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Dünaamiliste süsteemide modelleerimine

Identification for control in a non-linear system world

Eduard Petlenkov, Automaatikainstituut, 2015

Control Design

Nominal Model

Regulator

© Lennart Ljung

Control Design

True System

Regulator

© Lennart Ljung

Control Design

Nominal Model

Regulator

Model error model

© Lennart Ljung

Control Design

Nominal Model

Regulator

Model error modelTrue system

© Lennart Ljung

Control Design

Nominal Model

Regulator

Model error model

Nominal closed loop system

© Lennart Ljung

Model Error Models

e = y – ymodel

u e

© Lennart Ljung

Identification for Control

n Identifiction for control is the art and technique to design identification experiments and regulator design methods so that the model error model matches the nominal closed loop system in a suitable way

© Lennart Ljung

A Data Set/ nonlinear process

Output

Input

© Lennart Ljung

Feedback Neural Networks. Identification of dynamic systems (1)


External feedback:

))(,),1(),(,),((

))

)(

)1()(

)(

((()( 211122

ntytymtutufnty

tymtu

tu

WFWFty

nn ---=

=Q+Q+

úúúúúúúú

û

ù

êêêêêêêê

ë

é

-

--

××××=

!!

"

"

11Eduard Petlenkov, Automaatikainstituut, 2015

Elman network

Internal feedback:

úúú

û

ù

êêê

ë

é=

rrr

r

r

ww

wwW

!"#"

!

1

111

wherer is the number of neurons on the recurrent layer;

îíì

-==

)1()(),),(),(()(

tYtXBWtXtUftY

h

nn

Feedback Neural Networks. Identification of dynamic systems (2)

Identification with ANNs (1)


U

sY

mY

is the input of the system

is the output of the system

is the output of the model

ms YYE -=

0®E



1. Getting experimental training data: Input signal isgiven to the system and the corresponding outputreaction is measured. Obtained input-output data isused to train a NN-based model;

2. Choosing NN’s structure: number of inputs, outputs,hidden layers, neurons; activation functions;

3. Choosing initial weighting coefficients (usuallyrandom).



4. Calculation of NN outputs corresponding to the training input set;

5. Comparison of outputs with reference outputs from the training set – calculation of model error;

6. Calculation of uptadeted weighting coefficientsusing selected training algorithm;

Repeat steps 4-6 untill a predefined number of iterations (epochs) is reached or until the desired accuracy is achieved.

Dünaamiliste süsteemide juhtimine tehisnärvivõrkudega

NN-based control of dynamic systems


min))2()1(())()(()(1

222

1

®-+--+++-+= åå=

ÙÙ

=

uN

jj

N

Nj

m jtujtujtyjtrtJ l

[ ]úúú

û

ù

êêê

ë

é

+-

+-×+-=

)()(

)()()1()(

2

1

1

Ntytr

Ntytrqqtutu

m

m

Nu!"

Predictive Control

Inverse model


)1()(:1 -®- tUtYf

))(,),2(),(,),(()1( 1 rtUtUqtYtYftU ---=- - !!

Inverse model


Inverse model based control

))2()2(),1(),(()1( 1 ---=- - tutytytyftuHere )()( twty =


))2()1(),2(),1(()( ----= tututytyfty

Adaptive inverse model based control


Adaptive Predictive Control

Mustrite tuvastamine ja klassifitseerimine tehisnärvivõrkudega

NN-based pattern recognition and classification


2424

Self-organizing networks

Self-orgamizing network is capable of adjusting its parameters on the basis of inputs and chosen optimization criteria.

It doesn’t have any reference outputs

Kohonen map

MjtWxdN

iijij ,,1,))((

1

2 !=-=å=


Näide 1: Character recognition(1)

úúúúúúúúú

û

ù

êêêêêêêêê

ë

é

=

10001100011111110001010100101000100

A

úúúúúúúúúúúúúúúúú

û

ù

êêêêêêêêêêêêêêêêê

ë

é

=

1

0101000100

!

LetterA

úúúúúúúúú

û

ù

êêêêêêêêê

ë

é

=

01111100011000101111100011000101111

B


û

ù


ë

é

=

0

1000101111

!

LetterB

Näide 1: Character recognition (2)


û

ù


ë

é

=

01

10010001100010111010

!!

"

"

Inputarv (klasside) kujundite erinevate

arv väljundite

100

000000000000000010001

=

ïïïïïï

þ

ïïïïïï

ý

ü

úúúúúúúúúúúú

û

ù

êêêêêêêêêêêê

ë

é

=

!!!"

"

Etalon

úúúúúúúúúúúúúúúúúúúúú

û

ù

êêêêêêêêêêêêêêêêêêêêê

ë

é

=

100100111101101

4Number

Näide 2: number recognition

Etalon_1=[1 0 0 0 0 0 0 0 0 0]';Etalon_2=[0 1 0 0 0 0 0 0 0 0]';Etalon_3=[0 0 1 0 0 0 0 0 0 0]';Etalon_4=[0 0 0 1 0 0 0 0 0 0]';Etalon_5=[0 0 0 0 1 0 0 0 0 0]';Etalon_6=[0 0 0 0 0 1 0 0 0 0]';Etalon_7=[0 0 0 0 0 0 1 0 0 0]';Etalon_8=[0 0 0 0 0 0 0 1 0 0]';Etalon_9=[0 0 0 0 0 0 0 0 1 0]'; Etalon_0=[0 0 0 0 0 0 0 0 0 1]';

net = newff(minmax(P),[25 10],{'logsig' 'logsig'},'traingda')

• net.trainParam.goal = 0

• net.trainParam.epochs=10000

Näide 2: Number recognition– NN structure

Näide 2: Number recognition– testing on the training set

1 2 3 4 5 6 7 8 9 0

1 0.9999 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2 0.0000 0.9999 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

3 0.0000 0.0000 0.9999 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

4 0.0000 0.0000 0.0000 0.9999 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

5 0.0000 0.0000 0.0000 0.0000 0.9999 0.0000 0.0000 0.0000 0.0000 0.0000

6 0.0000 0.0000 0.0000 0.0000 0.0000 0.9999 0.0000 0.0000 0.0000 0.0000

7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9999 0.0000 0.0000 0.0000

8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9999 0.0000 0.0000

9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9999 0.0000

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9999

Näide 2: Number recognition– testing on the validation set

1 2 3 4 5 6 7 8 9 0

1 0,0668 0,0006 0,0066 0,0026 0,0089 0,0152 0,0034 0,0000 0,0050 0,0001

2 0,0019 0,1548 0,0014 0,0005 0,0108 0,0005 0,0044 0,0001 0,0000 0,0013

3 0,0145 0,0002 0,1495 0,0001 0,0032 0,0343 0,0081 0,0001 0,0039 0,0001

4 0,0003 0,0026 0,0001 0,0102 0,0004 0,0000 0,0014 0,0047 0,0008 0,0092

5 0,0000 0,0003 0,0000 0,0000 0,0177 0,0000 0,0000 0,0000 0,0000 0,0000

6 0,0001 0,0013 0,0020 0,0002 0,0005 0,0611 0,0002 0,0000 0,0065 0,0081

7 0,0002 0,0000 0,0002 0,0003 0,0001 0,0000 0,0154 0,0009 0,0000 0,0000

8 0,0000 0,1030 0,0000 0,0000 0,0000 0,0000 0,0000 0,5868 0,0000 0,0000

9 0,0002 0,0002 0,0003 0,0008 0,0580 0,0056 0,0006 0,0010 0,2799 0,0051

10 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0186

Databases for benchmark problems

v MNISTdatabase ofhandwrittendigitsØ 26x26pxØ 60000labeledimagesand10000test

images

v Semeion - 1593handwrittendigitsfromaround80personsØ 16x16px

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