Homework 3
ECE147A
October 19, 2009
For problems that are taken from the text book, I’ll just give you excerpts from the solution manual. Those pages
are going to be marked with a black frame.
Problem 1
In order to obtain the transfer function from
v
1
:=
u
1

u
◦
1
to
z
3
:=
x
3

x
◦
3
, we need to linearize the state space
model around (
x
◦
, u
◦
). As usual, we define deviation variables
z
:=
x

x
◦
and
v
:=
u

u
◦
and compute the
Jacobians:
∂
(right hand side)
∂x
=

3(
u
1

x
1
)
2
0

1
0
0
1
1

1
dr
dx
3
(1)
∂
(right hand side)
∂u
=
3(
u
1

x
1
)
2
1
0
0
0
0
(2)
The term
dr
dx
3
is a little tricky, but since we are going to insert the equilibrium
x
◦
3
, all we need to know is the
slope of
r
(0), which from the figure is easily obtained as 2. The linearized system thus looks like this:
˙
z
=

27
0

1
0
0
1
1

1
2

{z
}
=:
A
z
+
27
1
0
0
0
0

{z
}
=:
B
v
(3)
Since we are only interested in the transfer function from
v
1
to
z
3
, we set
c
T
=
0
0
1
and
v
2
= 0 (which is
the same as disregarding the second column in the
B
matrix) and obtain:
H
(
s
) =
Z
3
(
s
)
V
1
(
s
)
=
c
T
(
sI

A
)

1
B
1
=
27
s
s
3
+ 25
s
2

52
s
+ 27
.
(4)
5 points for the model, 1 point for the equilibrium, 7 points for the linearization if you got
it all,
including
the definiton of deviation variables.
1
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9005
3. Consider the circuit shown in Fig. 9.58;
u
1
and
u
2
are voltage and current sources, respectively,
and
R
1
and
R
2
are nonlinear resistors with the following characteristics:
Resistor 1 :
i
1
=
G
(
v
1
) =
v
3
1
Resistor 2 :
v
2
=
r
(
i
2
)
;
where the function
r
is de°ned in Fig. 9.59.
(a) Show that the circuit equations can be written as
_
x
1
=
G
(
u
1
°
x
1
) +
u
2
°
x
3
_
x
2
=
x
3
_
x
3
=
x
1
°
x
2
°
r
(
x
3
)
:
Suppose we have a constant voltage source of 1 Volt at
u
1
and a constant current source
of 27 Amps; i.e.,
u
o
1
= 1
,
u
o
2
= 27
. Find the
equilibrium state
x
o
= [
x
o
1
; x
o
2
; x
o
3
]
T
for the
circuit.
For a particular input
u
o
, an equilibrium state of the system is de°ned to be any
constant state vector whose elements satisfy the relation
_
x
1
= _
x
2
= _
x
3
= 0
:
Consequently, any system started in one of its equilibrium states will remain there inde°
nitely until a di/erent input is applied.
(b) Due to disturbances, the initial state (capacitance, voltages, and inductor current) is slightly
di/erent from the equilibrium and so are the independent sources; that is,
u
(
t
)
=
u
o
+
°u
(
t
)
x
(
t
0
)
=
x
o
(
t
0
) +
°x
(
t
0
)
:
Do a smallsignal analysis of the network about the equilibrium found in (a), displaying
the equations in the form
°
_
x
1
=
f
11
°x
1
+
f
12
°x
2
+
f
13
°x
3
+
g
1
°u
1
+
g
2
°u
2
:
(c) Draw the circuit diagram that corresponds to the linearized model. Give the values of the
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 Spring '10
 VolkanRodoplu
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