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description
Model System Pengaturan
Dr. Ir. M. Sabri, MT
Komponen Sistem Kontrol
System atau proses (harus dikendalikan)
Actuator (mengubah sinyal kontrol untuk sinyal daya)
Sensors (menyediakan pengukuran keluaran sistem)
Input Referensi (menggambarkan output yang diinginkan)
Kesalahan deteksi (membentuk kesalahan kontrol)
Kontroller (beroperasi pada kesalahan kontrol untuk membentuk sinyal kontrol, Kadang-kadang disebut kompensator)
Karakteristik Sistem Umpan balik
Perhatikan sistem kontrol kecepatan berikut
Beban
Kl Motor
Km
Amp
Ka
Sensor kecepatan Ks
referensi
kecepatan
u +
_
gangguan
Momen puntir
wo
system loop terbuka
Lajur umpan balik
wr +
+
Td
Tm
Karakteristik System Loop terbuka
Keakuratan dari sistem loop terbuka tergantung pada kalibrasi gain dan pengetahuan gangguan sebelumnya (pilih kontrol u untuk memberikan diinginkan wo).
Masalah:
nonlinear atau berbagai gain waktu
Berbagai gangguan yang tidak diketahui
Beban
Kl Motor
Km
Amp
Ka
u
Gangguan
momen puntir
wo
system loop terbuka
+
+
Td
Tm
dllma
dmlo
TKuKKK
TTK
)(w
Karakteristik Loop Tertutup Perhatikan kes dengan umpan balik
d
slma
lr
slma
lmao
dlosrlma
dmlo
TKKKK
K
KKKK
KKK
or
TKKKKK
TTK
11
)(
)(
ww
ww
w
Beban
Kl Motor
Km
Amp
Ka
Sensor kecepatan Ks
Kecepatan rujukan
u +
_
Gangguan
momen puntir
wo
system loop terbuka
Feedback Path
wr +
+
Td
Tm
Karakteristik Loop tertutup
jika Ka sangat besar sehingga seperti ,
kemudian,
Ks adalah gain sensor dalam satuan volts per rad/s.
Hubungan input/output tidak terlalu sensitive terhadap gangguan atau perubahan dalam gain sistem
slmaslma KKKKKKKK 1
d
sma
r
s
o TKKKK
11 ww
rad/s volts 0
Karakteristik Loop tertutup
Sistem salah
Kesalahan kontrol adalah
lagi, jika gain loop, Ka Km Kl Ks besar, kemudian kesalahannya kecil.
d
slma
slr
slma
d
slma
slr
slma
slma
osr
TKKKK
KK
KKKK
TKKKK
KK
KKKK
KKKK
Ke
11
1
111
)(
w
w
ww
Catatan: Definisi Gain
Gain depan: Ka Km Kl
Gain umpan balik: Ks
Gain loop: Ka Km Kl Ks
Gain loop tertutup : gain depan
1 + gain loop
Chapter 2 mathematical models of systems
2.1 Introduction
Controller Actuator Process
Disturbance
Input r(t)
desired output
temperature
Output T(t)
actual
output
temperature
Control
signal
Actuating
signal
u k u a c
Fig. 2.1
temperature
measurement
Feedback signal b(t)
+
- ()
e(t)=
r(t)-b(t)
1) Easy to discuss the full possible types of the control systemsin terms of the systems mathematical characteristics.
2) The basis analyzing or designing the control systems.
For example, we design a temperature Control system :
The key designing the controller how produce uk.
2.1.1 Why
Chapter 2 mathematical models of systems
2.1.3 How get
1) theoretical approaches 2) experimental approaches
3) discrimination learning
2.1.2 What is
Mathematical models of the control systems the mathematical
relationships between the systems variables.
Different characteristic of the process different uk:
T(t)
uk
T1
T2
uk12 uk11 uk21
For T1
12
11
k
k
u
u
For T1
22
21
k
k
u
u
Chapter 2 mathematical models of systems
2.2.1 Examples
2.2 Input-output description of the physical systems differential
equations
2.1.4 types
1) Differential equations
2) Transfer function
3) Block diagramsignal flow graph
4) State variables(modern control theory)
The input-output descriptiondescription of the mathematical
relationship between the output variable and the input variable of the
physical systems.
Chapter 2 mathematical models of systems
ur
uc
R L
Ci
define: input ur output uc we have
rccc
crc
uudt
duRC
dt
udLC
dt
duCiuu
dt
diLRi
2
2
rccc uu
dt
duT
dt
udTTT
R
LTRCmake 12
2
2121:
Example 2.1 : A passive circuit
Chapter 2 mathematical models of systems
Example 2.2 : A mechanism
y
k
f
F
m
Define: input F output y. We have:
Fkydt
dyf
dt
ydm
td
ydm
dt
dyfkyF
2
2
2
2
Fk
ydt
dyT
dt
ydTThavewe
Tf
mT
k
fmakeweIf
1:
:
12
2
21
2,1
Compare with example 2.1: ucy; urF analogous systems
Chapter 2 mathematical models of systems
Example 2.3 : An operational amplifier (Op-amp) circuit
ur
uc
R1
C
R2
R4
R1
R3
i3
i1
i2
+-
Input ur output uc
)3........(....................).........(1
)2...(........................................
)1)......(()(1
223
3
112
2342333
iRuR
i
R
uii
iiRdtiiC
iRu
c
r
c
(2)(3); (2)(1); (3)(1)
r
rCRRR
RR
R
RRc
cCR udt
duu
dt
du)( 4
32
324
1
32
)(:
)(;;: 432
32
1
324
rr
cc u
dt
duku
dt
duThavewe
CRRR
RRk
R
RRTCRmake
Chapter 2 mathematical models of systems Example 2.4 : A DC motor
ua
w1
Ra
La
ia
M
w3
w2 ( J
3, f
3)
( J1, f
1)
( J2, f
2)
Mf
i1
i2
Input ua output 1
)4.....(
)3.....(....................
)2.....(....................
)1....(
11
1
ww
w
fdt
dJMM
CE
iCM
uEiRdt
diL
ea
am
aaaaa
a
(4)(2)(1) and (3)(1):
MCC
RM
CC
Lu
C
CC
fR
CC
JR
CC
fL
CC
JL
me
a
me
aa
e
me
a
me
a
me
a
me
a
1
)1()( 111 www
Chapter 2 mathematical models of systems
): (
..........................
......
......
:
321211
21
22
21
321
21
22
21
321
21
www iiifromderivedbecan
torqueequivalentii
MM
nt coefficie frictionequivalentii
f
i
fff
inertia of momentequivalentii
J
i
JJJ
here
f
Make:
constant-timeelectricfrictionCC
fRT
constant-timeelectric-mechanicalCC
JRT
constant- timemagnetic-electricR
LT
me
af
me
am
a
ae
-.......
.......
............
Chapter 2 mathematical models of systems
ae
mme uCdt
dT
dt
dTT
12
2
www
Assume the motor idle: Mf = 0, and neglect the friction: f = 0,
we have:
)(11
)1()( 111
MTMTTJ
uC
TTTTTT
mmeae
fmfeme
www
The differential equation description of the DC motor is:
Chapter 2 mathematical models of systems
Example 2.5 : A DC-Motor control system
+
t r i ggerUf
ur
-M
M
+
-rect i f i er
DC
mot or
t echomet er
l oad
ua-
uk
R3
R1
R1
R2 R
3
w
Input urOutput ; neglect the friction:
(4)MTMTTJ
uCdt
dT
dt
dTT
(3)uku(2)u
(1)uukuuR
Ru
mmeae
mme
kaf
frfrk
)......(11
...................... .....................
..................................)......()(
2
2
2
11
2
www
w
Chapter 2 mathematical models of systems
2134we have
)(1
)1( 211
212
2
MMTJ
Tu
Ckkkk
dt
dT
dt
dTT e
mr
eCmme e
www
2.2.2 steps to obtain the input-output description (differential
equation) of control systems
1) Determine the output and input variables of the control systems.
2) Write the differential equations of each systems components in
terms of the physical laws of the components.
* necessary assumption and neglect.
* proper approximation.
Chapter 2 mathematical models of systems
2.2.3 General form of the input-output equation of the linear
control systemsA nth-order differential equation:
mnrbrbrbrbrb
yayayayay
mmmmm
nnnnn
.........)1(1)2(
2)1(
1)(
0
)1(1
)2(2
)1(1
)(
3) dispel the intermediate(across) variables to get the input-output
description which only contains the output and input variables.
4) Formalize the input-output equation to be the standard form:
Input variable on the right of the input-output equation .
Output variable on the left of the input-output equation.
Writing polynomial according to the falling-power order.
Suppose: input r output y
Chapter 2 mathematical models of systems
2.3 Linearization of the nonlinear components
2.3.1 what is nonlinearity
The output is not linearly vary with the linear variation of the
systems (or components) input nonlinear systems (or
components).
2.3.2 How do the linearization
Suppose: y = f(r)
The Taylor series expansion about the operating point r0 is:
))(()(
)(!3
)()(
!2
)())(()()(
00)1(
0
30
0)3(
20
0)2(
00)1(
0
rrrfrf
rrrf
rrrf
rrrfrfrf
00 :)()(: rrrandrfrfymake
equationionlinearizatrrfywehave ............)(: 0'
Chapter 2 mathematical models of systems
Examples:
Example 2.6 : Elasticity equation kxxF )(
25.0;1.1;65.12:suppose 0 xpointoperatingk
11.1225.01.165.12)()( 1.00'1' xFxkxF
equationionlinearizatxF
xxxFxF
..............11.12 :is that
)(11.12)()( :have we 00
Example 2.7 : Fluxograph equation
pkpQ )(
Q Flux; p pressure difference
Chapter 2 mathematical models of systems
equationionlinearizatp
p
kQ
p
kpQbecause
...........2
: thus
2)(' :
0
2.4 Transfer function
Another form of the input-output(external) description of control
systems, different from the differential equations.
2.4.1 definition
Transfer function: The ratio of the Laplace transform of the
output variable to the Laplace transform of the input variable,with
all initial condition assumed to be zero and for the linear systems,
that is:
Chapter 2 mathematical models of systems
)(
)()(
sR
sCsG
C(s) Laplace transform of the output variable
R(s) Laplace transform of the input variable
G(s) transfer function
* Only for the linear and stationary(constant parameter) systems.
* Zero initial conditions.
* Dependent on the configuration and the coefficients of the
systems, independent on the input and output variables.
2.4.2 How to obtain the transfer function of a system
1) If the impulse response g(t) is known
Notes:
Chapter 2 mathematical models of systems
)()( tgLsG
1)()()( if ,)(
)()( sRttr
sR
sCsG
Because:
We have:
Then:
Example 2.8 : )2(
)5(2
2
35)( 35)( 2
ss
s
sssGetg t
2) If the output response c(t) and the input r(t) are known
We have: )(
)()(
trL
tcLsG
)()()( tgLsCsG
Chapter 2 mathematical models of systems
Example 2.9:
responseUnit step
sssssCetc
functionUnit step s
ttr
t
.........
)3(
3
3
11)( 1)(
........1
R(s) )(1)(
3
Then:
3
3
1
)3(3
)(
)()(
ss
ss
sR
sCsG
3) If the input-output differential equation is known
Assume: zero initial conditions;
Make: Laplace transform of the differential equation;
Deduce: G(s)=C(s)/R(s).
Chapter 2 mathematical models of systems
Example 2.10:
432
65
)(6)(5)(4)(3)(2
)(6)(5)(4)(3)(2
2
2
ss
s
R(s)
C(s) G(s)
sRssRsCssCsCs
trtrtctctc
4) For a circuit
* Transform a circuit into a operator circuit.
* Deduce the C(s)/R(s) in terms of the circuits theory.
Chapter 2 mathematical models of systems
Example 2.11: For a electric circuit:
ucur C1 C2
R1
R2
uc( s)
1/ C1s 1/ C2s
R1
R2
ur( s)
2112222111
r
c
r
rc
CR; TCR; TCRT
sTTTsTTsU
sUsG
sUsTTTsTT
sCR
sCsU
sCR
sCR
sCR
sCsU
:here
1)(
1
)(
)()(
)(1)(
1
1
1
)(
)1
(//1
)1
(//1
)(
12212
21
12212
21
22
2
22
11
22
1
Chapter 2 mathematical models of systems
Example 2.12: For a op-amp circuit
ur u
c
R1
R2
R1
+-
C R2 1/ Cs
ur u
c
R1
R1
+-
...... ; :here
.................)1
1(
11
)(
)()(
21
2
1
2
1
2
ntime constaIntegral tCRR
Rk
ller.PI-Contros
k
CsR
CsR
R
sCR
sU
sUsG
r
c
Chapter 2 mathematical models of systems
5) For a control system
Write the differential equations of the control system, and Assume
zero initial conditions;
Make Laplace transformation, transform the differential equations
into the relevant algebraic equations;
Deduce: G(s)=C(s)/R(s).
Example 2.13
+
t r i ggerUf
ur
-M
M
+
-rect i f i er
DC
mot or
t echomet er
l oad
ua-
uk
R3
R1
R1
R2 R
3
w
the DC-Motor control system in Example 2.5
Chapter 2 mathematical models of systems
In Example 2.5, we have written down the differential equations
as:
(4)MMTJ
Tu
Cdt
dT
dt
dTT
(3)uku(2)u
(1)uukuuR
Ru
em
ae
mme
kaf
frfrk
)......(1
................... ....................
.........................).........()(
2
2
2
11
2
www
w
Make Laplace transformation, we have:
(4)sMJ
TsTTsU
CessTsTT
(3)sUksU(2)ssU
(1)sUsUksU
mmeamme
kaf
frk
)......()(1
)()1(
.....).........()( ......).........()(
...........................................)]........()([)(
2
2
1
Chapter 2 mathematical models of systems
(2)(1)(3)(4), we have:
)()(1
)()]1
1([ 21212 sM
J
TsTTsU
Ckks
CkksTsTT mmer
eemme
- ......
- ........... :
constanttimeelectricmechanicalCC
JRT
constanttimemagnetic electricR
LThere
me
am
a
ae
)1
1(
1
)(
)()(
212
21
emme
e
r
CkksTsTT
Ckk
sU
ssG
Chapter 2 mathematical models of systems
2.5 Transfer function of the typical elements of linear systems
A linear system can be regarded as the composing of several
typical elements, which are:
2.5.1 Proportioning element
Relationship between the input and output variables:
)()( tkrtc
Transfer function: ksR
sCsG
)(
)()(
Block diagram representation and unit step response:
R( s) C( s)k
1k
t
r ( t ) C( t )
t
Examples:
amplifier, gear train,
tachometer
Fungsi Transfer
Dr. Ir. M. Sabri, MT
Chapter 2 mathematical models of systems
2.5.2 Integrating element
Relationship between the input and output variables:
constant timeintegralTdttrT
tc I
t
I
:..........)(1
)(
0
Transfer function: sTsR
sCsG
I
1
)(
)()(
Block diagram representation and unit step response:
1
R( s) C( s)
1
t
r ( t ) C( t )
t
sTI
1
TI
Examples:
Integrating circuit, integrating
motor, integrating wheel
Chapter 2 mathematical models of systems
2.5.3 Differentiating element
Relationship between the input and output variables:
dt
tdrTtc D
)()(
Transfer function: sTsR
sCsG D
)(
)()(
Block diagram representation and unit step response:
Examples:
differentiating amplifier, differential
valve, differential condenser
R( s) C( s)TDs
1 TD
t
r ( t ) C( t )
t
2.5.4 Inertial element
Chapter 2 mathematical models of systems
Relationship between the input and output variables:
)()()(
tkrtcdt
tdcT
Transfer function: 1)(
)()(
Ts
k
sR
sCsG
Block diagram representation and unit step response:
Examples:
inertia wheel, inertial load (such as
temperature system) 1
R( s) C( s)
k
t
r ( t ) C( t )
tT
1Ts
k
Chapter 2 mathematical models of systems
2.5.5 Oscillating element
Relationship between the input and output variables:
10 )()()(
2)(
2
22 tkrtc
dt
tdcT
dt
tcdT
Transfer function: 10 12)(
)()(
22
TssT
k
sR
sCsG
Block diagram representation and unit step response:
Examples:
oscillator, oscillating table,
oscillating circuit
R( s) C( s)
12
122 TssT C( t )
k
t
1
t
r ( t )
2.5.6 Delay element
Chapter 2 mathematical models of systems
Relationship between the input and output variables:
)()( tkrtc
Transfer function: skesR
sCsG
)(
)()(
Block diagram representation and unit step response:
Examples:
gap effect of gear mechanism,
threshold voltage of transistors
R( s) C( s)
1
t
r ( t )
ske
kC( t )
t
2.6 block diagram models (dynamic)
Portray the control systems by the block diagram models more
intuitively than the transfer function or differential equation models.
2.6.1 Block diagram representation of the control systems
Chapter 2 mathematical models of systems
Examples:
Si gnal
( var i abl e)G( s)Component
( devi ce)
Adder ( compar i son)
E( s) =x1( s)+x
3( s) - x
2( s)
X( s)
X3( s)
X2( s)
+
-
+X1( s) E( s)
Example 2.14
Chapter 2 mathematical models of systems
For the DC motor in Example 2.4
In Example 2.4, we have written down the differential equations as:
)4.....( )3.....(....................
)2.....(.................... )1....(
ww
w fdt
dJMMCE
iCMuEiRdt
diL
ea
amaaaaa
a
Make Laplace transformation, we have:
(8)sMsMfsJ
ssfssJsMsM
(7)sCsE
(6)sICsM
(5)RsL
sEsUsIsUsEsIRssIL
ea
am
aa
aaaaaaaaa
)]......()([1
)( )()()()(
..............................................................................).........()(
.............................................................................).........()(
.............)()(
)( )()()()(
Chapter 2 mathematical models of systems
Draw block diagram in terms of the equations (5)(8):
Ua( s)
aa RsL
1Cm
Ia( s) M( s)
Ea( s)
Ce
)(s
fsJ
1
)(sM
-
-
Consider the Motor as a whole:
1)(
1
2 ffemme
e
TsTTTsTT
C
1)(
)(1
2
ffemme
mme
TsTTTsTT
TsTTJ
Ua( s) )(s
)(sM
-
Chapter 2 mathematical models of systems Example 2.15 The water level control system in Fig 1.8:
Desi red
wat er l evel
ampl i f i er Mot or Gear i ng Val veWat er
cont ai ner
Fl oat
Act ual
wat er l evel
Feedback si gnal hf
I nput hi
Out put h
-
e ua w Q
1k 1
1
2 sTsTT
C
mme
e
s
ek s211
3
sT
k
12
4
sT
k
)(1
)1(
2sM
sTsTT
sTJ
T
mme
em
End
?