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Chapter 2 Identification

(Empirical Modeling)


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Process Control Prof. Cai Wenjian 2

Lecture 3

1. Fundamentals of Empirical Modeling2. Identification from Step Response


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What is Process Modeling?

Constructing process model from experimentally obtained

input/output data, with no recourse to physical nature andproperties of system.

What are three problems in control engineering?

Given input and model find

output?

Given output and model

find input?

Given input and output find

model?


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Purpose of Modeling

Improve Process Understanding: Simulation on dynamic and steady-state

process behavior before plant is constructed, model based simulation caninvestigate process transient without disturb process.

Train plant operating personnel: Interfacing a process simulator with

standard process control equipment to create a realistic training environment,

train plant operators to run complex units and deal with emergency situations

Develop control strategy for new processes: Process dynamic model

allows alternative control strategies to be evaluated. For model-based

control strategies, process model is part of the control law.

Optimize process operating conditions: Use steady-state model torecalculate optimum operating conditions to maximize profit or minimize

cost. Use steady-state process model and economic information to

determine most profitable operating conditions.


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Models Classification

Empir ical model:obtained by fitting experimentaldata.

Hybrid model:combination of the two; values

of some parameters in a theoretical model arecalculated from experimental data.

Theoretical model:developed using the

principles of chemistry, physics, and biology.


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Process Control Dr. Cai Wenjian 6

Procedure of Empirical Modeling

Empirical modeling (identification) consists of followingsteps:

1. Problem Definition Step 1. Problem Definition

2. Model Formulation Step 2. Model Formulation

3. Input function selection Step 3. Input function Selection

4. Parameter Estimation Step 4. Parameter Estimation

5. Model Validation Step 5. Model Validation

Flow Chart of Process Identification


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Step 1. Problem Definition

Different interpretation, resulting different models

same aspect but various angles, vary degree of complexity.

how simple or complex, will model have to be?

model only useful with tool available for solution

which aspects of process most relevant and be contained in model?

Impossible to represent all aspect of the physical process,

capture those aspects that most relevant to problem at hand.

what do we intend to use the model for?

Some modeling solved analytically, others by numericalmethods

how can we test the adequacy of model?

how much time do we have for the modeling exercise?



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Step 2. Model Formulation

In empirical modeling, we analysis of input/output data to detect

model form capable of explaining the observed behavior

( )1

Ls

p

Kg s e

Ts

2( )

( 1)

Ls

p

Kg s e

Ts

Four parameter model

1 2

( )( 1)( 1)

Ls

p

Keg s

s s

2

sL

p

eg s

as bs c

Five parameter model Procedure of Empirical Modeling

1 2

( 1)( )

( 1)( 1)

Ls

p

K s eg s

s s

2

( 1) sL

p

s eg s

as bs c

Three parameter model


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Step 3. Input function Selection

Process information of output data dependent on input function.

Input should provide output rich in useful information and easily extracted.

Typical input functions used in process identification are:

Sine waves

Impulse#Pulse (rectangular or arbitrary)#

White noise

Pseudo random binary sequences

Step#

Relay#



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Step 4. Parameter Estimation

After candidate model selected, estimate unknownparameters.

Fitting experimental data to a predetermined model

form by finding the parameter values which provide

the best fit.

Estimating unknown parameters carried in time

domain or in the frequency domain.Procedure of Empirical Modeling


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Step 5. Model Validation

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Final step involves checking how the empirical model fits data it supposed to

represent. Comparing model predictions with additional process data, and

evaluating the fit.

Time domain, response to certain signal

Frequency domain, Nyquist Plot. Procedure of Empirical Modeling


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Flow Chart of Process Identification


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Basic Requirement of Step Response

Obtain process model from a transient response experiment.

inject step input at the process

measure response

Requirement:

stable process

Amplitude of step input must be determined before test

sufficiently large so response is easily visible above noise level

as small as possiblenot to disturb the process more than necessary

keep the dynamics linear.


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Graphic Method( )

1p

Kg s

s

/( ) (1 )ty t y e

AyK steady-state gain

Time Constant

Deepest Slop

/( )

( )max

tydy t edt

ydy t

dt

tan to (0) y

y t


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Graphic Method

Ls

p e

Ts

KsG

1

)(

when times greater than the time delay

)1()( /)( Lt

eyty

A

yK

steady-state gain

Time Constant

Time Delay L:


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Two Points Method( )

1

Ls

p

Kg s e

Ts

)1()( TLt

eKAty

y()% 28.4 39.3 55 59.3 63.2 77.7 86.5

time(t) T/3+L T/2+L 0.8T+L 0.9T+L T+L 1.5T+L 2T+L

t1 and t2, the time when response

with value 28.4% and 63.2%

)(5.1 12 ttT

)3(5.0 21 ttL

Process model and step response

t1= T/3+L t2=T+L


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Lecture 4

Identification from Step Response (2)


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Log Method ( )1

Ls

p

Kg s e

Ts

when times greater than the time delay

)1()( /)( Lteyty

yy AK K A

/)( Ltkey

yy

ln y y L t

y

steady-state gain

Time Constant and Time Delay L:


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Log Method

Plot against t: a straight line with

slope of

intercept y-axis atL / .

meet the t-axis at the point t=L.

1/ ln

y y L t

y


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Area Method

0 0 0

( )/

0

[ ( ) ( )] ( ) [ ( ) ( )]

[1 (1 )]

L

Lar

Lt L T

L

y y t dt y dt y y t dtAT

K K K

Kdt K e dtT L

K

Average residence time Taris computed form the area ofA0


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Area Method

1

0

/

01 ]1[)(

KTedteKdttyAT

TtTar

0 01 1( )

arT

ar

e y t dt AeA eAT L T T

K K K K

Measure and compute areaA1

under step response up to time Tar

Than TandL can be estimated as

arT T L


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Numerical Integration

Integral of a continuous time signal over 0, ft approximate

00

0

( ) ( ) ( )f

lti

l s i

i

y d y t y t lT q y t

Ts: integration step size,

l: length factor of the integrator (a natural number);

-1q

1 ( ) ( ).sq y t y t T

unit delay operator

a0al: coefficients

For trapezoidal integration rule, filter coefficients:

0 , , 1, 12

sl i s

TT i l


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FOPDT Process

Lthbdyatyt

101

0

1 1 1

( ) ( )

( ) [ ]

[ ]

t

T

t y t

t y d h th

a b L b

( ) ( )t t e Define

1

1( ) ( )

Lsb

Y s e U ss a


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Least Squares Solution

,

TT 1

32

3

1

1

1

/

L

b

a

Least Squares Solution

TakingNinput output samples

Parameter solved

Must start t L1 11

2 2

1

1

( ) ( )

( ) ( ), ,

( ) ( )N N

t ta

t tb L

b

t t


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Working Example

8

1

( ) ( 1)g s s


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Two Points Method

28.4%, t1 = 6.2s63.2%, t2 = 8.64s

T = 1.5(t2 - t1)

= 1.5(8.64 - 6.2) = 3.66sL = 0.5(3t1 - t2)

= 0.5(3*6.2 -8.64) = 4.98s

4.981( )3.66 1

sg s e

s

K = y/u = 1


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Log Method

( )

ln( )

i iy L

y T

y t

T

t

4.27;

/ 1.65

4.27 /1.65 2.59

L

L T

T

4.271( )2.59 1

sg s e

s

K = y/u = 1

For i=1N, calculate the

equation;

Draw a straight line which

best approximate the curve


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Area method

0 1A 1 1 1 / 2 7.9989T y i y i

1 2A 1 / 2 1.4960T y i y i

1 1.496 4.0665eAT eK

07.9989 4.0665 3.9324

AL T

K

3.93241( )4.0665 1

sg s es

K = y/u = 1


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Least Squares Method

303 matrix

301 vector

Sampling: (start after apparent time delay) 4 to 34,

Sampling time: 1s

1

0.2843

0.2200

0.2875

T T

7653.02875.0/2200.02875.0,0.2843 3111 Lba

4.76531

1

4.7653

0.2875

( ) 0.2843

1.0113( )

1 3.5174s 1

Ls s

Ls s

b

g s e es a s

Kg s e e

Ts

N = 30


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Verification (time domain)


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Verification (frequency domain)


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Lecture 5

Relay Feedback Method


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Identification from Relay Feedback

Procedure

Closed-loop I dentif ication

Generating Sustained Oscillation Generating Sustained Oscillation

Determine Critical Parameters Determine Critical Parameters

Approximate Transfer Functions Fourier series Analysis


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Generating Sustained Oscillation

Increased uby h, (e.g., 5%)

Bring the system to steady-state

Repeat to generate sustained oscillation

AfterL, y increase, relay switches to

opposite position,

Output-input phase lag: -, limit cycle

with periodPu.



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Determine Critical Parameters

u

u

P

2

Limit cycle periodPu, obtain

H, height of the relay;

a, amplitude of oscillation.

L, difference between e andy



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Fourier series Analysis

tate sin)(

0

1

( ) cos sin

2

n n

n

Au t A n t B n t

2

0

2

0

sin)(

cos)(

ttdntuB

ttdntuA

n

n

1 sin)( nn

tnBtu

,6,4,2,0

,5,3,1,4

n

nn

h

Bn

Input signal (e(t)) to relay: sinusoidal wave:

Output u(t) of relay: square wave:

u(t): odd-symmetric

(N(a): unbiased and symmetric)

A0 andAn zero


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Describing function

Transfer function of nonlinear system.

only the first Fourier coefficient

2 2

1 1( )B A

N aa

ahaN 4)(

For ideal relay,A1 = 0,B1 = 4h/,


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Ultimate Gain

1 ( ) ( ) 0u

N a g

1 4( )( )

u

u

hK N ag j a

Relay feedback oscillation frequency

corresponds to limit of stability:

Ultimate gain (Ku)becomes:

( ) 1u u

K g j

Results:


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Steady-state Gain

uyKp

Kp: compare input and output values at two different steady-states:

changes in u: small enough so it represents the linearized gain. highly nonlinear

processes, changes as small as 10-3 to 10-6percent of full range

such small changes only feasible for mathematical steady-state gains

impractical to obtain reliable steady-state gains from plant data. (how to improve?)


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Transfer function Modeling

Once the model is selected, model parameters back-calculated from ultimate gain

and ultimate frequency equations.

( ) 1u u

K g j

arg ( )ug j


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Three Parameter Model

Both TorKp needed to solve for time constant, if L available

( )1

p Ls

p

Kg s e

Ts

2( )

( 1)

p Ls

p

Kg s e

Ts

2

1 11

up jL

u u p u

u

KK e K K T

jT

1tanu uL T

tan( )u

u

LT

2

1up

u

TK

K

2

tan( ) / 2

1

u

u

u

p

u

LT

TK

K


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Four Parameter Model

Kp assume to be known

Ku and L needed to solve for two time constants

1 2

( )( 1)( 1)

Ls

p

Keg s

s s

1 1

1 2

2 2

1 2

tan ( ) tan ( )

1

[1 ( ) ][1 ( ) ]

u u u

p

u u u

L

K

K

1 2and


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Example: Wood and Berry column

Transfer function between top composition and reflux flow.

True value: Ku=2.1, u=1.608

Relay feedback test:Ku=1.71, u=1.615

Parameters calculated for

Model 1: (assumeL=1 is known)

Model 2: (assumeL=1 is known)Model 3: (assumeL=1 andKp are known)

12.8( )

16.8 1

s

pg s e

s

13.2( )14.8 1

spg s e

s

2

1.12( )

(0.59 1)

s

pg s e

s

12.8( )

(13.5 1)(0.0009 1)

s

p

eg s

s s


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Fourier Transform Method

y(t) and u(t) are piece wise continuous and periodic,Laplace transform

0

1( ) ( )

1

Pst

PsY s y t e dt

e

0

1( ) ( )

1

P

st

PsU s u t e dt

e

0

0

( )( )

( )( ) ( )

P

st

P st

y t e dtY s

G sU s u t e dt

1 1

2 2

( )( )

( )

Y j c jd G j a jb

U j c jd

1

0

( ) cos( )

P

c y t t dt

1

0

( ) sin( )

P

d y t t dt

2

0

( ) cos( )

P

c u t t dt

2

0

( ) sin( )

P

d u t t dt

1 2 1 2

2 22 2

1 2 1 2

2 2

2 2

( )

( )

( )

( )

c c d d a

c d

c d d cb

c d


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Working Example

8

1( )

( 1)g s

s

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


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Three Parameter Model

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

4.5L

0.7442a

61 ?u

P

41.7183

u

hK

a

Reading:

4.51( )2.4463 1

s

pg s e

s

2

0.57uu

P

2

1

1

2.4463

p

p u

u

K

K KT


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Model verification

0 5 10 15 20 25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Step Response

Time (sec)

Amplitud

e

sys

G


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Summary

1. Time-saving, easy to use, automatically extractsprocess response at an important frequency, facilitatessimple push-button tuning

2. Test under closed-loop to keeps the process in linear

region, good on highly nonlinear processes3. No requirement for careful choice of sampling rate,

useful in initializing a more sophisticated adaptivecontroller.

4. Can be modified to cope with disturbances andperturbations to process.