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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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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
Number of measurements
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. Preliminary Example
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. Preliminary Example
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
Number of measurements
N
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2. Preliminary Example
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. Preliminary Example
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
Number of measurements
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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. Preliminary Example
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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2. Preliminary Example
2.k Extension to arbitrary currents (contd)
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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2. Preliminary Example
2.k Extension to arbitrary currents (contd)
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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2. Preliminary Example
2.k Extension to arbitrary currents (contd)
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. Preliminary Example
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
3 Nonparametric models of LTI systems
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3. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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
=
3 Nonparametric models of LTI systems
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3. Nonparametric models of LTI systems
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
-------------------------------------------= =
3 Nonparametric models of LTI systems
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3. Nonparametric models of LTI systems
3.b Periodic excitations - errors-in-variables problem (contd)
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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3. Nonparametric models of LTI systems
3.b Periodic excitations - errors-in-variables problem (contd)
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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3. Nonparametric models of LTI systems
3.b Periodic excitations - errors-in-variables problem (contd)
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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[
-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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3.f Nonlinear systems - introduction (contd)
:- 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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3.f Nonlinear systems - introduction (contd)
:- 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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3.f Nonlinear systems - introduction (contd)
:
- 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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3.g Nonlinear systems - noisy output measurements (contd)
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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3.g Nonlinear systems - noisy output measurements (contd)
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
3.k Multivariable systems - periodic excitations (contd)
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3.k Multivariable systems periodic excitations (cont d)
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. Nonparametric models of LTI systems
3.k Multivariable systems - periodic excitations (contd)
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3.k Multivariable systems periodic excitations (cont d)
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
3. Estimation parametric plant model with estimatednonparametric noise model
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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
3. Estimation parametric plant model with estimatednonparametric noise model
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3.l MIMO experiment: identification d-axis synchronous machine (contd)
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
3. Estimation parametric plant model with estimatednonparametric noise model
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3.l MIMO experiment: identification d-axis synchronous machine (contd)
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Nonparametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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( )
4. Parametric models of LTI systems
4.h Simulation example: 4th order discrete-time Butterworth filter (contd)
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
4. Parametric models of LTI systems
4.h Simulation example: 4th order discrete-time Butterworth filter (contd)
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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. Parametric models of LTI systems
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. Parametric models of LTI systems
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
X
X
X
X
X
X
XXX
X
XXX
X
X
XXXXX
X
XXXX
X
XX
X
XX
XXXXXXXXXXXXXXXXXXX
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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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
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. Parametric models of LTI systems
4.o Multivariable systems
Left matrix fraction (LMF) description
with , andG ,( ) A 1 ,( )B ,( )= A ny ny B ny nu
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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. Parametric models of LTI systems
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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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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
5.a Arbitrary excitations generalised output error (contd)
e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=
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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( )=
5. Improved nonparametric models for LTI systems
5.a Arbitrary excitations generalised output error (contd)
( )
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
=
5. Improved nonparametric models for LTI systems
5.a Arbitrary excitations generalised output error (contd)
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
5. Improved nonparametric models for LTI systems
5.a Arbitrary excitations generalised output error (contd)
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
5. Improved nonparametric models for LTI systems
5.a Arbitrary excitations generalised output error (contd)
H ( )
e t( ) Y k( ) Gk( )U k( ) Tk( ) V k( )+ +=
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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=
5. Improved nonparametric models for LTI systems
5.a Arbitrary excitations generalised output error (contd)
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
5.d Arbitrary excitations errors-in-variables (contd)
nc t( )
Controller
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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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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. Improved nonparametric models for LTI systems
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( ),
5. Improved nonparametric models for LTI systems
5.f Periodic excitations steady state (contd)
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( )
5. Improved nonparametric models for LTI systems
5.f Periodic excitations steady state (contd)
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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( )=
5. Improved nonparametric models for LTI systems
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