EE 530
PROBLEM SET 2
DUE: 6 DEC 2006
Reading assignment: Ch 3
The goal of the second problem set is to:
1. Gain experience using stability analysis tools.
2. Design and study the stability properties of elementary MRAC systems.
Problem 5:
(16 points)
1. (8 points) Consider a nonlinear system with statespace model
˙
x
1
=

x
1
+
x
2
(
x
1
+
c
)
˙
x
2
=

x
2
1

x
1
c
where
c
is a nonzero constant. Investigate the stability properties of the equilibrium state at the origin using
Lyapunov’s direct method. In particular, determine whether or not the origin is an asymptotically stable
equilibrium state.
2. (8 points) Repeat part 1 when
c
(
t
):
±
+
→±
is a bounded timevarying function.
Problem 6:
(25 points)
1. (6 points) Consider the transfer function
G
(
s
)=
b
1
s
+
b
0
s
2
+
a
1
s
+
a
0
.
(a) (3 points) For what conditions on
b
0
,b
1
,a
0
and
a
1
is
G
(
s
) strictly positive real ?
(b) (3 points) Suppose that
b
1
= 0 and
b
o
²
= 0, can
G
(
s
) be positive real ? Justify your answer.
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 Fall '06
 SCHIANO

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