chapter 2 emprical model.pdf
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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 steadystate
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 modelbased
control strategies, process model is part of the control law.
Optimize process operating conditions: Use steadystate model torecalculate optimum operating conditions to maximize profit or minimize
cost. Use steadystate 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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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?
Procedure of Empirical Modeling

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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#
Procedure of Empirical Modeling

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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 1.21
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 steadystate 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
steadystate 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
steadystate gain
Time Constant and Time Delay L:

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Log Method
Plot against t: a straight line with
slope of
intercept yaxis atL / .
meet the taxis 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
Closedloop 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 steadystate
Repeat to generate sustained oscillation
AfterL, y increase, relay switches to
opposite position,
Outputinput phase lag: , limit cycle
with periodPu.
Identification from Relay Feedback

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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
Identification from Relay Feedback

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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): oddsymmetric
(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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Steadystate Gain
uyKp
Kp: compare input and output values at two different steadystates:
changes in u: small enough so it represents the linearized gain. highly nonlinear
processes, changes as small as 103 to 106percent of full range
such small changes only feasible for mathematical steadystate gains
impractical to obtain reliable steadystate gains from plant data. (how to improve?)

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Transfer function Modeling
Once the model is selected, model parameters backcalculated 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 1001
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 1001
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. Timesaving, easy to use, automatically extractsprocess response at an important frequency, facilitatessimple pushbutton tuning
2. Test under closedloop 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.