
Y k( ) G Wk q,( )U k( )=
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Parametric models of LTI systemsRelation between input/output DFT spectra
Arbitrary signals
with
,
where
and
Y k( ) G Wk q,( )U k( ) TG Wk q,( )+=
G W q,( ) B W q,( )A W q,( )------------------
brWrr 0=nbå
arWrr 0=naå
------------------------------= = TG W q,( )IG W q,( )A W q,( )--------------------
igrWr
r 0=nigå
arWrr 0=naå
-------------------------------= =
Wz 1– nig
max na nb,( ) 1–=
s s, nigmax na nb,( ) 1–>
îïíïì
=
TG Wk q,( )0 for periodic and time-limited signals
O N 1 2/–( ) arbitrary signalsîíì
=
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Parametric models of LTI systemsFull equivalence time domain - frequency domain
Frequency domain
leakage begin and end conditions
Time domain
Transient effects due to initial conditions
Û
Y k( ) G Wk q,( )U k( ) TG Wk q,( )+=
y t( ) G q q,( )u t( ) TG t q,( )+=
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Experimental illustration: octave band-pass filter
Periodic multisine excitation and random noise excitation
plant model na 6 nb, 4= =
measured FRF(multisine excit.)
error model withtransient nig
6=
error model
• error model
without transient
(multisine excit.)
(arbitr. excit.)
(arbitr. excit.)
18/67
Parametric models of LTI systemsParametrizations
• Rational form
with
• Partial fraction expansion
for
with
G W q,( ) B W q,( )A W q,( )------------------
brWrr 0=nbå
arWrr 0=naå
------------------------------= = qT a0a1¼anab0b1¼bnb
[ ]=
G W q,( )Lr
W lr–----------------r p–=r 0¹
p
åSr
W sr–----------------r 1=
q
å W W w,( )+ += W s s,=
G z 1– q,( )Lrz 1–
1 lrz 1––----------------------
r p–=r 0¹
p
åSrz 1–
1 srz 1––----------------------
r 1=
q
å W z 1– w,( )+ +=
qT s1¼sqRe l1( )Im l1( )¼Re lp( )Im lp( )¼[=
S1¼SqRe L1( )Im L1( )¼Re Lp( )Im Lp( )w0¼wnw]
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Parametric models of LTI systemsParametrizations (cont’d)
• State space representation for proper transfer functions ( )
with
• Pole/zero representation
disadvantage: ill conditioned for multiple poles/zeroes
• Systems with time delay
nb na£
G s q,( ) C sInaA–( ) 1– B D+=
G z 1– q,( ) z 1– C Inaz 1– A–( ) 1– B D+=
qT vecT A( ) BT C D[ ]=
G W q,( ) KW zr–( )
r 1=nbÕ
W lr–( )r 1=naÕ
----------------------------------------=
G W q,( ) e ts– B W q,( )A W q,( )------------------=
G z 1– q,( ) z t Ts¤– B z 1– q,( )A z 1– q,( )---------------------=
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Outline
Introduction
Data: what is going on in between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions
Noisy data
Model
Cost
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Basic problem for BL-setupSetup
Measurements
,
Model
, with
G0 W( )
MU k( )
U k( )
MY k( )
Y k( )
Np k( )
U0 k( )
Ng k( )
Y k( ) Y0 k( ) NY k( )+=
U k( ) U0 k( ) NU k( )+=k 1 ¼ F, ,=
Y0 k( ) G0 Wk( )U0 k( )= G Wk( )B Wk q,( )A Wk q,( )--------------------
brWkr
r 0=nbå
arWkr
r 0=naå
-------------------------------= =
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Match model and data: choice of the cost functionIntuitive choice
- : measured FRF
- : uncertainty
Works amazingly well in many situations
Problem: good measurements in the presence of input noise
VF q Z,( ) 1F---
G Wk( ) G Wk q,( )– 2
sG2 k( )
------------------------------------------------k 1=Få=
G Wk( )
sG2 k( )
G Wk( )M ¥®
limSYU Wk( )SUU Wk( )--------------------- G0 Wk( ) 1
1 SMUMUWk( ) SUU Wk( )¤+--------------------------------------------------------------= =
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Alternative
under the constraint
Parameters to be estimated:: real parameters
: complex parameters
VF q Z,( ) 1F---
U k( ) Up k( )– 2
sU2 k( )
------------------------------------Y k( ) Yp k( )– 2
sY2 k( )
----------------------------------+k 1=
F
å=
1F---
Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö H
k 1=
F
åsY
2 k( ) 0
0 sU2 k( )
1–Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö
=
Yp k( ) G Wk q,( )Up k( )= k 1 2 ¼ F, , ,=
q na nb 1+ +
Up Yp, 2F
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Generalized problem
correlated input output noise
under the constraint
Parameters to be estimated:: real parameters
: complex parameters
VF q Z,( ) 1F---
Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö H
k 1=
F
åsY
2 k( ) sYU2 k( )
sUY2 k( ) sU
2 k( )
1–Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö
=
Yp k( ) G Wk q,( )Up k( )= k 1 2 ¼ F, , ,=
q na nb 1+ +
Up Yp, 2F
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Generalized problem
under the constraint
This is the maximum likelihood estimator
- Gaussian distributed noise
- Known covariance matrix
VF q Z,( ) 1F---
Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö H
k 1=
F
åsY
2 k( ) sYU2 k( )
sUY2 k( ) sU
2 k( )
1–Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö
=
Yp k( ) G Wk q,( )Up k( )= k 1 2 ¼ F, , ,=
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Elimination of
Symmetric formulation
Up Yp,
VF q Z,( ) 1F---
Y k( ) G Wk q,( )U k( )– 2
sY2 k( ) sU
2 k( ) G Wk q,( ) 2 2Re sYU2 k( )G Wk q,( )( )–+
-------------------------------------------------------------------------------------------------------------------------k 1=
F
å=
G B A¤=
VF q Z,( ) 1F---
A Wk q,( )Y k( ) B Wk q,( )U k( )– 2
sY2 k( ) A Wk q,( ) 2 sU
2 k( ) B Wk q,( ) 2 2Re sYU2 k( )A Wk q,( )B Wk q,( )( )–+
------------------------------------------------------------------------------------------------------------------------------------------------------------------------k 1=
F
å=
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Special case 1: identifying from the measured FRF
Put, ,
and
, and
VF q Z,( ) 1F---
Y k( ) G Wk q,( )U k( )– 2
sY2 k( ) sU
2 k( ) G Wk q,( ) 2 2Re sYU2 k( )G Wk q,( )( )–+
-------------------------------------------------------------------------------------------------------------------------k 1=
F
å=
Y k( ) G Wk( )= U k( ) 1=
sU2 k( ) 0= sY
2 k( ) sG2 k( )=
VF q Z,( ) 1F---
G Wk( ) G Wk q,( )– 2
sG2 k( )
------------------------------------------------k 1=Få=
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Special case 2: the input is exactly known
Put
,
VF q Z,( ) 1F---
Y k( ) G Wk q,( )U k( )– 2
sY2 k( ) sU
2 k( ) G Wk q,( ) 2 2Re sYU2 k( )G Wk q,( )( )–+
-------------------------------------------------------------------------------------------------------------------------k 1=
F
å=
sU2 k( ) 0= sYU
2 k( ) 0=
VF q Z,( ) 1F---
Y k( ) G Wk q,( )U k( )– 2
sY2 k( )
------------------------------------------------------k 1=Få=
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Outline
Introduction
Data: what is going on between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions
Noisy data
Model
Cost
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Noise modelsTime domain
and
Frequency domain
and
v t( ) H q( )e t( ) Th t( )+=
H q( )e t( )»E v r( )v s( ){ } Rvv r s–( )=
V k( ) H k( )E k( ) TH k( )+=
H k( )E k( )»E V k( )HV l( ){ } sV
2 k( )dkl O N 1–( )+=
cost frequency domain
VHO N 1–( )
sV2 k( )
O N 1–( )
1–
V
VH0
sV2 k( )
0
1–
V
cost time domain
vTRvv k( )
Rvv 0( )
Rvv k( )
1–
v
H 1– e( )T 0
10
1–
H 1– e( )
or
»
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Noise models
cost function frequency domain
- nonparametric noise model
- no interference with plant estimate
- periodic excitation --> very simple extraction
- arbitrary excitation --> more complicated
VH0
sV2– k( )
0
V
sV2 k( )
cost function time domain
- simultaneous identificationparametric plant/noise model
- Errors-in-Variablesalso parametric model excitation
- Only noise on outputclassic prediction error identif.
H 1– e( )T 0
10
1–
H 1– e( )
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Noise model frequency domain
Cost function
2nd order moments of the noise needed: to be extracted from the data
Prior analysis
separate signals and noiseextract a nonparametric noise model
1) periodic excitations
2) arbitrary excitations
VF q Z,( ) 1F---
Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö H
k 1=
F
å=sY
2 k( ) sYU2 k( )
sUY2 k( ) sU
2 k( )
1–Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö
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Noise model, prior analysisperiodic excitation
Identify , and
Additional information: the signals are periodic
, ,
and
sU2 k( ) sY
2 k( ) sYU2 k( )
u t( )
u 1[ ] t( ) u 2[ ] t( ) u l[ ] t( )... ...t
U k( ) 1M-----
U l[ ] k( )l 1=Må= Y k( ) 1
M-----Y l[ ] k( )
l 1=Må=
sU2 k( ) 1
M 1–-------------- U l[ ] k( ) U k( )– 2l 1=Må= sY
2 k( ) 1M 1–-------------- Y l[ ] k( ) Y k( )– 2
l 1=Må=
sYU2 k( ) 1
M 1–-------------- Y l[ ] k( ) Y k( )–( ) U l[ ] k( ) U k( )–( )l 1=Må=
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Noise model, prior analysisperiodic excitation
Propertiesconsistency: periods are enough
efficiency: periods are enough
‘loss’ in efficiency
normality: is enough
Recent results2 periods + overlapping windows are enough
additional loss in efficiency: about 15% (compared to , no overlap)
M 4=
M 6=
CqSMLM 2–M 3–--------------CqML=
M 7=
M 6=
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Example of a nonparametric prior noise analysisThe flexible robot arm
Data from Jan Swevers, KULeuven, PMA
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Raw data
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Segment the record10 periods
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Variance analysis
frequency
frequency
Signal
std.dev
Signal
std.dev
Output
Input
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Variance analysisFRF
FRF
std. dev.
Frequency
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Estimated model
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Noise model frequency domain
Cost function
2nd order moments of the noise needed: to be extracted from the data
Prior analysis
separate signals and noiseextract a nonparametric noise model
1) periodic excitations
2) arbitrary excitations
VF q Z,( ) 1F---
Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö H
k 1=
F
å=sY
2 k( ) sYU2 k( )
sUY2 k( ) sU
2 k( )
1–Y k( ) Yp k( )–
U k( ) Up k( )–è øç ÷æ ö
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Nonparametric noise model, prior analysisarbitrary excitation
Simplification required: only noise on the output
G0 W( )
U k( )
MY k( )
Y k( )
Np k( )
U0 k( )
Ng k( )
MU k( )
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Basic idea
Basic idea: eliminate and
coherence (+ leakage errors)
More advanced methodssolve set of equations at multiple frequencies
smooth --> Taylor
with
u0 t( ) y t( )linear systemy0 t( )
v t( )
G0 k( )U0 k( ) T0 k( )
SYY f( )SYU0
f( ) 2
SU0U0f( )-----------------------– sv
2 f( )=
G0 k( ) T0 k( ),
Y k( ) G0 k( )U0 k( ) T0 k( ) V k( )+ += k n– ¼ k ¼ k n+, , , ,
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Noise model, prior analysisarbitrary excitation
Example
SystemEstimate G0H0noise
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Parametric noise modelSimultaneous analysis - General problem
Estimate plant and noise model together
Extract a parametric noise model
Additional constraints needed- NO cross-correlation between and
- input: filtered white noise --> a parametric input model is also estimated
Errors-in-variables problem --> out of the scope of this course
G0 W( )
MU k( )
U k( )
MY k( )
Y k( )
Np k( )
U0 k( )
Ng k( )
MU Np MY,
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Parametric noise modelSimultaneous analysis - Simplified problem
Input is exactly known
Estimate parametric plant and noise model together
No signal model needed
Classical prediction error method --> time domain identification
G0 W( )
MY k( )
Y k( )
Np k( )
U0 k( )
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Outline
Introduction
Data: what is going on between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions
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Time domain identification
- Use discrete time models:
- Assume that the input is exactly known: ,
- Use a parametric noise model:
- The cost function becomes:
, with
Actuator y1
+
y kTs( )
+Generator ZOH G s( )
G z 1– q,( )
sU2 k( ) 0= sYU
2 k( ) 0=
sY2 k( ) H z 1– q,( ) 2=
VF q Z,( ) 1F---
Y k( ) G zk1– q,( )U k( )– 2
H zk1– q,( ) 2-------------------------------------------------------
k 0=N 1–å= zk
1– ej2pk
N---------–=
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Time domain identification (Cont’d)
Interpretation in the time domain:
model
cost function
,
y t( ) G0 q( )u t( ) H q( )e t( )+=
y t|t 1–( ) H 1– q q,( )G q q,( )u t( ) 1 H 1– q q,( )–( )y t( )+=
e t q,( ) y t( ) y t|t 1–( )–=
VN q Z,( ) 1N----
e t q,( )2k 0=N 1–å=
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Time domain identification (Cont’d)
, or
1) and -->
problem that is linear-in-the-parameters (ARX)
2) and -->
problem that is linear-in-the-parameters (ARMAX)
3) and -->
problem that is nonlinear-in-the-parameters (Output Error)
4) and -->
problem that is nonlinear-in-the-parameters (Box-Jenkins)
V 1N----
e t( )2t 0=N 1–å= V 1
N----
Y k( )B zk
1– q,( )
A zk1– q,( )
-----------------------U k( )–2
H zk1– q,( ) 2--------------------------------------------------------
k 0=N 1–å=
G q q,( ) B q q,( )A q q,( )------------------= H q( ) 1
A q q,( )----------------= A q q,( )y t( ) B q q,( )u t( ) e t( )+=
G q q,( ) B q q,( )A q q,( )------------------= H q( ) C q q,( )
A q q,( )-----------------= A q q,( )y t( ) B q q,( )u t( ) C q q,( )e t( )+=
G q q,( ) B q q,( )A q q,( )------------------= H q( ) 1= y t( ) B q q,( )
A q q,( )----------------u t( ) e t( )+=
G q q,( ) B q q,( )A q q,( )------------------= H q( ) D q q,( )
C q q,( )-----------------= y t( ) B q q,( )A q q,( )----------------u t( ) e t( )D q q,( )
C q q,( )-----------------+=
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Outline
Introduction
Data: what is going on between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions
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ValidationClassic
- Cross-correlation ? --> test
- Auto-correlation residuals or prediction errors white? --> test
Nonparametric noise models
--> check the actual value >< theoretic value
Remark: in classical prediction error framework
estimated from residuals --> includes model errors
Compare FRF modelled transfer function with measured FRF
Rue t( ) 0=
e q k,( )se k( )---------------
V e q k,( )2
se2 k( )
------------------k 1=Nå= c2 N nq–( )~
E V{ } N nq–=
sV2 2 N nq–( )=
se k( )
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Linear identification framework
Parametric noise modelClassical prediction error frame work
Non-parametric noise modelFrequency domain identification
Preprocessing- non-parametric noise model
Estimates- parametric plant model- parametric noise model
(nonlinear and disturbing noise)
Estimates- parametric plant model
Properties- consistent- efficient- normal
Properties- consistent- efficient- normal
Validation- nonlinearity is NOT detected
Validation- nonlinearity is detected- alternative validation scheme
Happy but ‘unconscious’ user Happy but ‘conscious’ user
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Time domain versus frequency domain identification
- Transforming data from time to frequency domain does not create or delete information!
- There exists a full equivalence between both approaches
- Practical issues are decisivesome information easier accessible in one domain than in the other(non causal) prefiltering in frequency domainimproved SNR --> simpler generation of starting valuescombining different sampling frequencies --> wide frequency range
- Use periodic excitations if possible --> access to a nonparametric noise model
Some of these aspects will be illustrated on the examples
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Outline
Introduction
Data: what is going on between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions
56/67
Example 1Identification d-axis synchronous machine
3x220V3x220V
HP
1430
A/D
-con
verto
r
HP
1445
AW-g
ener
ator
HP
1430
A/D
-con
verto
r
HP
1430
A/D
-con
verto
r
HP
1430
A/D
-con
verto
r
HP
1445
AW-g
ener
ator
MX
Icon
trolle
r
ComputerVXI
measurementsystem
Armature Field
Currentcontroller
Thyristorrectifier
+
-
Currentcontroller
+
-
Thyristorrectifier
ia
if
efea
57/67
Identification d-axis synchronous machine
M 8=N 65536=0.01 Hz 230 Hz,[ ]
Current Armature Voltage Armature
Voltage fieldCurrent field
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Estimation parametric plant model with estimatednonparametric noise model
Measurement example: identification d-axis synchronous machine (cont’d)
Z s q,( )Brsr
r 0=nbå
arsrr 0=nb 1–å
-----------------------------=
BrT Br=
nb 6=
measured FRF
noise variance
difference modelledand measured FRF
Z11 dB( ) Z12 dB( )
Z22 dB( )
f (Hz) f (Hz)
f (Hz)
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Estimation parametric plant model with estimatednonparametric noise model
Measurement example: identification d-axis synchronous machine (cont’d)
Cost function much too large --> model errors
2 3.97e5 625.83 3.21e4 620.24 8.15e3 614.65 3.12e3 609.06 2.18e3 603.47 2.14e3 597.8
nb VSML q Z,( ) VTheoretic
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Observations
- a very wide frequency range is covered [0.01 Hz, 230 Hz]
- improper models can be used (more zeros than poles)
- model errors are easily detected
- only a small number of frequencies is excited
- a high SNR on these lines --> averaging and filtering effect- generation of initial estimates
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averaging and filtering: Elimination of non excited frequenciesOriginal signal
Signal + noise (freq. domain)original averaged (10 times) averaged and filtered
Additive noise (time domain)original averaged (10 times) averaged and filtered
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Example 2Nuclear magnetic resonance (NMR) spectroscopy
Nuclear Magnetic Resonace (NMR) scanner:
· ~ Tesla static magnetic field,· ~ MHz oscillating field perpendicular to the static field· response measured in two orthogonal directions x and y
Þ complex signal x(t) + jy(t)
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Nuclear magnetic resonance (NMR) spectroscopy (con’t)
absolute value demodulated signal x(t) + jy(t)(averaged over 64 measurements)
time (s)
abs(
resp
onse
)
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Nuclear magnetic resonance (NMR) spectroscopy (con’t)
measuredspectrum•
model
residualmeas.-model
noise variance
frequentie (Hz)
Am
plitu
de(d
B)
T z 1– q,( )brz r–
r 0=n 1–å
arz r–r 0=nå
------------------------------=
ar br, CÎ
n 9=
signal model = sum of complexdamped exponentials
NMR spectrum muscle
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Nuclear magnetic resonance (NMR) spectroscopy (con’t)
Lag (in samples)
Aut
ocor
rela
tion
resi
dual
s
• autocorrelation
50% uncertaintybound (fraction
95% uncertaintybound (fraction
Whiteness test residuals
VSML q Z,( ) 584=
Vnoise 502=
Vnoise1 2/ 22=
outside = 51.6%)
outside = 5.2%)
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Observations
Transfer functions with complex coefficients
No model errors observed
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Outline
Introduction
Data: what is going on between the samples
Model: parametric models of LTI-systems
Cost functionFrequency domain formulationNoise modelsTime domain formulation
Validation
Examples
Conclusions