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Nonlinear Systems and Control
Lecture # 15
Positive Real Transfer Functions
Connection with Lyapunov Stability
– p.1/22
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View Full Document Definition:
A
p
×
p
proper rational transfer function matrix
G
(
s
)
is
positive real
if
poles of all elements of
G
(
s
)
are in
Re
[
s
]
≤
0
for all real
ω
for which
jω
is not a pole of any element of
G
(
s
)
, the matrix
G
(
jω
) +
G
T
(
−
jω
)
is positive
semidefinite
any pure imaginary pole
jω
of any element of
G
(
s
)
is a
simple pole and the residue matrix
lim
s
→
jω
(
s
−
jω
)
G
(
s
)
is positive semidefinite Hermitian
G
(
s
)
is called
strictly positive real
if
G
(
s
−
ε
)
is positive real
for some
ε >
0
– p.2/22
Scalar Case (
p
= 1
):
G
(
jω
) +
G
T
(
−
jω
) = 2
Re
[
G
(
jω
)]
Re
[
G
(
jω
)]
is an even function of
ω
.
The second condition of the definition reduces to
Re
[
G
(
jω
)]
≥
0
,
∀
ω
∈
[0
,
∞
)
which holds when the Nyquist plot of of
G
(
jω
)
lies in the
closed righthalf complex plane
This is true only if the
relative degree
of the transfer function
is
zero or one
Note: for
G
(
s
) =
n
(
s
)
d
(
s
)
, the relative degree is deg
d
deg
n
.
– p.3/22
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View Full Document G
(
jω
) =
1
jω
+ 1
40
35
30
25
20
15
10
5
0
Magnitude (dB)
10
2
10
1
10
0
10
1
10
2
90
45
0
Phase (deg)
Bode Diagram
Frequency
(rad/sec)
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Nyquist Diagram
Real Axis
Imaginary Axis
Bode plot
Nyquist plot
– p.4/22
G
(
jω
) =
1
(
jω
)
2
+
jω
+ 1
80
60
40
20
0
20
Magnitude (dB)
10
2
10
1
10
0
10
1
10
2
180
135
90
45
0
Phase (deg)
Bode Diagram
Frequency
(rad/sec)
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1.5
1
0.5
0
0.5
1
1.5
Nyquist Diagram
Real Axis
Imaginary Axis
Bode plot
Nyquist plot
– p.5/22
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View Full Document Lemma:
Suppose
det [
G
(
s
) +
G
T
(
−
s
)]
is not identically
zero. Then,
G
(
s
)
is strictly positive real if and only if
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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.
 Spring '08
 CHOI

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