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Lecture 1. Feedback Control System
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Feedback Control System
Figure 1. Schematic diagram of a heat exchanger
Schematic Diagram : A Heat Exchanger ( without controlsystem)
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Feedback Control System
Model Equations : Assuming W(s)=0
)(1
)(1
1)( 1 sW
s
KsT
ssT s
p
)()()()( 1 sWsGTsGsT spd
In general
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Schematic Diagram : A Heat Exchanger ( with Feedback
control system)
Feedback Control System
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Feedback Control System
Block Diagram Diagram
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Block Diagram General
Feedback Control System
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Closed-loop Equation
CAB4523 Multivariable Process Control 74/11/2014
)(1
)(1
sDGGGG
GsY
GGGG
GGGY
mpvc
dsp
mpvc
pvc
Feedback Control System
)(1
sDGGGG
GY
mpvc
d
)(1
sYGGGG
GGGY sp
mpvc
pvc
Servo-problem
Regulator-problem
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Stability Analysis
The characteristic equation
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01 mpvc GGGG
A feedback control system is stable if and only if all
roots of the characteristic equation are negative or
have negative real parts. Otherwise, the system is
unstable.
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ECB4034 - Chemical Process Instrumentation and Control 10
Routh Stability Criterion
Uses an analytical technique for determining whetherany roots of a polynomial have positive real parts.
Characteristic equation
00111 asasasa n
nn
n
where an>0. According to the Routh criterion, if any of
the coefficients a0, a1, aK, an-1 are negative or zero, then at
least one root of the characteristic equation lies in the RHP,and thus, the system is unstable.
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ECB4034 - Chemical Process Instrumentation and Control 11
The first two rows of the Routh Array are comprised of
the coefficients in the characteristics equation. The
elements in the remaining rows are calculated from
coefficients from the using the formulas:
1
3211
n
nnnn
aaaaab
1n
5nn4n1n2
aaaaab
1
21311
b
baabc nn
1
31n5n12
b
baabc
(n+1 rows must be constructed
n = order of the characteristic eqn.)
Routh Stability Criterion
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ECB4034 - Chemical Process Instrumentation and Control 12
Routh Stability Criterion:
A necessary and sufficient condition for all roots of thecharacteristic equation to have negative real parts is
that all of the elements in the left column of the Routh
array are positive.
Example 1:Determine the stability of a system that
has the characteristic equation
0135 234 sss
Solution: Because thesterm is missing, its coefficient is
zero. Thus the system is unstable.
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Example 2The transfer functions comprising a feedback
control system are given below. Check whether the
closed-loop response is stable.
CAB4523 Multivariable Process Control 134/11/2014
ss
c3
11)(G
)18(
3.2)(G
s
sp
15
5.2)(G
ssv
12
1)(G
s
sm
Routh Stability Criterion:
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ECB4034 - Chemical Process Instrumentation and Control 15
The Routh array is:
Routh Stability Criterion:
Row
1 240 45 5.75
2 198 2.25
3 42.27 5.75
4 -24.68
5 1
2727.42198
)25.2240()45198(
68.2427.42
)75.5198()25.227.42(
Since element in row 4 is negative the feedback control
system is unstable
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Tuning of Feedback Controllers
Tuning Methods: There are numerous tuningmethods one of the most popular methodZeigler Nichols
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Controller Parameters
Controller Kc I D
P
PI
PID
2
cuK
2.2
cuK
2.1
uT
7.1
cuK
2
uT
8
uT
Zeigler-Nichols controller setting
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Ziegler-Nichols Closed-loop
MethodKnown as cont inuou s cyc l ing method. It is based on thefollowing trial-and-error procedure:
Step 1.After the process has reached steady state, the integraland derivative actions are eliminated by setting Tdto zeroand Tito the largest possible value.
Step 2.Set Kcequal to a small value and place the controller inthe automatic mode.
Step 3.Introduce a small momentary set point change. Graduallyincrease Kcin small increments until continuous cyclingoccurs. The numerical value of Kcthat produces the effectis called the ultimate gain, Kcuand the period of thecorresponding sustained oscillation is referred to as the
ultimate period, Pu.Step 4.Calculate the PID controller settings using the Ziegler-
Nichols tuning relations as given in the Table.
Step 5.Evaluate the Ziegler-Nichols controller settings byintroducing small set point change and observing theclosed-loop response. Fine-tune the settings, if necessary.
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This method is an open- loop method. An open-loop transient is induced by a step change in thesignal to the valve. The Cohen-Coon method issummarized in the following steps:
Step 1. With the controller in manual, introduce asmall step change in the controller outputthat goes to the valve and record thetransient, which is the process reactioncurve.
Step 2. Draw a straight line tangent to the curve atthe point of inflection. The intersection ofthe tangent line with the time axis is theapparent transport lag, Td.
Cohen-Coon Tuning
Method
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Typical process reaction curve showing graphical
construction to determine first-order with transport lag
model.
Tangent line, slope =Bu/T= S
Bu
0
0 t
M
0
0 t
Input
Cohen-Coon Tuning
Method
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Step 3.
The apparent first-order time constant, is obtained
from
where Buis theultimate value of Bat large tand Sis
the slope of the tangent line. The steady state gain is
given by
where Mis the magnitude of the input signal. The time
delay is given by
Cohen-Coon Tuning
Method
S
BT u
MB
K up
Td
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Step 4. Using the values Kp, Tand Tdfrom step 3, the
controller settings are found from the relations given in
the Table.
Controller KcI
D
P
T
T
T
T
K
d
dp 31
1
- -
PI
T
T
T
T
K
d
dp 1210
91
TT
TTT
d
dd
/209
/3030
-
PID
T
T
T
T
K
d
dp 43
41
TT
TTT
d
dd
/813
/632
TTT
d
d/211
4
Cohen-Coon Tuning
Method