Hautus引理 （Hautus lemma）是在控制理论 以及状态空间 下分析线性时不变系统 时，相当好用的工具，得名自Malo Hautus[1] [2] [3]
有许多有关引理的不同型式。
可控制性Hautus引理提到若给定一方阵
A
∈
M
n
(
ℜ
)
{\displaystyle \mathbf {A} \in M_{n}(\Re )}
B
∈
M
n
×
m
(
ℜ
)
{\displaystyle \mathbf {B} \in M_{n\times m}(\Re )}
(
A
,
B
)
{\displaystyle (\mathbf {A} ,\mathbf {B} )}
可控制性 针对所有的
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
rank
[
λ
I
−
A
,
B
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}
针对所有
A
{\displaystyle \mathbf {A} }
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
rank
[
λ
I
−
A
,
B
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}
可稳定性Hautus引理提到若给定一方阵
A
∈
M
n
(
ℜ
)
{\displaystyle \mathbf {A} \in M_{n}(\Re )}
B
∈
M
n
×
m
(
ℜ
)
{\displaystyle \mathbf {B} \in M_{n\times m}(\Re )}
(
A
,
B
)
{\displaystyle (\mathbf {A} ,\mathbf {B} )}
可稳定性 针对所有
A
{\displaystyle \mathbf {A} }
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
ℜ
(
λ
)
≥
0
{\displaystyle \Re (\lambda )\geq 0}
rank
[
λ
I
−
A
,
B
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}
可侦测性Hautus引理提到若给定一方阵
A
∈
M
n
(
ℜ
)
{\displaystyle \mathbf {A} \in M_{n}(\Re )}
C
∈
M
m
×
n
(
ℜ
)
{\displaystyle \mathbf {C} \in M_{m\times n}(\Re )}
(
A
,
C
)
{\displaystyle (\mathbf {A} ,\mathbf {C} )}
可侦测性 针对所有
A
{\displaystyle \mathbf {A} }
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
ℜ
(
λ
)
≥
0
{\displaystyle \Re (\lambda )\geq 0}
rank
[
λ
I
−
A
,
C
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {C} ]=n}
^ Malo Hautus . [2017-12-10 ] . （原始内容存档 于2018-11-29）. ^ Belevitch, V. Classical Control Theory. San Francisco: Holden–Day. 1968. ^ Popov, V. M. Hyperstability of Control Systems. Berlin: Springer-Verlag. 1973: 320.
![{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7673844d5ec19ec48726647d01cf9afd0ac4dab6)
![{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {C} ]=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b015732d71a79463693542d06b869041f3cc0f2)