Identification: Estimation of unknown plant-parameters ... · 2016 Copyright S. Zacher 4 Estimation of plant-parameters with an error-minimization Generally are known three identifications-methods

www.zacher-automation.de 2016 Copyright S. Zacher 1

Prof. Dr. S. Zacher

Automation-Letter Nr. 17

31.03.2016

There are known two approaches to the design of adaptive systems:

a) Identification of the unknown plant with the model, which is build like an adaptive filter. Such systems consist from the model

and the algorithm, which minimized the error between output of plant and model by the same inputs. The controller is not

involved to the identification, so it is a kind of the parametric adaptive control.

b) The unknown plant parameters are constants (LTI), the controller itself is also an LTI-process with adjustable gain and lag. The

assumption is, that the plant parameters are constant long enough for the steps, measurements and the step responses to

achieve the controller. This kind of adaptive control will be called feedback adaptive control.

Identification: Estimation of unknown plant-parameters with an error-minimization

S. Zacher: Adaptive Control. 2013, Script, Hochschule Darmstadt, FB EIT, MSE


Abstract, Urheberrechts- und Haftungshinweis

This publication has copyright. All rights are reserved by S. Zacher. The further development or use of the

publication without reference of author is not allowed.

For the purposes of this publication, in industry, in the laboratory and in other practical cases as well as for

any damage that may arise over the dynamic systems of incomplete or incorrect information, the author

assumes no liability.

Described is the identification of the unknown plant with algorithms, which minimized the error between output

of plant and model by the same inputs.

Generally is the output of the plant-model is .

The unknown parameters could be calculated from the following equation, if the inverse matrix M-1 exists and if

the output of the model describes the plant without error e (white noise):

If there is a white noise on the output of the plant, the model equation includes the error:

The minimization of the least mean error leads to the expression

The inverse of this matrix equation is known as Pseudoinverse or Moore -Penrose Inverse. For this equaition

is the MATLAB-command pinv to be applied, which includes also the function arx of the System-Identification-

Toolbox. Instead of pseudoinverse it is possible to use so called RLS-algorithm, which is described in this letter

with examples and solutions.

Px

M

xP 1

M

PxE plant

M

XTT

MMMP1)(


I N H A L T :

Estimation of plant-parameters with an error-minimization …………………….… page 4

1. SLE (Solution of Linear Equations) ………………………………………………. page 8

2. LMS (Least Mean Squares) ………………………………………………………... page 24

3 RLS (Recursive Least Squares) ……..…………………………………………….. page 38

4. Exercises ……………………………………………………………………………….page 42


Estimation of plant-parameters with an error-minimization

Generally are known three identifications-methods for a plant separated from the control system:

a) Theoretical description with the experimental adjustment of model parameters

With knowledge about physical nature of the plant you describe the plant with differential equations or systems

differential equations. Then you solve this equations for known inputs (step, pulse, wave) with known methods

and adjust the model parameters to the experimental measurements:

in time domain

in s-domain by Laplace-Transformation

In frequency domain .

b) Experimental Identification upon supposed plant dynamics

Without knowledges about physical nature of the plant you choose the model of the plant like one of the

followings: transfer function, recursive equation, Volterra-model, ANN (artificial neural network) etc. The you do

the same as in previous point, i.e. you apply inputs, measure outputs and adjust the model parameters to the

experimental measurements.

http://www.zacher-international.com/Automation_Letters/03_Hinweise_Identifikation.pdf

c) Block-Box Appoach

Without knowledges about physical nature of the plant and without supposing the model its dynamics you

apply inputs, measure outputs and adjust the vector of model parameters to minimize the difference between

plant and model.

In the following are some methods of last approach (identification with an error-minimization) based on

Pseudoinverse or Moore-Penrose Inverse described:

1. SLE (Solution of lineare equations)

2. LMS (Least Mean Squars)

3. RLS (Recursive Least Squars or recursive LMS)







Estimation of plant-parameters with an error-minimization

How to win the information from the input step and step

response is shown below for k = N = 7.

The system of the equations above could be shown in

the vector and matrix form:


Estimation of model-parameters with an error-minimization


Estimation of model-parameters with an error-minimization

Instead of pseudoinverse it is possible to use so called RLS-algorithm, which will be described in the following

examples.


1 SLE (Solution of Linear Equations)
































2 LMS (Least Mean Squares)




























3 RLS (Recursive Least Squares)








4 Exercises

Solutions: www. zacher-international.com/Automation_Letters/LMS_Solutions.zip

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Identification: Estimation of unknown plant-parameters ... · 2016 Copyright S. Zacher 4 Estimation of plant-parameters with an error-minimization Generally are known three identifications-methods