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Copyright Hanyang University
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. Control
Ch. 1 Introduction
Ch. 2 Basic Control Actions
Ch. 3 Root-locus
Ch. 4 Frequency Response
Ch. 5 Compensator Design
Ch. 6 Modern Control
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Ch. 1 IntroductionControl : to decide input to get the desired outputControl System : collection of components to have desired response
(Plant, Controller, Sensor, Actuator)Process(Plant) : the object of control
Open-loop ControlClosed-loop Control
- Closed-loop Control : Measure and feedback the output andcompare with the desired output (feedback control)
-
-
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- Open-loop ControlNo feedback of the output.Feedforward using the prior knowledge of the system
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Example of Control Systems- Liquid level control
- Driving a car
- Toaster
OpenlooporClosedloop ?
.
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Motor speed controlOpen-loop
Closed-loop
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Traffic light controlOpen-loop : fixed intervalClosed-loop : change interval depending on the traffic
Open-loop Closed-loop (feedback) Simple less expensive Inaccurate Sensitive to
parameter changeexternal disturbance
Stable
Complicated Expensive Accurate less Sensitive to
parameter changeexternal disturbance
May be unstable
Special considerations required
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Control System Design Procedure
Use a detailed model to predict the real responses and performance
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mathematical model- Represent the characteristics and behavior of a system in a
mathematical form- Have to be detail to represent the real characteristics- If too complicated, not good for controller design simple model- Compromise between the two
Modeling procedure- Derive the governing equations- convert to Laplace Transform, Transfer function
State-space eqn.Block diagramSignal processing model
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Linearization
pointoperating:),( 00 yx
Taylor Series Expansion of f(x) around ),( 00 yx
2
02
2
00 )(
!2
1)()()(
0
xx
x
fxx
x
fxfxf
xx
y 0y
)(
)(ofionapproximatLinear:)(
00
00
0
xxKyy
xfyxxx
fyy
xx
Neglect higher-order terms
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Transfer functionLinear, Time-invarient, differential eqn Transfer function, G(s)
)(
)()(
input
outputsG
:
describes characteristics of the system, independent of input
Example :
kbsmssF
sYsG
2
1
)(
)()(
)()()( sFsGsY
fkyybym
)()()( 2 sFsYkbsms
Let initial conditions zero
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n-th order differential equations => transfer functionxbxbxbxbyayayaya mmmmnnnn
1)1(1)(01)1(1)(0
nn
nn
mm
mm
asasasa
bsbsbsb
sX
sYsG
1
1
10
1
1
10
)(
)()(
: zero (): pole ()
)(
)(
)())((
)())((
)(
21
21
0
0
sD
sN
pspsps
zszszsA
asa
bsbsG
n
m
n
n
m
m
0)( sN izs
0)( sD ips
dif s
integs
1
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Operations of Block Diagram Cascade Connection
Feedback connection
Error ,
)()()( 2 sXsGsY )()()( 1 sFsGsX
)()()()( 21 sFsGsGsY
)()()()(
)()()()(
sEsGsHsF
sYsHsFsE
)()()(1
)(
)()()(
sFsHsG
sG
sEsGsY
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: Closed-loop Transfer function
Basic Blocksconstant :
integrator :
differentiator :
Summer(summing point) :
)()(1
)(
)(
)(
sHsG
sG
sF
sY
)()( sKFsY )()( tKfty
)(
1
)( sFssY
t
dttfty0
)()(
)()( ssFsY )()( tfty
21)( FFsY )()( 21 tftfy
)(sG
)()( sHsG
: feed forward T. F: open-loop T.F.
Where,
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State-Space ApproachState variable : min number of variables to represent the states of asystem
Ex :
In Matrix form,
Fkzzbzm
zyzx
Fuzx
Let
2
1
equationstate1
1)(1
212
21
2
1
um
xm
kx
m
kx
xx
um
kzzbm
zx
zx
1Output xy
2
1
2
1
2
1
01
10
10
x
xy
u
mx
x
m
b
m
kx
x
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State-space representation
VectorStatex
x
x
2
1
,
State equation Laplace transform transfer functionL
zero i. c.
)()()(
)()()(
)()()(
1
1
sUDBAsICsY
sBuAsIsX
sBusXAsI
Transfer function :
DuCxyBuAxx
0,01,10
,10
DC
m
B
m
b
m
kA
)()()(
Bu(s)AX(s)sX(s)
sDusCXsY
DuCxy
BuAxx
DBAsICsu
sYsG 1)(
)(
)()(
Space spanned by state vectors State space
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Ch. 2 Basic Control Actions Basic elements of feedback control system
actuator : motor, hydraulic(pneumatic) motor, heatersensor : position encoder, potentiometer, LVDT
velocity tachometerpressure transducer, accelerometer, themometer
controller : computes control input uusing the error signal and sends out tothe actuator
Basic industrial controllers1.Two-position (on-off)
On-off with differential gap
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2.Proportional Control(P-control)
3. Integral Control
4.P-I Control (Proportional plus Integral)
5.P-D Control (Proportional plus Derivative)
6.P-I-D Control
)( yyKu dp
dtyyKu di )(
)1
1()(sT
Ks
KKsG
i
pi
pc
)1()( sTKsKKsG ddpc
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)1
1(
)(
sTsT
K
s
KsKKsG
d
i
p
idpc
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Derivative Control increases damping. improve stability
P-control under external disturbances
pp KbsJs
bsJs
KbsJssDsE
2
2
2
1
1
1
)()(
Assume reference input, r(t)=0 ( 0)( sR )
step disturbances
TsD d)(
Steady state error
p
dd
ps
sss
K
T
s
Ts
KbsJs
ssEe
20
0
1lim
)(lim
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PI control under external disturbances
i
p
p
i
ip
T
KsKbsJs
s
bsJssT
sTKbsJs
sD
sE
23
2
2
1)1(1
1
)(
)(
s
TsD
sR
d
)(
0)(
0lim23
0
s
Ts
T
KsKbsJs
se d
i
p
p
sss
* Integral Control reduces the steady-state error,But may cause instability.
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Stability criterion
Closed-loop T. F. )()(
)(
)(
0
0
sA
sB
asa
bsb
sR
sCnn
m
m
A(s)=0 closed-loop polesStability Real[Closed-loop poles] < 0 : asymptotically stable
0 : stableRouths Stability Criterion
Determine stability without computing the Poles
0110
n
nn asasa Condition 1 : 0
i
a If any 0ia , there exists at least one root with positive or zero
real partCondition 2 : decide the number of unstable poles
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Example
)10)(1(
107
ss
s
F
Y , sF1 tt eety 10
3
2
3
11)(
Dominant pole: located close to the imaginary axis :
Can predict the response of a higher-order system by looking at thedominant pole
Fast, well-damped system
21 n
n
4.0
time)(settling4
st
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Example
i) P-I control )1()(s
KKsG ic
innn
in
KKsKss
KsK
R
Y2223
2
)1(2
)(
For stability, innn KKK 22 )1(2 0,
12
ini KK
K
KK
ii) Integral controls
KsG ic )(
022223 ninn Ksss
niK 20
iii) P-control KsGc )( 0)1(2
22 nn Kss
1K
22
2
2 nn
n
ss
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Steady-state Error Open-loop T.F.
)1()1(
)1()1()(
1
sTsTs
sTsTKsG
p
N
ma
: Type N system
Type : no. of free integrators
)(lim
)(1
1
)(
)(
0ssEe
sGsR
sE
sss
Step input
ssR
1)( 1)( tr
ps
ssKGsGe 1
1
)0(1
1
)(1
1
lim0 consterrorpositionstatic:)0()(lim
0GsGK
sp
Type 0 system
)1(
)1()()0(
s
i
pT
sTKsGKGK
"2Type
"1Type
system0Type
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Type 1 or higher system )0(GKp
higheror1type:0
0type:1
1
ss
ss
e
Ke
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Performance Index
Quantitative measure of the performance of a control system
0
)( dtefJ 1. I.S.E (Integral square Error)
0
2dteJ
2. I.T.S.E (Integral of Time-multiplied square Error)
0
2dtteJ
3. I.A.E (Integral Absolute Error)
0
dteJ 4. I.T.A.E (Integral of Time-Multiplied Absolute Error)
0
dtetJ
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5.others
dteJor
dteeJ
]e[
][
0
0
22
minimize J by choosing an appropriate controller and controlparameter.
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Ch. 3. Root-Locus
Rouths stability Criterion : determine only stabilityRoot Locus method : plots trace of C.L. poles on s-plane as K varies.
stability + help to select controller and gain K+ predict the system dynamics
Root LocusLocus of C. L. poles in the s-plane as gain K is varied
Example
For stability 0K i) K1 , s=-1 1 Kj
Kss
K
sR
sC
2)(
)(2
Kss
Kss
11,
02
21
2
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Root LocusFeedback Control system(negative feedback)
)()(1
)(
)(
)(
sHsG
sG
sR
sC
closed-loop poles C.E : 0)()(1 sHsG 1)()(
T.Floop-Open
sHsG traces of s satisfying the C.E.
Root Locuss : complexG(s)H(s) : complex function
Angle Condition ),21,0,(K360180)()( KSHsG Magnitude Condition 1)()( sHsG Traces of s satisfying the two conditions
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Note K = 0 K =
Departs from pole arrives at zeroSource Sink
Repel R.L. Attract R.L. pole pole real axis break away pt.
zero zero real axis break in pt.
break-away & break in locus real axis 90
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Double Pole Example
2)4)(2(
)1(11
sss
sKGH
Example2
)(1
s
asK
314
)1()4420(
180,60asymptotes3
14
A
mn
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Example
)4(
)1)(21(
122
2
ss
ssK
Example
-5 -4 -3 -2 -1 0 1 2 3 4-2
- 1 . 5
-1
- 0 . 5
0
0 . 5
1
1 . 5
2
R e a l A x i s
ImagAxis
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Ch. 4 Frequency Response
Steady-state response of a system to a sinusoidal input
- Frequency response is easily obtainedno need for modelingno need for characteristic eqn. .
no need for poles/zeros
At steady-state linear system : freq. of y is the same as freq. of xnonlinear system : y contains many freqs. depending
on amp of x
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T.F.)(
)()(
sX
sYsG
Input tAtx sin)( Output )sin()( tBty
shiftphase)(
)()(
ratio)ituderatio(magnampitude)(
)()(
jX
jYjG
jX
jYjG
A
B
T.Fsinusoidal:)()(
js
sGjG
Frequency Response Graphical expression : )( jG plot1.Bode diagram : Already covered2.Polar plot3.Log-magnitude vs. phase plot
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Polar plot (Nyquist plot))(Im)(Re)( jGjjGjG
i) 9011j
1)(
1)(
jjG
ssG
ii) jjGssG )()( iii)
2
2
2
2222
2
1
2
1
11
1
1
1)(
1
1)(
YY
T
Tj
TTjjG
sTsG
YX
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Nyquist stability Criterion
Char. Eq. : 0)()(1 sHsG
RHP)in0GH1of(zerospolesCLunstableofno.:ZplaneGHin1-ofntencirclemewise-clockofno.:N
GHofpolesunstableofno.:P
PNZ Example
)1)(1(
)()(21
sTsT
KsHsG
0-(iii)
0
090R(ii)
1800
0)(i
js
GH
eRs
GHKGH
js
j
stable0
1)-ofntencircleme(no0
0
Z
N
P
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Minimum phase systemNo O.L. poles or zeros in RHP
)1)(1(
)1)(1(
21 sTsTs
sTsTKGH ba
0P
For stability, N=0 Z=0polar plot 1
Relative stability1. Polar plot GM / PM
180
1
GMKm
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2. Gain Margin & Phase Margin in R-locus
K
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Gain margin : gain
180
)(
1
jwG
Kn
Phase margin : Phase lag
1)()(180
jwGn jwG Stable
0
0
n
nK
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Magnitude-phase plot
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Bandwidth
Cutoff frequency( 0w )db
jR
jC
jwR
jwC3
)(
)(
)(
)(
0
0 for 0ww cutoffw Frequency for which
707.02
1,3log20 input
outputdb
input
output Bandwidth = Range of frequency where system tracks
input sinusoids. Measure of speed of response ~rise time High bandwidth : better tracking properties for input of wide
frequency range But, too high bandwidth Noise problem
Higher cost
Closed-loop T.FG
G
1
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Ch. 5 Compensator DesignEffect of additional poles : root locus .Effect of additional zero : root locus .
Compensator : A controller to modify dynamics of a system to meetspecs.
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Lead compensator
)0,10(1
1
)( pzps
zs
Ts
Ts
KsG cc
Bode diagram of lead compensator ( )1.0,1 cK m : Maximum phase lead angle at nww
1
1sin m
Lead compensator : phase margin , Lead compensator :
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Example
4.18
41.47.41)(
s
ssGc
)4.18)(2(
)41.4()(
sss
sKsGGc
Frequency (rad/sec)
Phase(deg);Magnitude(d
B)
Bode Diagrams
-150
-100
-50
0
50
100
From: U(1)
10-2 10-1 100 101 102 103-200
-150
-100
-50
0
50
To:Y
(1)
Gc
G
GcG
Gc
G
GcG
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Lag compensator
),1(1
1
)( zpps
zsK
Ts
Ts
KsG ccc
Lag compensator : static error coeff. Steady state error
Frequency (rad/sec)
Phase(deg);Magnitude(dB)
Bode Diagrams
-20
-15
-10
-5
0From: U(1)
10-4 10-3 10-2 10-1 100-60
-40
-20
0
To:Y(1)
Example
]
Type 1 system : step input 0sse
-0.24+j0.86
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Ramp input 11 v
ssK
e (too big) 1vK
ccs
cs
v K
sssT
s
Ts
sKsGssGK
)15.0)1(
1
1
1
lim)()(lim00
vK cK .To reduce sse , increase cK to 5 for example
If)15.0)(1(
5,5)( 1
sss
GGGsG cc
Lag compensator control
)15.0)(1)(1100(
)110(5)()(
100
1
)10
1(5.0
1100
)110(5)(
ssss
ssGsG
s
s
s
ssG cc
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Frequency (rad/sec)
Phase(deg);Magnitude(dB)
Bode Diagrams
-200
-100
0
100 From: U(1)
10-2 10-1 100 101 102 103-300
-200
-100
0
To:Y(1)
Gc
G
GcG
Gc
G
GcG
Summary
ps
zskGc
)( Lead compensator : adds phase lead at crossoverw
increase without decreasing nw Lag compensator : adds low frequency gain
rcompensatolagpz
rcompendatoleadpz
:
:
Improve sse Disturbance reject
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PID controlPID Turning
)1
1()( dsi
pc TsT
KSG
Parameter : dip TTK ,, trial & errorZieglerNichols rule
Plant step response parameter
1) (if no integrator) plant response parameter
LLL
T
L
L
T
LT
TTK dp
0.521.2PID
03.0
0.9PI
0P
i
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2)P() Kp 0 output crp KK
crP :
crcrcr
crcr
cr
lp
PPK
PK
K
TTK
0.1250.50.6PID
02.1
10.45PI
00.5P
i
1.Ziegler-Nichols turning 25 overshoot
2.plant model tuning3. .
gain initial tuning trial & error
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Example
0
21
0
1
3
02
2
1
2
1
cP
ux
x
x
x
2
0
1,
3
02
AB
BA
ableuncontroll0
lecontrollab0)det(
cP 2x
Observability ()Output y state variable x
observable0)det(
nn:
1
Q
CA
CA
C
Q
n
22 3xx
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Optimal Control
CXY
BuAXX
min ft
dttuXgJ0
),,( Control Law : state feedback ( state )
Kxu
x
xKKxKxKxKu
n
nnn
1
12211
Closed-loop systemXAXBKABKXAXX *)(
Problem J feedback gain Ki
e.g. desired 0dX 0X
ft T
dtXXJ0
)( XXPXX
dt
d TT )( P symmetric):n(n
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Performance Index
2
2
2
2
221211
2
12
2
)0()0(
K
KK
PPP
PXXJ T
2,1 min2 JKWhen
Optimal Control with Energy term
Control gain Pbr
KT 1
whereP
is obtain fromequationRiccati:0
1 QPPbb
rPAPA
TT
factorgweightin:r)(:..
:
energy
2
0dtruXQXJIP
KXuControl
buAXX
T
1
1)0(xAssume
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Ch. 6 Modern Control1.Linear Quadratic Control System model
0),()()( tttuBtxAtx
Performance index
T
t
TT
T
dttuRtutxQtx
TxTSTxtJ
0
))()()()((2
1
)()()(2
1)( 0
0,0,0)( RQTS
Riccati EducationQSBRBSASSAS
TT
1
Algebraic Riccati Equation(ARE) (When T )QSBRBSASSA
TT 1
0
Feedback control gainSBRK
T1
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2. Adaptive Control () .
1) Direct Adaptive control1. -Output error controller gain adjust.
2) Indirect Adaptive control- Plant model parameter adjust controller gain .
1) Gain scheduling2) Model Reference Adaptive Systems3) Self-Tuning Regulator4) Dual Control
Controller
DesiredReference model
Process
Adjustmentrule
ym
yr u
e
y
+-
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3. Intelligent ControlControl strategy and decision are made at a symbolic level throughperception and reasoning and learning.
Signal Symbol Control
Pattern recognition Neural network
Rule based control Fuzzy logic control
A. Neural NetworkBiological neurons are believed to be the structural constituents of the
brain. A neural network can:
Learn by adapting its synaptic weights to changes in the
surrounding environments
Handle imprecise, fuzzy, noisy, and probabilistic information
Generalize from known tasks or examples to unknown ones
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Membership Function
Neural NetworkControl
B. Fuzzy logic control Control Decision system Boolean Logic Fuzzy logic
Fuzzy Set Membership Function Fuzzy Logic
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Fuzzy controller Fuzzifier Fuzzy Inference Engine Fuzzy Rule Base Defuzzifier
Merits of Fuzzy Controllera. Linguistic control rulesb. Highly nonlinearc. Fewer rules
OutputDefuzzifier
FuzzyInference Engine
Fuzzy
Rule Base
Input 1
FuzzifierInput 2
< Block Diagram of fuzzy logic controller >