Bab 2 Control Theory
18
1 Gene Franklin David Powell Abbas Emami-Naeini Addison-Wesley Feedback Control of Dynamic Systems Feedback Control of Dynamic Systems Feedback Control Chapter 1 Sheet 2 4 rd edition 2002 SR 1 € 60 Simple Feedback Systems Simple Feedback Systems Feedback Control Chapter 1 Sheet 4 thermostat house gas valve furnace heat loss desired temperature room temperature - more examples ? Simple Feedback Systems Simple Feedback Systems Feedback Control Chapter 1 Sheet 5 reference sensor position sensor ship disturbance reference position output control output + - Block diagram actuators
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Merupakan teori pengontrolan
Transcript of Bab 2 Control Theory
1
Gene Franklin
David Powell
Abbas Emami-Naeini
Addison-Wesley
Feedback Control of Dynamic SystemsFeedback Control of Dynamic Systems
Feedback Control
Chapter 1 Sheet 2
4 rd edition 2002
SR 1
€ 60
Simple Feedback SystemsSimple Feedback Systems
Feedback Control
Chapter 1 Sheet 4
thermostat housegas valve furnace
heat loss
desired
temperature
room
temperature
-
more examples ?
Simple Feedback SystemsSimple Feedback Systems
Feedback Control
Chapter 1 Sheet 5
reference
sensor
position
sensor
ship
disturbance
reference
position output
control
output+
-
Block diagram
actuators
2
Feedback Control
Chapter 1 Sheet 11
History of controlHistory of control
Liquid level & flow control
supply
float
Feedback Control
Chapter 1 Sheet 12
Cornelis Drebbel : egg hatching control systemCornelis Drebbel : egg hatching control system
Feedback Control
Chapter 1
Fly-ball governerFly-ball governer
James Watt
REGULATOR
Sheet 13
3
Feedback Control
Chapter 2 Sheet 2
Dynamics of Mechanical SystemsDynamics of Mechanical Systems
maF !Equations of motion
m u
x
x!!
xb !friction force
xmxbu !!! !"m
ux
m
bx !# !!!
m
uv
m
bv !#!
Cruise controlnewton
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
stepresponsestepresponse
position
velocity
input
Chapter 2 Sheet 3
Feedback Control
Feedback Control
Chapter 2 Sheet 17
Room temperatureRoom temperature
q1
Ti
To
R1 R2
q2Ci
!TC R R
T Tii
o i! #FHG
IKJ "
1 1 1
1 2b g
To
4
Feedback Control
Chapter 2 Sheet 18
Fluid flowFluid flow
h
p1
wout
win Mass balance
!m w win out! "
!hA
w win out! "1
$b g
A
m h.A.! $
Feedback Control
Chapter 2 Sheet 19
Tank system!h
Aw win out! "
1
$b g
Non linear equation
h
p1
wout
win
A
wR
pout !1
1
p p p1 0! # %%p small
p p pp
p0 00
1
2# & #% %
!hA
wAR
pAR p
pin! " "1 1 1
20
0$ $ $%
% %p g h! $
Feedback Control
Chapter 2 Sheet 20
LinearizationLinearization
x
y = f(x)y
y f x x x x
y y y
! ! #
! #
b g with 0
0
%
%
y f x x f x K x
Kf x
x
x
x
! # ! #
!
0 0
0
% %
%
%
b g b gb g % %y K xx!
x0
y0
5
Feedback Control
Chapter 3 Sheet 1
Dynamic ResponseDynamic Response
Chapter 3Chapter 3
Feedback Control
Chapter 3 Sheet 2
LaplaceLaplace
Linear system:• superposition
• convolution
systemsystemu y
input output
u y
u y
u u y y
1 1
2 2
1 1 2 2 1 1 2 2
'
'
# ' #( ( ( (
Feedback Control
Chapter 3 Sheet 3
h(t,))h(t,))*(t) y(t)
impulse response
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time
y(t)
f t f t db g b g b g! ""+
+
z ) * ) )Dirac pulse
Impulse response
6
Feedback Control
Chapter 3 Sheet 4
h(t,))h(t,))u(t) y(t)
input response
y t u h t db g b g b g! ""+
+
z ) ) )
y t u t h db g b g b g! ""+
+
z ) ) )
convolution integral y h u! ,
input-output behaviour
impulse response
Feedback Control
Chapter 3 Sheet 5
h(t,))u(t) y(t)
input response
y t u h t db g b g b g! ""+
+
z ) ) )
H(s)H(s)U(s) Y(s)
Y s H s U sb g b g b g! .
Y s y e dsb g b g!"+
+"z ) ))Laplace
Feedback Control
Chapter 3 Sheet 6
H(s)H(s)U(s) Y(s)
Laplace transformLaplace transform X s x e dsb g b g!+
"z% ) ))
0
!y t ky t u tb g b g b g# !Dynamic system
H sk s
b g !#1
sY s kY s U sb g b g b g# !
Transfer function
7
Feedback Control
Chapter 3 Sheet 7
Laplace transformLaplace transform X s x e dsb g b g!+
"z% ) ))
0
time function s-function
Dirac pulse
Unit step 11
s
Ramp
Exponential
Sinusoid
-
*
--
-
t
ts
es a
ts
at
b g 1
1
1
2
2 2
#
#sin
Feedback Control
Chapter 3 Sheet 11
. /. /. /. /. /s 2 s 4
Y ss s 1 s 3
# #!
# #
. / 31 2 CC CY s
s s 1 s 3! # #
# #
Partial fraction expansion real polesPartial fraction expansion real poles
1
2
3
8C
3
3C
2
1C
6
!
! "
! "
example
. / t 3t8 3 1y t e e
3 2 6
" "! " "with
Feedback Control
Chapter 3 Sheet 12
Partial fraction expansion with complex rootsPartial fraction expansion with complex roots
Example: Y ss s s
b ge j
!# #
1
12
Y sC
s
C s C
s sb g ! #
#
# #1 2 3
2 1
. /. /
1 12 2
2 2 3142
s1 s 1 1Y s
s ss s 1 s
# ##! " ! "
# # # #
y t e t e tt tb g ! " "
FHG
IKJ
" "1 3 3 3
12
121
213
12
cos sin
8
Feedback Control
Chapter 3 Sheet 22
Negative feedbackNegative feedback
U (s)R(s)
G1(s)G1(s)
G2(s)G2(s)
Y1 (s)
+
Y2 (s)
. / . / . /
. / . / . / . /
. / . / . /
2
2 2 1
1 1
U s R s Y s
Y s G s G s U s
Y s G s U s
! "
!
!
Y sG s
G s G sR s1
1
1 21b g b g
b g b g b g!#
Forward gain divided by 1 plus loop gain
Feedback Control
Chapter 3 Sheet 29
Response versus pole locationResponse versus pole location
H sb s
a sb g b gb g!
b s zeros
a s poles
b gb g
! '
! '
0
0
Transfer function
H ss
h t e tb g b g!#
' ! "1
00
STABLE: poles < 0
time constant: )0
!1
example:
Feedback Control
Chapter 3 Sheet 30
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
time 1) !
0
1
e
t ! )
First-order system responseFirst-order system response
Natural response = impulse response
response
te"0
9
Feedback Control
Chapter 3 Sheet 32
Pole locations in the s-planePole locations in the s-plane
Im
Re
0 1 2 3 4 5-15
-10
-5
0
5
Time (secs)
Am
plit
ude
0 1 2 3 4 5-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (secs)
Am
plit
ude
0 1 2 3 4 5-1
-0.5
0
0.5
1
Time (secs)
Am
plit
ud
e
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
Time (secs)
Am
plit
ud
e
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Time (secs)
Am
plit
ude
0 1 2 3 4 5-1
-0.5
0
0.5
1
Time (secs)
Am
plit
ud
e
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Time (secs)
Am
plit
ude
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Time (secs)
Am
plit
ude
0 1 2 3 4 50
2
4
6
8
10
12
14
Time (secs)
Am
plit
ude
Feedback Control
Chapter 3 Sheet 33
Complex polesComplex poles
s j d! " #0 -
Complex pairs s j s j s sd d n n# " # # ! # #0 - 0 - 1- -b gb g 2 22
H ss s
n
n n
b g !# #
-
1- -
2
2 22
0
-d
Re
Im2
-n 0 1-
- - 1
2 1
!
! "
! "
n
d n 1 2
1sin
Feedback Control
Chapter 3 Sheet 34
Response of system with complex polesResponse of system with complex poles
H ss s
n
n n
b g !# #
-
1- -
2
2 22
H ss
n
n n
b gb g e j
!# # "
-
1- - 1
2
2 2 21
. / td d
2h t 1 e cos t sin t
1
"03 415 6! " - # -5 6" 17 8
Transfer function
step response
10
Feedback Control
Chapter 3 Sheet 35
Step response of second order system with complex polesStep response of second order system with complex poles
0 2 4 6 8 100
0.5
1
1.5
2
time
19! 0.0
0.1
0.2
0.3
0.4
0.5
0.7
1.0
-n=1
Feedback Control
Chapter 3 Sheet 36
Pole location damping ratioPole location damping ratio
ReReRe
Im Im Im
45:
30:17 5. :
1 ! 0 707. 1 ! 0 5. 1 ! 0 3.
Feedback Control
Chapter 3 Sheet 37
Oscillatory time responsesOscillatory time responses
0 5 10 15 20-1
-0.5
0
0.5
1
time
h(t)
e t"0
" "e t0
11
Feedback Control
Chapter 3 Sheet 39
Time domain specificationsTime domain specifications
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time
y(t)
tr
;1%
ts
Mp
tp
tr rise time
ts settling time
tp peak-time
Mp overshoot
90%
10%
Feedback Control
Chapter 3 Sheet 40
SpecificationsSpecifications
For second order systems:
t
t
t
M e
rn
p
n
sn
p
!
!"
! ;
!
""
18
1
4 61%
2
1 2
.
.
-
<
- 1
1-<1
1
for
Feedback Control
Chapter 3 Sheet 41
Overshoot versus damping ratio 1Overshoot versus damping ratio 1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1
Mp
1 !1
2
1 !1
22
1 ! 1
12
Feedback Control
Chapter 3 Sheet 42
SpecificationsSpecifications
t t Mr s p
-
1 1
0
nr
p
s
t
M
t
=
=
=
1 8
4 6
.
.
d i
-n Re ReRe
Im ImIm2
0
Feedback Control
Chapter 3 Sheet 43
Specifications in the s-planeSpecifications in the s-plane-
1 1
0
nr
p
s
t
M
t
=
=
=
1 8
4 6
.
.
d i
Re
Im
example:
t s
M
t s
r
p
s
>
>
>
0 6
10
3
. .
%
.
-5 -4 -3 -2 -1 0 1
3
2
1
0
-1
-2
-3
-
1
0
n rad s=
=
=
2 8
0 6
15
. .
.
.
Feedback Control
Chapter 3 Sheet 54
Routh’s stability criterionRouth’s stability criterion
Characteristic equation:
. / n1n2n
21n
1n asasasassa #####! "
"""
Routh 1874 Hurwitz 1895
Routh array:
all elements in first column positive
system stable
Routh array:
all elements in first column positive
system stable
13
Feedback Control
Chapter 4 Sheet 55
Routh’s stability criterionRouth’s stability criterion
. / n1n2n
21n
1n asasasassa #####! "
"""
Routh array: n2 4
n 11 3 5
n 21 2 3
n 31 2 3
2
1
0
s 1 a a
s a a a
s b b b
s c c c
s * *
s *
s *
"
"
"
!
!
!
!
" " " "
Feedback Control
Chapter 3 Sheet 56
Routh arrayRouth array
*s
*s
**s
cccs
bbbs
aaas
aa1s
0
1
2
3213n
3212n
5311n
42n
####
"
"
"
"
"
"
"
1
2131
21
31
11
1
541
51
4
12
1
321
31
2
11
b
baab
bb
aa
b
1c
a
aaa
aa
a1
a
1b
a
aaa
aa
a1
a
1b
"!"!
"!"!
"!"!
#
number of sign changes
=
number of poles in Right Half Plane
Feedback Control
Chapter 3 Sheet 57
exampleexample . / 4s4ss2s3s4ssa 23456 ######!
Routh array
4s
0s
43s
02s
40s
0424s
4131s
05761
25
123254
5
6
"
"
2 RHP poles-4 -3 -2 -1 0 1
-2
-1
0
1
2im
re
14
Feedback Control
Chapter 3 Sheet 60
ApplicationsApplications
KK . /. /5s1ss
1
##
Stability ?
characteristic equation: . /. / 0K5s1ss !###
Ks
s
K6s
51s
06
K301
2
3
"30K ?
0K @30K0 ??
Feedback Control
Chapter 3 Sheet 61
KK . /. /5s1ss
1
##
0 5 10 15 200
0.5
1
1.5
2
t
y 1
5
10
25
K
3 23 2 1 0
0 1 2 3
1 2 0 3
a s a s a s a 0
a ,a ,a ,a 0
a a a a 0
# # # !
@
" @
4 3 24 3 2 1 0
0 1 2 3 4
2 21 2 3 4 1 0 3
a s a s a s a s a 0
a ,a ,a ,a ,a 0
a a a a a a a 0
# # # # !
@
" " @
Feedback Control
Chapter 3 Sheet 66
Summary of Routh’s criterionSummary of Routh’s criterion
! first-order system
! second-order system
! third-order system
! fourth-order system
1 0
1 0
a s a 0
a ,a 0
# !
@
22 1 0
0 1 2
a s a s a 0
a ,a ,a 0
# # !
@
15
Feedback Control
Chapter 4 Sheet 1
Chapter 4Chapter 4
Basic properties of feedback
controllercontroller processprocess
sensorsensor
+
-
reference output
Feedback Control
Chapter 4 Sheet 2
Feedback control systemFeedback control system
controllercontroller processprocess
sensorsensor
+
-
control
input
output
Open loop control systemOpen loop control system
controllercontroller processprocessoutput
reference
input
reference
input
control
input
Feedback Control
Chapter 4 Sheet 3
Open loop speed controlOpen loop speed control
controllercontroller motormotoroutput
reference
speed
loadload
++
var y
A
Bload motor . /. /1s1s
A
21 #)#)controller K
ryA
1Kfor ss !'!
w
16
Feedback Control
Chapter 4 Sheet 4
Feedback control systemFeedback control system
controllercontrollerprocessprocess
11
+
-
reference
speed
sensor
A
B
+
+va y
r
. /yrKva "!control action
rAK1
AKyss #
!
Feedback Control
Chapter 4 Sheet 6
Feedback control systemFeedback control system
KK processprocess
11
+
-
A
B
+
+va yr
. /yrKva "! rAK1
AKyss #
!
. /. /1s1s
A
21 #)#)
w
w1AK
By
#!*w
1AK
Br
1AK
AKyss #
##
!
Feedback Control
Chapter 4 Sheet 11
ExampleExample
w
D(s)D(s) G(s)G(s)+
- +
+u
y
r
. /. / . /
1G s
s 1 10s 1!
# #
setpoint stepsetpoint step disturbace stepdisturbace step
17
Feedback Control
Chapter 4 Sheet 13
Propoprtional plus Integral ControlPropoprtional plus Integral Control
. / . / . /0
t
p i
t
u t K e t K e d! # ) )A
PI
. / . /. /
ip
U s KD s K
E s s! ! #
. /i
1D s K 1
T s
3 4! #5 6
7 8
KpKp
iK
s
+
+e(t) u(t)
t t
Feedback Control
Chapter 4 Sheet 14
Proportional plus Derivative FeedbackProportional plus Derivative Feedback
. / . / . /p d
de tu t K e t K
dt! #
. / . /. / p d
U sD s K K s
E s! ! #
PD
KpKp
dK s+
+e(t) u(t)
t t
. / . /dD s K 1 T s! #
Feedback Control
Chapter 4 Sheet 15
Proportional-Integral-Derivative FeedbackProportional-Integral-Derivative Feedback
PID
. / . / . / . /
0
t
p i d
t
de tu t K e t K e d K
dt! # ) ) #A
KpKp
iK
s +
+e(t)u(t)
dK s
+
. / . /di
1D s K 1 1 T s
T s
3 4! # #5 6
7 8
18
Feedback Control
Chapter 4 Sheet 16
Proportional control onlyProportional control only
set point disturbance
Feedback Control
Chapter 4 Sheet 17
Proportional Integral controlProportional Integral control
disturbance
K = 25K = 25
setpoint
PIPI
Feedback Control
Chapter 4 Sheet 18
Proportional Integral Derivative controlProportional Integral Derivative control
y
set point
K=25
Ti=10 s.
disturbance
