Week 8 - Block Diagram Reduction
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1 TRANSFER FUNCTION & CHAPTER 2 TRANSFER FUNCTION & BLOCK DIAGRAM 1 THE TRANSFER FUNCTION OF LINEAR SYSTEM Ratio of the LT of the output variable to the LT of the input variable, with all initial conditions assumed to be zero. Represents the relationship describing the dynamics of Represents the relationship describing the dynamics of the system under consideration. The transfer function, G(s) for a system representation in figure below is ) ( ) ( ) ( s R s C s G System Input Output r(t) c(t) 2
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Transcript of Week 8 - Block Diagram Reduction
1
TRANSFER FUNCTION &CHAPTER 2
TRANSFER FUNCTION & BLOCK DIAGRAM
1
THE TRANSFER FUNCTION OF LINEAR SYSTEM
Ratio of the LT of the output variable to the LT of the inputvariable, with all initial conditions assumed to be zero.
Represents the relationship describing the dynamics of Represents the relationship describing the dynamics of the system under consideration.
The transfer function, G(s) for a system representation in figure below is
)(
)()(
sR
sCsG System
Input Output
r(t) c(t)
2
2
THE TRANSFER FUNCTION OF LINEAR SYSTEM
Example 2-1
Find the transfer function represented by )()(2)(
trtcdt
tdc
Solution
2
1
)(
)()(
)()(2)(
ssR
sCsG
sRsCssCTaking Laplace transform
at both side (assume zero i.c.)
3
THE TRANSFER FUNCTION OF LINEAR SYSTEM
Exercise 2-1
Find the transfer function, G(s) corresponding to differential equation: equation:
Solution
rdt
dr
dt
rdc
dt
dc
dt
cd
dt
cd34573
2
2
2
2
3
3
34)()(
2 sssCG
573
3
)(
)()(
23
sss
ss
sR
sCsG
4
3
THE TRANSFER FUNCTION OF LINEAR SYSTEM
Exercise 2-2
Find the differential equation corresponding to the transfer function function
Solution
rdr
cdccd
2262
2
26
12
)(
)()(
2
ss
s
sR
sCsG
dtc
dtdt6
2
5
THE TRANSFER FUNCTION OF LINEAR SYSTEM
Exercise 2-3
Find the ramp response for a system whose transfer function is:
)(C
Solution)()()( sGsRsC
)8)(4()(
)()(
ss
s
sR
sCsG
)8)(4(
12
ss
s
s
where
)8)(4( sss
)8()4()8)(4(
11
s
C
s
B
s
A
sss
32
1
)8)(4(
1
0
sss
A16
1
)8(
1
4
sss
B32
1
)4(
1
8
sss
C
6
4
THE TRANSFER FUNCTION OF LINEAR SYSTEM
Thus,
tt eetc 84
32
1
16
1
32
1)(
• In general, a physical system that can be represented by a linear, time-invariant differential equation can be modeled linear, time invariant differential equation can be modeled as a transfer function.
7
Transfer function for feedback control system
The general feedback form (simplified model) is shown in figure below:
G(s)
H(s)
+
±
R(s) C(s)E(s)
B(s)
Plant & Controller
Error
Feedback
8
5
Transfer function for feedback control system
Forward path transfer function, G(s)
F db k th t f f ti H( )
)(
)()(
sR
sCsG
)()(
sBH Feedback path transfer function, H(s)
Open-loop transfer function, G(s)H(s)
)(
)()(
sCsH
)(
)(
)(
)(
)(
)()()(
sE
sB
sC
sB
sE
sCsHsG
Closed-loop transfer function, T(s)
)()()( sEsCsE
)(
)()(
sR
sCsT
9
Block diagram algebra
When multiple subsystems are interconnected, a few more schematic elements must be added to the block diagram.
There new elements are summing junctions and pickoff There new elements are summing junctions and pickoff points.
10
6
Block diagram algebra
There are three basic common forms, by which the subsystem are connected together; Cascade form Cascade form
Parallel form
Feedback form
11
Block diagram algebra
Cascade (series) form Transfer functions connected in series are combined by
multiplication. multiplication.
12
7
Block diagram algebra
Parallel Form Transfer functions connected in parallel are combined by
addition. addition.
13
Block diagram algebra
Feedback Form
14
8
Block diagram algebra
Moving blocks to create familiar forms – Block diagram for summing junctions
)()()()()( sGsXsGsRsC
To the left past a summing junction15
Block diagram algebra
Moving blocks to create familiar forms – Block diagram for summing junctions
)()()()( sXsGsRsC
To the right past a summing junction16
9
Block diagram algebra
Moving blocks to create familiar forms – Block diagram for pickoff points
To the left past a pickoff point17
Block diagram algebra
Moving blocks to create familiar forms – Block diagram for pickoff points
To the right past a pickoff point18
10
Block diagram reduction
Procedure for reduction of block diagram: 1. Reduce the cascade blocks
2 Reduce the parallel blocks2. Reduce the parallel blocks
3. Reduce the internal feedback loops
4. It is advisable to shift take-off points toward right and summing points toward left.
5. Repeat step 1 to step 4 until the simple form is obtained.
6 Find the transfer function of the overall system using the 6. Find the transfer function of the overall system using the formula C(s)/R(s).
19
Block diagram reduction
Example 2-2
Reduce the block diagram shown in figure below to a single transfer function transfer function.
20
11
Block diagram reduction
Solution 2.2…
21
Block diagram reduction
Example 2-3
Reduce the block diagram shown in figure below to a single transfer function transfer function.
22
12
Block diagram reduction
Solution 2.3…To the right pass a summing junction
23
Block diagram reduction
Solution 2.3…
Feedback
24
13
Block diagram reduction
Exercise 2-4
Find the equivalent transfer function, T(s) = C(s)/R(s), for the system shown below system shown below.
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