Hautus lemma explained

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, 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, 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

A \in M _{n(\Re)} and a B \in M _{n x} (\Re) the following are equivalent:

1. The pair

( A , B ) is controllable

1. For all

λ \in C it holds that \operatorname{rank}[ λ I - A , B ] = n

1. For all

λ \in C that are eigenvalues of A it holds that \operatorname{rank}[ λ I - A , B ] = n

Hautus Lemma for stabilizability

The Hautus lemma for stabilizability says that given a square matrix

A \in M _{n(\Re)} and a B \in M _{n x} (\Re) the following are equivalent:

1. The pair

( A , B ) is stabilizable

1. For all

λ \in C that are eigenvalues of A and for which \Re( λ )\ge 0 it holds that \operatorname{rank}[ λ I - A , B ] = n

Hautus Lemma for observability

The Hautus lemma for observability says that given a square matrix

A \in M _{n(\Re)} and a C \in M _{m x} (\Re) the following are equivalent:

1. The pair

( A , C ) is observable.

1. For all

λ \in C it holds that \operatorname{rank}[ λ I - A ; C ] = n

1. For all

λ \in C that are eigenvalues of A it holds that \operatorname{rank}[ λ I - A ; C ] = n

Hautus Lemma for detectability

The Hautus lemma for detectability says that given a square matrix

A \in M _{n(\Re)} and a C \in M _{m x} (\Re) the following are equivalent:

1. The pair

( A , C ) is detectable

1. For all

λ \in C that are eigenvalues of A and for which \Re( λ )\ge 0 it holds that \operatorname{rank}[ λ I - A ; C ] = n

References

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

Notes

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