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Affine Nonlinear Systems
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Affine Nonlinear Systems
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Affine systems are nonlinear systems that are linear in the input. They can be specified in multiple ways and can also be converted to other systems models.
A system specified using an ODE:
I
n
[
1
]
:
=
A
f
f
i
n
e
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
m
x
'
'
[
t
]
+
c
[
x
]
x
'
[
t
]
+
k
[
x
]
F
[
t
]
,
x
[
t
]
,
F
[
t
]
,
x
[
t
]
,
t
]
O
u
t
[
1
]
=
x
.
1
[
t
]
x
.
2
[
t
]
0
x
.
2
[
t
]
-
k
[
x
]
-
c
[
x
]
x
.
2
[
t
]
m
1
m
x
.
1
[
t
]
0
A system specified using its components:
I
n
[
2
]
:
=
A
f
f
i
n
e
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
{
{
f
1
[
x
1
,
x
2
]
,
f
2
[
x
1
,
x
2
]
}
,
{
{
g
1
1
[
x
1
,
x
2
]
}
,
{
g
2
1
[
x
1
,
x
2
]
}
}
,
{
h
1
[
x
1
,
x
2
]
}
}
,
{
x
1
,
x
2
}
]
O
u
t
[
2
]
=
x
1
f
1
[
x
1
,
x
2
]
g
1
1
[
x
1
,
x
2
]
x
2
f
2
[
x
1
,
x
2
]
g
2
1
[
x
1
,
x
2
]
h
1
[
x
1
,
x
2
]
0
Systems obtained from other systems models:
I
n
[
3
]
:
=
A
f
f
i
n
e
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
/
@
a
1
1
a
1
2
b
1
1
a
2
1
a
2
2
b
2
1
c
1
1
c
1
2
0
,
1
-
2
s
+
2
s
,
x
1
f
1
[
x
1
,
x
2
]
+
u
1
g
1
1
[
x
1
,
x
2
]
x
2
f
2
[
x
1
,
x
2
]
+
u
1
g
2
1
[
x
1
,
x
2
]
h
1
[
x
1
,
x
2
]
O
u
t
[
3
]
=
x
1
a
1
1
x
1
+
a
1
2
x
2
b
1
1
x
2
a
2
1
x
1
+
a
2
2
x
2
b
2
1
c
1
1
x
1
+
c
1
2
x
2
0
,
x
.
1
x
.
2
0
x
.
2
2
x
.
2
1
x
.
1
0
,
x
1
f
1
[
x
1
,
x
2
]
g
1
1
[
x
1
,
x
2
]
x
2
f
2
[
x
1
,
x
2
]
g
2
1
[
x
1
,
x
2
]
h
1
[
x
1
,
x
2
]
0
A
N
o
n
l
i
n
e
a
r
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
that is not input linear is approximated:
I
n
[
4
]
:
=
A
f
f
i
n
e
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
x
1
f
1
[
x
1
,
x
2
,
u
1
]
x
2
f
2
[
x
1
,
x
2
,
u
1
]
h
1
[
x
1
,
x
2
,
u
1
]
O
u
t
[
4
]
=
x
1
f
1
[
x
1
,
x
2
,
0
]
(
0
,
0
,
1
)
f
1
[
x
1
,
x
2
,
0
]
x
2
f
2
[
x
1
,
x
2
,
0
]
(
0
,
0
,
1
)
f
2
[
x
1
,
x
2
,
0
]
h
1
[
x
1
,
x
2
,
0
]
(
0
,
0
,
1
)
h
1
[
x
1
,
x
2
,
0
]
A linear
A
f
f
i
n
e
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
is exactly converted to linear systems models:
I
n
[
5
]
:
=
a
s
s
m
=
x
1
x
1
+
x
2
1
x
2
x
1
1
x
1
0
;
I
n
[
6
]
:
=
{
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
a
s
s
m
]
,
T
r
a
n
s
f
e
r
F
u
n
c
t
i
o
n
M
o
d
e
l
[
a
s
s
m
]
}
O
u
t
[
6
]
=
1
1
1
1
0
1
1
0
0
,
1
+
s
.
-
1
-
s
.
+
2
s
.
In general, affine models are approximated during conversion to linear systems models:
I
n
[
7
]
:
=
a
s
s
m
=
x
1
x
1
+
x
2
+
x
1
x
2
1
x
2
x
1
1
+
x
2
x
1
0
;
I
n
[
8
]
:
=
{
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
a
s
s
m
]
,
T
r
a
n
s
f
e
r
F
u
n
c
t
i
o
n
M
o
d
e
l
[
a
s
s
m
]
}
O
u
t
[
8
]
=
1
1
1
1
0
1
1
0
0
,
1
+
s
.
-
1
-
s
.
+
2
s
.
The conversion to a
N
o
n
l
i
n
e
a
r
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
is exact:
I
n
[
9
]
:
=
N
o
n
l
i
n
e
a
r
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
x
1
f
1
[
x
1
,
x
2
]
g
1
1
[
x
1
,
x
2
]
x
2
f
2
[
x
1
,
x
2
]
g
2
1
[
x
1
,
x
2
]
h
1
[
x
1
,
x
2
]
0
O
u
t
[
9
]
=
x
1
f
1
[
x
1
,
x
2
]
+
u
1
g
1
1
[
x
1
,
x
2
]
x
2
f
2
[
x
1
,
x
2
]
+
u
1
g
2
1
[
x
1
,
x
2
]
h
1
[
x
1
,
x
2
]
"
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