Module 2
Transfer Function and
Block Diagram Algebra
(4 hours)
2.1 Modeling of physical
systems
Model of system – the relationship between
variables in system
Differential Equation (mathematic model)
(parameter model)
Block Diagram and Transfer Function (parameter
model)
Response Curve (non parameter model)
2.1.1 Why do we must study the
model of control system?
________
An accurate mathematic model that
describes a system completely must
be determined in order to analyze
and control a dynamic system.
2.1.2 The Steps of Analyzing and
Studying a Dynamic System
Define the system and its components.
Formulate the mathematic model and list the
necessary assumptions.
Write the differential equations describing the
model.
Solve the equations for the desired output
variables.
Examine the solutions and the assumptions.
If necessary,reanalyze or redesign the system.
2.1.3 The Block Diagram Model
– which consists of block,arrow,
Differencing junction and pickoff point.
Pickoff point
A block diagram represents the flow of
information and the function performed by
each component in the system.
Arrows are used to show the direction of
the flow of information.
The block represents the the function or
dynamic characteristics of the component
and is represented by a transfer function.
The complete block diagram shows how the
functional components are connected and
the mathematic equations that determine the
response of each component,
2.2 Laplace Transform (Review)
asaFatfLG
.
G,The relationship between time
and frequency
2.3 System Model and Transfer Functions
(Transfer function,Laplace transform of
the input-output relation of a system)
SP1,Electric Circuit
System
R L
CUi Uoi iooo uu
dt
duRC
dt
udLC
2
2
1
1
2 R CsL Cs
)(sUi )(sUO
dti
C
u
u
dt
di
LiRu
o
oi
1
SP2,Mechanic Movement (Translation)
System with Spring – Mass – Damper
Displacement
Mass
ky
dt
dyb
Mechanic Movement System with Spring –
Mass – Damper
2
2
dt
yd
m
dt
dy
bkyr
rky
dt
dy
b
dt
yd
m2
2
kbsms2
1
)(sR )(sY
SP3,Mechanic Rotational System with Damper
c
Damper
dt
d
dt
dJcT,
TdtdcdtdJ2
2
csJs?2
1
)(sT )(s?
)( to r q u ein p u tT
SP4.
yc?
ky
yc?
ky
yc?
SP5,The armature – controlled DC Motor
)()( sIKsT amm?
sbKE
)()()( sIsLREsV aaaa
)()()()( sJssbsTsT dm
)()( sss
E
)()( sIKsT amm?
sbKE
)()()( sIsLREsV aaaa
)()()()( sJssbsTsT dm
)()( sss
aa sLR?
1
E
)(sIa)(sVa
)(sIa
mK )(sTm
bJs?
1
)(sTm
)(sTd
)(s?
bK
)(s? )(sE
s
1)(s? )(s?
)()( sIKsT amm?
sbKE
)()()( sIsLREsV aaaa
)()()()( sJssbsTsT dm
)()( sss
SP6,Complex Control System –
Position Tracking System
DC Motor
dT
i? o?
pK
pK
3? 2?
3K?
aV
a
mRK bJs?1
s
1
bK
TK
2.4,Block Diagram Algebra
SP7,Two Step R – C Filter Circuit
R1
C2
Ui(s) Uo(s)
C1
R2
)(1 sI )(2 sI
)(1 sU
sC2
1
)(sUo
sC1
1
)(1 sU
)(2 sI
1
1
R
)(sUi
)(1 sI
)(1 sU
2
1
R
)(2 sI
)(sUo
sC2
1
)(sUosC1
1
)(1 sU
)(2 sI
1
1
R
)(sUi )(1 sI
2
1
R
)(2 sI
1R sC2
1
1
11?CsR 1
1
22?CsR
21CsR
)(sUi )(sUo
)(sUi )(sUo
212211 )1)(1(
1
CsRCsRCsR
SP2.3.1
2.5 Signal Flow Graphs (SFG)
From SP2.3,we
know that the
transfer function
of system is as
follows:
Having the block diagram simplifies the analysis of
a complex system,Such an analysis can be even
further simplified by using a signal flow graph
(SFG),which looks like a simplified block diagram.
The composing of SFG:
Node (summing junction and pickoff point):
a,Addition of the signals on all incoming branches;
b,Transmission of the total node signal (the sum of
all incoming signals) to all outgoing branches.
Arrow (block and arrow which expresses the
direction of signal flowing)
2.5.1 SFG Algebra
1x 2x
3x
a b
1x 2x
a
b
1x
13 a b xx?
ab
1x 12 )( xbax
ba?
1x
2x
3x
4x
a
b
c
1x
2x
3x
a
b
c
1x
2x
ac
bc
214 b cxa cxx
1x
13 1 xbc
abx
2.5.2 The Mason Gain Rule
The overall transmittance (gain) can be
obtained from the Mason gain formula:;
nn
P
G
fedcba LLLLLL1
aL
-- the gain of each closed path;
cbLL -- the product of the gains of two non-
touching loops;
fed LLL
-- the product of the gains of three non-
touching loops.
nP
-- the gain of each forward path ;
-- the cofactor of,it is the determinant of
the remaining subgraph when the forward path that
produces is removed,Thus,does not
include any loops that touch the forward path in
question,is equal to unity when the forward path
touches all the loops in the graph or when the graph
contains no loops.
n?
n? n
P
n?
nP
SP8.
2512532135322512
5321
1 HGHGGGGGHGGGHGHG
GGGG
SP9,
sC2
1
)(sUo
sC1
1
)(1 sU
)(2 sI
1
1
R
)(sUi
)(1 sI
)(1 sU
2
1
R
)(2 sI
)(sUo
sCRsCRRsCsCRsCR
sCRsCR
sU
sU
i
o
2211212211
2211
1111111111
1
1111
)(
)(
2
22112122111
1
sCRCRsCRsCRsCR
dT
i? o?
pK
pK
3? 2?
3K?
aV
a
mRK bJs?1
s
1
bK
TK
SP10,
SP11,
Instructional objectives:
At the end of this lecture students
should be able to
develop dynamic models of physical
components
derive transfer function for them.
Transfer Function and
Block Diagram Algebra
(4 hours)
2.1 Modeling of physical
systems
Model of system – the relationship between
variables in system
Differential Equation (mathematic model)
(parameter model)
Block Diagram and Transfer Function (parameter
model)
Response Curve (non parameter model)
2.1.1 Why do we must study the
model of control system?
________
An accurate mathematic model that
describes a system completely must
be determined in order to analyze
and control a dynamic system.
2.1.2 The Steps of Analyzing and
Studying a Dynamic System
Define the system and its components.
Formulate the mathematic model and list the
necessary assumptions.
Write the differential equations describing the
model.
Solve the equations for the desired output
variables.
Examine the solutions and the assumptions.
If necessary,reanalyze or redesign the system.
2.1.3 The Block Diagram Model
– which consists of block,arrow,
Differencing junction and pickoff point.
Pickoff point
A block diagram represents the flow of
information and the function performed by
each component in the system.
Arrows are used to show the direction of
the flow of information.
The block represents the the function or
dynamic characteristics of the component
and is represented by a transfer function.
The complete block diagram shows how the
functional components are connected and
the mathematic equations that determine the
response of each component,
2.2 Laplace Transform (Review)
asaFatfLG
.
G,The relationship between time
and frequency
2.3 System Model and Transfer Functions
(Transfer function,Laplace transform of
the input-output relation of a system)
SP1,Electric Circuit
System
R L
CUi Uoi iooo uu
dt
duRC
dt
udLC
2
2
1
1
2 R CsL Cs
)(sUi )(sUO
dti
C
u
u
dt
di
LiRu
o
oi
1
SP2,Mechanic Movement (Translation)
System with Spring – Mass – Damper
Displacement
Mass
ky
dt
dyb
Mechanic Movement System with Spring –
Mass – Damper
2
2
dt
yd
m
dt
dy
bkyr
rky
dt
dy
b
dt
yd
m2
2
kbsms2
1
)(sR )(sY
SP3,Mechanic Rotational System with Damper
c
Damper
dt
d
dt
dJcT,
TdtdcdtdJ2
2
csJs?2
1
)(sT )(s?
)( to r q u ein p u tT
SP4.
yc?
ky
yc?
ky
yc?
SP5,The armature – controlled DC Motor
)()( sIKsT amm?
sbKE
)()()( sIsLREsV aaaa
)()()()( sJssbsTsT dm
)()( sss
E
)()( sIKsT amm?
sbKE
)()()( sIsLREsV aaaa
)()()()( sJssbsTsT dm
)()( sss
aa sLR?
1
E
)(sIa)(sVa
)(sIa
mK )(sTm
bJs?
1
)(sTm
)(sTd
)(s?
bK
)(s? )(sE
s
1)(s? )(s?
)()( sIKsT amm?
sbKE
)()()( sIsLREsV aaaa
)()()()( sJssbsTsT dm
)()( sss
SP6,Complex Control System –
Position Tracking System
DC Motor
dT
i? o?
pK
pK
3? 2?
3K?
aV
a
mRK bJs?1
s
1
bK
TK
2.4,Block Diagram Algebra
SP7,Two Step R – C Filter Circuit
R1
C2
Ui(s) Uo(s)
C1
R2
)(1 sI )(2 sI
)(1 sU
sC2
1
)(sUo
sC1
1
)(1 sU
)(2 sI
1
1
R
)(sUi
)(1 sI
)(1 sU
2
1
R
)(2 sI
)(sUo
sC2
1
)(sUosC1
1
)(1 sU
)(2 sI
1
1
R
)(sUi )(1 sI
2
1
R
)(2 sI
1R sC2
1
1
11?CsR 1
1
22?CsR
21CsR
)(sUi )(sUo
)(sUi )(sUo
212211 )1)(1(
1
CsRCsRCsR
SP2.3.1
2.5 Signal Flow Graphs (SFG)
From SP2.3,we
know that the
transfer function
of system is as
follows:
Having the block diagram simplifies the analysis of
a complex system,Such an analysis can be even
further simplified by using a signal flow graph
(SFG),which looks like a simplified block diagram.
The composing of SFG:
Node (summing junction and pickoff point):
a,Addition of the signals on all incoming branches;
b,Transmission of the total node signal (the sum of
all incoming signals) to all outgoing branches.
Arrow (block and arrow which expresses the
direction of signal flowing)
2.5.1 SFG Algebra
1x 2x
3x
a b
1x 2x
a
b
1x
13 a b xx?
ab
1x 12 )( xbax
ba?
1x
2x
3x
4x
a
b
c
1x
2x
3x
a
b
c
1x
2x
ac
bc
214 b cxa cxx
1x
13 1 xbc
abx
2.5.2 The Mason Gain Rule
The overall transmittance (gain) can be
obtained from the Mason gain formula:;
nn
P
G
fedcba LLLLLL1
aL
-- the gain of each closed path;
cbLL -- the product of the gains of two non-
touching loops;
fed LLL
-- the product of the gains of three non-
touching loops.
nP
-- the gain of each forward path ;
-- the cofactor of,it is the determinant of
the remaining subgraph when the forward path that
produces is removed,Thus,does not
include any loops that touch the forward path in
question,is equal to unity when the forward path
touches all the loops in the graph or when the graph
contains no loops.
n?
n? n
P
n?
nP
SP8.
2512532135322512
5321
1 HGHGGGGGHGGGHGHG
GGGG
SP9,
sC2
1
)(sUo
sC1
1
)(1 sU
)(2 sI
1
1
R
)(sUi
)(1 sI
)(1 sU
2
1
R
)(2 sI
)(sUo
sCRsCRRsCsCRsCR
sCRsCR
sU
sU
i
o
2211212211
2211
1111111111
1
1111
)(
)(
2
22112122111
1
sCRCRsCRsCRsCR
dT
i? o?
pK
pK
3? 2?
3K?
aV
a
mRK bJs?1
s
1
bK
TK
SP10,
SP11,
Instructional objectives:
At the end of this lecture students
should be able to
develop dynamic models of physical
components
derive transfer function for them.