非线性系统
x
˙
(
t
)
=
f
(
x
(
t
)
,
u
(
t
)
)
,
x
(
0
)
=
x
0
,
u
(
t
)
∈
R
m
,
x
(
t
)
∈
R
n
,
Rank
∂
f
(
x
,
u
)
∂
u
=
m
{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {f} (\mathbf {x} (t),\mathbf {u} (t)),\quad \mathbf {x} (0)=\mathbf {x} _{0},\quad \mathbf {u} (t)\in R^{m},\quad \mathbf {x} (t)\in R^{n},{\text{Rank}}{\frac {\partial \mathbf {f} (\mathbf {x} ,\mathbf {u} )}{\partial \mathbf {u} }}=m}
具有平坦性，假设存在输出
y
(
t
)
=
(
y
1
(
t
)
,
.
.
.
,
y
m
(
t
)
)
{\displaystyle \mathbf {y} (t)=(y_{1}(t),...,y_{m}(t))}
满足以下条件：
信号
y
i
,
i
=
1
,
.
.
.
,
m
{\displaystyle y_{i},i=1,...,m}
x
i
,
i
=
1
,
.
.
.
,
n
{\displaystyle x_{i},i=1,...,n}
u
i
,
i
=
1
,
.
.
.
,
m
{\displaystyle u_{i},i=1,...,m}
u
i
(
k
)
,
k
=
1
,
.
.
.
,
α
i
{\displaystyle u_{i}^{(k)},k=1,...,\alpha _{i}}
y
=
Φ
(
x
,
u
,
u
˙
,
.
.
.
,
u
(
α
)
)
{\displaystyle \mathbf {y} =\Phi (\mathbf {x} ,\mathbf {u} ,{\dot {\mathbf {u} }},...,\mathbf {u} ^{(\alpha )})}
状态
x
i
,
i
=
1
,
.
.
.
,
n
{\displaystyle x_{i},i=1,...,n}
u
i
,
i
=
1
,
.
.
.
,
m
{\displaystyle u_{i},i=1,...,m}
y
i
,
i
=
1
,
.
.
.
,
m
{\displaystyle y_{i},i=1,...,m}
y
i
(
k
)
,
i
=
1
,
.
.
.
,
m
{\displaystyle y_{i}^{(k)},i=1,...,m}
y
{\displaystyle \mathbf {y} }
ϕ
(
y
,
y
˙
,
y
(
γ
)
)
=
0
{\displaystyle \phi (\mathbf {y} ,{\dot {\mathbf {y} }},\mathbf {y} ^{(\gamma )})=\mathbf {0} }
若上述条件至少有在局部成立，则（可能是虚拟的）输出则称为平坦输出，系统即为平坦系统。
线性时不变系统
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
,
x
(
0
)
=
x
0
{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t),\quad \mathbf {x} (0)=\mathbf {x} _{0}}
x
,
u
{\displaystyle \mathbf {x} ,\mathbf {u} }
可控制性 。因此线性时不变系统中这二种性质是等效的，可以互换。
平坦性的特性可以用在分析非线性动态系统，以及合成其控制器上。在解决轨迹规划问题和渐近设定点追随控制时特别好用。
M. Fliess, J. L. Lévine, P. Martin and P. Rouchon: Flatness and defect of non-linear systems: introductory theory and examples. International Journal of Control 61(6 ), pp. 1327-1361, 1995 [1] （页面存档备份 ，存于互联网档案馆 ）
A. Isidori, C.H. Moog et A. De Luca. A Sufficient Condition for Full Linearization via Dynamic State Feedback. 25th CDC IEEE, Athens, Greece, pp. 203 - 208, 1986 [2]
