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In control theory and in particular when studying the properties of a linear time-invariant system in state space form, the Hautus lemma , named after Malo Hautus , can prove to be a powerful tool. This result appeared first in [ 1] and.[ 2] Today it can be found in most textbooks on control theory.
The main result
There exist multiple forms of the lemma.
Hautus Lemma for controllability
The Hautus lemma for controllability says that given a square matrix
A
∈
M
n
(
ℜ
)
{\displaystyle \mathbf {A} \in M_{n}(\Re )}
and a
B
∈
M
n
×
m
(
ℜ
)
{\displaystyle \mathbf {B} \in M_{n\times m}(\Re )}
the following are equivalent:
The pair
(
A
,
B
)
{\displaystyle (\mathbf {A} ,\mathbf {B} )}
is controllable
For all
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
it holds that
rank
[
λ
I
−
A
,
B
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}
For all
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
that are eigenvalues of
A
{\displaystyle \mathbf {A} }
it holds that
rank
[
λ
I
−
A
,
B
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}
Hautus Lemma for stabilizability
The Hautus lemma for stabilizability says that given a square matrix
A
∈
M
n
(
ℜ
)
{\displaystyle \mathbf {A} \in M_{n}(\Re )}
and a
B
∈
M
n
×
m
(
ℜ
)
{\displaystyle \mathbf {B} \in M_{n\times m}(\Re )}
the following are equivalent:
The pair
(
A
,
B
)
{\displaystyle (\mathbf {A} ,\mathbf {B} )}
is stabilizable
For all
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
that are eigenvalues of
A
{\displaystyle \mathbf {A} }
and for which
ℜ
(
λ
)
≥
0
{\displaystyle \Re (\lambda )\geq 0}
it holds that
rank
[
λ
I
−
A
,
B
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}
Hautus Lemma for observability
The Hautus lemma for observability says that given a square matrix
A
∈
M
n
(
ℜ
)
{\displaystyle \mathbf {A} \in M_{n}(\Re )}
and a
C
∈
M
m
×
n
(
ℜ
)
{\displaystyle \mathbf {C} \in M_{m\times n}(\Re )}
the following are equivalent:
The pair
(
A
,
C
)
{\displaystyle (\mathbf {A} ,\mathbf {C} )}
is observable
For all
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
it holds that
rank
[
λ
I
−
A
;
C
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ;\mathbf {C} ]=n}
For all
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
that are eigenvalues of
A
{\displaystyle \mathbf {A} }
it holds that
rank
[
λ
I
−
A
;
C
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ;\mathbf {C} ]=n}
Hautus Lemma for detectability
The Hautus lemma for detectability says that given a square matrix
A
∈
M
n
(
ℜ
)
{\displaystyle \mathbf {A} \in M_{n}(\Re )}
and a
C
∈
M
m
×
n
(
ℜ
)
{\displaystyle \mathbf {C} \in M_{m\times n}(\Re )}
the following are equivalent:
The pair
(
A
,
C
)
{\displaystyle (\mathbf {A} ,\mathbf {C} )}
is detectable
For all
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
that are eigenvalues of
A
{\displaystyle \mathbf {A} }
and for which
ℜ
(
λ
)
≥
0
{\displaystyle \Re (\lambda )\geq 0}
it holds that
rank
[
λ
I
−
A
;
C
]
=
n
{\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ;\mathbf {C} ]=n}
References
^ Belevitch, V. (1968). Classical Network Theory . San Francisco: Holden–Day.
^ Popov, V. M. (1973). Hyperstability of Control Systems . Berlin: Springer-Verlag. p. 320.