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In-Depth Information
The following state variables are defined x
1
D V
C
and x
2
D i
L
.FromEq.(
2.8
)it
holds
I
L
I
C
I
R
D 0)I
C
D I
L
I
R
)I
C
D x
2
h.x
1
/
(2.10)
Moreover, it holds
I
L
R E C V
L
C V
C
D 0)E D V
L
C V
C
C i
L
R)E D L
dI
L
dt
C V
C
C I
L
R)
1
L
ŒV
C
Rx
2
C E)x
2
D
1
L
Œx
1
Rx
2
C
u
(2.11)
E D Lx
2
C V
C
C x
2
R)x
2
D
where
u
D E. Additionally, from Eq. (
2.6
) it holds
dV
c
.t/
dt
1
C
I
c
.t/
(2.12)
D
By replacing Eqs. (
2.10
)into(
2.12
) one obtains
dV
c
.t/
dt
1
D
C
Œx
2
h.x
1
/
(2.13)
From Eqs. (
2.11
) and (
2.13
) one has the description of the system is state-space form
1
x
1
D
C
Œx
2
h.x
1
/
(2.14)
1
x
2
D
L
Œx
1
Rx
2
C
u
The associated equilibrium is computed from the condition
x
1
D 0 and
x
2
D 0
which gives
h.x
1
/ C x
2
D 0
x
1
Rx
2
C
u
D 0
(2.15)
which gives
E
x
1
R
h.x
1
/ C
D 0
(2.16)
E
x
1
R
x
2
D
Therefore, finally the equilibrium point is computed from the solution of the relation
E
R
1
R
x
1
(2.17)
h.x
1
/ D
Example 3.
Spring-mass system (Fig.
2.3
).
It holds that
mx D F F
f
F
sp
)
mx D F b x kx
(2.18)
