automatic control theory.ppt
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Automatic control theory
A Courseused for analyzing and
designing a automatic control system
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Chapter 1 I ntroduction
Figure 1.1
* Operating principle
* Feedback control
1) A water-level control system
21 centuryinformation age, cybernetics(control theory), system
approach and information theory , three science theory mainstay(supports)in 21 century.
1.1 Automatic control
A machine(or system) work by machine-self, not by manual operation.
1.2 Automatic control systems
1.2.1 examples
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Chapter 1 I ntroduction
Wat er exi t
wat er
ent rance
f l oat
l ever
Fi gure 1. 2
* Operating
principle* Feedback
control
2) A temperature Control system(shown in Fig.1.3)
M
+
e
ua=k( u
r- u
f)
ur
uf
ampl i f i er
t hermo
met er
Gear
assembl y
cont ai ner
Fi gure 1. 3
* Operating principle
* Feedback
control(error)
Another example of the water-level
control is shown in figure 1.2.
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Chapter 1 I ntroduction
3) A DC-Motor control system
M
M
+
-
+
r egul at or
t r i gger
rect i f i er
DC
mot or
t echomet er
l oad
e
Uf( Feedback)
ur
Fi g. 1. 4
ua
Uk=k( u
r- u
f)
* Principle
* Feedback control(error)
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Chapter 1 I ntroduction
4) A servo (following) control system
ser vopot ent i omet er
M
+
-
I nput
Tr
out put
Tc
ser vomechani sm
ser vo mot or
ser vomodul at or
l oad
* principle
* feedback(error)
Fig. 1.5
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Chapter 1 I ntroduction
* principle
* feedback(error)
gover nment
( Fami l y pl anni ng commi t t ee)
census
soci et y
excess
procreat e
Desi re
popul at i on popul at i on+
- Pol i cy orstatutes
Fig. 1.6
5) A feedback control system model of the family planning
(similar to the social, economic, and political realm(sphere or field))
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Chapter 1 I ntroduction
x2
x3
Si gnal
( var i abl e)xxx
Component s
( devi ces)
+
-
+x1 eAdders ( compar i son)
e=x1+x
3- x
2
x
Fig. 1.7
Example:
1.2.2 block diagram of control systems
The block diagram description for a control system :Convenience
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Chapter 1 I ntroduction
amplifier Motor Gearing Valve
Actuator
Water
container
Processcontroller
Float
measurement
(Sensor)
Error
Feedback
signal
resistance comparator
Desiredwater level
Input
Actual
water level
Output
Fig. 1.8
For the Fig.1.1, The
water level control
system:
Figure 1.1
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Chapter 1 I ntroduction
For the Fig. 1.4, The DC-Motor control system
Desi red
rot ate speedn
Regul at or Tri gger Rect i f i er DCmot or
Techomet er
Actuator
Processcont rol l er
measurement ( Sensor)
comparat or
Act ual
rot ate speedn
Error
Feedback si gnal
Reference
i nput ur
Output n
Fig. 1.9
auk
ua
uf
e
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Chapter 1 I ntroduction
1.2.3 Fundamental structure of control systems
1) Open loop control systems
Cont rol l er Act uat or Process
Di st urbance
( Noi se)
I nput r(t)
Ref erence
desi red out put
Out put c( t )
( act ual out put )
Cont rol
si gnal
Act uat i ng
si gnal
uk
uact
Fi g. 1. 10
Features: Only there is a forward action from the input to the
output.
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Chapter 1 I ntroduction
2) Closed loop (feedback) control systems
Cont rol l er Act uat or Process
Di sturbance
( Noi se)
I nput r( t )
Ref erence
desi red out put
Out put c( t )
( act ual out put )
Cont rol
si gnal
Actuat i ng
si gnal
uk
uact
Fi g. 1. 11
measur ementFeedback si gnal b( t )
+-
( +)
e( t ) =
r ( t ) - b( t )
Features:
1) measuring the output (controlled variable) . 2) Feedback.
not only there is a forward action , also a backward actionbetween the output and the input (measuring the output and
comparing it with the input).
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Chapter 1 I ntroduction
Notes: 1) Positive feedback; 2) Negative feedbackFeedback.
1.3 types of control systems
1) linear systems versus Nonlinear systems.
2) Time-invariant systems vs. Time-varying systems.
3) Continuous systems vs. Discrete (data) systems.
4) Constant input modulation vs. Servo control systems.
1.4 Basic performance requirements of control systems
1) Stability.
2) Accuracy (steady state performance).
3) Rapidness (instantaneous characteristic).
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Chapter 1 I ntroduction
1.5 An outline of this text
1) Three parts:mathematical modeling; performance analysis;
compensation (design).
2) Three types of systems:l inear continuous; nonlinear continuous; l inear discrete.
3) three performances:stability, accuracy, rapidness.
in all: to discuss the theoretical approaches of the control
system analysis and design.
1.6 Control system design process
shown in F ig.1.12
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Chapter 1 I ntroduction
1. Establish control goals
2. Identify the variables to control
3. Write the specifications
for the variables
4. Establish the system configuration
Identify the actuator
5.Obtain a model of the process,
the actuator and the sensor
6.Describe a controller and select
key parameters to be adjusted
7. Optimize the parameters and
analyze the performance
Performance does not
Meet the specifications
Finalize the design
Performance
meet the
specifications
Fig.1.12
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Chapter 1 I ntroduction
1.7 Sequential design example: disk drive read system
Actuator
motor
Arm
SpindleTrack a
Track b
Head slider
Rotation
of arm Disk
Fig.1.13 A disk drive read system
A disk drive read system Shown in F ig.1.13
Configuration Principle
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Chapter 1 I ntroduction
Sequential design:
here we are concerned with the design steps 1,2,3, and 4 of F ig.1.12.
(1) Identify the control goal:
(2) Identify the variables to control:
Position the reader head to read the date stored on a track on the disk.
the position of the read head.
(3) Write the initial specification for the variables:
The disk rotates at a speed of between 1800 and 7200 rpm and the read head
flies above the disk at a distance of less than 100 nm.
The initial specification for the position accuracy to be controlled:
1 m (leas than 1 m ) and to be able to move the head from track a to track b
within 50 ms, if possible.
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Chapter 1 I ntroduction
(4) Establish an initial system configuration:It is obvious : we should propose a closed loop system, not
a open loop system.
An initial system configuration can be shown as in Fig.1.13.
Control
device
Actuator
motor
Read
arm
sensor
Desiredhead
position
error Actualhead
position
Fig.1.13 system configuration for disk drive
We will consider the design of the disk dr ive fur ther in the after -
mentioned chapters.
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Chapter 1 I ntroduction
Exercise: E1.6, P1.3, P1.13
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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
uk uac
Fig. 2.1
temperature
measurement
Feedback signalb(t)
+
-()
e(t)=
r(t)-b(t)
1) Easy to discuss the full possible types of the control systemsin terms of thesystems mathematical characteristics.
2) The basisanalyzing or designing the control systems.
For example, we design a temperature Control system :
The keydesigning the controller how produce uk.
2.1.1 Why
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Chapter 2 mathematical models of systems
2.1.3 How get1) theoretical approaches 2) experimental approaches
3) discrimination learning
2.1.2 What is Mathematical models of the control systemsthe mathematical
relationships between the systems variables.
Different characteristic of the processdifferentuk:
T(t)
uk
T1
T2
uk12uk11
uk21
For T1
12
11
k
k
u
u
For T1
22
21
k
k
u
u
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Chapter 2 mathematical models of systems
2.2.1 Examples
2.2 Input-output description of the physical systemsdifferential
equations
2.1.4 types
1) Differential equations2) 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.
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Chapter 2 mathematical models of systems
ur
uc
R L
C
i
define: input ur output ucwe have
rccc
crc
uu
dt
duRC
dt
udLC
dt
duCiuudt
diLRi
2
2
rccc uu
dt
duT
dt
udTTT
R
LTRCmake 12
2
2121:
Example 2.1 : A passive circuit
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Chapter 2 mathematical models of systems
Example 2.2 : A mechanism
y
k
f
F
m
Define: input Foutput y. We have:
Fkydt
dyf
dt
ydm
td
ydm
dt
dyfkyF
2
2
2
2
Fk
ydt
dyT
dt
ydTThavewe
Tf
m
Tk
f
makeweIf
1:
:
12
2
21
2,1
Compare with example 2.1: ucy; urF analogous systems
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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
r
CRRR
RR
R
RR
c
c
CR udt
du
udt
du
)( 432
32
4 1
32
)(:
)(;;: 432
32
1
324
r
r
c
c
udt
du
kudt
du
Thavewe
CRRR
RRk
R
RRTCRmake
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Chapter 2 mathematical models of systemsExample 2.4 : A DC motor
ua
w1
Ra La
ia
M
w3
w2 ( J
3, f
3)
( J1
, f1
)
( J2
, f2
)
Mf
i 1i
2Input ua output 1
)4.....(
)3.....(....................)2.....(....................
)1....(
11
1
fdt
dJMM
CEiCM
uEiRdt
diL
ea
am
aaaaa
a
(4)(2)(1) and (3)(1):
MCC
RM
CC
Lu
C
CCfR
CCJR
CCfL
CCJL
me
a
me
aa
e
mea
mea
mea
mea
1
)1()( 111
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Chapter 2 mathematical models of systems
):(
..........................
......
......
:
321211
21
22
21
321
21
2221
3
21
21
iiifromderivedbecan
torqueequivalent
ii
MM
ntcoefficiefrictionequivalentii
f
i
fff
inertiaofmomentequivalent
ii
J
i
JJJ
here
f
Make:
constant-timeelectricfriction
CC
fRT
constant-timeelectric-mechanicalCC
JRT
constant-timemagnetic-electricR
LT
me
af
me
am
a
ae
-.......
.......
............
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Chapter 2 mathematical models of systems
a
e
mme uCdt
dT
dt
dTT 12
2
Assume the motor idle: Mf= 0, and neglect the friction: f= 0,
we have:
)(11
)1()( 111
MTMTT
J
u
C
TTTTTT
mmeae
fmfeme
The differential equation description of the DC motor is:
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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 R3
w
Input urOutput ; neglect the friction:
(4)MTMTT
J
u
Cdt
dT
dt
dTT
(3)uku(2)u
(1)uukuuR
Ru
mmeae
mme
kaf
frfrk
)......(11
...........................................
..................................)......()(
2
2
2
112
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Chapter 2 mathematical models of systems
2134we have
)(1
)1( 211
212
2
MMTJ
Tu
Ckkkk
dt
dT
dt
dTT e
mr
eCmme e
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.
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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 variableon the right of the input-output equation .
Output variableon the left of the input-output equation.Writing polynomialaccording to the falling-power order.
Suppose: input routput y
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Chapter 2 mathematical models of systems
2.3 Linearization of the nonlinear components2.3.1 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 linearizationSuppose: 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
rrrfrrrfrrrfrfrf
00 :)()(: rrrandrfrfymake
equationionlinearizatrrfywehave ............)(: 0'
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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:isthat
)(11.12)()(:havewe 00
Example 2.7 : Fluxograph equation
pkpQ )(
QFlux; ppressure difference
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Chapter 2 mathematical models of systems
equationionlinearizatpp
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 definitionTransfer 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:
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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 thesystems, independent on the input and output variables.
2.4.2 How to obtain the transfer function of a system
1) If the impulse responseg(t) is known
Notes:
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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
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Chapter 2 mathematical models of systems
Example 2.9:
responseUnit step
sssssCetc
functionUnit stepsttr
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 knownAssume: zero initial conditions;
Make: Laplace transform of the differential equation;
Deduce: G(s)=C(s)/R(s).
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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 thecircuits theory.
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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
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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 tCR
R
Rk
ller.PI-Contros
k
CsR
CsR
R
sCR
sU
sUsG
r
c
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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 gger
Uf
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
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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
Make Laplace transformation, we have:
(4)sMJ
TsTTsU
CessTsTT
(3)sUksU(2)ssU
(1)sUsUksU
mmeamme
kaf
frk
)......()(1
)()1(
.....).........()(......).........()(
...........................................)]........()([)(
2
2
1
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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
constanttimemagneticelectricR
L
There
me
am
a
ae
)1
1(
1
)(
)()(
212
21
emme
e
r
CkksTsTT
Ckk
sU
ssG
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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
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Chapter 2 mathematical models of systems
2.5.2 Integrating element
Relationship between the input and output variables:
constanttimeintegralTdttrT
tc I
t
I
:..........)(1
)(
0
Transfer function:sTsR
sCsGI1
)()()(
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
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Chapter 2 mathematical models of systems
2.5.3 Differentiating element
Relationship between the input and output variables:
dt
tdrTtc D
)()(
Transfer function: sTsRsCsG D )()()(
Block diagram representation and unit step response:
Examples:differentiating amplifier, differential
valve, differential condenser
R( s) C( s)T
Ds
1 TD
t
r ( t ) C( t )
t
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2.5.4 Inertial element
Chapter 2 mathematical models of systems
Relationship between the input and output variables:
)()()(
tkrtcdt
tdcT
Transfer function:1)(
)()( Tsk
sRsCsG
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 )
t
T
1Tsk
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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: 1012)(
)(
)( 22
TssT
k
sR
sC
sG
Block diagram representation and unit step response:
Examples:
oscillator, oscillating table,
oscillating circuit
R( s) C( s)
12
1
22 TssT C( t )
k
t
1
t
r ( t )
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2.5.6 Delay element
Chapter 2 mathematical models of systems
Relationship between the input and output variables:
)()( tkrtc
Transfer function: skesRsCsG )()()(
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
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2.6 block diagram models (dynamic)
Portray the control systems by the block diagram models moreintuitively 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 ( compari son)E( s) =x
1( s)+x
3( s)- x
2( s)
X( s)
X3( s)
X2( s)
+
-
+X1( s) E( s)
-
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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....(
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
)()()()()(
..............................................................................).........()(.............................................................................).........()(
.............)()(
)()()()()(
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Chapter 2 mathematical models of systems
Draw block diagram in terms of the equations (5)(8):
Ua
( s )
aa RsL
1C
m
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
-
-
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Chapter 2 mathematical models of systemsExample 2.15 The water level control system in Fig 1.8:
Desi r ed
wat er l evel
ampl i f i er Motor 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 Q
1k 1
1
2 sTsTT
C
mme
e
s
ek s211
3
sTk
12
4
sT
k
)(
1
)1(
2sM
sTsTT
sTJ
T
mme
em
-
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Chapter 2 mathematical models of systems
The block diagram model is:
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