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Consider a nonlinear, continuous-time system
y
˙
=
A
(
x
)
+
B
u
(
x
)
u
,
{\displaystyle {\dot {y}}=A(x)+B_{u}(x)u,}
y
=
C
y
(
x
)
+
D
y
u
(
x
)
u
,
{\displaystyle y=C_{y}(x)+D_{yu}(x)u,}
x
∈
R
n
{\displaystyle x\in \mathbb {R} ^{n}}
u
∈
R
n
u
{\displaystyle u\in \mathbb {R} ^{n_{u}}}
y
∈
R
n
y
{\displaystyle y\in \mathbb {R} ^{n_{y}}}
x
∈
R
n
{\displaystyle x\in \mathbb {R} ^{n}}
u
∈
R
n
u
{\displaystyle u\in \mathbb {R} ^{n_{u}}}
y
∈
R
n
y
{\displaystyle y\in \mathbb {R} ^{n_{y}}}
A
,
B
u
,
C
y
,
D
y
u
{\displaystyle A,B_{u},C_{y},D_{yu}}
A
(
0
)
=
0
{\displaystyle A(0)=0}
C
y
(
0
)
=
0
{\displaystyle C_{y}(0)=0}
B
u
,
D
y
u
{\displaystyle B_{u},D_{yu}}
[
A
(
x
)
B
u
(
x
)
C
y
(
x
)
D
y
u
(
x
)
]
{\displaystyle {\begin{bmatrix}A(x)&B_{u}(x)\\C_{y}(x)&D_{yu}(x)\end{bmatrix}}}
[
A
B
u
C
y
D
y
u
]
{\displaystyle {\begin{bmatrix}A&B_{u}\\C_{y}&D_{yu}\end{bmatrix}}}
[
B
p
D
y
p
]
△
(
x
)
[
I
−
D
q
p
△
(
x
)
]
6
−
1
{\displaystyle {\begin{bmatrix}B_{p}\\D_{yp}\end{bmatrix}}\bigtriangleup (x){\begin{bmatrix}I-D_{qp}\bigtriangleup (x)\end{bmatrix}}6-1}
[
C
q
D
q
u
]
{\displaystyle {\begin{bmatrix}C_{q}&D_{qu}\end{bmatrix}}}
