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2.5 The Transfer Function of Linear Systems
Differential equations
Transfer function
Block diagramSignal flow graph
State variables(modern control theory)
Types of mathematical models of systems
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The transfer function of a linear system is defined as the ratio
of the Laplace transform of the output variable to the Laplace
transform of the input variable, with all initial conditions assume
to be zero.
Linear system
input
r(t)
output
y(t)
)(
)()(
sR
sYsG
2.5 The Transfer Function of Linear Systems
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u Ri ur c
According to Kirchhoffs voltage law:
rcc uu
dt
duRC ur c
uinput output
Example: RC network
0
1
t
cu idtC
iCdt
duc 1 idt
duC c
)()()1( sUsURCs rc 1
1
)(
)(
RCssU
sU
r
c
The Transfer Function
-
Note:
Only for the linear time-invariant systems (constant
parameter).
Zero initial conditions.
Dependent on the configuration and the coefficients of the
systems,
Independent on the input and output variables.
The Transfer Function:
-
Steps to obtain the transfer function
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).
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Steps to obtain the 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.
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.
Review:
-
The linear time-invariant system is described by the following
equation (General form)
rbdt
drbr
dt
dbr
dt
db
yadt
dyay
dt
day
dt
d
mmm
m
m
m
nnn
n
n
n
11
1
10
11
1
1
nn
nn
mm
mm
asasas
bsbsbsb
sR
sY
1
1
1
1
1
10
)(
)(
The transfer function is
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2.6 Block Diagram Models
Block diagrams consist of unidirectional, operational blocks
that represent the transfer function of the variables of
interest.
G(s)
R C
)(
)()(
sR
sCsG
The block diagram models are more intuitive than the transfer
function or differential equation models and may
provide control engineers with a better understanding of
the composition and interconnection of the components of
a system.
-
Signal(variable)
G(s)Component(device)
Adder (comparison)E(s)=x1(s)+x3(s)-x2(s)
X(s)
X3(s)
X2(s)
+
-
+X1(s) E(s)
Block diagram representation of the control systems
)(sX
Branch point or Pickoff Point )(sX
Comparator performs addition and subtraction
-
Combining Serial Blocks
G1G2 X Y
G1 G2 X Y U
)()()()()()( 122 sXsGsGsUsGsY
1G
2G
nGX Y
YXnGGG 21
sGsGsX
sYs
n
i
i
1
cascaded blocks
sGsGsGsX
sYs 21
-
Combining Parallel Blocks
G1
G2
X Y
21 GG X Y
)())()(()()()()()( 2121 sXsGsGsXsGsXsGsY
1 2
( ) ( ) ( )Y s
s G s G s G sX s
-
Combining Parallel Blocks
2G
1G
nG
1Y
2Y
nY
YX
YXnGGG 21
n
i
i sGsGsX
sYs
1
-
Closing a feedback loop
GH
G
1X Y
G
H
X Y
E
)]()()()[(
)(*)()(
sYsHsXsG
sEsGsY
)()()]()(1)[( sXsGsHsGsY
)()()(1
)()( sX
sHsG
sGsY
-
Moving a comparator to make the block diagram
reduction process simpler
ahead of a block Past block
Y
G X Z
Z G X
G Y
+
G X Z
Y
G Z
1/G Y
X
Block Diagram Reduction
-
G X Y
Y
G X Y
G Y
ahead of a block past block X
X
G Y
G X Y
1/G X
Block Diagram Reduction
Moving a branch point to make the block diagram
reduction process simpler
-
The neighboring branch points y
x1 x2
y
x1 x2
x1
x3
y
x2
The neighboring camparators x3
x1
x2
y
231321 XXXXXXY
Neighboring camparator and branch point can not be
interchanged!
Interchanging make the block diagram reduction process
simpler
Block Diagram Reduction
-
Block Diagram Reduction
1G 2G 3G 4G
1H
2H
3H
R Y
-
1G 2G 3G 4G
1H
2H
3H
R Y
1G 2G 3G 4G
1H
4
2
G
H
3H
R Y
Block Diagram Reduction
-
1G 2G 3G 4G
1H
4
2
G
H
3H
R Y
1G 2G
4
2
G
H
3H
R Y143
43
1 HGG
GG
Block Diagram Reduction
-
1G 2G
4
2
G
H
3H
R Y143
43
1 HGG
GG
1G
3H
R Y232143
432
1 HGGHGG
GGG
Block Diagram Reduction
-
1G
3H
R Y232143
432
1 HGGHGG
GGG
R Y34321232143
4321
1 HGGGGHGGHGG
GGGG
Block Diagram Reduction
