process control chp 5

7/27/2019 Process Control Chp 5

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Process Control

CHAPTER V

CONTROL SYSTEMS


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For the mathematical analysis of control systems

its possible to consider the controller as a

simple computer. For example, a proportional

controller may be thought of as a device which

receives the error signal and puts out a signal

proportional to it. Similarly, the final control

element may be considered as a device which

produces corrective action (related with output

signal of the controller) on the process. In

industry final control elements are operated

electronically in most of the cases. However,

here a pneumatic (needs air to operate) valve

is chosen in order to understand the details.


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In the system, there exists a heat exchanger which

is used to cool down the hot stream. The

temperature of the outlet stream is continuously

measured and changing the flow rate of the

cooling water temperature is controlled.

If temperature rises the pressure in mercury filled

bulb (which senses temperature) increases andpressure in the Bourdon helix increases causing

it to unwind. The motion of the helix moves the

pen across the chart and moves the baffle

towards the nozzle. This baffle motion is made

proportional to the pen motion by proper

linkage.


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The baffle motion results in a proportional

increase in pressure in nozzle and valve

top which result an increase in cooling

water flow.

As the baffle is moved toward the nozzle,

the pressure P in the nozzle increases

because the area for air discharge isreduced. The nozzle pressure becomes

equal to the supply pressure when the

nozzle is closed by the baffle and thesystem is so designed that the nozzle

pressure falls linearly as the baffle to

nozzle distance is increased.


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With the increase in pressure the plug

moves downward and supplies the flow of

cooling water through the valve.

In general, the flow rate of the fluid through

the valve depends upon the upstream and

downstream fluid pressures and the sizeof the opening through the valve.

In this system, we assume that at steady

state, the flow is proportional to the valve

top pneumatic pressure. A valve with this

relation is called a linear valve.


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Transfer function for a control valve

From the previous experimental studiesconducted on pneumatic valves it has been

found that the relationship between flow rateand valve top pressure for a linear valve canbe represented by a first-order transferfunction.

In many practical systems, the time constant ofthe valve is very small compared with the time

constants of other components of the controlsystem. Therefore, the transfer function of thevalve can be approximated by a constant.

1)()(

sK

sPsQ

v

v

vK

sP

sQ

)(

)(


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TRANSFER FUNCTIONS FOR CONTROLLERS

The transfer functions in the following part are developedfor pneumatic controllers. For electronic controllers,the same equations are applicable by changing (P)with suitable representation of the signal.

i. Proportional control (P Control)

The proportional controller produces an output signal whichis proportional to the error .

Where;

p: output pressure signal from controllerKc: gain or sensitivity (adjustable)

: error, ( = set point-measured variable)

psusually adjustable to obtain the required output

sc pKp


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Kc: controller gain, proportionality constant

i. The controller gain can be adjusted to

make the controller output changes assensitive as desired to deviations in

error.

ii. The sign can be chosen to make thecontroller output increase (or

decrease) as the error signal

increases.

iii. Its usually adjusted after the controller

has been installed.


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In order to obtain transfer functions deviation

variables should be introduced.

P=p-ps

is already a deviation variable (at t=0, s =0)

therefore,

c

c

c

KssP

sKsP

tKtP

)()(

)()(

)()(


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Disadvantage; a steady-state error (offset)occurs after a set-point change or a sustaineddisturbance. This can be eliminated by manually

resetting either the set-point or steady-statevalue after an offset occurs. However, thisapproach is inconvenient because;

to have the change an operator intervention is

required new value must usually be found by trial and

error.

Advantage; for control applications where offsetscan be tolerated, proportional control is attractivebecause of its simplicity.


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ii. Proportional-integral control (PI Control)

This mode of control is described by the relationship,

where,

Kc: gain

I: integral timeps: constant set-up

this term is proportional to the integral of the error. The Kcand Ivalues are adjustable.

t

s

I

cc pdtK

Kp0

dtK

t

I

c 0


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Advantage: In integral control offset will beeliminated. Thus, when integral control is usedp automatically changes until it attains the

value required to make the steady-state errorzero.

Although elimination of offset is usually animportant control objective, integral control isnot used by itself because little control action

takes place until the error signal has persistentfor some time. In contrast, proportional controlaction takes immediate corrective action assoon as an error is detected. Therefore,integral control action is normally used in

conjunction with proportional control as theproportional-integral (PI) controller.

Disadvantage: Integral control action tends toproduce oscillatory responses of the controlledvariable which affects stability.


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sK

s

sP

s

sKsKsP

dt

K

Kpp

I

c

I

c

c

t

I

c

cs

11

)(

)(

)()()(

0


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iii. Proportional-derivative control (PD Control)

This mode of control is described by the relationship:

where,

Kc: gain

D: derivative time

Ps: constant

in this term correction is proportional to the derivative of the error.

Derivative control action tends to improve the dynamic response of thecontrolled variable by decreasing the process settling time (i.e., the time it

takes the process to reach steady-state).

In application , PD control algorithm is physically undesirable since it can not

be implemented exactly using analog or digital components.

sDcc pdtdKKp

dt

dK

Dc


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)1()(

)(

))()(()()(

0

0

sKs

sP

tssKsKsP

dt

dKKP

ppP

Dc

tDcc

Dcc

s


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iv . Proportional-derivative-integral control

This mode of control is a combination of the previous models and is

given by;

where, KC, Dand Iare adjustable.

The transfer function for this mode of control is given as

t

s

I

CDCC pdt

K

dt

dKKp

0

s

sK

s

sP

I

DC

11

)(

)(


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Example for proportional-integral control.

Where, unit step change in error i.e., (t)=1

Example for proportional-derivative control.Where, (t)=A.t

s

I

CC

s

t

I

C

C

ptK

Ktp

pdt

K

Ktp

)(

)1()1()( 0

sDCC

sDCC

pAKAtKtp

pdt

AtdKAtKtp

tAt

)(

)()(

.)(


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Deviation

variable

Time (min)

no control

proportional control

proportional-integral

control

proportional-integral-derivative

control


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If disturbance occurs the value of the

controlled variable starts to rise. Without

control this variable continues to rise to anew steady-state value.

With proportional action only, the control

system can stop the rise of the controlled

variable and ultimately bring it to rest at a

new steady-state value. The difference

between this new steady-state value and

the original value is called offset.


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The addition of integral action eliminates

the offset, the controlled variable

ultimately returns to the original value.The disadvantage of this action is a more

oscillatory behavior.

With the addition of derivative action the

response will be improved. The rise of

the controlled variable is stopped more

quickly and its returned rapidly to the

original value with little or no oscillation.


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First-order Systems

1)(

,

)(

)(

00

1

00

1

01

s

KsG

Kab

aa

tfa

by

dt

dy

a

a

tbfya

dt

dya

p

p

: time constant

Kp: steady state gain


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i. They have capacity to store material, energy or

momentum.

ii. There exists a resistance with the flow of mass,

energy or momentum associated with pumps,

valves, weirs and pipes.

iii. A first order system is self-regulating

: (time constant) is a measure of the time

necessary for the process to adjust to a change in

input.

p

984

953

5.862

2.63

p

p

p

p

Time

elapsed

y(t) as % of ultimate

value


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iv. At steady state

to have the same change in the output:

a small change in input is required if Kpis large (very

sensitive)

a large change in input is required if Kpis small.

pKinputoutput

)()(


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Second-order systems

)(012

2

2 tbfyadt

dya

dt

yda

i. Multicapacity processes (i.e., two or more first order systems)

ii. Inherently second order systems (having inertia)

iii. A processing system with its controller


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Example: Obtaining the transfer function for afirst order system with a capacity for mass

storage.

Consider the tank shown in figure. Thevolumetric flow in is Fi and the outlet

volumetric flow rate is Fo. In the outlet stream

there is a resistance to flow, such as a pipe,valve or weir.

Fi

Fo

h

R

Fi

h

Fo


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Assume that the effluent flow rate Fo is related

linearly to the hydrostatic pressure of the liquid

level h, through the resistance R:

The total mass balance gives;

flowtoresistance

flowforforcedriving0

R

hF

R

hFFF

dt

dhA ii 0


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1)(

)()(

processtheofgainstatesteady

processtheofconstanttime

where

statesteadyat

,

,

s

K

sF

sHsG

RK

AR

FFF

hhh

FRhdt

hdAR

RFh

RFhdt

dhAR

p

p

i

P

p

siii

s

s

sis

i


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The cross-sectional area of the tank A is

a measure of its capacitance to store

mass. Thus the larger the value of A, the

larger the storage capacity of the tank.

Since time constant is defined as AR, we

can say that,(time constant)=(storage capacitance) x

(resistance to flow)

process control chp 5

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