from which we obtain the following differential equation
˙
v
R
+ 1
.
5
v
3
R
+
v
R
=
v
i
When
v
i
= 14, we can find a steady state solution for the circuit by considering ˙
v
R
= 0, and thus
1
.
5
v
3
R
+
v
R
= 14, from which we obtain
v
R
= 2. Hence, we can linearize the differential equation of
12
CHAPTER 1.
the system around the solution (
v
R
= 2
, v
i
= 14). Since ˙
v
R
=
f
(
v
R
, v
i
) =

1
.
5
v
3
R

v
R
+
v
i
we have
Δ ˙
v
R
=
∂f
∂v
R
v
R
=2
v
i
=14
Δ
v
R
+
∂f
∂v
i
v
R
=2
v
i
=14
Δ
v
i
=
(

4
.
5
v
2
R

1)
v
R
=2
Δ
v
R
+ (1)Δ
v
i
=

19Δ
v
R
+ Δ
v
i
Exercise 1.13.
Defining
x
1
=
φ
,
x
2
=
˙
φ
,
x
3
=
s
,
x
4
= ˙
s
, we have the following equations
˙
x
1
=
x
2
˙
x
2
=
g
L
sin
x
1

1
L M
(

Fx
4
+
μ
(
t
)) cos
x
1
˙
x
3
=
x
4
˙
x
4
=

F
M
x
4
+
μ
(
t
)
M
We see that
x
i
= 0,
i
= 1
, . . . ,
4,
μ
(
t
) = 0 is a solution. Linearizing the system about this solution,
we get
Δ ˙
x
1
Δ ˙
x
2
Δ ˙
x
3
Δ ˙
x
4
=
0
1
0
0
g
L
0
0
F
L M
0
0
0
1
0
0
0

F
M
Δ
x
1
Δ
x
2
Δ
x
3
Δ
x
4
+
0

1
L M
0
1
M
Δ
μ.
Exercise 1.14.
The states for the simple pendulun are shown in Figure 1.6 and 1.7, where the dashed line
corresponds to the linearized system, and the solid to the nonlinear one.
When
θ
0
=
π/
18 and
θ
0
=
π/
12 (Figure 1.6) the states are similar for the nonlinear and the linearized model. The system
was linearized about the solution
x
= [0
,
0]
T
which is close to the initial condition. In the case when
θ
0
=
π/
6 and
θ
0
=
π/
3 the linearized system is not a good approximation of the nonlinear one. This
can be observed from the evolution of the states in Figure 1.7).
13
Figure 1.6: (i)
θ
0
=
π/
18, (ii)
θ
0
=
π/
12
Figure 1.7: (i)
θ
0
=
π/
6, (ii)
θ
0
=
π/
3
14
CHAPTER 1.
Chapter 2
Exercise 2.1.
(a) In Problem 1.2(a), let
x
1
=
y
1
, x
2
= ˙
y
1
, x
3
=
y
2
, and
x
4
= ˙
y
2
. Also, let
u
1
=
f
1
and
u
2
=
f
2
,
and let (
v
1
, v
2
) =
v
T
denote the output vector with
v
1
=
y
1
=
x
1
and
v
2
=
y
2
=
x
3
. From Problem
1.2(a) we obtain
˙
x
1
˙
x
2
˙
x
3
˙
x
4
=
0
1
0
0

[
(
K
1
+
K
)
M
1
]

[
(
B
1
+
B
)
M
1
]
K
M
1
B
M
1
0
0
0
1
K
M
2
B
M
2

[
(
K
+
K
2
)
M
2
]

[
(
B
+
B
2
)
M
2
]
x
1
x
2
x
3
x
4
+
0
0
1
M
1
0
0
0
0

1
M
2
u
1
(
t
)
u
2
(
t
)
=
A
x
1
x
2
x
3
x
4
+
B
u
1
(
t
)
u
2
(
t
)
and
v
1
v
2
=
1
0
0
0
0
0
1
0
x
1
x
2
x
3
x
4
=
C
x
1
x
2
x
3
x
4
.
(b) Same as in item (a), except consider
u
T
= (
u
1
, u
2
) = (
f
1
,
5
f
2
)
T
as the system input and
(8
y
1
+ 10
y
2
) as the (scalarvalued) system output. Then
˙
x
1
˙
x
2
˙
x
3
˙
x
4
=
A
x
1
x
2
x
3
x
4
+
0
0
1
M
1
0
0
0
0

1
M
2
u
1
(
t
)
u
2
(
t
)
=
A
x
1
x
2
x
3
x
4
+
0
0
1
M
1
0
0
0
0

5
M
2
f
1
(
t
)
f
2
(
t
)
=
A
x
1
x
2
x
3
x
4
+
B
f
1
(
t
)
f
2
(
t
)
15
16
CHAPTER 2.
You've reached the end of your free preview.
Want to read all 90 pages?
 Fall '13