Ch-10 Introduction to Design

8/8/2019 Ch-10 Introduction to Design

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INTRODUCTION TOINTRODUCTION TODESIGNDESIGN


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In most situations the design proceedsIn most situations the design proceeds

on a trial and error basis whereinon a trial and error basis wherein

analysis techniques are repeatedlyanalysis techniques are repeatedly

applied.applied.


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Methods of specifying performanceMethods of specifying performance

of a control systemof a control system

By a set of specifications in timeBy a set of specifications in timedomain and/or in frequency domaindomain and/or in frequency domain

such as peak overshoot, settling time,such as peak overshoot, settling time,

gain margin, steadygain margin, steady--state error, phasestate error, phase

margin etcmargin etc

By optimality of a certain function,By optimality of a certain function, egeg,,

an integral function.an integral function.


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The collection of devices to serve the purpose isThe collection of devices to serve the purpose is

the plant of the control system.the plant of the control system.

The choice of plant components is dictated notThe choice of plant components is dictated not

only by performance specifications but also byonly by performance specifications but also bysize, weight , available power supply, cost etc.size, weight , available power supply, cost etc.

In some cases, it may be possible to meet the givenIn some cases, it may be possible to meet the given

specifications merely by gain adjustment.specifications merely by gain adjustment.

In most practical cases, the gain adjustment does notIn most practical cases, the gain adjustment does not

provide the desired result and increasing the gainprovide the desired result and increasing the gain

reduces the steady state error but results inreduces the steady state error but results in

oscillatory transient response or, even in instability.oscillatory transient response or, even in instability.


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It is necessary to introduce some kind ofIt is necessary to introduce some kind ofcorrective subsystems so that the chosen plantcorrective subsystems so that the chosen plantmeet the given specifications.meet the given specifications. These subsystemsThese subsystems

are known as Compensatorsare known as Compensators and their job is toand their job is tocompensate for the deficiency in thecompensate for the deficiency in theperformance of the plant.performance of the plant.

Design problem statement:Design problem statement:

Given a plant and a set of desiredGiven a plant and a set of desired

specifications, design suitable compensators sospecifications, design suitable compensators so

that the system meets the given specifications.that the system meets the given specifications.


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There are basically two approaches to the controlThere are basically two approaches to the controlsystem design problemsystem design problem

Select the configuration of the overall systemSelect the configuration of the overall systemby introducing compensators and then chooseby introducing compensators and then choosethe parameters of the compensators to meetthe parameters of the compensators to meet

the given specifications on performance.the given specifications on performance.

For given plant ,find an overall system thatFor given plant ,find an overall system that

meets the given specifications and thenmeets the given specifications and thencompute the necessary compensators.compute the necessary compensators.


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Preliminary considerationsPreliminary considerations

Define the performance specificationsDefine the performance specifications

Select configuration for overall systemSelect configuration for overall system

Compensators may be realized by electrical,Compensators may be realized by electrical,

mechanical, pneumatic , hydraulic or othermechanical, pneumatic , hydraulic or other

components and the choice of the componentscomponents and the choice of the components

type to be used depends upon the systemtype to be used depends upon the system

structure.structure.

Realization by electrical components is quiteRealization by electrical components is quitecommon in many control systems.common in many control systems.

Determine the parameter of compensatorsDetermine the parameter of compensators


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Practically most compensators require a trialPractically most compensators require a trialand error parameter adjustment to achieve atand error parameter adjustment to achieve at

least an acceptable performance if it is notleast an acceptable performance if it is not

possible to satisfy exactly all the desiredpossible to satisfy exactly all the desiredperformance specifications.performance specifications.


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R +

-K

1

s(s+1)

CExample

KssK

)s(R)s(Cand

)1s(sKG 2 ++=+=


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K2

1

,Kn ==

If K = 100n = 10 rad / sec, = 0.05

Mp= 85 %

ess for ramp input = 2/n = 0.01 unit

.sec84%)2( == nst


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While the steady state error is acceptable

Mp is not.Damping () to be increased to say, 0.5

10.52

1=1=2Then nn =

ess for ramp input = 2/n

= 1unit

ess increases to 1 - not acceptable.

change the amplifierconfiguration.


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R +

-KA

1

s(s+2)(s+6)

C

Example


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LEAD COMPENSATORLEAD COMPENSATOR

Lead Compensator is characterized by aLead Compensator is characterized by a

zero at s=1/zero at s=1/ and a pole at s=and a pole at s=--1/1/

Constructing transfer functionConstructing transfer function

(S+1/)

GC(S) =(S+ZC)

=

(S+1/)

(S+Pc)= z= zcc//ppcc 0


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The zero is closer to the origin than the pole.The zero is closer to the origin than the pole.

-Pc= ==--1/1/ -ZC=1/1/

jS-Plane

Lead compensation speeds up the tr. Res. andLead compensation speeds up the tr. Res. and

increase the margin of stability. Also increaseincrease the margin of stability. Also increase

the error const. to a limited extentthe error const. to a limited extent


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ei

R1

R2 e0

C

Electrical Lead Network


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LEAD COMPENSATOR (Contd.)LEAD COMPENSATOR (Contd.)

ei

R1

R2

C

A=1/e0

Phase-lead network amplifier


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We realize the transfer function of the leadWe realize the transfer function of the lead

network using the electric network.network using the electric network.

We assume the impedance of the source to beWe assume the impedance of the source to bezero and the output load impedance to bezero and the output load impedance to be

infinite for deriving the transfer function.infinite for deriving the transfer function.


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EE00(s)(s)

EEii(s)(s)=

R2

R2+R1/sC

R2+1/sC

=

(s+1/R1C)

s +1

[R2/(R1+ R2)] R1C

Defining,

= R1C and ==R2/ (R1+ R2) < 1

(S+1/)

GC(S) =

(S+ZC)

=

(S+1/)

(S+Pc)= z= zcc//ppcc 0

This function can be recognized to bethe general form of lead compensator

G(s) = 1+sT1+sT


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In Frequency domain

s=j

G(j) = 1+jT

1+ jT

= G(j) =-tan-1 (T) + tan-1 (T)


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So = 1+ 2 T2T- T

For to be maximum , tan i.e. should bemaximum

d/d =0 m=1/(T)

tanm= 21

1-

Let =tan

= G(j) =-tan-1 (T) + tan-1 (T)

M[Gc(j m)]=


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090

45

0

db

log

=1/T =1/T. .

=1/T

20 db/decade

0 db


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Lead compensator provides:

Phase lead between output and input

Fast transient response introducing zero

Provides increased system bandwidth and

gain at high frequency


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Lag compensatorLag compensator

It has a pole atIt has a pole at --1/1/ and a zero atand a zero at --1/1/ with thewith the

zero located to the left of the pole on thezero located to the left of the pole on thenegative real axis.negative real axis.

Constructing transfer functionConstructing transfer function

GC(S)

= z= zcc//ppcc >1>1=

(S+ZC)

(S+Pc) (S+1/ )=

(S+1/)


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The zero is closer to the origin than the pole.The zero is closer to the origin than the pole.

-Zc= ==--1/1/-PC=-1/1/

j

S-Plane


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ei

R1

R2e0

C

ELECTRICAL LAG NETWORK


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EE00(s)(s)EEii(s)(s)

=

R2

+1/sC

R2+1

1/sC

=

s

(s+1/R2C)

+1

[(R1+R2)/R2] R2C

R1+

[(R1+R2)/R2]

1

*


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= z= zcc//ppcc >1>1

Defining,

= R2C and ==(R1+ R2)/ R2 > 1

The transfer function of the network turns out to

GC(S)

(S+1/ )

=

(S+1/)=

(S+ZC)

(S+Pc)

1

*

;

GC (s) =1+ sT

1+sT


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In Frequency domain, s=j

G(j) = 1+ j T

1+ j T

= G(j) =-tan-1 (T) + tan-1 (T)

>1, So will be negative always.

Hence output lags input : lag network


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Let =tan

So =1+ 2 2 T2

T- T

For to be maximum , tan i.e. should bemaximum

d/d =0 =1/T

tanmax=2

1-


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LAGLAG--LEAD COMPENSATORLEAD COMPENSATOR

The lagThe lag--section has one real pole and one realsection has one real pole and one real

zero with the pole to the right of zero.zero with the pole to the right of zero.

The leadThe lead--section has one real pole and onesection has one real pole and one

real zero with but the zero to the right of zero.real zero with but the zero to the right of zero.


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LAGLAG--LEAD COMPENSATORLEAD COMPENSATOR

(S+1/ 11

)

(S+1/11)

(S+1/ 22

)

(S+1/22)

*GC(S) = >1,>1,


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-Zc2= ==--1/1/22

-PC1=-1/1/ 11

j

S-Plane

-PC1=-1/1/ 22

-Zc2

===

--1/1/11


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ei

R1 R2e0

C2

C1

ELECTRICAL LEAG-LAD NETWORK


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Ch-10 Introduction to Design