lecture 1 feedback control system sept 2012

8/12/2019 Lecture 1 Feedback Control System Sept 2012

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CAB4523 Multivariable Process Control 14/11/2014

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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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)



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Block Diagram Diagram


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Block Diagram General



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Closed-loop Equation


)(1

)(1

sDGGGG

GsY

GGGG

GGGY

mpvc

dsp

mpvc

pvc


)(1

sDGGGG

GY

mpvc

d

)(1

sYGGGG

GGGY sp

mpvc

pvc

Servo-problem

Regulator-problem


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Stability Analysis

The characteristic equation


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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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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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.


ss

c3

11)(G

)18(

3.2)(G

s

sp

15

5.2)(G

ssv

12

1)(G

s

sm



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The Routh array is:


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


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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CAB4523 M lti ariable Process Control 214/11/2014

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

lecture 1 feedback control system sept 2012

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