lecture-slides chapter 02
DESCRIPTION
Modern Control SystemsTRANSCRIPT
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Mathematical Modelsof
Systems
Chapter 2
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Introduction
Differential Equations of Physical Systems
Linear Approximation of Physical Systems
The Laplace Transform
The Transfer Function of Linear Systems
Block Diagram Models
Signal-Flow Graphs Models
Outline
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
To understand and control complex physical systems, we need their mathematical models.
To obtain mathematical models, we need the relationship between the system variables.
As the systems under consideration are dynamic in nature, then this relationship is in the form of
differential equations.
Introduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
In general, the linear differential equation of an nth-order system is written:
1st order linear ordinary differential equation:
2nd order linear ordinary differential equations:
In this course we treat only LINEAR ORDINARY DIFFERNTIAL EQUATIONS
Differential Equations
)()()()()(
11
1
1tftya
dt
tdya
dt
tyda
dt
tydon
n
nn
n
)()()(
tftyadt
tdyo
)()()()(
12
2
tftyadt
tdya
dt
tydo
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Methods of Modeling Linear System:
Transfer Function Method (only linear systems)
State-Variable Method (both linear and nonlinear systems)
Most dynamic systems have nonlinear behavior:
Linearization by proper assumptions and approximations
Modeling of Physical Systems
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The motion of a Mechanical system:
Translation
Rotation
Combination of above
Modeling of Mechanical Systems
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Translational motion:
Newtons 2nd law of motion
Example: A mass M under the action of force f(t).
Modeling of Mechanical Systems
M
)(ty
)(tf
dt
tdvM
dt
tydMtMatf
)()()()(
2
2
MaFext
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Linear Spring:
Hooks law:
Viscous Damper:
Modeling of Mechanical Systems
)()( tKytf
Bvdt
tdyBtf
)()(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Mass-spring-damper system
Modeling of Mechanical Systems
2
2)()(
)()(dt
tydM
dt
tdyBtKytf
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Rotational motion
Eulers 2nd law of motion
Example: A body with inertia J under the action of a torque (t).
Modeling of Mechanical Systems
Jext
dt
tdJ
dt
tdJtJt
)()()()(
2
2
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Torsional Spring:
Viscous damper:
Modeling of Mechanical Systems
)()( tKt
Bdt
tdBt
)()(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
A disk in a viscous medium and supported by a shaft
Modeling of Mechanical Systems
)()()()( tJtttds
2
2)()(
)()(dt
tdJ
dt
tdBtKt
Jext
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Resistor:
Inductor:
Capacitor:
Modeling of Electrical Systems
R
L
C
)(tv
)(tv
)(tv
)(tI
)(tI
)(tI
R
tvtI
)()(
dttvL
tI )(1
)(
dt
tdvCtI
)()(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Kirchhoff s laws:
Current law:
The algebraic sum of all currents entering a node is zero.
Voltage law:
The algebraic sum of all voltage drops around a complete closed loop is zero.
Example of RLC circuit:
Modeling of Electrical Systems
dttvLdt
tdvC
R
tvtr )(
1)()()(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Spring-Mass-Damper system:
RLC circuit:
Rotational motion:
This is called as velocity voltage analogy (force-current analogy)
Analogy
dt
tdJtBdttKt
)()()()(
dt
tdvMtBvdttvKtf
)()()()(
dt
tdvC
R
tvdttv
Ltr
)()()(
1)(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Linearization(Linear Approximation)
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
A linear system satisfies the following properties:
Superposition
Homogeneity
Example:
Test whether is linear.
Linearization
Linearsystem
)()(2211tuatua )()(
2211tyatya
5xf(x)
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Examples of physical systems
Linearization
)()()()(
2
2
tutydt
tydLC
dt
tdyRC
)()()(
2
2
tudt
tdyB
dt
tydM
dt
tdyB
)(
M
)())(()()(
2
2
tutyfdt
tdyB
dt
tydM
B
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Examples of physical systems
A nonlinear system can be described by a linear model for a small range of input values around an operating
point.
Linearization
)())(()()(
2
2
tutyfdt
tdyB
dt
tydM
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
To find the linear model of a nonlinear system f(y)
We expand f(y) into a Taylor series around the operating point or equilibrium point (yo, f(yo)):
If the variation around the operating point, is small, then we may neglect the higher-order terms:
This approximation results in a linear (straight line) relationship
Linearization
oo yy
o
yy
o
ody
fdyy
dy
dfyyyfyf
2
22
!2!1)()(
oyyy
ycyf )(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Linearization of differential equations
Example: Pendulum oscillator model
Linearization around the equilibrium point
This approximation is reasonably accurate for
Linearization
sin)( MgLT
)()()(oo
od
dTTT
o
o0
44
MgLT )(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Laplace Transform (LT)
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
LT is a mathematical tool that:
Transforms many time (t) domain functions f(t) into algebraic functions F(s) of a complex domain (s).
Provides an algebraic way to solve linear time invariant differential equations.
Can be used to predict the system performance without actually solving system differential equations.
Laplace Transform (LT)
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Solution of differential equations
Obtain linearized differential equation.
Obtain the Laplace transform of the differential equation.
Solve the algebraic equation by the inverse Laplace transform.
Laplace Transform (LT)
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Laplace Transform of a function f(t):
is the Laplace transform operator
(s) is a complex variable:
f(t) is a function of time (t) with f(t)=0 for t
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Theorem 1: Multiplication by a constant
Theorem 2: Sum and differences
Theorem 3: Differentiation
Theorems of Laplace Transform
)()( skFtkf
)()()()(2121sFsFtftf
)0()()(
fssFdt
tdf
0
2
2
2)(
)0()()(
tdt
tdfsfsFs
dt
tfd
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Step function:
Unit step function:
LTs of Simple Functions
0
00)(
tc
ttf
s
ce
s
cdtcedtetfsF
st
t
st
t
st
000
)(
s
csF )(
ssF
1)(
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Ramp function:
Exponential function:
Sinusoidal function:
LTs of Simple Functions
0
00)(
tct
ttf 2)(
s
csF
0
00)(
te
ttf
at assF
1)(
22)(
s
sF
0sin
00)(
tt
ttf
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Table of LTs
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
First order linear differential equation
Let
Now, as:
LT of equation (1) is:
LTs of Differential Equations
oo
ayytf ,)0(,0)(
)1()()()(
tftyadt
tdyo
)0()()(
)()(
yssYdt
tdy
sYty
s
ysY
o
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Second order linear differential equation (Spring-Mass-Damper)
Let
LT of equation (1) is:
LTs of Differential Equations
)1()()()()(
2
2
tftKydt
tdyB
dt
tydM
0)(
,)0(,0)(
0
t
odt
tdyyytf
KBsMs
yBMssY
o
2
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The inverse LT of F(s) is:
This is a complex integral and is rarely used.
For simple functions, we directly refer to the LTs table.
For complex functions, we first perform the partial-fraction expansion on F(s) and then use the LTs table.
Inverse Laplace Transform
j
j
stdsesF
jtfsF
2
11
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Consider the following Laplace Transform function:
N(s) and D(s) are the polynomials of (s).
Characteristic equation:
Roots (s1, s2, sn) of this characteristic equation are called the poles of the system.
Distinct poles
Repeated poles
Partial-Fraction Expansion
sD
sNsG
o
n
n
n
n
nasasasassD
1
1
2
2
1
1
KBsMs
yBMssY
o
2
0sD
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Case 1: Distinct poles
Consider the function
Write G(s) in terms of partial-fraction expansion:
Determine the coefficient k1 and k2
Partial-Fraction Expansion
31
2
ss
s
sD
sNsG
2
11
1
1
ssD
sNsk
31
21
s
k
s
ksG
2
13
3
2
ssD
sNsk
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The simplified function is:
Now taking the inverse LT:
Partial-Fraction Expansion
32
1
12
1
sssG
tt eetg 35.05.0
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Case 2: Repeated poles
Consider the function
Write G(s) in terms of partial-fraction expansion:
Determine the coefficient k1 and k2
Partial-Fraction Expansion
21
2
s
s
sD
sNsG
11
1
2
2
ssD
sNsk
2
21
11
s
k
s
ksG
11
1
2
1
ssD
sNs
ds
dk
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The simplified function is:
Now taking the inverse LT:
Partial-Fraction Expansion
21
1
1
1
sssG
tt teetg
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Solve the following 2nd order linear ODE
With us(t) as unit step function and following initial conditions:
Example
tuty
dt
tdy
dt
tyds
5232
2
2,10
0
tdt
tdyy
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Transfer Function
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The ratio of the Laplace Transform of the output variable to the Laplace Transform of the input variable, with all initial
conditions to be zero.
Consider the spring mass damper system: input is r(t), output is y(t).
Transfer Function
sRsKYsBsYsYMs 2
trtKy
dt
tdyB
dt
tydM
2
2
KBsMssR
sYsG
2
1
sInput
sOutputsG
tr
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Write the transfer function of the following circuit, where:
Input: source voltage v1 Output: voltage drop across capacitor v2
Transfer Function
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
DC Motor
Converts DC electrical energy into rotational mechanical energy
Transfer Function DC Motor
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
DC Motor
Input: voltage (field, armature)
Output: speed of shaft, position of the shaft
Transfer Function DC Motor
current armature
current field
ntdisplacemerotor
voltagefield
voltagearmature
ti
ti
t
tv
tv
a
f
f
a
sV
ssG
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Two types of control of dc motor
Field control
(variable field voltage and fixed armature voltage)
Armature control
(variable armature voltage and fixed field voltage)
Transfer Function DC Motor
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Field control of dc motor
The angular displacement is proportional to the field voltage
Transfer Function DC Motor
voltagefield
ntdisplacemeangular
sV
ssG
f
sVsGsf
sVRsLBJss
Ks
f
ff
m
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The air gap-flux is proportional to the field current
The motor torque Tm is assumed to be related linearly to and the armature current:
In case of field control, armature current is kept constant:
Km is the motor constant
Transfer Function DC Motor
tiKtff
titKtTam
1
titiKKtTaffm 1
tiKtTfmm
sIKsTfmm
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Relating armature voltage to armature current:
In s-domain
Thus the motor torque is:
Transfer Function DC Motor
dt
tdiLtiRtv
f
ffff
sVsLR
sIf
ff
f
1
sVsLR
KsT
f
ff
m
m
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The load torque in time domain:
In s-domain:
Now:
Transfer Function DC Motor
tBdt
tdJtT
L
sBJsssTL
ff
m
fRsLBJss
K
sV
ssG
sTsTsTdLm
dt
tdt
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Transfer function
Block diagram
Transfer Function DC Motor
sLRsVsI
fff
f
1
m
f
mK
sI
sT
BJs
s
sTL
ss
s 1
sTsTsTdmL
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Armature control of dc motor
The angular displacement is proportional to the armature voltage
Transfer Function DC Motor
voltageArmature
ntdisplacemeangular
sV
ssG
a
sVsGsa
sVKKBJssLRs
Ks
a
mbaa
m
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
The air gap-flux is proportional to the field current
The motor torque Tm is assumed to be related linearly to and the armature current:
In case of armature control, field current is kept constant:
Km is the motor constant
Transfer Function DC Motor
tiKtff
titKtTam
1
titiKKtTaffm 1
tiKtTamm
sIKsTamm
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Relating armature voltage to armature current:
Where vb is the back emf voltage:
In s-domain
Thus the motor torque is:
Transfer Function DC Motor
tvdt
tdiLtiRtv
b
a
aaaa
sLR
sKsVsI
aa
ba
a
sKsVsLR
KsT
ba
aa
m
m
tKtvbb
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Transfer function of armature controlled dc motor
Transfer Function DC Motor
mbaa
m
aKKBJssLRs
K
sV
ssG
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Transfer Function DC Motor
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Gear ratio:
Relate shaft torques:
Transfer Function Gear Trains
2
1
N
Nn
2
1
N
N
T
T
L
m
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Tachometer
An electromechanical device that converts mechanical energy into electrical energy.
Input: shaft angular velocity
Output: voltage
Transfer Function Tachometer
t
s
sVsG K
2
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Consider an incompressible fluid in a tank:
Determine the transfer function which relates head to inflow
Mass balance:
mass flow in mass flow out = accumulation rate of mass in tank
Transfer Function Fluid System
inletat rate flow volumetric:
in tank fluid of head:)(
outletat rate flow volumetric:)(q
fluid ofdensity :
area sectional-cross uniform:
o
iq
th
t
A
inflow
head
sQ
sHsG
i
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Consider an incompressible fluid in a tank:
Determine the transfer function which relates head to inflow
Energy balance:energy in energy out = accumulation of energy in tank
Transfer Function Thermal System
peratueoutlet tem:
eratueinlet temp:
sourceheat fromheat :
inlet andoutlet at rate flow volumetric:
heat specific:
fluid ofdensity :
area sectional-cross uniform:
o
i
T
T
q
C
A
eTemperaturInlet
eTemperaturOutlet
sT
sTsG
i
o
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Block Diagram Models (BDM)
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
So far:
Dynamic systems are represented by mathematical models:
Set of simultaneous differential equations in time domain.
Set of linear algebraic equations in the s-domain.
Transfer function:
Mathematically relating the output variable to the input variable in the s-domain.
Block Diagram Model (BDM)
Graphical technique for modeling control systems.
Graphical relationship between the variables of interest.
Introduction
s
ssG
Input
Output sG sInput sOutput
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Block Diagram Model Usage
BDM provides a better understanding of the composition and interconnection of the components of a system.
BDM describes the input-output relationship throughout the system with the help of transfer functions.
Introduction
ControllerProcess
orPlant
Feedback
ActuatorRef.Input
ActualOutput
Measured output
Error
_
+ Actuatingsignal
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Linear spring
Transfer function:
Block diagram model:
Introduction
K
sF
sXsG
K sF sX
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Field control DC motor
Transfer function:
Block diagram model:
Introduction
ff
m
fRsLBJss
K
sV
ssG
sVf
s
ff
m
RsLBJss
K
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Elements of BDM
Blocks
The rectangular box that contains a component of a system.
Signals
Arrowed lines from one block to another representing input/output variables.
Comparators (summing point)
Junction point for signals comparison.
Block Diagram Model
ControllerProcess
orPlant
Feedback
ActuatorRef.Input
ActualOutput
Measured output
Error
_
+ Actuatingsignal
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Comparator (summing point)
To perform simple mathematical operations (addition or subtraction)
Block Diagram Algebra
+
+ sR
sY
sYsRsE
_
+ sR
sY
sYsRsE
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Block
To represent the transfer function of a component of a system or the system as a whole.
Transfer function
Block Diagram Algebra
sG sU sY
sU
sYsG
sUsGsY
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Combining blocks in cascade
Block Diagram Algebra
sXsGsGsX1213
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Combining blocks in Parallel
Block Diagram Algebra
sXsGsGsX1212
sX1
sX2
sGsG21
sX1
sX2
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Feedback control system
Transfer function:
Forward-path TF:
Loop TF:
Block Diagram Algebra
sHsG
sG
sR
sY
1
sHsGsL
sGsGsGsGpac
sGa
sGp sG c
sH
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Eliminating a feedback loop
Unity feedback loop
Block Diagram Algebra
G1
G
1X
2X
1X
2XG
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Moving a summing point to the right of a block
Block Diagram Algebra
213GXGXX 213 XXGX
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Moving a summing point to the left of a block
Block Diagram Algebra
213XGXX
G
XX
G
X2
1
3
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Moving a takeoff point (pickoff point)to the left of a block
Block Diagram Algebra
12GXX
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Moving a takeoff point (pickoff point)to the right of a block
Block Diagram Algebra
G
XX
2
1
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Example: reduce the following block diagram and determine the transfer function
Block Diagram Reduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Example: reduce the following block diagram and determine the transfer function
Block Diagram Reduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Block Diagram Reduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Block Diagram Reduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Block Diagram Reduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Block Diagram Reduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Example: reduce the following block diagram and determine the transfer function
Block Diagram Reduction
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Multiple Inputs
1. Set all inputs except one equal to zero
2. Determine the output signal due to this one non-zero input
3. Repeat the above steps for each of the remaining inputs in turn
4. The total output of the system is the algebraic sum (superposition) of the outputs due to each of the inputs.
Block Diagram Reduction
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Signal-Flow Graphs (SFG)
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Signal-flow graphs (SFG)
A graphical representation of control systems (a simplified version of Block diagram model)
The cause-and-effect relationship among the variables of a set of linear algebraic equations (like we have in case of linear
control systems)
A diagram consisting of nodes that are connected by several directed branches.
Introduction
sVsGsf
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Node (junction point):
To represent the variables of the system
Branch (line segment):
To connect the nodes according to the cause-and-effect equations
Branch is a unidirectional line segment (from input toward the output)
Signal-Flow Graph Basic Elements
sVf
s sG
sVsGsf
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Basic properties
SFG applies only to linear systems
The equations must be in algebraic form (in s-domain) in the form of cause-and-effect relationship.
Example:
For N equations;
Signal-Flow Graph Basic Properties
sYsGsYsGsY
sYsGsYsGsY
sYsGsYsGsY
3342244
4432233
3321122
NjsYsGsYkkj
N
k
j1,
1
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Input node (Source)
A node that has only outgoing branches (example: node Y1)
Output node (Sink)
A node that has only incoming branches (example: Y4)
Path
A branch or a continuous sequence of branches that can be traversed from one node to another node
Forward path
A path that starts at an input node and ends at an output node with no node traversed more than once (example: Y1 to Y2 to Y3 )
Loop
A path that originates and terminates on the same node with no other node traversed more than once. (four loops in example)
Signal-Flow Graph Terms
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Path gain The product of the branch gains encountered in traversing a
path.
For example, the path gain for the path Y1-Y2-Y3-Y4 is G12G23G34
Forward-path gain The path gain of a forward path
Loop gain The path gain of a loop
Non-Touching loops Two loops are non-touching if they do not have a common
node
Signal-Flow Graph Terms
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
2 Forward paths:
4 Loops:
Non-touching loops:
L1 and L3, L1 and L4, L2 and L3, L2 and L4
Signal-Flow Graph Terms
43211GGGGP
87652GGGGP
332HGL
221HGL
774HGL
663HGL
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Series connection of branches
Parallel branches
Feedback control system
Signal Flow Graphs Algebra
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Gain Formula
The linear dependence between input variable and output variable
Pk = Gain of the kth path from input variable to output variable
= Determinant of the SFG
k = Cofactor of the path Pk
N = the total number of forward paths between input and
output variable
Signal Flow Graphs Gain Formula
NN
N
k
kkPPP
P
T
22111
input
output
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
= 1 (sum of all loop gains)
+ (sum of the gain products of all combination of two non-touching loops)
(sum of the gain products of all combination of three non-touching loops)
+
k = with all the loops touching the kth forward path put to zero
Signal Flow Graphs Gain Formula
nontoching
pmn
pmn
nontoching
mn
mn
n
nLLLLLL
,,,
1
-
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Find the system transfer function by using the gain formula of SFG
Signal Flow Graphs Example
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Field-controlled motor
Draw the signal-flow graph for the above block diagram
Signal Flow Graph DC motor control
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Armature-controlled motor
Signal Flow Graph DC motor control
sD
sD
-
ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Determine the transfer function of the system from the following signal flow graph, by using Masons Gain
formula.
Also draw the equivalent block diagram
Signal Flow Graphs Exercises (E2.22)
NN
N
k
kkPPP
P
sR
sYsT
22111
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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali
Determine the following transfer function:
Determine a relationship for the system that will make Y2(s)independent of R1(s)
Draw the equivalent block diagram
Signal Flow Graphs Problems (P2.33)
sR
sYsT
1
2