Hautus lemma

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 ${\displaystyle \mathbf {A} \in M_{n}(\Re )}$ and a ${\displaystyle \mathbf {B} \in M_{n\times m}(\Re )}$ the following are equivalent:

1. The pair ${\displaystyle (\mathbf {A} ,\mathbf {B} )}$ is controllable
2. For all ${\displaystyle \lambda \in \mathbb {C} }$ it holds that ${\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}$
3. For all ${\displaystyle \lambda \in \mathbb {C} }$ that are eigenvalues of ${\displaystyle \mathbf {A} }$ it holds that ${\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}$

Hautus Lemma for stabilizability

The Hautus lemma for stabilizability says that given a square matrix ${\displaystyle \mathbf {A} \in M_{n}(\Re )}$ and a ${\displaystyle \mathbf {B} \in M_{n\times m}(\Re )}$ the following are equivalent:

1. The pair ${\displaystyle (\mathbf {A} ,\mathbf {B} )}$ is stabilizable
2. For all ${\displaystyle \lambda \in \mathbb {C} }$ that are eigenvalues of ${\displaystyle \mathbf {A} }$ and for which ${\displaystyle \Re (\lambda )\geq 0}$ it holds that ${\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A} ,\mathbf {B} ]=n}$

Hautus Lemma for observability

The Hautus lemma for observability says that given a square matrix ${\displaystyle \mathbf {A} \in M_{n}(\Re )}$ and a ${\displaystyle \mathbf {C} \in M_{m\times n}(\Re )}$ the following are equivalent:

1. The pair ${\displaystyle (\mathbf {A} ,\mathbf {C} )}$ is observable
2. For all ${\displaystyle \lambda \in \mathbb {C} }$ it holds that ${\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A}$ ;\mathbf {C} ]=n}
3. For all ${\displaystyle \lambda \in \mathbb {C} }$ that are eigenvalues of ${\displaystyle \mathbf {A} }$ it holds that ${\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A}$ ;\mathbf {C} ]=n}

Hautus Lemma for detectability

The Hautus lemma for detectability says that given a square matrix ${\displaystyle \mathbf {A} \in M_{n}(\Re )}$ and a ${\displaystyle \mathbf {C} \in M_{m\times n}(\Re )}$ the following are equivalent:

1. The pair ${\displaystyle (\mathbf {A} ,\mathbf {C} )}$ is detectable
2. For all ${\displaystyle \lambda \in \mathbb {C} }$ that are eigenvalues of ${\displaystyle \mathbf {A} }$ and for which ${\displaystyle \Re (\lambda )\geq 0}$ it holds that ${\displaystyle \operatorname {rank} [\lambda \mathbf {I} -\mathbf {A}$ ;\mathbf {C} ]=n}

References

• Sontag, Eduard D. (1998). Mathematical Control Theory: Deterministic Finite-Dimensional Systems. New York: Springer. ISBN 0-387-98489-5.
• Zabczyk, Jerzy (1995). Mathematical Control Theory – An introduction. Boston: Birkhauser. ISBN 3-7643-3645-5.

1. Belevitch, V. (1968). Classical Network Theory. San Francisco: Holden–Day.
2. Popov, V. M. (1973). Hyperstability of Control Systems. Berlin: Springer-Verlag. p. 320.