B_lecture16 Bode Compensation and PID Controller Automatic control System
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Transcript of B_lecture16 Bode Compensation and PID Controller Automatic control System
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The Design of Feedback Control System
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Introduction
The performance of a feedback control system is primary importance.
What is a suitable control system?
-- It is stable. -- It results in an acceptable response to input commands.
-- It is less sensitive to system parameter changes. -- It results in a minimum steady-state error for input.
-- It is able to reduce the effect of undesirable disturbances.
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)(ty
t0
overshoot
pt st
Performance specifications
performance specifications in the time domain
Overshoot
Setting time
Steady-state error
%
st
sse
-
c
h
Glg20
G
0
r b
rM
)0(707.0 M
performance specifications in the frequency domain
Closed-loop Open-loop
Resonant peak Gain-crossover frequency
Resonant frequency Gain margin
Bandwidth Phase margin
c
r
b
rM
/ hh L
Performance specifications
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o
j
n
n
T1
Typical complex domain indices are represented by the
location of the dominant poles
%100%21
e
n
st
5.3
or
Tts 3
Performance specifications
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What is compensation or correction of a control system ?
)1)(1()( :exampleFor
sTss
KsG
stable becan system loop-closed thismake
:getcan wecriterion, Hurwitz-Routh toAccording 0)T 0(K 1
11
TT
TK
.or ngonly varyi stable benot can system
loop-closed thisCriterion, Hurwitz-Routh toAccording )1(
)( :ifBut 2
TK
Tss
KsG
Solution:
Example
TTss
sKsG
)1(
)1()( :make weIf
2
This closed-loop system can be stable.
We make the system stable by increasing a component.
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Compensation & Compensator
Increasing a component ,which makes the systems performance to be improved, other than only varying
the systems parameters, this procedure is called
the compensation or correction of the system.
The compensating device may be electric,
mechanical, hydraulic, pneumatic, or some other
type of device or network and is often called
a compensator.
A compensator is an additional component that is
inserted into a control system to compensate for a
deficient performance.
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Compensation & Compensator
The compensator can be placed in a suitable location within the
structure of the system.
The compensator placed in forward path is called a cascade (or
series) compensator.
)(sGC )(sG)(sR )(sY
Controll
er
controlled
process
r.compensato a is 1)s( stable, becan system the
, 1)s( component increase to,)1(
)( :Example2
Tss
KsG
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Compensation & Compensator
Similarly, the other compensation schemes are called feedback, output, input and disturbance compensation.
)(sGC
)(sG
)(sH
)(sR )(sY
)(sG
)(sH
)(sGC
)(sR )(sY
)(sG
)(sH
)(sGC
)(sR )(sY
)(sG)(sGC
)(sR)(sY
)(sGn)(sN
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In the following sections, we will assume that the process has been
improved as much as possible and that the G(s) representing the
process is unalterable.
For frequency response methods, we are concerned with altering the system so that the frequency response of the compensated system will satisfy the system specifications.
Alternatively the design of a control system can be accomplished in the s-plane by root locus methods. For the case of the s-plane, the designer wishes to alter and reshape the root locus so that the roots of the system will lie in the desired position in the s-plane.
We shall consider the addition of so-called phase-lead , phase-lag and phase lag-lead compensation network ,and describe the design of the network by frequency response techniques.
Approaches to Compensation
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Consider the first-order compensator with the transfer function
The design problem then becomes the selection of parameters ,
in order to provide a suitable performance.
T and
)1( 1
1)(
Ts
TssGc
TTjGc11 tantan)(
Tm
1
1
1arcsin
m
The maximum value of the phase lead occurs at frequency
The maximum phase lead is
Phase-lead Compensation Network
The frequency is the geometric mean of
and . Tp /1Tz /1m
lg10)(lg20 mjG
L
[+20]
T
1
m
0
0 T1
T1
cG
lg10
-
)1( 901
1arcsin0 00
m
The above equation is very useful for calculating a necessary ratio
between the pole and zero of a compensator in order to provide a required
maximum phase lead.
)1( 1
1)(
Ts
TssGc
1R
2R
C
1V 2V
1
11
1
1
)(
)()(
1
21
2
1
21
2
1
2
Ts
Ts
CsRRR
R
CsR
RR
R
sV
sVsGc
2
21
R
RR C
RR
RRT
21
21
Example : Phase-lead electric network compensation
Phase-lead Compensation Network
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Summary of Effects of Phase-lead Compensation
Advantages and disadvantages of Phase-lead controller on performance are :
1. Improving damping and reducing maximum overshoot.
2. Improving h(Lh) and . 3. Increasing Wc.
4. Reducing setting time because of increasing Wc
5. Possibly accentuating noise at higher frequencies.
Black curve 1 controlled process G(s) Bode plot Red curve controller GC(s) Bode plot Green curve 2 Compensated system GC(s)G(s) Bode plot
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Let us consider a single-loop feedback control system, where
)(sG)(sGC
)(sR )(sY
)11.0()(
ss
KsG
We want to have steady-state error ess=0.01 for an unit ramp input. Furthermore, we desire that the phase margin of the system be at least
450 and the gain crossover frequency be at least 40 rad/s.
100 01.01
KK
ess
Example: A phase-lead compensator design for
a second-order system using the Bode diagram
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The first step is to plot the Bode diagram of the
uncompensated transfer function.
10 31
44
dB6
decdB /20
decdB /40
Glg20
022
88
sradc / 3109.17
ionsspecificat hesatisfy tt don' and c
Example: A phase-lead compensator design for
a second-order system using the Bode diagram
-
)11.0(
100)(
sssG
101136.0
104544.0)(
s
ssGc
Black line : 20log ( )G j
Green line : 20log ( ) ( )cG j G j
10 31
44
dB6
decdB /20
decdB /40
Glg20
022
88
)1( 1
1)(
Ts
TssGc
88T
1T ,22
T
1T 21
here
08.49
/44
101136.0)11.0(
)104544.0(100)()(
srad
sss
ssGsG
c
c
Example: A phase-lead compensator design for
a second-order system using the Bode diagram
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Blue line : controlled process Bode plot 20log ( ) and ( )G j G j
Green line : Compensated system Bode plot 20log ( ) ( ) and ( ) ( )c cG j G j G j G j
Example: A phase-lead compensator design for
a second-order system using the Bode diagram
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Consider the first-order compensator with the transfer function
The design problem then becomes the selection of parameters and
T, in order to provide a suitable performance.
TTjGc11 tantan)(
The phase of this compensator is always negative. Thus it is called a
phase-lag compensator.
Phase-lag Compensation Network
)(sG)(sGC
)(sR )(sY
1
11
c
TsG s
Ts
)( lg20
)( lg20
0
)(
1
11
1
T
TTT
T
L
L
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Phase-lag Compensation Network
lg20
L
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Summary of Effects of Phase-lag Compensation
1 T
1
T
Black curve 1 controlled process G(s) Bode plot Red curve controller GC(s) Bode plot Green curve 2 Compensated system GC(s)G(s) Bode plot
Advantages and disadvantages of Phase-lag controller on performance are :
1. Improving damping and reducing maximum overshoot.
2. Improving h(Lh) and . 3. Filtering out high-frequency noise (lessening noise at higher frequencies).
4. Decreasing Wc.
5. Increasing settling time because of decreasing Wc .
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The phase lag-lead compensator
The phase-lead compensator improves settling time,
phase margin and increase the bandwidth.
However, phase-lag compensator when applied properly
improves phase margin but usually results in a longer
settling time.
Therefore, each of these control schemes has its advantages, disadvantages,and limitations, and there are many systems that cannot be satisfactorily compensated by either scheme acting alone.
It is natural, therefore, whenever necessary, to consider
using a combination of the lead-lag compensator, so
that the advantages of both schemes are utilized.
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The transfer function of a lag-lead compensator can be written as
sT
saT
sT
sTsGsGsG ccc
2
2
1
121
1
1
1
1)()()(
)10 ,1(
lead lag
It is usually assumed that the two break frequencies of the lag portion are
lower than the two break frequencies of the lead portion.
1
1
T1
1
T 2
1
T 2
1
T
||lg20)( GL
0
The phase lag-lead compensator
G)(
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The phase lag-lead compensator
1R
2R
2Cie
i
0e1C
Phase lag-lead electric network compensation
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PID controllers in the frequency domain
The PID controller provides a proportional term, an integral term,
and a derivative term.
We have the PID controller transfer function as
sKs
KKsG D
IPc )(
If we set , we have the PI controller 0DK
s
KKsG IPc )(
If we set , we have the PD controller 0IK
sKKsG DPc )(
Effects are similar to phase lag-lead compensation.
Effects are similar to phase-lag compensation.
Effects are similar to phase-lead compensation.
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PD controller
Effects are similar to phase-lead compensation
)( :function transfer sKKsG Dpc
C(s) G(s)
R(s)
pK
sKD
)(sGC)2(
)( :Assuming
2
n
n
sssG
)2(
)()()( :is system dcompensate theoffunction transfer loop-open The
2
n
DPnc
ss
sKKsGsG
D
P
K
Ks :at zero loopopen a adding toequivalent is controller PD that theshowsIt
Advantages and disadvantages of PD controller on the performance are :
1. Improving damping and reducing maximum overshoot.
2. Improving h(Lh) and . 3. Increasing Wc.
4. Reducing setting time because of increasing Wc .
5. Possibly accentuating noise at higher frequencies.
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s
KKsG Ipc1
)( :functionTransfer
)2()( :Assuming
2
n
n
sssG
C(s) G(s)
R(s)
pK
sK I
1
)(sGC
)2(
)(
)2(
)1
(
)()(
:is system dcompensate theoffunction transfer loopopen The
2
22
n
IPn
n
IPn
css
KsK
ss
sKK
sGsG
0s :at pole a and :at zero loopopen a adding toequivalent is controller PI P
I
K
Ks
PI controller
Effects are similar to phase-lag compensation
Advantages and disadvantages of PI controller on the performance are :
1. Improving damping and reducing maximum overshoot.
2. Improving h(Lh) and . 3. Filtering out high-frequency noise (lessening noise at higher frequencies).
4. Decreasing Wc.
5. Increasing settling time because of decreasing Wc .
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Transfer function: sKs
KK(s)G DIpc 1
PID controller have advantages both of PI and PD.
G(s) R(s) C(s)
- + pK
sKI
1
sKD)(sGC
PID controller
Effects are similar to phase lag-lead compensation
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Circuits of PI , PD and PID
_
+
C
R1 u
r u0
PI controller
R2
)1
1(21
2
)(
)(0
CsRR
R
UU
s
s
R
_
+ C
R1 u
r u0
PD controller
R2 )1( 11
1
2
)(
)(0 sCRR
R
UU
s
s
R
_
+ C
1
R1 ur
u0
PID controller
C2 R2
?)(
)(0 s
s
RUU