1 Chapter 2 1. Parametric Models. 2 Parametric Models The first step in the design of online...

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Chapter 2Chapter 2

1.1. Parametric ModelsParametric Models

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Parametric ModelsParametric 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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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 ModelsParametric ModelsAnother parameterization of the above scalar plant is

In the general case, the above parametric model is of the form

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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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We can derive SPM from DPM if we use the fact that is a

constant vector and redefine 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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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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bilinear static parametric model (B-SPM)

bilinear dynamic parametric model (B-DPM)

where are signals available for measurement

at each time t, and are the unknown parameters.

The transfer function is a known stable transfer function.

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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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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 ModelsParametric Modelsbilinear 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 ModelsParametric ModelsPI ProblemPI Problem

For the SPM and DPM:

Given the measurements , generate , the estimate

of the unknown vector , at each time t. The PI algorithm

updates 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 .PIPI

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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.

PIPIl

z

z

( )

( )

t

t

*

*

( )

( )

t

t

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For the SSPM::

Given the measurements generate estimates

of , (and hence the estimates

, respectively)at each time t the same way as in the case of SPM

and DPM.

PIPIx

u

ˆ( )

ˆ

Tm

T

A A

B

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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 ModelsParametric ModelsExample 1: Mass-Spring-Dashpot SystemExample 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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Measurements:

To avoid of derivatives , we filter bothsides with the stable filter

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Another possible parametric model is:

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Parametric ModelsParametric ModelsExample 2: Cart with two inverted pendulumsExample 2: Cart with two inverted pendulums

1:y

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To avoid of derivatives , we filter bothsides with the stable filter ,

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If in the above model we know that is nonzero, redefining the

constant parameters as we obtain the

following B-SPM:

where,

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Parametric ModelsParametric ModelsExample 3: second-order autoregressive moving Example 3: second-order autoregressive moving average (ARMA) modelaverage (ARMA) model

This model can be rewritten in the form of the SPM as:

where,

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Parametric ModelsParametric ModelsExample 3: second-order autoregressive moving Example 3: second-order autoregressive moving average (ARMA) modelaverage (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 ModelsParametric ModelsExample 4: nonlinear systemExample 4: nonlinear system

Filtering both sides of the equation with the filter , we can express the system in the form of the SPM

where,

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Parametric ModelsParametric ModelsExample 5: Example 5: dynamical system in the transfer function formdynamical system in the transfer function form

where the parameter b is known and a, c are the unknown parameters. Rewrite it as:

where,

divide to

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Parametric ModelsParametric ModelsExample 6: Example 6: DPM modelDPM model

If we want W(s) to be a design transfer function with a pole, say at we write:

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Parametric ModelsParametric ModelsExample 6: Example 6: DPM modelDPM model

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Parametric ModelsParametric ModelsExample 7: Example 7: SSPM modelSSPM model

where, 1 1 11 12 11 12

21 22 21 222 2

, , ,x u a a b b

x u A Ba a b bx u

where,

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Parametric ModelsParametric ModelsExample 8: Example 8: n-n-th order-SISO LTI systemth order-SISO LTI system

where,

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Parametric ModelsParametric ModelsExample 8: Example 8: n-n-th order-SISO LTI systemth order-SISO LTI system

Filtering by

where,

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Exercises Exercises

From reference 1, chapter 2,

Choose 5 problems from 8 problems.

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END

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