Control based system Identification and GOBF – ARMA

7/31/2019 Control based system Identification and GOBF ARMA

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Control based system Identificationand GOBF ARMA modeling

Presented by:ElsaShilpaArun JosephLalu Seban(Group VI)


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Overview

Introduction

White box modeling

System Identification Deterministic modeling GOBF filter

Correlation Analysis

Stochastic modeling ARMA model

Results and Discussions

Reference


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Introduction

Mathematical model of any system is used for

Process behavior analysis simulation studies

Process design

Process control - prediction

Process optimization

Operator training

Safety system design


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White box modeling

Mathematical model using first principle

law of conservation of mass, energy or momentum

electrical networks KVL, KCL

mechanical s/ms Newtons 2nd law

rate of accumulation = rate in rate out

differential equations of order n

need great understanding in the physics or chemistryof the system under consideration

domain knowledge

complex process with assumptions


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System Identification

Identifying the system by correlating known input and output data

Simple methods for lower order systems

Standard test signals : Difficult to model nonlinear and higherorder system

Test signal characteristics

Excite the system to maximum

Excite all frequencies

PRBS

INPUT

OUTPUT

SYSTEM

?


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Contd.

0 50 100 150 200 250 300 350 400 450 500

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Sampling Time

In

put

Sample PRBS Signal


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Deterministic modeling GOBF filter

Step1: assume an OE model structure

Generate an estimate of G(q) (deterministic component) usingGOBF

Step 2: estimateH(q) (stochastic component) for residuals

EstimateH(q) using ARMA noise model

( ) ( ) ( ) ( )y k G q u k v k= +

~ ~

( ) ( ) ( ) ( )v k y k G q u k =

(1)

(2)

~ ~

( )v k H e= (3)


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GOBF Advantages:

Capture the dynamics with acceptable accuracy with relativelyfewer number of parameters

Time delays can be easily estimated and incorporated into the

model

GOBF Filter :

The output is given as:

=

=

1

1

2

)1(1),(

i

j j

j

i

i

ipq

qp

pq

ppqf

88

~

1 1 2 2( ) ( ) ( ) ........ ( )f f n fny k l u k l u k l u k= + + +

( , ) ( )fi iu f q p u k =

(4)

(5)

(6)

Contd.


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In vector-matrix notation, the output is given by

where =[l1 l2 ln] is the parameter vector

the output vector given by

Regression matrixXis,

~Ty X=

~ ~ ~ ~

1 ...n n Ny y y y+

=

1 2

1 2

1 2

( ) ( 1) . . . (1)

( 1) ( ) . . . (2)

. . . . . .

. . . . . .

( 1) ( 2) . . . ( )

f f fm

f f fm

f f fm

u n u n u

u n u n u

X

u N u N u N n

+ +

=

(7)

(8)

Contd.


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Correlation Analysis

Confidence interval for 95% confidence limit

Auto correlation :

1

1( ( ) )( ( ) )

( )(0) (0)

N k

N t

ur

uu rr

u t u r t k r k

= +

=

1.96

N

1010

(9)

(10)


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( )

( )

C q

D q

( )GOBF

G q)(ku

~

( )y k

)(ke

+

+

~

gobf 1 2 3 1 2(k/k-1) y (k) - d r (k-1) - d r (k-2) - d r (k-3 ) c e (k-1) c e (k-2)y = + +

(11)

Stochastic modeling ARMA model


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Results and Discussions

0 50 100 150 200 250 300 350 400 450 5000.84

0.86

0.88

0. 9

0.92

0.94

0.96

Sampling Time (Min)

MolarLiquid

Composition

Plant output

Output data sequence of distillation column for the PRBS input(Model from Luyben and Kuo (2008))


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Contd.

Comparison of plant output to GOBF model output

0 50 100 1 50 200 25 0 30 0 35 0 400 450 50 00.84

0.86

0.88

0. 9

0 .92

0.94

0.96

S a m p l in g T i me ( M i n )

MolarLiquidCom

position

P lan t ou tpu t

G O B F mo d e l o u tp u t


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Contd.

Comparison of plant output to GOBF-ARMA model output

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 00 .7 8

0 .8

0 .8 2

0 .8 4

0 .8 6

0 .8 8

0 .9

0 .9 2

0 .9 4

0 .9 6

0 .9 8

S a m p l in g T im e ( M i n )

MolarLiquidComposition

P l a n t o u t p u t

G O B F - A R M A m o d e l o u t pu t


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Contd.

Cross correlation Percentageprediction error

For analyzing the performance ofmodel

GOBF model : 4.6

GOBF-ARMA model : 0.05

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0-0 .0 5

-0 .0 4

-0 .0 3

-0 .0 2

-0 .0 1

0

0 .0 1

0 .0 2

0 .0 3

0 .0 4

0 .0 5

L a g (M i n )

CCF

(Xb

/Vs

)

~2

1

2

1

(( ( ) ( ))

100

( ( ) )

n

i i

i

n

i mean

i

y k y k

PPE

y k y

=

=

=


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Conclusion

Combination of GOBF ARMA modeling can be used foridentification of high order process

Effective method for reducing order

Problem statement : Reactive Distillation column Plant output and model output analyzed using PPE


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References

1) Abhijit S. B., Patwardhan S. C., and Ravindra D. G., Closed loop

identification using direct approach and high order ARX/GOBF-ARXmodels,Journal of Process Control, volume 21, pp. 1056-1071, (2011).

1) Billings S.A., and Zhu Q.M., Nonlinear model validation using correlation

tests,International journal of control, 60(6), pp. 1107-1120, (1994).

1) Garnier and Wang L., Identification of Continuous-time Models fromSampled Data, Springer-Verlag, London, (2008).

1717


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Control based system Identification and GOBF – ARMA

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