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FUNDAMENTALS OF
CONTROL SYSTEMS
LECTURE 5:FEEDBACK CONTROL SYSTEM
CHARACTERISTICS
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OPEN-AND CLOSED-LOOPCONTROL SYSTEMS
An open-loop system operates withoutfeedback and directly generates the output inresponse to an input signal.
c ose - oop sys em uses a measuremen othe output signal and a comparison with thedesired output to generate an error signal that
is applied to the actuator.
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R(s)G(s) C(s)
Open-loop controlsystem
R(s) + Ea(s)
G(s)
H(s)
C(s)
-
systems = s s
)()()(1
1)(
)()()(1
)()(
)]()()()[()()()(
sRsHsG
sE
sRsHsG
sGsC
sCsHsRsGsEsGsC
a
a
+=
+
=
==
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Sensitivity toParameter Variations
A process, represented by G(s), is subject to a changing
environment, aging, ignorance of the exact values of the processparameters. In the open-loop system, all these errors and changesresult in a changing and inaccurate output.
However, a closed-loop system senses the changes in the outputue o process c anges an a emp s o correc e ou pu .
Consider a change in the process as: )()( sGsG +
In the open-loop case, the change in theoutput is: )()()( sRsGsC =
In the closed-loopsystem:
)())()(1))(()()()(1(
)()(
)()())()((1
)()()()(
sRsHsGsHsGsHsG
sGsC
sRsHsGsG
sGsGsCsC
+++
=
++
+=+
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)())()(1(
)()(
),()()()(If
2sR
sHsG
sGsC
sHsGssG
+
=
The change in the output of theclosed-loop system is reduced by afactor [1+G(s)H(s)]2, which isgreater than one over the range offre uencies of interest.
System Sensitivity: Ratio of the percentage change inthe system transfer function to the percentage change ofthe process transfer function.
)()(
)()(
)(
)(
)(sG
sG
sTsT
SsR
sC
sT
==
In the limit, for
small incrementalchanges
T
G
G
T
GG
TT
S
=
=
Clearly, sensitivity of the open-loop system is equalto one.
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GHGH
G
G
GHT
G
G
TS
GH
GsT T
G+
=
+
+
=
=
+=
1
1
)1()1(
1
1)(
2
Result: Sensitivity of the system may be reduced belowthat of the open-loop system by increasing G(s)H(s) over
the frequency range of interest.The sensitivity of the feedback system to changes inthe feedback element H(s) is
GH
GH
GHG
H
GH
G
T
H
H
TSTH
+
=
+
+
=
=
1)1(1
2
When GH is large, the sensitivity approaches unity and thechanges in H(s) directly affect the output. Therefore it isimportant to use feedback components that will not vary withenvironmental changes.
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DISTURBANCE SIGNALS IN A
FEEDBACK CONTROL SYSTEM
A disturbance signal is an unwanted input signal that
affects the systems output signal. As a steel barapproaches the rolls, the rolls turn unloaded. However,when the bar engages with the rolls, the load on the rollsincreases immediately to a large value. This loading effectcan e approx ma e y a s ep c ange o s ur ancetorque.
Steel bar
Rolls
Conveyor
1Ra
Ia(s)Km
Tm(s)
+
Td(s)
TL(s) 1Js+B
Kb
Va(s)
+ Speed
W(s)
Motor back emf
Open-loop speed control system
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The error for the system is E(s)= R(s)-w(s) and
R(s)=wd(s). For simplicity in calculation, let R(s)=0, andexamine E(s)=-w(s).The change in speed due to the loaddisturbance is
( ))(
1)()( sT
RKKBJsssE d
abm
++==
( ) )()(lim)(lim)( 0 =+=== oabmstd
RKKB
DssEtEs
DsT
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KmRa
Tm(s)+
Td(s)
TL(s) 1Js+B
Kb
Va(s)+ SpeedW(s)
+Ka
Amplifie
r
Closed-loop speed controlsystem
Kt
Tachometer
( )abta
ma
d
KKKsHBJs
sGR
KKsG
sTsHsGsG
sGssE
+=+
==
+==
)(,1
)(,)(
)()()()(1
)()()(
21
21
2
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( )( ) sD
KKKRKBJss
sRs
DsT
batam
d
+++
=
==
1)(
0)(and)(If
( )( )D
KKKRKBsst
batamst
++==
)(lim)(lim0
When the amplifier gain Ka is sufficientlyhigh,
tma
bmac
c
tma
a
KKK
KKBR
DKKK
R
+=
=
)(
)(
)()(
0
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The Cost of Feedback
Increased number of components and complexity.
Loss of gain: Open-loop gain is G(s) and is reduced toG(s)/(1+G(s)) in a unity negative feedback system. We may needan amplifier to increase the gain of the feedforward path.
. - ,
the closed-loop system may not be always stable.
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EXAMPLE
R(s) + E(s) K
D(s)
+
+ )12(
1
+ssC(s)
Plot the unit-step input
response and response tounit-step disturbance ofthe system
)(12
1)(
12
)()(1
)()(
)(1
)(
)()()()()(
22sD
KsssR
Kss
K
sDsKG
sGsR
sKG
sKG
sDsTsRsTsC d
+++
++=
++
+=
+=
ii)K=100.
Compare the results.
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Disturbance response for both K values is acceptable, butstep-input response has less overshoot for K=50 and thesettling time is one second. For K=100, overshoot is
greater and settling time is about 0.65 seconds.
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54
)3)(1(2 ++
++
ss
ssK)3)(1(
1
++ ss
D(s)
+
+ Compare the open-loop and closed-loopcontrol systems fori) sensitivity relative
C(s)
- o n -s epdisturbance
Rover
)3)(1(
1
++ ss+
++ C(s)
Closed-loop control of MarsRover
D(s)
R(s)K
T sK
s s
T s K s s K
o
c
( )
( )
=+ +
=+ + +
2
2
4 5
4 3
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For K=2, T s T s T s
s so c( ) ( ) ( )= = =
+ +
2
4 52
Hence we can compare the sensitivity of the open-loop and closed-loop systems for the same transferfunction.
S dTdK
KT
SdT
dK
K
T
s s
s s K
KT o
o
K
T c
c
c
0 1
4 3
4 3
2
2
= =
= =+ +
+ + +
To examine the effect of thesensitivity at low frequencies,let s=jw
Sj
K jK
Tc = +
+ +
( )
( )
3 4
3 4
2
2
For K=2, the sensitivity at low frequencies, w
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The effect of disturbance can be determined by setting
R(s)=0 and letting D(s)=1/s. For the open-loopsystem, the steady-state value is
y s s s ss
( ) lim( )( )
=+ +
=0
1
1 3
1 1
3
y s s s K s K s
( ) lim( )
=+ + +
=+0 2
1
4 3
1 1
3
The output of the closed-loop system with a unit stepdisturbance is
ForK=2,
y( ) = 15