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 (after Malo L. J. Hautus), also commonly known as the Popov-Belevitch-Hautus test or PBH test,[1][2] can prove to be a powerful tool.

A special case of this result appeared first in 1963 in a paper by Elmer G. Gilbert,[1] and was later expanded to the current PBH test with contributions by Vasile M. Popov in 1966,[3][4] Vitold Belevitch in 1968,[5] and Malo Hautus in 1969,[5] who emphasized its applicability in proving results for linear time-invariant systems.

Statement

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.

Notes

1. Hespanha, Joao (2018). Linear Systems Theory (Second ed.). Princeton University Press. ISBN 9780691179575.
2. Bernstein, Dennis S. (2018). Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas (Revised and expanded ed.). Princeton University Press. ISBN 9780691151205.
3. Popov, Vasile Mihai (1966). Hiperstabilitatea sistemelor automate [Hyperstability of Control Systems]. Editura Academiei Republicii Socialiste România.
4. Popov, V.M. (1973). Hyperstability of Control Systems. Berlin: Springer-Verlag.
5. Belevitch, V. (1968). Classical Network Theory. San Francisco: Holden–Day.