Gevorderde Systeemidentificatie

8/12/2019 Gevorderde Systeemidentificatie

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System Identification

Rik Pintelon

Vrije Universiteit Brussels, dept. ELEC

Version 25 November 2011


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Table of Contents

1. Preliminary Example 3

3. Nonparametric models of LTI systems 294. Parametric models of LTI systems 715. Improved nonparametric models for LTI systems 1136. Estimation parametric plant model with known noise model 1467. Estimation parametric plant model with estimated nonparametric noise model 174

8. Estimation parametric plant and/or noise models 2009. Guidelines 23210. Books on system identification 242


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1. Preliminary Example

2.a Measurement of the value of a resistor 4

2.b Least squares estimator of the output error problem 62.c Maximum likelihood estimator of the errors-in-variables problem 72.d Properties estimators 102.e Stochastic convergence estimators 112.f Asymptotic variance estimators 12

2.g Asymptotic normality estimators 152.h Asymptotic efficiency estimators 172.i Robustness estimators 192.j Summary properties 212.k Extension to arbitrary currents 232.l References 28


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2.a Measurement of the value of a resistor

0

5

0 50 100

Measuredresista

nce()

0

2

0 50 100

Voltage(V

)

0

2

0 50 100

Current(A

)

k

kk

i k( ) i0 ni k( )+=

u k( ) u0 nu k( )+=

ni k( ) N0 i2,( )

nu k( ) N0 u2

,( )

u k( ) i k( )

R k( ) u k( ) i k( )=

u k( )

i k( )

R0 1 =

V

A


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2.a Measurement of the value of a resistor (contd)

Observation: the estimate seems not to converge asQuestion: what is going wrong?

0

1

2

1 10 100 1000 10000

EstimatedvalueR

Number of measurements

RSA N( ) 1

N---- R k( )

k 1=

N

1

N----

u k( )

i k( )---------

k 1=

N

= =

N


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2.b Least squares estimator of the output error problem

Observation: the estimate converges to a too small value asQuestion: what is going wrong?

0

1

2

1 10 100 1000 10000

EstimatedvalueR


VLS R z,( ) u k( ) Ri k( )( )2

k 1=

N

=

RLS N( ) arg min

R

VLS R z,( )u k( )i k( )

k 1=

N

i2 k( )k 1=N

---------------------------------= =

N


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2.c Maximum likelihood estimator of the errors-in-variables problem

Call the vector of the measurements

Gaussian iid random variables probability density function of the measurements

Constrained negative log-likelihood function

s.t.

Maximum likelihood cost function

Eliminate , and

z

z u1( ) u2( ) u N( ) i1( ) i2( ) i N( ), , , , , , ,[ ]T=

z

fz z R up ip, ,( ) fu k( ) u k( ) R up ip, ,( )fi k( ) i k( ) R up ip, ,( )k 1=N

=

1

2ui( )N--------------------------exp

1

2---

u k( ) up( )2

u2

----------------------------i k( ) ip( )2

i2

-------------------------+

k 1=

N

( )=

fz z R up ip, ,( )log constant 1

2---

u k( ) up( )2

u2

----------------------------i k( ) ip( )2

i2

-------------------------+

k 1=

N

+= up Rip=

VML R up ip z, , ,( ) 1

2---

u k( ) up( )2

u2

----------------------------i k( ) ip( )2

i2

-------------------------+

k 1=

N

up Rip( )+=

up ip RML N( )


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2.c Maximum likelihood estimator of the errors-in-variables problem (contd)

Observation: estimate converges to the true value as

RML N( )

1

N---- u k( )

k 1=

N

1

N---- i k( )

k 1=

N

-----------------------------=

0

1

2

1 10 100 1000 10000

E

stimatedvalueR


N


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2.c Maximum likelihood estimator of the errors-in-variables problem (contd)

Observation: estimate is different for each realisation stochastic process

Questions: - point wise convergence ?

- for almost all realisations?

- ?

- minimal uncertainty?

RMLr[ ] N( )

1

N---- u r[ ] k( )

k 1=

N

1N---- i r[ ] k( )

k 1=

N

----------------------------------=

r 1 2 , ,=

0.6

0.8

1

1.2

1.4

0 20 40 60 80 100

Resistance(

)

N

RMLr[ ]

N( )N lim RML

r[ ]=

RMLr[ ] RML=

RML R0=


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2.d Properties estimators

stochastic convergence asymptotic variance asymptotic distribution asymptotic efficiency robustness

Question: how to predict these properties?

RML N( )

1

N

---- u k( )k 1=

N

1N---- i k( )

k 1=

N

-----------------------------=

0

1

2

1 10 100 1000 10000

Estimatedvalu

eR


RLS N( )u k( )i k( )

k 1=N

i2 k( )

k 1=

N

---------------------------------=

RSA N( ) 1

N----

u k( )

i k( )---------

k 1=

N

=RSA N( )

RLS N( )

RML N( )


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2.e Stochastic convergence estimators

Basic tool: thelaw of large numbers

then as

Simple approach

for

no convergence Least squares

convergence to the wrong value =inconsistent Maximum likelihood

convergence to the true value = consistent

S N( ) 1

N---- x k( )

k 1=

N

= S N( ) S N( ){ } 1

N---- x k( ){ }

k 1=

N

= N

RSA N( ) 1

N----

u k( )

i k( )---------

k 1=

N

= 1

N---- u k( ){ }

1

i k( )--------

k 1=

N

= i k( ) N i0 i2,( )

RLS N( )

1

N---- u k( )i k( )

k 1=

N

1

N---- i2 k( )

k 1=

N

--------------------------------------=

1

N---- u k( ){ }E i k( ){ }

k 1=

N

1

N---- i2 k( ){ }

k 1=

N

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

u0i0

i02 i

2+----------------

R0

1 i2 i0

2+-----------------------= =

RML N( )

1

N---- u k( )

k 1=

N

1

N---- i k( )k 1=N

-----------------------------=

1

N---- u k( ){ }

k 1=

N

1

N---- i k( ){ }k 1=N

--------------------------------------- u0

i0

----- R0= =


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2.f Asymptotic variance estimators

Basic tool: linearisation estimate around the limit value

Least squares

the asymptotic ( ) variance

Notes:- valid for any voltage and current signal-to-noise ratio

- asymptotic variance is an

R

LS N( )

1

N---- u k( )i k( )

k 1=

N

1

N---- i2 k( )

k 1=

N

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

u0i0 w 1[ ]+

i02 w 2[ ]+-------------------------

R0

1 i2 i02+----------------------- 1 w 1[ ]

u0i0---------

w 2[ ] i2

i02 i

2+---------------------+

= =

w 1[ ]1

N---- u0ni k( ) i0nu k( ) ni k( )nu k( )+ +( )k 1=

N

= with w 1[ ]{ } 0=

w 2[ ]1

N---- 2i0ni k( ) ni

2 k( )+( )k 1=

N

= with w 2[ ]{ } i2=

N

var NRLS N( )( )

R02

1 i2 i0

2+( )2------------------------------var N w 1[ ]

u0i0---------

w 2[ ] i2

i02 i

2+---------------------

( )

O N 1( )


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2.f Asymptotic variance estimators (contd)

Maximum likelihood

the asymptotic ( ) variance

Notes:-valid for any voltage and current signal-to-noise ratio

- asymptotic variance is an

- expected value does not exist- variance does not exist

RML N( )

1

N---- u k( )

k 1=

N

1

N

---- i k( )k 1=

N

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

u0 w 1[ ]+

i0 w 2[ ]+--------------------- R0 1

w 1[ ]

u0---------

w 2[ ]

i0---------+

= =

w 1[ ]1

N---- nu k( )

k 1=

N

= with w 1[ ]{ } 0=

w 2[ ]1

N---- ni k( )

k 1=

N

= with w 2[ ]{ } 0=

N

var NRML N( )( ) R02var N w 1[ ]

u0---------

w 2[ ]

i0---------

( ) R02 i

2

i02

------u

2

u02

------+ =

O N 1( )

RML N( )

RML N( )


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2.f Asymptotic variance estimators (contd)

0.001

0.01

0.1

1

0.01 0.1 1

Standard deviation noise

Standa

rddeviationonR

std RLS( )

stdR

ML( )

0.001

0.01

0.1

1

0.01 0.1 1

RMSerror

Standard deviation noise

rms RML( )

rms RLS( )

MSE R( ) E R R0

( )2

{ } var R( ) R{ } R0

( )2+= =

uu0-----

ii0----=

uu0-----

ii0----=


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2.g Asymptotic normality estimators

Basic tool: linearisation estimate around the limit value + the central limit theorem

then as

Least squares and maximum likelihood estimators

Application CLT to , , are asymptotically normally distributed

, are asymptotically normally distributed

S N( ) 1

N---- x k( )

k 1=

N

= S N( ) S N( ){ }

varS N( )( )-------------------------------------- AsN0 1,( ) N

RLSN( )R0

1 i2 i0

2+----------------------- 1

w 1[ ]

u0i0---------

w 2[ ] i2

i02 i

2+---------------------+

RMLN( ) R0 1 w 1[ ]

u0

---------w 2[ ]

i0

---------+

w 1[ ] w 2[ ] w 1[ ] w 2[ ]

RLSN( ) RMLN( )


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2.g Asymptotic normality estimators (contd)

0

25

0.5 1 1.5 2 0

25

0.5 1 1.5 2 0

25

0.5 1 1.5 2

pdfR

R R R

pdfR

pdfR

N 10= N 100= N 1000=

RLS

R

ML


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2.h Asymptotic efficiency estimators

Basic tool: comparison asymptotic uncertainty with the Cramr-Rao lower bound

Gaussian iid random variables probability density function of the measurements

with

he Cramr-Raolower bound

with

the corresponding Fisher information matrix

z

fz z ( ) fu k( ) u k( ) ( )fi k( ) i k( ) ( )k 1=N

=

1

2ui( )N--------------------------exp

1

2---

u k( ) Rip( )2

u2

------------------------------i k( ) ip( )2

i2

-------------------------+

k 1=

N

( )=

ip R,[ ]T=

Cov( ) Fi 1 0( ) Fi 0( ) 2 fz z ( )log

02-------------------------------

=

Fi 1 0( ) 1

N----

i2 u0

i2

i02

------

u0i

2

i02------ R02

i2

i02------

u2

u02------+

= Fi 1 R0( )

R02

N------

i2

i02

------u

2

u02

------+ =


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2.h Asymptotic efficiency estimators (contd)

Conclusion: ML-estimate is asymptotically efficient

std RML( )

+ CR R0( ) Fi 1 2/ R0( )=

0.001

0.01

0.1

1

0.01 0.1 1Standard deviation noise

uu0-----

ii0----=


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2.i Robustness estimators

Basic question: are the (asymptotic) properties of the estimator sensitive to the (noise)assumptions made to construct the estimator?

Maximum likelihood estimator- basic assumption: zero mean Gaussian iid noise on current and voltage

measurements- relax assumption: zero mean non-Gaussian mixing noise on current and

voltage measurementsconsequences on the properties:

- consistency, asymptotic normality, convergence rate still valid- asymptotic efficiency is lost- expression asymptotic variance still valid for independently distributed noise

Least squares estimator

- basic assumption: zero mean iid noise on voltage measurements, andcurrent known exactly- relax assumption:(i) zero mean non-Gaussian mixing noise on voltage measurements, andcurrent known exactly still consistent(ii) also noise on input inconsistent


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2.i Robustness estimators (contd)

Note:

0.6

0.8

1

1.2

1.4

0 100 200 300 400 500

Resistance()

N

RML N( )

RLS N( )

RSA N( )

u0 1 V=

i0 1 A=

nu k( ) 0.5 V 0.5 V,[ ]uniformlyni k( ) 0.5 A 0.5 A,[ ]uniformly

u k( ) u0 nu k( )+=

i k( ) i0 ni k( )+=

RSA N( ){ } R0i0

2 3i---------------

1 3i i0+

1 3i i0-----------------------------

log 1.1


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2.j Summary properties

Goal asymptotic analysis- what happens with the estimate if more data are gathered?- hypothesis: the asymptotic behaviour reflects the finite sample behaviour- true finite sample behaviour is in general very difficult to establish

(exception: linear least squares) Consistency

- convergence in stochasticsense to the true value- does not exclude divergence for some realisations

- guarantees with high probability that the estimate is close to the true value- consistency does not imply asymptotic unbiasedness- proof: law of large numbers

Asymptotic unbiasedness

- asymptotically the expected value equals the true value- does not guarantee that the estimate is close to the true value- in general very difficult to verify (exception: linear least squares)

Asymptotic normality- allows to construct uncertainty bounds with a given confidence level- proof: linearisation around limit value + central limit theorem


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2.j Summary properties (contd)

Asymptotic variance- measure of the convergence rate- construction of uncertainty bounds- proof: linearisation around limit value

Asymptotic efficiency- minimal uncertainty within the class of asymptotically unbiased estimators- in practice applied to the class of consistent estimators- Cramr-Rao lower bound does not always exist (e.g. uniformly distributed

noise), or may be too conservative (minimal uncertainty may be higher thanthe CR-bound)- proof: comparison asymptotic variance with Cramr-Rao lower bound

Robustness- sensitivity (asymptotic) properties to the noise assumptions made toconstruct the estimator

- proof: validity law of large numbers and central limit theorem under relaxednoise assumptions


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2.k Extension to arbitrary currents

Zero mean Gaussian iid random variables maximum likelihood cost function

Elimination , , , gives

which is a weighted nonlinear least squares problem

- starting values via linear least squares- iterative Newton-Gauss minimization procedure

u k( ) u0 k( ) nu k( )+=

i k( ) i0 k( ) ni k( )+=

ni k( ) N0 i2 k( ),( )

nu k( ) N0 u2 k( ),( )

u k( )

i k( )

R0 1 =

VML R up ip z, , ,( ) 1

2---

u k( ) up k( )( )2

u2 k( )

----------------------------------i k( ) ip k( )( )2

i2 k( )

--------------------------------+

k 1=

N

k up k( ) Rip k( )( )k 1=

N

+=

up k( ) ip k( ) k k 1 2 N, , ,=

VML R z,( ) 1

2---

u k( ) Ri k( )( )2

u2 k( ) R2i

2 k( )+--------------------------------------

k 1=

N

=


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2.k Extension to arbitrary currents (contd)

Consistency

Difficulty: no explicit expression for the minimizer of exists

Strong law of large numbers applied to

Expected value cost function is minimal in consistent

R VML R z,( )

VN R z,( ) VML R z,( ) N=

VN R z,( ) VN R( ) VN R z,( ){ } 1

2N-------

u0 k( ) Ri0 k( )( )2{ }u

2 k( ) R2i2 k( )+

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

k 1=

N

1

2---+= =

VN R( ) R R0=

2 P li i E l


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Asymptotic variance and asymptotic normality

Mean value theorem on derivative cost function

with ,

Use(by definition of )

as (consistency)

as (law of large numbers)

then

with

Asymptotic variance

Asymptotic normality

(central limit theorem on )

VN' R z,( ) VN' R0 z,( ) VN'' R1 z,( ) R R0( )+= R1 tR 1 t( )R0+= t 0 1,[ ]

VN' R z,( ) 0= R

R1 R0 N

VN

'' R1

z,( ) VN

'' R0

( ) N

R R0 R VN'' 1 R0( )VN' R0 z,( )= R{ } 0=

var R( ) var R( ) VN'' 2 R0( ) VN' R0 z,( )( )2{ }=

N R R0( ) AsN0 var NR( ),( ) VN' R0 z,( )

2 P li i E l


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Asymptotic efficiencyGaussian iid random variables probability density function of the measurements

with

he Fisher information matrixequals

Conclusion

- the ML estimate is in general inefficient- asymptotically efficient if current is known exactly

z

fz z ( ) fu k( ) u k( ) ( ) fi k( ) i k( ) ( )k 1=N

=

12( )N u k( )i k( )k 1=

N

-------------------------------------------------------exp 1

2--- u k( ) Rip k( )( )2

u2 k( )

------------------------------------- i k( ) ip k( )( )2i

2 k( )--------------------------------+

k 1=

N

( )=

ip1( ) ip2( ) ip N( ) R, , , ,[ ]T=

Fi 0( ) 2 fz z ( )log

02-----------------------------

= Fi 1 R0( ) 1

N----VN''

1 R0( )=

VN' R0 z,( )( )2{ } VN'' R0( ) N= var R( ) Fi 1 R0( )=

2 P li i E l


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Notes- inefficiency ML estimate is not in contradiction with general ML properties because aninfinitenumber of parameters are estimated

- uncertainty on does not decrease as

- in practice a good approximation of the variance of is given by

where

with

- the variance of the estimate follows as a by product of the Newton-Gaussminimization procedure

ip1( ) ip2( ) ip N( ), , , N

R

1

N----VN''

1 R0( ) R z0,( )

R0---------------------

T R z0,( )

R0---------------------

1 R z,( )

R------------------

T R z,( )

R------------------

1=

R z,( )

1[ ] R z,( )

2[ ] R z,( )

N[ ] R z,( )

= k[ ] R z,( ) u k( ) Ri k( )

u2 k( ) R2i

2 k( )+------------------------------------------=

2 Preliminary Example


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2.l References

Billingsley, P. (1995). Probability and Measure. Wiley, New York.Brillinger, D. R. (1981). Time Series: Data Analysis and Theory. McGraw-Hill, New York.Caines, P. E. (1988). Linear Stochastic Systems. Wiley, New York.Chow, Y. S. and H. Teicher (1988). Probability Theory: Independence, Interchangeability,Martingales (2nd edition).Springer-Verlag, New York.

Fuller, W. A. (1987). Measurement Error Models. Wiley, New York.Lukacs, E. (1975). Stochastic Convergence. Academic Press, New York.Pintelon, R. and J. Schoukens (2012). System Identification: a Frequency Domain

Approach.IEEE Press, Piscataway, New Jersey (second edition).

3 Nonparametric models of LTI systems


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3. Nonparametric models of LTI systems

3.a Problem statement 30

3.b Periodic excitations - errors-in-variables problem 323.c Measurement example: octave bandpass filter 373.d Arbitrary excitations - output error problem 383.e Simulation example: FRF and variance estimate using arbitrary input 413.f Nonlinear systems - introduction 43

3.g Nonlinear systems - noisy output measurements 483.h Simulation example: FRF estimate in the presence of nonlinear distortions 523.i Nonlinear systems - noisy input/output measurements 543.j Measurement example: the open loop gain of an operational amplifier 563.k Multivariable systems - periodic excitations 57

3.l MIMO experiment: identification d-axis synchronous machine 613.m MIMO experiment: vibrating mechanical structure 653.n Multivariable systems - random excitations 673.o MIMO simulation example: discrete-time system 683.p References 70



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3.a Problem statement

Noisy observations

- periodic signals

and

- arbitrary signals

and

Ng k( )

Ug k( )

MU k( )

U k( ) Y k( )

MY k( )

Np k( )

G0 j( )

Ng k( ) generator noise

Np k( ) process noise

MU k( ) input measurement noise

MY k( ) output measurement noise

Y k( ) Y0 k( ) NY k( )+=

U k( ) U0 k( ) NU k( )+=

Y0 k( ) G0 jk( )U0 k( )=

U0 k( ) Ug k( )= NYk( ) G0 jk( )Ng k( ) Np k( ) MY k( )+ +=

NUk( ) Ng k( ) MUk( )+=

Y0 k( ) G0 jk( )U0 k( )=U0 k( ) Ug k( ) Ng k( )+=

NY k( ) Np k( ) MY k( )+=

NU k( ) MU k( )=



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3.a Problem statement (contd)

Nonparametric representation

Frequency response function,

(Co-)variances circular complex distributed noise

,

and

G0 jk( ) k 1 2 , ,=

Y2 k( ) var NY k( )( ) NY k( )

2{ }= =

U2 k( ) var NU k( )( ) NU k( )

2{ }= =

YU2 k( ) covar NY k( ) NU k( ),( ) NY k( )NU k( ){ }= =

k 1 2 , ,=

NY2 k( ){ } 0=

NU2 k( ){ } 0=

NY k( )NU k( ){ } 0=



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3.b Periodic excitations - errors-in-variables problem

Noisy input-output measurements

Observe consecutive periods of the steady state response

Sample means

,

Sample (co-)variances

, ,

M

m 1= 2 M

NT M N= data points

N data points

Y k( ) 1

M----- Y m[ ] k( )

m 1=

M

= U k( ) 1

M----- U m[ ] k( )

m 1=

M

=

Y2

k( ) 1

M M 1( )------------------------ Y m[ ] k( ) Y k( )

2

m 1=

M

= U2

k( ) 1

M M 1( )------------------------ U m[ ] k( ) U k( )

2

m 1=

M

=

YU2

k( ) 1

M M 1( )------------------------ Y m[ ] k( ) Y k( )( ) U m[ ] k( ) U k( )( )m 1=M

=



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3.b Periodic excitations - errors-in-variables problem (contd)

Maximum likelihood estimate frequency response function

is independent (over ), circular complex normally distributed Gaussian probability density function of the measurements

with

, and

Gaussian ML cost function

Eliminate ,

Z m[ ] Y m[ ] k( ) U m[ ] k( ),[ ]T= mZ

fZ

Z( ) fZ

m[ ] Z m[ ] ( )m 1=

M

=

1

2M det CZ( )m 1=M

--------------------------------------------exp Z m[ ] Zp( )HCZ1 Z m[ ] Zp( )m 1=

M

( )=

ZpT G jk( ),[ ]T= Zp

Yp k( )

Up k( )= CZ Cov Z

m[ ]( ) Y

2 k( ) YU2 k( )

YU2 k( ) U

2 k( )= =

VML G jk( ) Zp Z, ,( ) Z m[ ] Zp( )HCZ1 Z m[ ] Zp( )m 1=M

1 G jk( ),[ ]Zp+=

Zp

GML jk( ) Y k( )U k( )-----------

M 1 Y m[ ] k( )m 1=

M

M 1 U m[ ] k( )m 1=

M

-------------------------------------------= =



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Maximum likelihood estimate frequency response function (contd)

Properties ML estimate FRF

- consistent ( )

- asymptotically circular complex normally distributed- asymptotically efficient

- robust w.r.t. independence and normality assumptions- expected value exists bias expression

- variance does not exist

Cramr-Rao lower bound

where

GML jk( ) Y k( )

U k( )-----------

1

M----- Y m[ ] k( )

m 1=

M

1

M----- U m[ ] k( )

m 1=

M

---------------------------------------= =

M

varG jk

( )( ) G0

jk

( ) 2

Y2 k( )

Y0 k( ) 2-----------------

U2 k( )

U0 k( ) 2------------------ 2Re

YU2 k( )

Y0 k( )U0 k( )-------------------------( )+

X

2 O 1

M-----( )=



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Maximum likelihood estimate frequency response function (contd)Relative bias on FRF estimate

where

if

Asymptotic ( ) variance

with

b k( ) G jk( ){ }

G0

jk( )

---------------------------- 1 exp U0 k( )

2

U2

k( )

------------------( ) 1

YU2 k( )

U

k( )Y

k( )--------------------------

U0 k( ) U k( )

Y0

k( ) Y

k( )------------------------------

= =

b k( ) 1 410< min U0 k( )

U k( ) Y0 k( )

Y k( ),( ) 10 dB>

M GML jk( ) G0 jk( ) G k( )+

G k( ) G0 jk( ) 1

M-----

NYm[ ] k( )

Y0 k( )-----------------m 1=M

1

M-----

NUm[ ] k( )

U0 k( )-----------------m 1=M

=

varG k( )( ) G0 jk( ) 2

Y2 k( )

Y0 k( )2

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

U2 k( )

U0 k( )2

------------------ 2Re

YU2 k( )

Y0 k( )U0 k( )-------------------------( )+

=



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Maximum likelihood estimate frequency response function (contd)Uncertainty bounds

with

% confidence region = circle with radius

Notes:- uncertainty bound accurate if

- radius

- decrease uncertainty FRF estimate: large for a given

- increase frequency resolution : small for a given

GML jk( ) G0 jk( ) AsNc0 G2

k( ),( )

G2

k( ) GML jk( )2 Y

2k( )

Y k( )2

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

U2

k( )

U k( )2

---------------- 2Re

YU2

k( )

Y k( )U k( )

---------------------( )+

=

1 p( )log G k( )

U0 k( ) U

k( ) 20 dB>

0.95= 3 G k( )

M NT M N=

s N M NT M N=



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p y

3.c Measurement example: octave bandpass filter

- multisine excitation , and- points per period, periods

-120

-100

-80

-60

-40

-20

0

0 300 600 900 1200 1500 1800

Amp

litude

(dB)

Frequency (Hz)

-180

-90

0

90

180

0 300 600 900 1200 1500 1800

P

hase

(deg.)

Frequency (Hz)

[[[

[[[[[[

[

[[

[[[

[

[

[

[

[[

[

[[

[

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[[[[[[[

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[

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[

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[[

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[

[

-100

-80

-60

-40

-20

0

20

0 300 600 900 1200 1500 1800

Amplitude(dB)

Frequency (Hz)

[[[

[

[[[[[[[

[

[[[

[[[[[[[

[[[[[[[[[

[

[

[[

[[[

[[[[[[

[

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[

[

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[

[

-100

-80

-60

-40

-20

0

20

0 300 600 900 1200 1500 1800

Amplitude(dB)

Frequency (Hz)

kf0 k 1 3 9 11 17 19 737, , , , , , ,= 0 fs 211 2.38 Hz=N 2048= M 8=



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p y

3.d Arbitrary excitations - output error problem

Noisy output measurements, known input Divide the input-output records in segments

Leakage reduction (Schoukens et al.,2006)- as small as possible for a given

- (better than Hann window!)

Sample auto- and cross-power spectra

, ,

M

m 1= 2 M

NT M N= data points

N data points

M NT

YDm[ ] diffY m[ ]( )= UD

m[ ] diffU m[ ]( )=,

SYD k( ) 1

M----- YD

m[ ] k( )2

m 1=

M

= SUD k( ) 1

M----- UD

m[ ] k( )2

m 1=

M

=

S

YD

UD k( )

1

M----- YDm[ ]

k( )UDm[ ]

k( )m 1=

M

=



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p y

3.d Arbitrary excitations - output error problem (contd)

Least squares estimator

Minimise w.r.t.

Sample variance

Asymptotic ( ) properties LS estimate FRF forknown fixed input- consistent

- asymptotically circular complex normally distributed- asymptotically efficient- expected value and variance exist- not robust w.r.t. input measurement noise

VLS G jk 1 2+( ) Z,( ) YDm[ ] k( ) G jk 1 2+( )UD

m[ ] k( ) 2

m 1=

M

=

G jk 1 2+( )

GLS jk 1 2+( ) SYDUD k( )

SUD k( )--------------------=

Y2 k 1 2+( ) M

2 M 1( )---------------------- SYD k( )

SYDUD k( )2

SUD k( )-------------------------

=

M



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3.d Arbitrary excitations - output error problem (contd)

Least squares estimator (contd)Uncertainty bounds for fixed input

with

% confidence region = circle with radius

Notes:- decrease leakage error: small for a given

- decrease uncertainty FRF estimate: large for a given- frequency resolution FRF estimate

GLS jk( ) G0 jk( ) AsNc0 G2

k( ),( )

G2

k 1 2+( ) YD

2k( ) M

SUDUD k( )------------------------

2 Y2

k 1 2+( ) M

SUDUD k( )-----------------------------------------= =

1 p( )log G k( )

M NT M N=

M NT M N=

s N



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3.e Simulation example: FRF and variance estimate using arbitrary input

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-4

-2

0

2

4

input

number of points

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6

-4

-2

0

2

4

6

output

number of points

N 5000= M, 2=



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3.e Simulation example: FRF estimate using arbitrary input (contd)

Note: variance estimate averaged over 30 frequencies

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-30

-25

-20

-15

-10

-5

0

5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-30

-25

-20

-15

-10

-5

0

5

10

15

FRF Variance output noise

estimate

true value

var(G)



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3.f Nonlinear systems - introduction

Class of excitation signals with the samepower spectrum:- Gaussian noise- periodic Gaussian noise- random phase multisine

with, and independent of

user defined, and

power is independent of

y0 t( )

yS t( )

y0 t( )yR t( )

Gaussian noise WienerSystems

u t( ) GBLA j( ) =

G0 j( ) GB j( )+

u t( ) F 1 2/

Ukej 2kfs N( )t

k F= k 0,

F

=

F N 2< F N N

Uk ej Uk{ } 0=

N 1 u2 nTs( )n 0=N 1

N



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3.f Nonlinear systems - introduction (contd)

Wiener systems:

- steady state response has same period as input (PISPO)- saturation, discontinuities (e.g. relays) are allowed- no chaos nor bifurcations

y0 t( )

yS t( )

y0 t( )yR t( )

Gaussian noise WienerSystems

u t( ) GBLA j( ) =

G0 j( ) GB j( )+



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:- true underlying linear system (if it exists)

:- bias contribution- depends on input power spectrum and odd nonlinear distortions only- smooth function of the frequency

:- best linear approximationor related linear dynamic systemor LTI secondorder equivalent

- can be approximated very well by a rational form in

yS t( )

y0 t( )yR t( )u t( ) GBLA s( ) =

G0 s( ) GB s( )+

Ys k( )

Y0 k( )U k( )

GBLA jk( )YR k( )

G0 s( )

GB s( )

GBLA s( )

s



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:- periodic signal

- depends on input power spectrum and the even and odd nonlinearities- zero mean

- uncorrelated with - but not independent of -

- not normally distributed

yS t( )

y0 t( )yR t( )u t( ) GBLA s( ) =

G0 s( ) GB s( )+

Ys k( )

Y0 k( )U k( )

GBLA jk( )YR k( )

ys t( )

u t( )



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:

- uncorrelated with but notindependent of

-

- is asymptotically uncorrelated over (cumulant mixing over )

- asymptotically normally distributed

- is a smooth function of the frequency

Note: expectations are taken over the different random realisations of the excitation

yS t( )

y0 t( )yR t( )u t( ) GBLA s( ) =

G0 s( ) GB s( )+

Ys k( )

Y0 k( )U k( )

GBLA jk( )YR k( )

Ys k( )

Ys k( )U k( ){ } 0= Ys k( ) U k( )

Ys k( ){ } 0=

Ys k( ) k k

Ys k( ) 2{ }



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3.g Nonlinear systems - noisy output measurements

:

- independent of-

- asymptotically normally distributed

- asymptotically independent over

- is a smooth function of the frequency

Question- distinction between and ?

GBLA s( )

Gaussian noiseyS t( )

yR t( )

ny t( )

y t( )y0 t( )u0 t( )

NY k( )

U0 k( )

NY k( ){ } 0=

k

NY k( )2{ }

ys t( ) ny t( )



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3.g Nonlinear systems - noisy output measurements (contd)

FRF measurement using random phase multisine

FRF measurement using Gaussian noise

periodstransient

1 2 P

U0p[ ]

U0= Ysp[ ]

Ys= NYp[ ], ,

GML GBLA Ys U0 N Y U0+ +=

varGML( ) var NYp[ ]( ) P U0

2( )=

segments

1 2 P

U0p[ ]

Ysp[ ]

NYp[ ], ,

GLS GBLA SYsU0 S

U0U0 SNYU0 SU0U0+ +=

varGLS( ) var NYp[ ]

( ) var Ysp[ ]

( )+( ) PSU0U0( )=



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FRF measurement using random phase multisine (contd)

...

Mr

ealizations

transient Pperiods

...

...

...

G m p,[ ] Y m p,[ ]

U0

m[ ]------------- GBLA

YSm[ ]

U0

m[ ]----------

NYm p,[ ]

U0

m[ ]--------------+ += =

G 1[ ] n2 1[ ],

G 2[ ]

n2 2[ ],

G M[ ]

n2 M[ ],

GML n2,

ML2

G 1 1,[ ] G 1 2,[ ] G 1 P,[ ]

G 2 1,[ ]

G 2 2,[ ]

G 2 P,[ ]

G 2 P,[ ]

G M 1,[ ]

G M 2,[ ]

G M P,[ ]



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FRF measurement using random phase multisine (contd)Sample mean and sample variance of each realisation

Sample mean and sample variance of the mean values

Properties

Note the asymmetric averaging of and

G m[ ] 1

P--- G

m p,[ ]

p 1=

P

=

n2 m[ ] 1

P P 1( )--------------------- G

m p,[ ] G m[ ] 2

p 1=

P

= n2 1

M2------- n

2 m[ ]m 1=

M

=,

GML1

M

----- G m[ ]

m 1=

M

=

ML2 1

M M 1( )------------------------ G

m[ ]GML

2

m 1=

M

=

n2

{ } varN

Y

( )

MP U02---------------------=

ML2{ }

varYs( )

M U02

-----------------varNY( )

MP U02

---------------------+=

Ys NY


3 h Si l i l FRF i i h f li di i


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3.h Simulation example: FRF estimate in the presence of nonlinear distortions

Multisine

- , , period length-

Gaussian noise- record length same measurement time

- segment length same averaging of the stochastic NL distortions- same power spectrum

u t( )5tanh

x

5---

x t( ) z t( )

Gaussian noise

1 2 1 2 P1

2

M

multisine

M

M P N( ) data points

ny t( )y t( )

M 4= P 2= N 12500=stdu t( )( ) 1=

M P N

N P


3 h Si l ti l FRF ti t i th f li di t ti ( td)


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3.h Simulation example: FRF estimate in the presence of nonlinear distortions (contd)

Note:

- random noise is times more averaged for the multisine experiment

0 200 400 600 800 1000 1200 1400 1600 1800 2000-60

-50

-40

-30

-20

-10

0

10

0 200 400 600 800 1000 1200 1400 1600 1800 2000-60

-50

-40

-30

-20

-10

0

10

Random phase multisine Gaussian noise

FRFtotal variance (noise + NL distortions)

noise variance

P 2=


3 i Nonlinear systems noisy input/output measurements


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3.i Nonlinear systems - noisy input/output measurements

Noisy input/output measurements

Two cases1. reference signal is not available

2. reference signal is available

Gaussian noise

GBLA s( )

yS t( )

yR t( )

ny t( )

y t( )y0 t( )u0 t( )

nu t( )

u t( )

A s( )r t( )

y t( ) yR t( ) yS t( ) ny t( )+ +=

u t( ) u0 t( ) nu t( )+=

r t( )

r t( )


3 i Nonlinear systems noisy input/output measurements (contd)


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3.i Nonlinear systems - noisy input/output measurements (cont d)

Case 1- different realisations of periods each

- for each realisation: sample mean and sample variance input/output spectraover the periods FRFs + noise variance (see section 3.b, p. 32-35)

- sample mean and sample variance FRFs over the realisations (see section3.g, p. 51)

Case 2- different realisations of periods each

- calculate the input, output, and reference DFT spectra of each period- projection step

or

- variance analysis of the set of projected input/output DFT spectra

, ,

- calculate uncertainty on FRF (see section 3.b, p. 35)

M P

M P

URm p,[ ] U

m p,[ ]

R m[ ]

--------------= YRm p,[ ] Y

m p,[ ]

R m[ ]

-------------=, URm p,[ ] U

m p,[ ]

ej R m[ ]

---------------= YRm p,[ ] Y

m p,[ ]

ej R m[ ]

---------------=,

UR2

varNU( ) YR2

varNY( ) var Ys( ), YRUR2

covarNY NU,( )


3 j Measurement example: the open loop gain of an operational amplifier


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3.j Measurement example: the open loop gain of an operational amplifier

101

102

103

104

105

-80

-40

0

40

80

Frequency (Hz)

Amplitude(dB)

Open loop gain

101

102

103

104

105

-80

-40

0

40

80

Frequency (Hz)

Amplitude(dB)

Open loop gain

101

102

103

104

105

-100

-60

-20

20

Frequency (Hz)

Phase()

Open loop gain

101

102

103

104

105

-100

-60

-20

20

Frequency (Hz)

Phase()

Open loop gain

v- 6.1 mVrms=

v- 12.3 mVrms=

M 25 P, 5= =


3 k Multivariable systems - periodic excitations


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3.k Multivariable systems - periodic excitations

Difficulty: from one experiment one cannot calculate the frequency response matrixSolution: perform experiments

with:

:

, : -th input/output signal of the -th experiment

regular? FRM

Problem: which experiments guarantee that is regular and well conditioned?

U k( )G jk( )

Y k( )

nu 1ny nu

ny 1

nu

Y k( ) G jk( )U k( )=

Y k( ) ny nu

U k( ) nu nu

U p q,[ ] k( ) Y p q,[ ] k( ) p q

U k( )

G jk( ) Y k( )U 1 k( )=

U k( )


3 k Multivariable systems - periodic excitations (contd)


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3.k Multivariable systems periodic excitations (cont d)

Guaranteed well conditioned MIMO experiments1. cyclic shifts of multisines with interleaved frequency grid,

for example,

Notes:- in theory one experiment is enough

- generators interact perform experiments

- loss in frequency resolution of a factor

- elimination interaction generators by precompensation

nu nu

nu 3=

U k( )

U1 k( ) U3 k( ) U2 k( )

U2 k( ) U1 k( ) U3 k( )

U3 k( ) U2 k( ) U1 k( )

=

U1

U2

U3

nu

nu


3.k Multivariable systems - periodic excitations (contd)


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Guaranteed well conditioned MIMO experiments (contd)2. with an orthogonal matrix

,

for example,

Notes:- maximal frequency resolution

- different power for each input

- nonlinear systems

with , ,

- elimination interaction generators by precompensation

U k( ) U k( )W= W

Wp q,[ ]1

nu

---------exp j2 p 1( ) q 1( )

nu-----------------------------------------( )= p q, 1 2 nu, , ,=

nu 3=

U k( ) U k( )

3

-----------

1 1 1

1 ej

23

------e

j2

3------

1 ej2

3------

ej

23

------

=

U k( ) U k( )diag A1 k( ) A2 k( ) Anu k( ), , ,[ ]( )W=

k( ) diag U1 k( ) U2 k( ) Unu k( ), , ,[ ]( )Wdiag 1 ej2 k( ) e jnu k( ), , ,[ ]( )=

ej

rk( ){ } 0= r 2 3 nu, , ,= k 1 2 F, , ,=


3.k Multivariable systems - periodic excitations (contd)


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Variance analysis- over consecutive periods disturbing noise- over different realisations of full MIMO experiments stochastic NL

distortions + disturbing noise

3. Estimation parametric plant model with estimatednonparametric noise model


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3.l MIMO experiment: identification d-axis synchronous machine

3x220V3x220V

HP1430A/D-convertor

HP1445AW-generator

HP1430A/D-convertor

HP1430A/D-convertor

HP1430A/D-convertor

HP1445AW-generator

MX

Icontroller

Computer

VXImeasurement

system

Armature Field

Currentcontroller

Thyristorrectifier

+

-

Currentcontroller

+

-

Thyristorrectifier

ia

if

efea



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3.l MIMO experiment: identification d-axis synchronous machine (contd)

-80

-60

-40

-20

0

0.0

05

0.0

10.1 1 1

0

1

00

3

00

Ia(dBA)

f(Hz)

-80

-60

-40

-20

0

0.005

0

.01

0.1 1 1

0

100

300

f(Hz)

If

(dBA)

-120

-100

-80

-60

-40

0.0

05

0.0

10.1 1 1

0

1

00

3

00

f(Hz)

Ea(dBV)

-120

-100

-80

-60

-40

0.005

0

.01

0.1 1 1

0

100

300

f(Hz)

Ef(dBV)

M 8=

N 65536=

0.1 Hz 230 Hz,[ ]

periods



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102

101

100

101

102

120

80

40

0

102

101

100

101

102

120

80

40

0

102

101

100

101

102

120

80

40

0

102

101

100

101

102

120

80

40

0

measurednoise variance

Z11dB( ) Z12dB( )

Z22dB( )

f(Hz)

Z21dB( )

f(Hz)

impedance



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0.01 0.1 1 10 100 0

25

50

75

100

0.01 0.1 1 10 100 0

25

50

75

100

0.01 0.1 1 10 100 0

25

50

75

100

0.01 0.1 1 10 100 0

25

50

75

100

Z11( )arg ( ) Z12( ) ( )arg

Z22( ) ( )arg

f(Hz)

Z21( ) ( )arg

f(Hz)


3.m MIMO experiment: vibrating mechanical structure


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u1

u2

r1

r2

y1

y2

nu ny 2= =

Shaker Al-plate

r1, r2: full random orthogonal multisines


3.m MIMO experiment: vibrating mechanical structure (contd)


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0 200 400 600 800 1000-100

-50

0

50G11

0 200 400 600 800 1000-100

-50

0

50G12

0 200 400 600 800 1000-100

-50

0

50G21

0 200 400 600 800 1000-100

-50

0

50G22

FRF

total variance

noise variance

M 25 P, 100= =


3.n Multivariable systems - random excitations


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FRM measurement,

where

Noise variance measurement,

segments

1 2 M

data pointsM N

M nu

G jk 1 2+( ) SYDUD k( )SUDUD1

k( )=

SXY1

M----- X m[ ]Y m[ ]H

m 1=

M

=

M nu 1+CY k 1 2+( )

M

2 M nu( )------------------------ SYDYD k( ) S

YDUD k( )SUDUD

1k( )SYDUD

Hk( )( )=

Cov vec G jk 1 2+( )( )( ) 1

M-----SUDUD

Tk( ) C YD k( )

2

M-----SUDUD

Tk( ) C Y k 1 2+( )= =


3.o MIMO simulation example: discrete-time system


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0 0.5 1 1.5 2-40

-20

0

20

0 0.5 1 1.5 2-20

-10

0

10

20

0 0.5 1 1.5 2-40

-20

0

20

40

0 0.5 1 1.5 2-20

0

20

40

0 0.5 1 1.5 2-40

-20

0

20

40

0 0.5 1 1.5 2-40

-20

0

20

40

true value

estimate

ny 3=

nu 2=

M nu 1+=

N 5000=

FRM


3.o MIMO simulation example: discrete-time system (contd)


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true value

estimate

0 1 2-5

0

5

10

0 1 2-10

-5

0

5

10

0 1 2-5

0

5

10

0 1 2-10

-5

0

5

10

0 1 2-20

-10

0

10

20

0 1 2-10

0

10

20

0 1 2-5

0

5

10

0 1 2-10

0

10

20

0 1 2-10

0

10

20

ny 3=

nu 2=

M nu 1+=

N 5000=

noise power spectrum

averaged over30 frequencies


3.p ReferencesBendat J S and A G Piersol (1980) Engineering Applications of Correlations and


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Bendat, J. S. and A. G. Piersol (1980). Engineering Applications of Correlations andSpectral Analysis. Wiley, New York.Brillinger, D. R. (1981). Time Series: Data Analysis and Theory. McGraw-Hill, New York.Dobrowiecki, T. and J. Schoukens (2007). Measuring a linear approximation to weaklynonlinear MIMO systems,Automatica, vol. 43, no. 10, pp. 1737-1751.Guillaume, P., I. Kollar and R. Pintelon (1996). Statistical Analysis of Nonparametric

Transfer Function Estimates, IEEE Trans. Instrum. Meas.,vol. IM-45, no. 2, pp. 594-600.Guillaume, P., R. Pintelon and J. Schoukens (1996). Accurate Estimation of MultivariableFrequency Response Functions, Proceedings of the 13th IFAC Triennial World Conference,San Fransisco (USA), June 30 - July 5, pp. 423-428.Pintelon, R. and J. Schoukens (2012). System Identification: a Frequency Domain

Approach.IEEE Press: Piscataway, New Jersey (second edition).Pintelon R. and J. Schoukens (2001). Measurement of frequency response functions usingperiodic excitations, corrupted by correlated input/output errors, IEEE Trans. Instrum. andMeas.,vol. 50, no. 6, pp. 1753-1760.Pintelon, R., Y. Rolain and W. Van Moer (2003). Probability density function for frequency

response function measurements using periodic signals, IEEE Trans. Instrum. and Meas.,vol. 52, no. 1, pp. 61-68.Schoukens, J., R. Pintelon, T. Dobrowiecki, and Y. Rolain (2005). Identification of linearsystems with nonlinear distortions,Automatica, vol. 41, no. 2, pp. 491-504.Schoukens, J., Y. Rolain, and R. Pintelon (2006). Analysis of window/leakage effects in

frequency response function measurements.Automatica, vol. 42, no. 1, pp. 27-38.

4. Parametric models of LTI systems

4.a Band-limited versus zero-order-hold measurement set up 724 b Band limited versus zero order hold signal assumption 77


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4.b Band-limited versus zero-order-hold signal assumption 77

4.c Measurement example: first order system 784.d Band-limited versus zero-order-hold noise models 834.e Lumped versus distributed LTI systems 914.f Lumped systems 934.g Relation between input/output DFT spectra - plant model only 954.h Simulation example: 4th order discrete-time Butterworth filter 984.i Simulation example: 2nd order continuous-time system 1024.j Measurement example: octave bandpass filter 1034.k Measurement example: reduction of iron 104

4.l Relation between input/output DFT spectra - plant and noise model 1054.m Plant and noise model structures 1064.n Parametrizations LTI systems 1084.o Multivariable systems 1114.p References 112


4.a Band-limited versus zero-order-hold measurement set up


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Zero-order-hold (ZOH)

input known exactly output error problem actuator is part of the model perfect ZOH set up

- , from DC to absolute calibration

-

Actuator

y1 t( )

+

y kTs( )

my t( )

yAA t( )

u t( )

+

np t( )

Generator/

udk( )

ZOH

uzoh t( )

Gy

s(

)Gc s( )

GZOH z 1( )

G s( )Gact s( )controller

Gact j( ) 1= Gy j( ) 1=

GZOH z 1( ) 1 z 1( )Z L 1 G s( ) s{ }{ }=


4.a Band-limited versus zero-order-hold measurement set up (contd)


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Band-limited (BL)

errors-in-variables problem actuator is not modelled perfect BL set up

- for relative calibration

- for

y1 t( )u1 t( )

+

+

+

y kTs( )

ng t( )

mu t( ) my t( )uAA t( ) yAA t( )

ug t( )

+

np t( )

Generator

udk( )

ZOH

uzoh t( )

Gu

s(

)

Gy

s(

)

u kTs( )

G s( )Gact s( )

Gu j( ) Gy j( )= f fs 2


4.g Relation between input/output DFT spectra - plant model only (contd)

Summary

Y k( ) G ( )U k( ) T ( )+=


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with

,

where

and

Y k( ) Gk ,( )U k( ) TG k ,( )+=

G ,( ) B ,( )

A ,( )------------------

brrr 0=nb

arrr 0=

na--------------------------= = TG ,( )

IG ,( )

A ,( )-------------------

igrr

r 0=

nig

arrr 0=

na---------------------------= =

z 1 nig

max na nb,( ) 1=

s s, nig max na nb,( ) 1>

=

TG k ,( )0 for periodic and time-limited signals

O N

1 2/

( ) arbitrary signals

=


4.h Simulation example: 4th order discrete-time Butterworth filter

input output


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Cut off frequency

0

0.2

0.4

0.6

0.8

1

0 100 200 300n (samples)

0

0.2

0.4

0.6

0.8

1

0 100 200 300n (samples)

excitationu(nT

s)

responsey(nT

s)

p p

fs 4


4.h Simulation example: 4th order discrete-time Butterworth filter (contd)


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Note:- random behaviour error on FRF due to random behaviour

[[[[[[[[[[

[

[[[[[[[[[[[

[[[[

[[[[

[

[[[[[[[

[

[

[[[[[[[[[[[[[[[[[

[

[[[[[[

[

[[[

[[[[[[[

-160

-120

-80

-40

0

40

0 0.1 0.2 0.3 0.4 0.5

amplitude(dB)

f/fs

[

[

[

[[[[[

[[[

[[

[[

[

[

[

[[

[[

[

[[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

-180

-90

0

90

180

0 0.1 0.2 0.3 0.4 0.5

phase()

f/fs

amplitude phase

true value

[

Y k( )

U k( )-----------

TG zk1 ,( ) U k( ) U k( )



difference true value and model


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-350

-300

-250

-200

-150

-100

-50

0

0 0.1 0.2 0.3 0.4 0.5

error(dB)

f/fs

without transient

with transient

difference true value and model




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-0.8

-0.6

-0.4

-0.2

0

0.2

0 50 100 150

transient

response

n (samples)

impulse response TG z 1( )

IG z 1( )

A z 1( )---------------=


4.i Simulation example: 2nd order continuous-time system

60


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true FRF

error model

without transient

error estimate

nig10=

with transient

-120

-90

-60

-30

0

30

60

0 10 20 30 40 50 60 70

amplitude(dB)

f (Hz)


4.j Measurement example: octave bandpass filter

plant modelna 6 nb, 4= =


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measured FRF(multisine excit.)

error model withtransient nig

6=

error model

error model

without transient

(multisine excit.)

(arbitr. excit.)

(arbitr. excit.)X

X

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-120

-100

-80

-60

-40

-20

0

20

0 0.5 1 1.5 2 2.5

am

plitude(dB)

f (kHz)


4.k Measurement example: reduction of iron

050

amplitude phase


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-30

-20

-10

0.01 0.1 1 10 100

Phase()

Frequency (Hz)

-50

0

0.01 0.1 1 10 100

Amp

litude(dB

)

Frequency (Hz)

-50

0

50

0.01 0.1 1 10 100

Amplitude(dB

)

Frequency (Hz)

-30

-20

-10

0

0.01 0.1 1 10 100

Phase

()

Frequency (Hz)

-domain modelwithout transient term

-domain modelwith transient term

model

error

+ sine measurement

standard deviation sine measurement

na nb 3= =nig5=

na nb 3= =

s

s


4.l Relation between input/output DFT spectra - plant and noise model

e t( )


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where, , ,

v t( )

y t( )u t( )

G ,( )

H ,( )

Y k( ) Gk ,( )U k( ) TG k ,( ) Hk ,( )E k( ) THk ,( )+ + +=

G ,( ) B ,( )A ,( )------------------= TG ,( )

IG ,( )A ,( )-------------------= Hk ,( )

C ,( )D ,( )-------------------= TH ,( )

IH ,( )D ,( )-------------------=


4.m Plant and noise model structures

Frequency domain

Y k( ) G ( )U k( ) H ( )E k( ) T ( ) T ( )+ + +=


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Model structures

ARX: , ,

ARMAX: , ,

ARMA: ,

BJ:

Notes:- AR = autoregressive

- MA = moving average- X = exogenous input- BJ = Box-Jenkins- hybrid BJ = CT plant + DT noise model- AR or MA or ARMA parametric noise modelling = time series analysis or

spectral estimation (DT and CT)

Y k( ) Gk ,( )U k( ) Hk ,( )E k( ) TG k ,( ) THk ,( )+ + +=

C 1= D A= TH 0=

D A= TH 0= nigmax na nb nc, ,( ) 1

G 0= TG 0=

D A


4.m Plant and noise model structures (contd)

DT domain

y t( ) G q ( )u t( ) H q ( )e t( )+=


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with the backward shift operator:

and where

CT domain

with the derivative operator:

and where

y t( ) G q ,( )u t( ) H q ,( )e t( )+

q z 1= qx t( ) x t 1( )=

G q( )x t( ) g r( )qrr 0=

( )x t( ) g r( )x t r( )r 0=

= =

y t( ) G p ,( )u t( ) H p ,( )e t( )+=

d

dt

-----= x t( ) dx t( )

dt

-----------=

G p( )x t( ) g ( )x t ( )d

0

=


4.n Parametrizations LTI systems

Plant model

Rational form


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with

Partial fraction expansion

for

with

G ,( ) B ,( )A ,( )------------------

brrr 0=nb

arrr 0=

na

--------------------------= = T a0a1anab0b1bnb[ ]=

G ,( )Lr

r---------------

r p=

r 0

p

Sr

r----------------

r 1=

q

W w,( )+ += s s,=

G z 1 ,( )Lrz

1

1 rz 1

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

r p=r 0

p

Srz

1

1 rz 1

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

r 1=

q

W z 1 w,( )+ +=

T 1qRe 1( )Im 1( )Re p( )Im p( )[=

S1SqRe L1( )Im L1( )Re Lp( )Im Lp( )w0wnw ]


4.n Parametrizations LTI systems (contd)

Plant model (contd)

State space representation for proper transfer functions ( )nb na


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p p p p ( )

with

Pole/zero representation

disadvantage: ill conditioned for multiple poles/zeroes

Systems with time delay

b a

G s ,( ) C sIna A( ) 1 B D+=

G z 1 ,( ) z 1 C Ina

z 1 A( ) 1 B D+=

T vecTA( ) BTCD[ ]=

G ,( ) K r( )r 1=

nb

r( )r 1=

na

------------------------------------=

G ,( ) e s B ,( )A ,( )------------------=

G z 1 ,( ) z Ts B z 1 ,( )

A z 1 ,( )--------------------=


4.n Parametrizations LTI systems (contd)

Noise model

Similar as plant model


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p

Note

Starting value problem parametrizations (see chapter 6)

- rational form linear least squares- state space subspace methods

- other parametrizations via rational form + back transformation


4.o Multivariable systems

Left matrix fraction (LMF) description

with , andG ,( ) A 1 ,( )B ,( )= A ny ny B ny nu


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111/243

Right matrix fraction (RMF) descriptionwith , and

Common denominatorwith , and

State spacesame form as SISO with , , , and

Partial fraction expansionare residue matrices

Notes:

- advantage LMF: - common denominator: rank residue matrices cannot be imposed(too many parameters if residue matrices are not of full rank)

- LMF, RMF, and state space: rank one residue matrices

( with )

G ,( ) B ,( )A 1 ,( )= A nu nu B ny nu

G ,( ) B ,( ) A ,( )= A 1 1 B ny nu

A n n B n nu C ny n D ny nu

Lr Sr, ny nu

Y k( ) G ,( )U k( )= A ,( )Y k( ) B ,( )U k( )=

F n n F 1 adjF( ) det F( )= det adjF( )( ) detF( )( )n 1=


4.p ReferencesBox, G. E. P. and G. M. Jenkins (1976).Time Series Analysis: Forecasting and Control.

Holden-Day, Oakland.Hannan, E. J. and M. Deistler (1988). Linear Systems.Wiley: New York.Henrici P (1974) Applied and Computational Complex Analysis Wiley: New York vol 1


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112/243

Henrici, P. (1974).Applied and Computational Complex Analysis.Wiley: New York, vol. 1.Kailath, T. (1980). Linear Systems. Prentice Hall, Englewood Cliffs.Ljung, L. (1999). System Identification: Theory for the User. Prentice Hall: Upper Saddle

River, New Jersey (second edition).Oldham, K. B., and J. Spanier (1974). The Fractional Calculus.Academic Press: New York.Peeters F., R. Pintelon, J. Schoukens and Y. Rolain (2001). Identification of Rotor-BearingSystems in the Frequency Domain. Part II: Estimation of Modal Parameters, MechanicalSystems and Signal Processing,vol. 15, no. 4, pp. 775-788.

Pintelon, R. and J. Schoukens (2012). System Identification: a Frequency DomainApproach.IEEE Press: Piscataway, New Jersey (second edition).Pintelon R. and J. Schoukens (1997). Identification of Continuous-Time Systems UsingArbitrary Signals,Automatica,vol. 33, no. 5, pp. 991-994.Pintelon R., J. Schoukens and G. Vandersteen (1997). Frequency Domain System

Identification Using Arbitrary Signals, IEEE Trans. Autom. Contr., vol. AC-42, no. 12, pp.1717-1720.Pintelon R., J. Schoukens, L. Pauwels, and E. Van Gheem (2005). Diffusion systems:stability, modeling and identification, IEEE Trans. Instrum. Meas.,vol. 54, no. 5, pp. 2061-2067.

Sderstrm, T. and P. Stoica (1989). System Identification.Prentice Hall: Englewood Cliffs.

5. Improved nonparametric models for LTI systems

5.a Arbitrary excitations generalised output error 1145.b Simulation example: comparison LPM and SA 121

5.c Measurement example: nonlinear electrical circuit 1235.d Arbitrary excitations errors-in-variables 1255 e Measurement example: flexural vibrations of a steel beam 127


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5.e Measurement example: flexural vibrations of a steel beam 1275.f Periodic excitations steady state 1305.g Measurement example: longitudinal vibrations of a plexiglass beam 140

5.h Periodic excitations transient regime 1415.i Measurement example: flexural vibrations of a steel beam 1425.j Summary local polynomial methods 1445.k References 145


5.a Arbitrary excitations generalised output error

e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=


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Goal: starting from and obtain a nonparametric estimate of

- the frequency response function- the noise covariance matrix

Difficulties: the presence of

- for estimating- and for estimating

Property:

- and are smooth functions of the frequency

G( )u t( ) y t( )

v t( )

H( ) V k( ) Hk( )E k( )=

Tk

( ) TH

k

( ) TG

k

( )+=

U k( ) Y k( ),

G( )CV k( ) Cov V k( )( )=

Tk( ) Gk( )

Tk( ) Gk( )U k( ) CV k( )

G( ), H( ), T( )


5.a Arbitrary excitations generalised output error (contd)

e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=


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115/243

Basic idea solution: local polynomial approximation of and

with Taylor series expansion

and

G( )u t( ) y t( )

v t( )

H( ) V k( ) Hk( )E k( )=

Tk( ) THk( ) TG k( )+=

G( ) T( )

Y k r+( ) Gk r+( )U k r+( ) Tk r+( ) V k r+( )+ +=

Gk r+( ) Gk( ) gs k( )rss 1=

R

O r N( ) R 1+( )( )+ +=

Tk r+( ) Tk( ) ts k( )rss 1=R N 1 2/ O r N( ) R 1+( )( )+ +=

Gk( ) gs k( )

ts k( )Tk( )

r n n 1+ 1 0 1 n 1 n, , , , , , , ,=

Gk( ) g1 k( ) g2 k( ) gR k( ) Tk( ) t1 k( ) t2 k( ) tR k( )=



( )

e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=


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Local polynomial model in the band :

where

and

with

G( )u t( ) y t( )

v t( )

H( ) V k( ) Hk( )E k( )=

Tk( ) THk( ) TG k( )+=

k n k n+,[ ]

Yn Kn Vn+=

Xn X k n( ) X k n 1+( ) X k( ) X k n+( )=

K k r+( ) K1 r( ) U k r+( )

K1 r( )= K1 r( )

1

r

rR

=



H ( )

e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=


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Local polynomial model

Linear least squares estimate

Residual linear least squares estimate

with

the degrees of freedom of the residuals

G( )u t( ) y t( )

v t( )

H( ) V k( ) Hk( )E k( )=

Tk( ) THk( ) TG k( )+=

Yn Kn Vn+=

YnKnH KnKnH( ) 1= G k( )

V n Yn Kn= CV k( ) 1

q---V nV n

H=

q 2n 1 R 1+( ) nu 1+( )+=

dof



Repeat the previous procedure at frequency for all the other frequencies , , k k 1+ k 2+


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Consequence

the FRF and noise covariance estimates are correlated over thefrequency

the correlation length is

k 1+kk n 1 k n k n 1+ k 1 k n+ k n 1+ + k n 2+ +

G

k( ) C

V k( )

2n



H ( )

e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=


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119/243

Bias errorlocal polynomial method(LPM)

Bias error spectral analysis approach

with for win = diff, half sine; and for win = hann

G( )u t( ) y t( )

v t( )

H( ) V k( ) Hk( )E k( )=

Tk( ) THk( ) TG k( )+=

G k( ){ } G0k( ) G0R 1+( ) k( )OintG n N( ) R 1+( )( ) OleakG n N( ) R 2+( )( )+ +=

CV k( ){ } CV k( ) CV1( ) k( )OintH n N( ) G0R 1+( ) k( )OintG n N( )2 R 1+( )( ) G0R 1+( ) k( )( )H+ +=

Gwin

k( ){ } G0k( ) winG01( ) k( )OintG M N( )2( ) OleakG M N( )2( )+ +=

CVwin

k( ){ } CV k( ) win

2----------CV

2( ) k( )OintH M N( )2( ) win

2----------G0

1( ) k( )OintG M N( )2( ) G01( ) k( )( )H+ +=

win 1 4= win 1 3=



H ( )

e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=


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Local polynomial approach Spectral analysis method

smaller bias error larger bias error

smaller uncertainty larger uncertainty

larger frequency resolution smaller frequency resolution

larger calculation time smaller calculation timev

correlation length correlation length (rect),

(diff, half sine), or (hanning)

G( )u t( ) y t( )

v t( )

H( ) V k( ) Hk( )E k( )=

Tk( ) THk( ) TG k( )+=

2n 0

1 2


5.b Simulation example: comparison LPM and SA

50 50

LPM and SA (hanning window): same dofnoise covariance estimates


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0 0.5 1 1.5 2100

50

0

Frequency (Hz)

FRF

(dB)

0 0.5 1 1.5 2100

50

0

Frequency (Hz)

FRF

(dB)

0 0.5 1 1.5 2100

50

0

50

Frequency (Hz)

FRF(dB)

0 0.5 1 1.5 2100

50

0

50

Frequency (Hz)

FRF(dB)

G0 G

SA G0 G

LPM G

0


5.b MIMO simulation example: comparison LPM and SA (contd)

LPM and SA (hanning window): same dofnoise covariance estimates 20 0


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0 0.5 1 1.5 260

40

20

0

Frequency (Hz)

Noise

var.(dB)

0 0.5 1 1.5 240

30

20

10

Frequency (Hz)

Noise

var.(dB)

0 0.5 1 1.5 240

30

20

10

0

Frequency (Hz)

N

oisevar.(dB)

0 0.5 1 1.5 260

40

20

0

Frequency (Hz)

N

oisevar.(dB)

true value SA LPM


5.c Measurement example: nonlinear electrical circuit

C 9.4 F=R 16 =


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White noise excitation with bandwidth of

samples at :

split in 20 blocks of samples each

spectral analysis and local polynomial estimates on each block of samples

sample means and sample variances over the 20 blocks

y3 t( )u t( ) y t( ) L 1 H=

1000 V 2

200 Hz

10400 s 20 MHz 215 600 Hz=

N 520=

N 520=


5.c Measurement example: nonlinear electrical circuit (contd)

20


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0 50 100 150 200 25080

60

40

20

0

20

Freq. (Hz)

FRF(dB)

0 50 100 150 200 250120

100

80

60

40

20

Freq. (Hz)

Noisevar.(

dB) hann, diff

LPM

hann, diff

LPM

BJ BJ

full lines: FRF

dashed lines: std FRF full lines: noise var.dashed lines: std noise var.


5.d Arbitrary excitations errors-in-variables

nc t( )

Controller


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Indirect methodfor measuring the FRF (two possible cases):

Linearplant and (non)linearactuator and controller:

Nonlinearplant andlinearactuator and controller:

Solution:

Reference signal needed + modeling from to

Plant

np t( )

r t( )

my t( )

y t( )

mu t( )

u t( )

Actuator

ng t( )

G0 jk( ) Syr jk( )Sur1 jk( )=

GBLA jk( ) Syr jk( )Sur1 jk( )=

r t( ) r t( ) z t( ) yT t( ) uT t( )[ ]T=


5.d Arbitrary excitations errors-in-variables (contd)

nc t( )

Controller


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126/243

Modeling from reference to input and output:

with

Note: applicable to both spectral analysis and local polynomial methods

Plant

np t( )

r t( )

my t( )

y t( )

mu t( )

u t( )

Actuator

ng t( )

Z k( ) Gry k( )Gru k( )

R k( ) THY k( )THU

k( )VZ k( )+ += Z k( ) Y k( )

U k( )=

G k( ) Gry k( )Gru1 k( )=


5.e Measurement example: flexural vibrations of a steel beam

0 x 0 Lx


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Beam parameters: length 61 cm, height 2.47 cm, width 4.93 mm, density 7800 kg/m3

Excitation in the band [0 Hz, 6 kHz]:

random binary sequence

random phase multisine

samples of the reference, force (input), and acceleration (output) signals at:

spectral analysis and local polynomial estimates

generalised output error and errors-in-variables estimates

u t( )

y t x,( )

N 50 1024=

s 10 MHz 29 19.53 kHz=


5.e Measurement example: flexural vibrations of a steel beam (contd)

50 50

OE-LPM EIV-LPM

random binary sequence


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0 2 4 6

-50

0

50

Frequency (kHz)

FRF

(dB)

0 2 4 6

-50

0

50

Frequency (kHz)

FRF

(dB)

0.92 0.93 0.94 0.95

40

60

80

Frequency (kHz)

FRF(

dB)

0.92 0.93 0.94 0.95

40

60

80

Frequency (kHz)

FRF(

dB)


5.e Measurement example: flexural vibrations of a steel beam (contd)

20 50


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4.12 4.14 4.16 4.18 4.2

-60

-40-20

0

Frequency (kHz)

FRF

(dB)

0.16 0.18 0.2-50

0

Frequency (kHz)

FRF

(dB)

+local polynomial approachmultisine

spectral analysis

FRF std(FRF)

olocal polynomial approachmultisine

spectral analysis


5.f Periodic excitations steady state

nc t( )

Controller


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Two possible cases:

Linearplant and (non)linearactuator and controller: Nonlinearplant andlinearactuator and controller:

Required signals:

(for multi-input and/or nonlinear plant only), and

Plant

np t( )

r t( )

my t( )

y t( )

mu t( )

u t( )

Actuator

ng t( )

G0 jk( ) Syr jk( )Sur1

jk( )=GBLA jk( ) Syr jk( )Sur1 jk( )=

r t( ) u t( ) y t( )


5.f Periodic excitations steady state (contd)

withZ k( ) Z0 k( ) HZk( )EZ k( ) THZ k( )+ += Z k( ) Y k( )

( )=


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Noise transients (leakage) errors

Smooth functions of the frequencies

Increase the variability of the FRF measurement

Introduce a correlation between consecutive signal periods

Important for lightly damped systems (e.g. mechanical vibrating structures)

k Z kU k( )

THZk( ),



noiseless signal noisy signal

P 2=DFT spectrum of consecutive periods


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1 2 3 DFT bin 1 2 3 DFT bin

noiseless signal noisy signal

signal noise transientnoise

Z k( ) Z0 k( ) HZk( )EZ k( ) THZ

k( )+ += HZk( )EZ k( ) THZ

k( )




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Basic idea solution: local polynomial expansion at the non-excited DFT lines

with Taylor series expansion for ( is not used!)

and

kP 5+kP 3+kP 1+kP 3 kP 1kP

DFT bin

kP 2+ kP 4+kP 2

THZ ( )

Z kP mi( ) HZkP mi( )EZ kP mi( ) THZ kP mi( )+= HZkP mi( )EZ kP mi( ) THZ kP mi( )

i 1 2 n, , ,= kP

THZkP mi( ) THZ kP( ) trk( ) mi( )

r

r 1=

R

1

PN------------O

mi

PN--------

R 1+( )

( )+ += THZkP( ) trk( )

THZ

k

( ) t1

k( ) t2

k( ) tR

k( )=


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