SYSTEMS Identification

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SYSTEMSIdentification

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

Reference System Identification Theory For The
User Lennart Ljung
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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.

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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.

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Linear models and sets of linear models
A complete model is given by
with
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
It is also common to assume e(t) is Gaussian.
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Linear models and sets of linear models
with
A particular model thus corresponds to
specification of the function G, H and fe.
We try to parameterize coefficients so
Where ? is a vector in Rd space.
We thus have
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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.

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A family of transfer function models
Equation error model structure
Adjustable parameters in this case are
Define
ARX model
So we have
where
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A family of transfer function models
Equation error model structure
We have
where
Now if we introduce
regression vector
Linear regression in statistic
Linear regression in statistic
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A family of transfer function models
Linear regression in statistic
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A family of transfer function models
ARMAX model structure
with
So we have
now
where
Let
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A family of transfer function models
Then we have
Or
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).
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A family of transfer function models
Now if we introduce
Pseudo linear regressions
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A family of transfer function models
Other equation error type model structures
ARARX model
With
We could use an ARMA description for error
ARARMAX model
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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
Then
With
So
OE model
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A family of transfer function models
Let
w(t) is never observed instead it is constructed
from u
So
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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
BJ model
This is Box and Jenkins model (1970)
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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
General model structure
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A family of transfer function models
Sometimes the dynamics from u to y contains a
delay of nk samples, so
So
But for simplicity
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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.
General model structure
B
FIR (finite impulse response)
AB
ARX
ABC
ARMAX
AC
ARMA
ABD
ARARX
ABCD
ARARMAX
BF
OE (output error)
BFCD
BJ (Box-Jenkins)
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A family of transfer function models
A pseudolinear form for general model structure
Predictor error is
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A family of transfer function models
So we have
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A family of transfer function models
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A family of transfer function models
Other model structure
Consider FIR model

• It is a linear regression (being a special
  case of ARX)

The model can be effectively estimated.

• It is a an output error model (being a
  special case of OE)

It is robust against noise.
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.
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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.

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State Space models
For most physical systems it is easier to
construct models with physical insight in
continuous time
? is a vector of parameters that typically
correspond to unknown values of physical
coefficients, material constants, and the like.
Let ?(t) be the measurements that would be
obtained with ideal, noise free sensors
We can derive the transfer operator from u to ?
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State Space models
Sampling the transfer function
Let
Then x(kTt) is
So x(kTT) is
We can derive the transfer operator from u to ?
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State Space models
Example 4.1 DC servomotor
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State Space models
Example 4.1 DC servomotor
Let La 0 so we have
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State Space models
Example 4.1 DC servomotor
Assume that the actual measurement is made with a
certain noise so
with v being white noise. The natural predictor
is
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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State Space models
A standard discrete time state space model.
Corresponding to
where
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 ?.
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State Space models
Noise Representation and the time-invariant
Kalman filter
A straightforward but entirly valid approach
would be
with e(t) being white noise with variance ?.
Note The ?-parameter in H(q, ?) could be partly
in common with those in G(q, ?) or be extra.
w(t) and v(t) are assumed to be sequences of
independent random variables with zero mean and
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State Space models
Noise Representation and the time-invariant
Kalman filter
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 white
noises.
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State Space models
Exercise(4G.2) Colored measurement noise
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State Space models
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State Space models
The conditional expectation of x(t) is
The predictor filter can thus be written as
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State Space models
Innovation representation
InnovationAmounts of y(t) that cannot be
predicted from past data
Let it e(t)
The innovation form of state space description
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State Space models
Innovation representation
The innovation form of state space description
Let suppose
Directly Parameterized Innovations form
Which one involve with lower parameters?
Both according to situation.
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State Space models
Innovation representation
It is ARMAX model
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State Space models
Example 4.2 Companion form parameterization
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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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Identifiability of some model structures
Some notation
It is convenient to introduce some more compact
notation
One step ahead predictor is
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Identifiability of some model structures
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
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Identifiability of some model structures
Identifiability properties
The problem is whether the identification
procedure will yield a unique value of the
parameter ?, and/or whether the resulting model
is equal to the true system.
Definition 4.6. A model structure M is globally
identifiable at ? if
This definition is quite demanding. A weaker and
more realistic property is
For corresponding local property, the most
natural definition of local identifiability of M
at ? would be to require that there exist an e
such that
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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
Let M be a model structure based on
one-step-ahead predictors for
Then define the set DT(S,M) as those ?-values in
DM for which SM (?)
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Identifiability of some model structures
A model structure is globally identifiable at ?
if and only if
Parameterization in Terms of Physical Parameters
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures
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Identifiability of some model structures