Chapter 2
1.
Parametric Models
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Parametric Models
The first step in the design of online parameter identification (PI)
algorithms is to lump the unknown parameters in a vector and
separate them from known signals, transfer functions, and other
known parameters in an equation that is convenient for parameter
estimation.
In the general case, this class of parameterizations is of the form
where
is the vector with all the unknown parameters and
are signals available for measurement.
We refer it as the linear "static "parametric model (SPM).
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Parametric Models
The SPM may represent a dynamic, static, linear, or nonlinear
system.
Example:
where x, u are the scalar state and input, respectively, and a, b are
the unknown constants we want to identify online using the
measurements of x, u .
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Parametric Models
4
Parametric Models
Another parameterization of the above scalar plant is
In the general case, the above parametric model is of the form
5
Parametric Models
Where
are signals available for measurement and
is a known stable proper transfer function, where q is either
the shift operator in discrete time (i.e., q = z) or the differential
operator (q = s) in continuous time. We refer to this model as the
linear "dynamic"parametric model (DPM).
The importance of the SPM and DPM is that the unknown parameter
vector
appears linearly.
So we refer to SPM and DPM as linear in the parameters
parameterizations.
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Parametric Models
We can derive SPM from DPM if we use the fact that
constant vector and redefine
is a
to obtain
In a similar manner, we can filter each side of SPM and DPM using a
stable proper filter and still maintain the linear in the parameters
property and the form of SPM, DPM. This shows that there exist an
infinite number of different parametric models in the form of SPM,
DPM for the same parameter vector
.
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Parametric Models
In some cases, the unknown parameters cannot be expressed in
the form of the linear in the parameters models. In such cases the
PI algorithms based on such models cannot be shown to converge
globally. A special case of nonlinear in the parameters models for
which convergence results exist is when the unknown parameters
appear in the special bilinear form
bilinear static parametric model (B-SPM)
or
bilinear dynamic parametric model (B-DPM)
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Parametric Models
bilinear static parametric model (B-SPM)
bilinear dynamic parametric model (B-DPM)
where
at each time t, and
The transfer function
are signals available for measurement
are the unknown parameters.
is a known stable transfer function.
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Parametric Models
state-space parametric models (SSPM)
In some applications of parameter identification or adaptive control
of plants of the form
whose state x is available for measurement, the following
parametric model may be used:
where
is a stable design matrix;
are the unknown
matrices; and
are signal vectors available for measurement.
The model may be also expressed in the form
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Parametric Models
state-space parametric models (SSPM)
It is clear that the SSPM can be expressed in the form of the
DPM and SPM.
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Parametric Models
bilinear state-space parametric models (B-SSPM).
Another class of state-space models that appear in adaptive
control is of the form
where B is also unknown but is positive definite, is negative
definite, or the sign of each of its elements is known.
The B-SSPM model can be easily expressed as a set of scalar
B-SPM or B-DPM.
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Parametric Models
PI Problem
For the SPM and DPM:
Given the measurements
of the unknown vector
updates
, generate
, the estimate
, at each time t. The PI algorithm
with time so that
approaches or converges to
.
Since we are dealing with online PI, we would also expect that if
changes, then the PI algorithm will react to such changes and
update the estimate
to match the new value of
.
PI
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Parametric Models
PI Problem
For the B-SPM and B-DPM:
Given the measurements
generate estimates
respectively, at each time t the same way as
in the case of SPM and DPM.
z
zl

PI
*
 (t )   
  (t )    * 

  
 (t )
 (t )
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Parametric Models
PI Problem
For the SSPM::
Given the measurements
of
generate estimates
, (and hence the estimates
, respectively)at each time t the same way as in the case of SPM
and DPM.
x 
 
u 
PI
(Aˆ  A m )T

T
ˆ
B




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Parametric Models
PI Problem
The online PI algorithms generate estimates at each time t, by
using the past and current measurements of signals. Convergence
is achieved asymptotically as time evolves.
For this reason they are referred to as recursive PI algorithms to be
distinguished from the non-recursive ones, in which all the
measurements are collected a priori over large intervals of time and
are processed offline to generate the estimates of the unknown
parameters.
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Parametric Models
Example 1: Mass-Spring-Dashpot System
Let us assume that M, f, k are the constant
unknown parameters that we want to
estimate online.
express in the form of SPM
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Parametric Models
Example 1: Mass-Spring-Dashpot System
Measurements:
To avoid of derivatives , we filter both
sides with the stable filter
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Parametric Models
Example 1: Mass-Spring-Dashpot System
Another possible parametric model is:
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Parametric Models
Example 2: Cart with two inverted pendulums
y : 1
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Parametric Models
Example 2: Cart with two inverted pendulums
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Parametric Models
Example 2: Cart with two inverted pendulums
To avoid of derivatives , we filter both
sides with the stable filter
,
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Parametric Models
Example 2: Cart with two inverted pendulums
If in the above model we know that
constant parameters as
is nonzero, redefining the
we obtain the
following B-SPM:
where,
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Parametric Models
Example 3: second-order autoregressive moving
average (ARMA) model
This model can be rewritten in the form of the SPM as:
where,
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Parametric Models
Example 3: second-order autoregressive moving
average (ARMA) model
If one of the constant parameters, i.e., , is nonzero. Then we can
obtain a model of the system in the B-SPM form as follows:
where,
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Parametric Models
Example 4: nonlinear system
Filtering both sides of the equation with the filter
express the system in the form of the SPM
, we can
where,
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Parametric Models
Example 5: dynamical system in the transfer function form
where the parameter b is known and a, c are the unknown parameters.
Rewrite it as:
divide to
where,
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Parametric Models
Example 6: DPM model
If we want W(s) to be a design transfer function with a pole, say
at
we write:
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Parametric Models
Example 6: DPM model
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Parametric Models
Example 7: SSPM model
x 1 
u1 
a11 a12 
b11 b12 
where, x    , u    , A  
, B 


a
a
b
b
x
u
 21 22 
 21 22 
 2
 2
where,
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Parametric Models
Example 8: n-th order-SISO LTI system
where,
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Parametric Models
Example 8: n-th order-SISO LTI system
Filtering by
where,
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Exercises
From reference 1, chapter 2,
Choose 5 problems from 8 problems.
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END
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