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

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

Hautus Lemma for stabilizability

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

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

Hautus Lemma for observability

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

The pair $(\mathbf {A} ,\mathbf {C} )$ is observable.
For all $\lambda \in \mathbb {C}$ it holds that $\operatorname {rank} [\lambda \mathbf {I} -\mathbf {A$
For all $\lambda \in \mathbb {C}$ that are eigenvalues of $\mathbf {A}$ it holds that $\operatorname {rank} [\lambda \mathbf {I} -\mathbf {A$

Hautus Lemma for detectability

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

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

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

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

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[:0-1] 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.

[:1-5] Belevitch, V. (1968). Classical Network Theory. San Francisco: Holden–Day.