Control of Nonlinear Dynamic Systems: Theory and Applications
J. K. Hedrick and A. Girard
© 2005
63
[]
g
f
g
ad
f
,
)
,
(
1
≡
g
f
f
g
ad
f
,
,
)
,
(
2
≡
…
[ ]
)
,
(
,
)
,
(
1
g
ad
f
g
ad
k
f
k
f
−
≡
Note: the “ad” is read “adjoint”.
Re-writing
controllability conditions for linear systems
using this notation:
m
m
u
B
u
B
u
B
Ax
Bu
Ax
x
+
+
+
+
=
+
=
...
2
2
1
1
&
Ax
x
f
=
)
(,
i
i
B
x
g
=
)
(
i
m
i
i
m
i
i
i
u
B
f
x
A
u
AB
x
A
x
A
x
∑
∑
=
=
−
=
+
=
=
1
2
1
2
,
&
&
&
How this came about…
A
x
f
=
∂
∂
,
0
=
∂
∂
=
∂
∂
x
B
x
g
So for example:
[
]
1
1
1
1
1
)
(
,
,
AB
B
x
Ax
Ax
x
B
B
Ax
B
f
−
=
∂
∂
−
∂
∂
=
=
If we keep going:
∑∑
==
+
=
+
=
m
i
m
i
i
i
f
i
i
u
B
ad
x
A
u
B
A
x
A
x
11
2
3
2
3
,
&
&
&
Notice how this time the minus signs cancel out.
…
−
−
−
−
+
=
+
=
=
m
i
m
i
i
i
n
f
n
n
i
i
n
n
n
n
n
u
B
ad
x
A
u
B
A
x
A
dt
x
d
x
1
1
1
)
(
,
)
1
(
Re-writing the controllability condition:
[ ] [ ] [ ] [ ]
[ ]
m
n
f
n
f
m
f
f
m
B
ad
B
ad
B
ad
B
ad
B
B
C
,
,...
,
,...
,
,...
,
,
,...,
1
1
1
1
1
−
−
=
The condition has not changed – just the notation.