# 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 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 detectabile
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 Control Theory. San Francisco: Holden–Day.
2. ^ Popov, V. M. (1973). Hyperstability of Control Systems. Berlin: Springer-Verlag. p. 320.