Frequency Domain Identificationhomepages.vub.ac.be/~pdreesen/edu/slides_freqdomain.pdf · 3/67 Outline Introduction Data: what is going on between the samples Model: parametric models

1/67

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

4/67

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




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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Periodic signals

if- steady state response- integer number of periods are observed

Y k( ) G Wk q,( )U k( )=

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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.)

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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



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


Parameters to be estimated:: real parameters

: complex parameters

VF q Z,( ) 1F---

Y k( ) Yp k( )–


k 1=

F

åsY

2 k( ) sYU2 k( )

sUY2 k( ) sU

2 k( )

1–Y k( ) Yp k( )–


=

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


This is the maximum likelihood estimator

- Gaussian distributed noise

- Known covariance matrix

VF q Z,( ) 1F---

Y k( ) Yp k( )–


k 1=

F

åsY

2 k( ) sYU2 k( )

sUY2 k( ) sU

2 k( )

1–Y k( ) Yp k( )–


=

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


-------------------------------------------------------------------------------------------------------------------------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


-------------------------------------------------------------------------------------------------------------------------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




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( )–


k 1=

F

å=sY

2 k( ) sYU2 k( )

sUY2 k( ) sU

2 k( )

1–Y k( ) Yp k( )–


33/67

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( )–


k 1=

F

å=sY

2 k( ) sYU2 k( )

sUY2 k( ) sU

2 k( )

1–Y k( ) Yp k( )–


42/67

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( )

43/67

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+, , , ,

44/67

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,

46/67

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




Validation

Examples

Conclusions

48/67

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=

50/67

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




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( )

53/67

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

54/67

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

55/67

Outline

Introduction




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

58/67

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)

59/67

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

60/67

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

62/67

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

)

64/67


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

65/67


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%)

66/67

Observations

Transfer functions with complex coefficients

No model errors observed

67/67

Outline

Introduction




Validation

Examples

Conclusions

Documents

Frequency Domain Identificationhomepages.vub.ac.be/~pdreesen/edu/slides_freqdomain.pdf · 3/67 Outline Introduction Data: what is going on between the samples Model: parametric models