SYSTEMS
Identification
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
Reference: “System Identification Theory For The User”
Lennart Ljung
lecture 4
Lecture 4
Models of linear time invariant
system
Topics to be covered include:

Linear models and sets of linear models.

A family of transfer function models.

State space models.

Identifiability of some model structures.
2
Ali Karimpour Vov 2009
lecture 4
Linear models and sets of linear models
Topics to be covered include:

Linear models and sets of linear models.

A family of transfer function models.

State space models.

Identifiability of some model structures.
3
Ali Karimpour Vov 2009
lecture 4
Linear models and sets of linear models
A complete model is given by
y(t )  G(q)u(t )  H (q)e(t )
with
f e (), thePDFof e

G (q)   g (k )q
k
k 1

H ( q )  1   h( k ) q  k
k 1
A particular model thus corresponds to specification of the function G, H and fe.
Most often fe not specified as a function, but first and second moments are specified as:
Ee (t )   xf e ( x)dx  0
Ee 2 (t )   x 2 f e ( x)dx  
It is also common to assume e(t) is Gaussian.
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Ali Karimpour Vov 2009
lecture 4
Linear models and sets of linear models
y(t )  G(q)u(t )  H (q)e(t )
with
f e (), thePDFof e

G (q)   g (k )q

H ( q )  1   h( k ) q  k
k
k 1
k 1
A particular model thus corresponds to specification of the function G, H and fe.
We try to parameterize coefficients so:
y(t )  G(q, )u(t )  H (q, )e(t )
Sets of models
f e ( x, ), thePDFof e(t ); e(t )white noise
Where θ is a vector in Rd space.
We thus have:


yˆ (t |  )  H 1 (q, )G(q, )u(t )  1  H 1 (q, ) y(t )
5
Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Topics to be covered include:

Linear models and sets of linear models.

A family of transfer function models.

State space models.

Identifiability of some model structures.
6
Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
AR part
Equation error model structure
y (t )  a1 y (t  1)  ...  ana y (t  na ) 
b1u (t  1)  ...  bnb u (t  nb )  e(t )
Adjustable parameters in this case are
Define
  a1 a2 ...an b1 b2 ...bn
a
b

X part
T
ARX model
Aq   1  a1q 1  ...  ana q  na
e
Bq   b1q 1  ...  bnb q nb
So we have:
y(t )  G(q, )u(t )  H (q, )e(t )
where
B(q )
1
G ( q,  ) 
, H ( q,  ) 
A(q)
A(q)
1
A
u
B
A
y
+
The ARX model structure
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Equation error model structure
y (t )  a1 y (t  1)  ...  ana y (t  na ) 
b1u (t  1)  ...  bnb u (t  nb )  e(t )
We have:
y(t )  G(q, )u(t )  H (q, )e(t )
where
G ( q,  ) 
B(q )
1
, H ( q,  ) 
A(q)
A(q)


yˆ (t |  )  H 1 (q, )G(q, )u(t )  1  H 1 (q, ) y(t ) last chapter
yˆ (t |  )  B(q)u(t )  [1  A(q)]y(t )
Now if we introduce
 (t )   y(t 1) ... y(t  na ) u(t 1) ...u(t nb )
yˆ (t |  )   T (t )   T (t )
  a1 a2 ...an b1 b2 ...bn
a
b

T
regression vector
8
Linear regression in statistic
Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Exercise(4E.1): Consider the ARX model structure
y (t )  a1 y (t  1)  ...  ana y (t  na ) 
b1u (t  1)  ...  bnb u (t  nb )  e(t )
where b1 is known to be 0.5. Write the corresponding predictor in the following
linear regression form.
yˆ (t |  )   T (t )   (t )
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
ARMAX model structure
AR part
y(t )  a1 y(t  1)  ...  ana y(t  na ) 
b1u (t  1)  ...  bnb u (t  nb )  e(t )  c1e(t  1)  ...  cnc e(t  nc )
with
Aq   1  a1q 1  ...  ana q  na
X part
MA part
B q   b1q 1  ...  bnb q  nb
C q   c1q 1  ...  cnc q  nc
So we have:
A(q) y(t )  B(q)u(t )  C (q)e(t )
now
y(t )  G(q, )u(t )  H (q, )e(t ) where
Let
B(q )
C (q)
G ( q,  ) 
, H ( q,  ) 
A(q)
A(q)
  a1 a2 ...an b1 b2 ...bn c1 c2 ...cn
a
b
c

T
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
y(t )  a1 y(t  1)  ...  ana y(t  na ) 
b1u (t  1)  ...  bnb u (t  nb )  e(t )  c1e(t  1)  ...  cnc e(t  nc )
Then we have


yˆ (t |  )  H 1 (q, )G(q, )u(t )  1  H 1 (q, ) y(t ) last chapter
yˆ (t |  ) 
Or
B(q)
A(q)
u (t )  [1 
] y (t )
C (q)
C (q)
C(q) yˆ (t |  )  B(q)u(t )  C(q)  A(q)y(t )
To start it up at time t = 0 and predict y(1) requires the knowledge of
One can consider the data as zero but there is a difference that decays cμt where μ is
the maximum magnitude of the zero of C(z).
Exercise(4G.1): Show that the effect from an erroneous initial condition in yˆ (11t |  )
Ali Karimpour Vov 2009
is bounded by cμt .
lecture 4
A family of transfer function models
We saw that
C(q) yˆ (t |  )  B(q)u(t )  C(q)  A(q)y(t )
To start it up at time t = 0 and predict y(1) requires the knowledge of
It is also possible to start the recursion at time max(n*, nb) and include the unknown
initial condition yˆ (k|θ), k = 1, 2,…, nc , in the vector θ. Then
yˆ (t |  )  B(q)u(t )  1  A(q)y(t )  C(q) 1y(t )  yˆ (t |  )
 (t , )
Now if we introduce
(t, )   y(t 1) ... y(t  na ) u(t 1) ...u(t nb )  (t 1, ) ... (t nc , )T
  a1 a2 ...an b1 b2 ...bn c1 c2 ...cn
a
yˆ (t |  )   T (t , )
b
c

T
Pseudo linear regressions
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Other equation error type model structures
A(q) y (t )  B(q)u (t ) 
AR part
With
D(q)  1 d1q1  ... dnd qnd
X part
1
e(t ) ARARX model
D(q)
AR part
We could use an ARMA description for error
e
A(q) y (t )  B(q)u (t ) 
AR part
X part
C (q)
e(t )
D(q)
ARMA part
ARARMAX model
C
D
u
B
+
1
A
y
The equation error model family
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Output error model structure
If we suppose that the relation between input and undisturbed output w can be
written as:
w(t )  f1w(t  1)  ...  f n f w(t  n f ) 
b1u (t  1)  ...  bnb u (t  nb )
Then
y(t )  w(t )  e(t )
With
F (q)  1  f1q 1  ...  f n f q
n f
So
y (t ) 
B(q)
u (t )  e(t )
F (q)
OE model
e
u
B
F
y
+
14
The output error model structure
Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Output error model structure
e
w(t )  f1w(t  1)  ...  f n f w(t  n f ) 
b1u (t  1)  ...  bnb u (t  nb )
u

+
The output error model structure
y(t )  w(t )  e(t )
Let
B
F
y
  b1 b2 ...bn f1 f 2 ... f n
b
f

T
w(t) is never observed instead it is constructed from u
yˆ (t |  )  w(t , ) 
B(q)
u (t )
F (q)
 (t, )  u(t 1) ...u(t nb )  w(t 1, ) ... w(t n f , )
So
yˆ (t |  )   T (t , )
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Box-Jenkins model structure
A natural development of the output error model is to further model the properties of
the output error. Let output error with ARMA model then
y (t ) 
B(q)
C (q)
u (t ) 
e(t )
F (q)
D( q )
This is Box and Jenkins model (1970)

BJ model

yˆ (t |  )  H 1 (q, )G(q, )u(t )  1  H 1 (q, ) y(t ) last chapter
e
D( q ) B ( q )
C ( q )  D( q )
yˆ (t |  ) 
u (t ) 
y(t )
C (q) F (q)
C (q)
C
D
u
B
F
y
+
16
The BJ model Ali
structure
Karimpour Vov 2009
lecture 4
A family of transfer function models
A general family of model structure
The structure we have discussed in this section may give rise to 32 different
model sets, depending on which of the five polynomials A, B, C, D, F are used.
For convenience, we shall therefore use a generalized model structure:
B(q)
C (q)
A(q) y (t ) 
u (t ) 
e(t )
F (q)
D( q )
General model structure
e
C
D
u
B
F
+
1
A
General model structure
y
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
e
B(q)
C (q)
A(q) y (t ) 
u (t ) 
e(t )
F (q)
D( q )
Sometimes the dynamics from u to y
contains a delay of nk samples, so
C
D
u
B
F
+
y
1
A
General model structure
Bq  bnk qnk  bnk 1qnk 1... bnk nb 1q(nk nb 1)  qnk B (q)
So
A(q) y (t )  q
 nk
B (q)
C (q)
u (t ) 
e(t )
F (q)
D( q )
But for simplicity
u(t 1)  u(t  nk )
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
The structure we have discussed in this section may give rise to 32 different
model sets, depending on which of the five polynomials A, B, C, D, F are used.
B(q)
C (q)
A(q) y (t ) 
u (t ) 
e(t )
General model structure
F (q)
D( q )
 D(q) A(q) 
D(q) B(q)
u(t )  1 
y(t )

C (q) F (q)
C (q) 

Some common black-box SISO models as special
cases of generalized model structure
yˆ (t |  ) 
Polynomial used
B
AB
ABC
AC
ABD
ABCD
BF
BFCD
Name of model Structure
FIR (finite impulse response)
ARX
ARMAX
ARMA
ARARX
ARARMAX
OE (output error)
BJ (Box-Jenkins)
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Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
A(q) y (t ) 
B(q)
C (q)
u (t ) 
e(t )
F (q)
D( q )
Predictor
yˆ (t |  ) 
General model structure
 D(q) A(q) 
D(q) B(q)
u(t )  1 
y(t )

C (q) F (q)
C (q) 

A pseudolinear form for general model structure
C(q) F (q) yˆ (t |  )  F (q)C(q)  D(q) A(q)y(t )  D(q) B(q)u(t )
Predictor error is:
 (t , )  y(t )  yˆ (t |  )
 (t , ) 

D(q) 
B( q )
A
(
q
)
y
(
t
)

u
(
t
)

C (q) 
F (q)

w(t , )
v(t , )
 (t , )  y(t )  yˆ (t |  ) 
D( q )
v(t , )
C (q)
20
Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
w(t , )
 (t , ) 

D(q) 
B( q )
A
(
q
)
y
(
t
)

u
(
t
)

C (q) 
F (q)

v(t , )
So we have:
w(t , ) 
B(q)
u (t )
F (q)
w(t, )  b1u(t 1)  ... bnb u(t  nb )  f1w(t 1, )  ... f n f w(t  n f , )
v(t , )  A(q) y(t )  w(t , )
v(t, )  y(t )  a1 y(t 1)  ... ana y(t  na )  w(t, )
 (t , )  y(t )  yˆ (t |  ) 
D( q )
v(t , )
C (q)
21
 (t, )  v(t, )  d1v(t 1, )  ... dn v(t  nd , )  c1 (t 1, )  ... cnAliKarimpour
(t 
nVov
)
c ,
2009
d
c
lecture 4
A family of transfer function models
w(t , ) 
B(q)
u (t ) w(t, )  b1u(t 1)  ... bn u(t  nb )  f1w(t 1, )  ... f n w(t  n f , )
b
f
F (q)
v(t , )  A(q) y(t )  w(t , )
 (t , )  y(t )  yˆ (t |  ) 
v(t, )  y(t )  a1 y(t 1)  ... ana y(t  na )  w(t, )
D( q )
v(t , )
C (q)
 (t, )  v(t, )  d1v(t 1, )  ... dn v(t  nd , )  c1 (t 1, )  ... cn  (t  nc , )
d
c
yˆ (t |  )  y(t )   (t , )
yˆ (t |  )  y(t )  v(t, )  d1v(t 1, )  ... dnd v(t  nd , )  c1 (t 1, )  ... cnc  (t  nc , )
yˆ (t |  )   T (t , )
 (t , )   y(t  1),... y(t  na ),u(t  1),...u(t nb ),  w(t  1, ),... w(t n f , ),
 (t  1, ),... (t nc , ),  v(t  1, ),... v(t n d , )

  a1 a2 ...an b1 b2 ...bn f1 f 2 ... f n c1 c2 ...cn d1 d 2 ...d n
a
b
f
c

T
d
22
Ali Karimpour Vov 2009
lecture 4
A family of transfer function models
Other model structure
Consider FIR model
n
G(q, )   bk q k
• It is a linear regression
k 1
(being a special case of ARX)
The model can be effectively estimated.
• It is a an output error model
It is robust against noise.
(being a special case of OE)
The basic disadvantage is that many parameters may be needed if the system has a
small time constant.
Whether it would be possible to retain the linear regression and output error
features, while offering better possibilities to treat slowly decaying impulse
responses.
n
G ( q,  )    k L ( q,  )
k 1
where
q k
L(q,  ) 
q 
23
Ali Karimpour Vov 2009
lecture 4
State space models
Topics to be covered include:

Linear models and sets of linear models.

A family of transfer function models.

State space models.

Identifiability of some model structures.
24
Ali Karimpour Vov 2009
lecture 4
State Space models
For most physical systems it is easier to construct models with physical insight
in continuous time:
x (t )  F ( ) x(t )  G( )u(t )
θ is a vector of parameters that typically correspond to unknown values of
physical coefficients, material constants, and the like.
Let η(t) be the measurements that would be obtained with ideal, noise free sensors
 (t )  Hx(t )
We can derive the transfer operator from u to η
 (t )  H  pI  F ( )1G( )u(t )  Gc ( p, )u(t )
25
Ali Karimpour Vov 2009
lecture 4
State Space models
x (t )  F ( ) x(t )  G( )u(t )
 (t )  Hx(t )
Sampling the transfer function
Let
u(t )  uk  u(kT )
Then x(kT+t) is
x(kT  t )  e
F ( ) t
x(kT )  
kT  t  (k  1)T
kT t
kT
e F ( ) ( KT t  )G( )uk d
So x(kT+T) is
x(kT  T )  e
F ( ) T
KT T

x(kT )   
e F ( ) ( KT T  )G ( )d  u k
 KT

x(kT  T )  AT ( ) x(kT )  BT ( )uk
We can derive the transfer operator from u to η
 (kT )  H qI  AT ( )1 BT ( )u(kT )  GT (q, )u(kT )
26
Ali Karimpour Vov 2009
lecture 4
State Space models
Example 4.1: DC servomotor
di (t )
u (t )  Ra i (t )  La
 s (t )
dt
d (t )
d (t )
s(t )  k1 (t )
 kv
dt
dt
Ta (t )  k2 (t )i(t )  kai(t )
d 2 (t )
d (t )
J

T
(
t
)

T
(
t
)

f
a
l
dt 2
dt
27
Ali Karimpour Vov 2009
lecture 4
State Space models
Example 4.1: DC servomotor
u (t )  Ra i (t )  La
di (t )
 s (t )
dt
d (t )
s (t )  kv
dt
d 2 (t )
d (t )
J

k
i
(
t
)

T
(
t
)

f
a
l
dt 2
dt
Let La ≈ 0 so we have
1 
0
 0 
 0 
d
x(t )  
x(t )  
u(t )  
Tl (t )



dt
0  1 /  
 /  
 /  
 (t )  1 0x(t )
Exercise1: Derive  ,  and 
 
Exercise2: Let     and supposeTl  0, thenderive AT ( ) and BT ( 28
)
Ali Karimpour Vov 2009
 
lecture 4
State Space models
Example 4.1: DC servomotor
 
x(t  T )  AT ( ) x(t )  BT ( )u(t ) where    
 
Assume that the actual measurement is made with a certain noise so:
y(t )   (t )  v(t )
with v being white noise. The natural predictor is:
yˆ (t |  )  GT (q, )u(t )  1 0qI  AT ( )1 BT ( )u(t )
This predictor parameterize using only two parameters. But ARX or OE model
contains four adjustable parameters.
But this method (2 parameters) is far more complicated than ARX or OE.
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Ali Karimpour Vov 2009
lecture 4
State Space models
A standard discrete time state space model.
x(t  1)  A( ) x(t )  B( )u (t )
y(t )  C ( ) x(t )  v(t )
Corresponding to
y(t )  G(q, )u(t )  v(t )
G(q, )  C ( )qI  A( ) B( )
1
where
A( )  e
F ( ) T
B( )  
0
T
e F ( ) G( )d
Although sampling a time-continuous is a natural way to obtain the discrete model
but for certain application direct discrete time is better since the matrices A, B and
C are directly parameterize in terms of θ.
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Ali Karimpour Vov 2009
lecture 4
State Space models
Noise Representation and the time-invariant Kalman filter
x(t  1)  A( ) x(t )  B( )u (t )
y(t )  C ( ) x(t )  v(t )
A straightforward but entirly valid approach would be:
v(t )  H (q, )e(t )
with {e(t)} being white noise with variance λ.
Note: The θ-parameter in H(q, θ) could be partly in common with those in G(q, θ)
or be extra.
x(t  1)  A( ) x(t )  B( )u (t )  w(t )
process noise
measurement noise
y(t )  C ( ) x(t )  v(t )
{w(t)} and {v(t)} are assumed to be sequences of independent random variables with
zero mean and
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Ali Karimpour Vov 2009
lecture 4
State Space models
Noise Representation and the time-invariant Kalman filter
x(t  1)  A( ) x(t )  B( )u (t )  w(t )
process noise
measurement noise
y(t )  C ( ) x(t )  v(t )
{w(t)} and {v(t)} are assumed to be sequences of independent random variables with
zero mean and
{w(t)} and {v(t)} may often be signals whose physical origins are known.
The load variation Tl(t) was a “process noise”.
The inaccuracy in the potentiometer angular
sensor is the “measurement noise”.
In such cases it may of course not always be realistic to assume that the signals are
32
white noises.
Ali Karimpour Vov 2009
lecture 4
State Space models
Exercise(4G.2) Colored measurement noise:
Suppose that
x(t  1)  A1 ( ) x(t )  B1 ( )u (t )  w1 (t )
y (t )  C1 ( ) x(t )  v(t )
(I)
where{w1 (t )} is white noise with variance R1 ( ), but {v(t )} is not white.
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Ali Karimpour Vov 2009
lecture 4
State Space models
For state space descriptions,
x(t  1)  A( ) x(t )  B( )u(t )  w(t )
y(t )  C ( ) x(t )  v(t )


E v(t )v (t )   R ( )
E w(t )v (t )   R ( )
E w(t ) wT (t )  R1 ( )
T
2
T
12
The conditional expectation of y(t), given data y(s), u(s), s≤t-1, is:
yˆ (t |  )  C ( ) xˆ (t , )
The conditional expectation of x(t), by Kalman filter is:
xˆ(t  1, )  A( ) xˆ(t , )  B( )u(t )  K ( )y(t )  C( ) xˆ(t , )
Here K(θ) is given by


K ( )  A( ) P ( )C T ( )  R12 ( ) C ( ) P ( )C T ( )  R2 ( )

1
where P ( ) is obtained as the psd solution of the stationary Riccati equation:
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Ali Karimpour Vov 2009
lecture 4
State Space models
For state space descriptions,
x(t  1)  A( ) x(t )  B( )u(t )  w(t )
y(t )  C ( ) x(t )  v(t )


E v(t )v (t )   R ( )
E w(t )v (t )   R ( )
E w(t ) wT (t )  R1 ( )
T
2
T
12
The conditional expectation of y(t), given data y(s), u(s), s≤t-1, is:
yˆ (t |  )  C ( ) xˆ (t , )
The conditional expectation of x(t), by Kalman filter is:
xˆ(t  1, )  A( ) xˆ(t , )  B( )u(t )  K ( )y(t )  C( ) xˆ(t , )
The conditional expectation of x(t) is:
1
xˆ(t, )  qI  A( )  K ( )C( ) B( )u(t )  K ( ) y(t )
The predictor filter can thus be written as:
1
yˆ (t, )  C( )qI  A( )  K ( )C( ) B( )u(t )  K ( ) y(t )
Exercise: Show that covariance matrix of state estimator error E x(t )  xˆ(t, )x(t )  xˆ(t, )T
is P ( )
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Ali Karimpour Vov 2009
lecture 4
State Space models
Innovation representation
Innovation=Amounts of y(t) that cannot be predicted from past data
y(t )  yˆ (t , )  y(t )  C ( ) xˆ(t , )  C( )x(t )  xˆ(t , )  v(t )
Innovation
Let it e(t)
xˆ (t  1, )  A( ) xˆ (t , )  B( )u (t )  K ( ) y(t )  C ( ) xˆ (t , )
yˆ (t |  )  C ( ) xˆ (t , )
xˆ (t  1, )  A( ) xˆ (t , )  B( )u (t )  K ( )e(t )
y(t )  C ( ) xˆ (t , )  e(t )
The innovation form of state space description
Exercise: Show that the covariance of e(t) is:
Ee(t )eT (t )  ( )  C( ) P( )CT ( )  R2 ( )
36
Ali Karimpour Vov 2009
lecture 4
State Space models
Innovation representation
The innovation form of state space description
xˆ (t  1, )  A( ) xˆ (t , )  B( )u (t )  K ( )e(t )
y(t )  C ( ) xˆ (t , )  e(t )
K ( )    A( ), C( ), R1 ( ), R2 ( ), R12 ( )
Let suppose
K ( )     Directly Parameterized Innovations form
1
n  (n  1)
2
1
p  ( p  1)
2
n p
K ( )    A( ), C( ), R1 ( ), R2 ( ), R12 ( )
K ( )    
n p
37
Which one involve with lower parameters? Both according toAlisituation.
Karimpour Vov 2009
lecture 4
State Space models
xˆ (t  1, )  A( ) xˆ (t , )  B( )u (t )  K ( )e(t )
Innovation representation
y(t )  C ( ) xˆ (t , )  e(t )


y(t )  C( )qI  A(q) B( ) u(t )  1  C( )qI  A(q) K ( ) e(t )
1
G(q, )
1`
H (q, )
y(t )  G(q, ) u(t )  H (q, ) e(t )
It is ARMAX model
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Ali Karimpour Vov 2009
lecture 4
State Space models
  a1 1 0
 b1 
A( )   a2 0 1
B( )  b2 
  a3 0 0
b3 
 k1 
K ( )  k 2  C ( )  1 0 0
 k3 
Example 4.2 Companion form parameterization
xˆ (t  1, )  A( ) xˆ (t , )  B( )u (t )  K ( )e(t )
y(t )  C ( ) xˆ (t , )  e(t )
Let
  a1 a2 a3 b1 b2 b3 k1 k2 k3 
C( )qI  A(q)
b1q 1  b2 q 2  b3q 3
B( ) 
1  a1q 1  a2 q 2  a3q 3
C( )qI  A(q)
k1q 1  k2 q 2  k3q 3
K ( ) 
1  a1q 1  a2 q 2  a3q 3
1
1
1  C( )qI  A(q)
1
K ( ) 
1
2
3
1  c1q  c2 q  c3q
1  a1q 1  a2 q 2  a3q 3
c1  a1  k1 , c2  a2  k2 , c3  a3  k3
y (t )  C ( )qI  A(q)  B( ) u (t )
1


 1  C ( )qI  A(q)  K ( ) e(t )
1`
A(q) y(t )  B(q) u(t )  C (q) e(t )
So we have an ARMAX
model with
na  nb  nc  3
 ?
39
Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
Topics to be covered include:

Linear models and sets of linear models.

A family of transfer function models.

State space models.

Identifiability of some model structures.
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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
Some notation
It is convenient to introduce some more compact notation
y(t )  G(q)u(t) H(q)e(t)
u(t )
T (q)  G(q) H (q) and  (t)  

e(t ) 
y(t )  T (q)  (t)
One step ahead predictor is:


yˆ (t | t 1)  H 1 (q)G(q)u(t )  1  H 1 (q) y(t )
Wu (q)  H 1 (q)G(q)


Wy (q)  1 H 1 (q)

W (q)  Wu (q) Wy (q)
yˆ (t | t  1)  W (q) z (t )

u(t ) 
z(t )  

y
(
t
)


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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
T (q)  G(q) H (q)
u(t )
and  (t)  

e
(
t
)


Wu (q)  H 1 (q)G(q)

W (q)  Wu (q) Wy (q)


u(t ) 
z (t )  

y
(
t
)



Wy (q)  1 H 1 (q)
Definition 4.1. A predictor model of a linear, time-invariant system is
a stable filter W(q).
Definition 4.2. A complete probabilistic model of a linear, time-invariant
system is a pair (W(q),fe(x)) of a predictor model W(q) and the PDF fe(x)
of the associated errors.
Clearly, we can also have models where the PDFs are only partially specified (e.g., by
the variance of e)
We shall say that two models W1(q) and W2(q) are equal if
W1 (ei )  W2 (ei ),
almost all 
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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
Identifiability properties
The problem is whether the identification procedure will yield a unique value of
the parameter θ, and/or whether the resulting model is equal to the true system.
Definition 4.6. A model structure M is globally identifiable at θ* if
M ( )  M (  ),
  DM     
Definition 4.7. A model structure M is strictly globally identifiable if it is
globally identifiable at all at  *  DM
This definition is quite demanding. A weaker and more realistic property is:
Definition 4.8. A model structure M is globally identifiable if it is
globally identifiable at almost all at  *  DM
For corresponding local property, the most natural definition of local
identifiability of M at θ* would be to require that there exist an ε such that
43
M ( )  M (  ),
  B(  ,  )     
Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
Use of the Identifiability concept
The identifiability concept concerns the unique representation of a given system
description in a model structure. Let such a description as:
S : y(t )  G0 (q)u(t )  H0 (q)e(t )
Let M be a model structure based on one-step-ahead predictors for
M : y(t )  G(q, )u(t )  H (q, )e(t )
Then define the set DT(S,M) as those θ-values in DM for which S=M (θ)
DT (S , M )    DM | G0 ( z)  G( z, ) , H0 ( z)  H ( z, ) almost all z
The set is empty in case S  M
Now suppose that S  M so that S=M(θ0)
M is globally identifiable at 0

DT (S , M )  0 
44
Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
A model structure is globally identifiable at θ* if and only if
G( z, )  G( z,  ) and H ( z, )  H ( z,  )
for almostall z     
Parameterization in Terms of Physical Parameters
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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
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Ali Karimpour Vov 2009
lecture 4
Identifiability of some model structures
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Ali Karimpour Vov 2009
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Identifiability of some model structures
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Ali Karimpour Vov 2009
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Identifiability of some model structures
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Ali Karimpour Vov 2009
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Identifiability of some model structures
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Ali Karimpour Vov 2009
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Identifiability of some model structures
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Ali Karimpour Vov 2009
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Identifiability of some model structures
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Ali Karimpour Vov 2009
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Identifiability of some model structures
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Identifiability of some model structures
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Ali Karimpour Vov 2009