Information Technology Reference
In-Depth Information
By defining the state variables x
1
D
w
, x
2
D V and setting current I as the external
control input one has
dx
1
dt
Dx
1
C x
2
dx
2
dt
D x
2
.x
2
a/.1 x
2
/ x
1
C
u
(4.32)
y D h.x/ D x
1
Equivalently one has
x
1
x
2
x
1
C x
2
x
2
.x
2
a/.1 x
2
/ x
1
0
1
u
D
C
(4.33)
y D h.x/ D x
1
The state variable
z
1
D h
1
.x/ D x
1
is defined. It holds that
@h
1
@x
1
f
1
C
@h
2
@x
2
f
2
D 1f
1
C 0f
2
D f
1
z
2
D L
f
h
1
.x/ D
(4.34)
L
f
h
1
.x/ D
@
z
2
@
z
2
@
@
@x
1
f
1
C
@x
2
f
2
D
@x
1
.x
1
C x
2
/f
1
C
@x
2
.x
1
C x
2
/f
2
)
L
f
h
1
.x/ D ./f
1
C f
2
)
L
f
h
1
.x/ D.x
1
C x
2
/ C Œx
2
.x
2
a/.1 x
2
/ x
1
(4.35)
Moreover, it holds
@h
1
@x
1
g
1
C
@h
2
@x
2
g
2
D 0g
1
C 1g
2
D g
2
D 1
L
g
h
1
.x/ D
(4.36)
@
z
2
@
z
2
@
@
L
g
L
f
h
1
.x/ D
@x
2
.x
1
C x
2
/g
2
)
L
g
L
f
h
1
.x/ Dg
1
C g
2
)L
g
L
f
h
1
.x/ D ./0 C 1)L
g
L
f
h
1
.x/ D
(4.37)
@x
1
g
1
C
@x
2
g
2
D
@x
1
.x
1
C x
2
/g
1
C
It holds that
z
1
D
z
2
z
2
D L
f
h
1
.x/ C L
g
L
f
h
1
.x/
u
(4.38)
or equivalently
z
1
D
z
2
z
2
Dfx
1
C x
2
C Œx
2
.x
2
a/.1 x
2
/ x
1
gC
u
(4.39)
