Problem 1.
(15 points)
Consider the following system
˙
ζ

2
ζ
=
θ
+2
ζu,
¨
θ
+
θ
2
±
1+
˙
θ
²
=
u
+
θ.
(This system does not arise from any physical application but its structure and its nonlinear
terms mimic phenomena that appear in mechanical, electrical and biochemical systems).
(a)
(5 points) Treating
ζ
as the output and
u
as the input, derive a state space representation
of the system.
(b)
(5 points) Let
u
= 0 and Fnd
all
the equilibrium points of the state space system.
(c)
(5 points) ±or
each
equilibrium found in (b), compute the linearization of the system
(in state space form, i.e., Fnd
F
,
G
,
H
,
J
).
Solution:
(a)
Calling
x
1
=
θ
,
x
2
=
˙
θ
,and
x
3
=
ζ
,weget
˙
x
1
=
x
2
,
˙
x
2
=
x
1

x
2
1
(1 +
x
2
)+
u,
˙
x
3
=
x
1
x
3
x
3
u,
and since
ζ
is the output, then
y
=
x
3
. This can be written as
˙
x
=
f
(
x, u
)
,
y
=
h
(
x, u
)
,
where
f
(
x, u
)=
x
2
x
1

x
2
1
(1 +
x
2
u
x
1
x
3
x
3
u
,h
(
x, u
x
3
.
(b)
±or Fnding the equilibria we set ˙
x
=0and
u
= 0. Then, we need to solve the equation
f
(
x,
0) = 0, i.e.,
0=
x
2
,
x
1

x
2
1
(1 +
x
2
)
,
x
1
x
3
.