SYSTEMS
Identification
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
Reference: “System Identification Theory For The User”
Lennart Ljung
lecture 81
Lecture 8
Convergence & Consistency
Topics to be covered include:





Conditions on the Data Set
Prediction-Error Approach
Consistency and Identifiability
LTI Models: A Frequency-Domain Description of the
Limit Model
The Correlation Approach
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Ali Karimpour Nov 2010
lecture 81
Introduction
Previously different methods to determine models from data are described.
To use these methods in practice, we need insight into their properties:
• How well will the identified model describe the actual system?
• Are some identification methods better than others?
• How should the design variables be chosen?
The questions depend to mapping:
Z N  ˆN  DM
(I)
1- Simulation.
Generate data Z N with known properties


Apply the mapping I
Evaluate the properties of θ̂N
2- Analysis.
Assume certain prpopertie s of Z N

Calculate the properties of θ̂N
3
Ali Karimpour Nov 2010
lecture 81
Lecture 8
Convergence & Consistency
Topics to be covered include:





Conditions on the Data Set
Prediction-Error Approach
Consistency and Identifiability
LTI Models: A Frequency-Domain Description of the
Limit Model
The Correlation Approach
4
Ali Karimpour Nov 2010
lecture 81
Conditions on the Data Set
Data generation configuration
Let
Z N  u(1) ,
y(1) , ..... u( N ) ,
y( N )
Analysis means:
Assume certain prpopertie s of Z N

Calculate the properties of θ̂N
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Ali Karimpour Nov 2010
lecture 81
Conditions on the Data Set
A Technical Condition D1


D1 : The data set Z  is such that for some filters d i(i ) (k )


y (t )   d (k )r (t  k )   d t( 2) (k )e0 (t  k )
k 1
(1)
t
k 0


u (t )   d (k )r (t  k )   d t( 4 ) (k )e0 (t  k )
k 0
( 3)
t
k 0
1. r (t )is a bounded, determinis tic, external input sequence.
2. e0 (t )is a sequence of independen t random variable with zero
mean and bounded moments of order 4   for some   0.


3. The family of filters d t(i ) (k )

k 1
, i  1, 2, 3, and 4; t  1, 2, ...
is uniformly stable.
4. The signals y(t)and u(t)are jointly quasi - stationary .
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Ali Karimpour Nov 2010
lecture 81
Conditions on the Data Set
Remind:
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Ali Karimpour Nov 2010
lecture 81
Conditions on the Data Set
A True System S
S1 : The data set Z  is generated acording to
S : y(t )  G0 (q)u(t )  H 0 (q)e0 (t )
e0 (t )is a sequence of independen t random variable
with zero mean and
bounded moments of order 4   for some   0.
H 0 (q) is an inversly stable, monic filter.
So S is the true system. Given a model structure
M:
G(q, ), H (q, ) |   DM 
It is necessaril y to know whether t he true system belongs to the set or not.

D T ( S , M )    DM | G (ei , )  G0 (ei ); H (ei , )  H 0 (ei );    
We say
DT (S , M )   or S  M

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Ali Karimpour Nov 2010
lecture 81
Conditions on the Data Set
When S1 holds, a more explicit version of conditions D1 can be given
Lemma 8.1. Suppose that S1 holds, and the input is chosen as
u (t )   F (q) y (t )  r (t )
such that there is a delay in either G0 or F and such that
are stable filters and that the disturbanc e is a quasi - stationary .
Then D1 holds.
Exercise1: Prove the lemma.
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Ali Karimpour Nov 2010
lecture 81
Conditions on the Data Set
Information Content in the Data Set.
Whether the data set Z allows us to distinguish between different models in
the set?
The data set is informative if it is capable of distinguishing between different
models.
Definition: A quasi-stationary data set Z is informative enough with respect to
the model set M* if , for any two models W1(q) and W1(q) in the set,
E W1 (q)  W2 (q)  z (t )  0
2
yˆ1 (t | t 1)

W1 (ei )  W2 (ei ) for almost all 
yˆ 2 (t | t 1)
Recall:
Definition: A quasi-stationary data set Z is informative if it is informative
enough with respect to the model set L* , consisting of all linear, time invariant
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models.
Ali Karimpour Nov 2010
lecture 81
Conditions on the Data Set
Remember
Theorem: A quasi-stationary data set Z is informative if the spectrum matrix
for z(t)=[u(t) y(t)]T is strictly positive definite for almost all .
Proof: We need to show
2
E W1 (q)  W2 (q)  z (t )  0

W1 (ei )  W2 (ei ) for almost all 
 ~
~
~ T i
2
~
i

W1 (q)  W2 (q)  W (q) E W (q) z (t )  0
W
(
e
)

(

)
W
(e )d  0
z


  u ( )  uy ( )
 z ( )  


(

)

(

)
yu
y




2-65
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Ali Karimpour Nov 2010
lecture 81
Lecture 8
Convergence & Consistency
Topics to be covered include:





Conditions on the Data Set
Prediction-Error Approach
Consistency and Identifiability
LTI Models: A Frequency-Domain Description of the
Limit Model
The Correlation Approach
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Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
In PEM
ˆN ( Z N )  arg min VN ( , Z N )
 DM
ˆN ( Z N )
Clearly it depends to
N 
?
VN ( , Z N )
For quadratic criterion and a linear, uniformly stable model structure M, we have
1
VN ( , Z ) 
N
N
N
1 2
 (t , )

t 1 2
 (t , )  1  Wy (q, )y (t )  Wu (q, )u (t )
Using D1 Condition:
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Uniformly Stable since M is stable
Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
Definition : A model structure M is said to be uniformly stable if the family of filters
W (q, ),  (q, ) and (d / d ) (q, );  DM is uniformly
stable and if the set DM is
compact. Note that  (q,  )  (d / d )W (q,  )
Lemma : Consider a uniformly stable, linear model structure M . Assume that the data
set Z  is subject to D1. then
1
VN ( , Z ) 
N
N
sup VN (θ , Z N )-V (θ )  0
 DM
where
remember
N
1 2
 (t , )

t 1 2
w. p.1 as N  
1 2
V (θ )  E  (t , )
2
1
E f (t )  lim
N  N
N
 Ef (t )
t 1
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Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
Lemma : Consider a uniformly stable, linear model structure M . Assume that the data
set Z  is subject to D1. then
w. p.1 as N   V (θ )  E 1  2 (t ,  )
2
sup VN (θ , Z N )-V (θ )  0
 DM
Since DM is compact
Minimizing argument ˆN of VN
Minimizing argument  * of V
It may happen that V ( ) does not have a unique/global minimum.


Dc  arg min V ( )   |   DM ,V ( )  min V ( )
 DM
 DM
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Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
Theorem : Let ˆN be defined by
ˆN ( Z )  arg min VN ( , Z )
N
N
 DM
1
VN ( , Z ) 
N
N
N
1 2
 (t , )

t 1 2
where  (t , ) is determined from a uniformly stable linear model structure M .
assume that the data set Z  is subject to D1 then :
ˆN  Dc
ˆN  Dc
w.p. 1 as N  
w.p. 1 as N  

inf ˆN    0
as N  
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Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
Example 8.1 Bias in ARX Structures
Suppose that the system is given by
y (t )  a0 y (t  1)  b0u (t  1)  e0 (t )  c0 e0 (t  1)
Where {u(t)} and {e0(t)} are independent white noises with unit variances. Let
the model structure be given by
yˆ (t |  )  ay (t  1)  bu (t  1),
a 
  
b 
The prediction-error variance is
V ( )  E  y (t )  ay (t  1)  bu (t  1)
2
V ( )  r0 (1  a 2  2aa0 )  b 2  2bb0  2ac0
2
b
 c0 (c0  a0 )  a0c0  1
r0  Ey 2 (t )  0
1  a02
Exercise2 : Proof I
(I)
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Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
Example 8.1 Bias in ARX Structures
The prediction-error variance is
c0 

*


a
a

 *   *    0 r0 
b   b0 
V ( )  r0 (1  a 2  2aa0 )  b 2  2bb0  2ac0
c02
V ( )  1  c 
r0
*
But for true value of parameters
a0 
0   
b0 
2
0
V ( 0 )  1  c02
When we apply PEM the estimates will converge, according to pervious theorem.
But one of the parameters is biased.
It is clear that bias is beneficial for the prediction performance of the model.
So it gives a strictly better predictor.
Exercise3 : Proof and explain the example by suitable simulation.
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Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
Example 8.2 Wrong Time Delay
Consider the system
y(t )  b0u(t  1)  e0 (t )
u(t )  d0u(t  1)  w(t )
where
{e(t)} and {w(t)} are independent white-noise sequence with unit variances.
Let the model structure be given by
yˆ (t |  )  bu (t  2),
The associated prediction-error variance is:
 b
Ey(t )  bu(t  2)  Eb0u(t 1)  bu(t  2)  Ee02 (t )
2
2
 E(b0 d0  b)u(t  2)  b0 w(t 1)  1
2
(b0 d 0  b) 2
2


b
0 1
2
1  d0
Exercise4 : Proof I
(I )
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Ali Karimpour Nov 2010
lecture 81
Prediction-Error Approach
Example 8.2 Wrong Time Delay
Consider the system
y(t )  b0u(t  1)  e0 (t )
where
u(t )  d0u(t  1)  w(t )
(b0 d 0  b) 2
2
E y (t )  bu (t  2) 

b
1
0
2
1  d0
2
Hence
Now the predictor
bˆN  b0 d0
w.p. 1 as N  
yˆ (t | t  1)  b0 d 0u(t  2)
Is a fairly reasonable one for
y(t )  b0u(t  1)  e0 (t )
where
u(t )  d0u(t  1)  w(t )
It yields the prediction error vari ance 1  b02 , compared to the value 1 for a correct
b02
model and the output var iance 1 
1  d 02
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Ali Karimpour Nov 2010
lecture 81
Lecture 8
Convergence & Consistency
Topics to be covered include:





Conditions on the Data Set
Prediction-Error Approach
Consistency and Identifiability
LTI Models: A Frequency-Domain Description of the
Limit Model
The Correlation Approach
21
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Karimpour Nov
Nov2010
2009
Ali
lecture 81
Consistency and Identifiability
Suppose that assumption S1 holds so that we have a true system.
Will PE identification recover the true plant?
The first condition:
So
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Exercise5: Prove the above-mentioned theorem.
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lecture 81
Consistency and Identifiability
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lecture 81
Consistency and Identifiability
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lecture 81
Consistency and Identifiability
Example 8.3 First Order Output Error Model
Suppose that the true system is given by
y (t )  a0 y (t  1)  b0u (t  1)  e0 (t )  c0 e0 (t  1)
Let we define a first-order output error model as:
bq 1
yˆ (t |  ) 
u (t )
1
1  aq
By pervious theorem the estimates of â and b̂ will converges to the true value of
a0 and b0 .
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AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 81
Lecture 8
Convergence & Consistency
Topics to be covered include:





Conditions on the Data Set
Prediction-Error Approach
Consistency and Identifiability
LTI Models: A Frequency-Domain Description of the
Limit Model
The Correlation Approach
26
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
Pervious theorem describe the limiting estimate  * , *  Dc , as the one that minimizes
the PE variance among all models in the structure M .
In the case S  M , this means that  *   0 is a true descriptio n of system,
But otherwise the model will differ from the true system.
In this section we shall develop some expression s that characteri ze this misfit between
the limiting model and the true system for the case of LTI models.
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2009
Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
An Expression for
1
1
V ( )  E  2 (t ,  ) 
2
4

   ( , )d

Where  (, ) is the spectrum of the prediction errors  (t , )
Under assumption S1 we have
y(t )  G0 (q)u(t )  H 0 (q)e0 (t )
where the noise source e0 has variance 0 . So
 (t , )  H 1 (q, )y(t )  G(q, )u(t )  H 1 (q, )G0 (q)  G (q, ) u (t )  H 0 (q)e0 (t )
 (t , )  H 1 (q, )G0 (q)  G (q, ) u (t )  H 0 (q)  H (q, ) e0 (t )  e0 (t )
 (t , )  H G0  G
1
 u (t ) 
H 0  H 
 e0 (t )

e0 (t )
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Karimpour Nov
Nov2010
2009
Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
An Expression for
1
1
V ( )  E  2 (t ,  ) 
2
4

   ( , )d

 (t , )  H G0  G
1
 u (t ) 
H 0  H 
  e0 (t )
e
(
t
)
 0 
Monic
So
Independent
Overbar is complex conjugate.
29
 eu is the cross spectrum between e0 and u.  eu  0 in the openloop operation.
AliKarimpour
Karimpour Nov
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2009
Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
An Expression for
1
1
V ( )  E  2 (t ,  ) 
2
4

   ( , )d

 (t , )  H G0  G
1
 u (t ) 
H 0  H 
  e0 (t )
e
(
t
)
 0 
Let
So
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Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
1
1
V ( )  E  2 (t ,  ) 
2
4

   ( , )d

We now have a characterization of
Dc  arg min V ( )

in the frequency domain.
G  G0  B
H  H 0
with indicated weightings.
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lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
Open Loop Case
 eu  0 in the openloop operation since u and e will be independen t.  B  0
If the noise model is fixed to H (q, )  H* (q), then
Now let   Dc 
*
G(ei , * ) is a clear square approximat ion of G0 (ei ),
with a frequency weighting Q* .
Q* can be interprete d as the model signal - to - noise ratio.
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lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
Open Loop Case
Consider an independently noise model with
Spectral factorization
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lecture 1
LTI Models: A Frequency-Domain Description of the
Limit Model
Open Loop Case
V ( )  E
V ( )  E
1 2
1
 (t ,  ) 
2
4

1 2
1
 (t ,  ) 
2
4
   ( , )d


   ( , )d

=
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Ali Karimpour Nov 2010
lecture 1
LTI Models: A Frequency-Domain Description of the
Limit Model
Open Loop Case
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Ali Karimpour Nov 2010
lecture 18
LTI Models: A Frequency-Domain Description of the
Limit Model
Closed Loop Case


y (t )   d (k )r (t  k )   d t( 2) (k )e0 (t  k )
k 1
(1)
t
k 0


u (t )   d (k )r (t  k )   d t( 4 ) (k )e0 (t  k )
k 0
V ( )  E
( 3)
t
1 2
1
 (t ,  ) 
2
4
k 0

   ( , )d

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Ali
lecture 1
LTI Models: A Frequency-Domain Description of the
Limit Model
Closed Loop Case
1
1
V ( )  E  2 (t ,  ) 
2
4

   ( , )d

=
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Ali Karimpour Nov 2010
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
Example 8.5 Approximation in the Frequency Domain
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Karimpour Nov
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2009
Ali
lecture 1
LTI Models: A Frequency-Domain Description of the
Limit Model
Example 8.5 Approximation in the Frequency Domain
39
Ali Karimpour Nov 2010
lecture 1
LTI Models: A Frequency-Domain Description of the
Limit Model
Example 8.5 Approximation in the Frequency Domain
40
Ali Karimpour Nov 2010
lecture 1
LTI Models: A Frequency-Domain Description of the
Limit Model
Example 8.5 Approximation in the Frequency Domain
41
Ali Karimpour Nov 2010
lecture 18
Lecture 8
Convergence & Consistency
Topics to be covered include:





Conditions on the Data Set
Prediction-Error Approach
Consistency and Identifiability
LTI Models: A Frequency-Domain Description of the
Limit Model
The Correlation Approach
42
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2009
Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
In section 7.5 we defined the correlation approach to
identification,with the special cases of PLR and IV methods.
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lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
Basic Convergence Result
Consider the function
1
f N ( , Z N ) 
N
where
N
  (t , ) 
t 1
F
(t , ) 
 F (t, )  L(q) (t, )
And correlation vector ξ(t,θ) is obtained by linear filtering of past data:
 (t , )  K y (q, ) y (t )  K u (q, )u (t )
44
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 18
LTI Models: A Frequency-Domain Description of the
Limit Model
45
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
46
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 81
LTI Models: A Frequency-Domain Description of the
Limit Model
Instrumental-variable Methods
47
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 18
The Correlation Approach
48
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 18
The Correlation Approach
49
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 81
The Correlation Approach
50
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali
lecture 18
LTI Models: A Frequency-Domain Description of the
Limit Model
51
AliKarimpour
Karimpour Nov
Nov2010
2009
Ali