Hautus lemma

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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 A∈Mn(ℜ) and a B∈Mn×m(ℜ) the following are equivalent:

  1. The pair (A,B) is controllable
  2. For all λ∈ℂ it holds that rank⁡[λI−A,B]=n
  3. For all λ∈ℂ that are eigenvalues of A it holds that rank⁡[λI−A,B]=n

Hautus Lemma for stabilizability

The Hautus lemma for stabilizability says that given a square matrix A∈Mn(ℜ) and a B∈Mn×m(ℜ) the following are equivalent:

  1. The pair (A,B) is stabilizable
  2. For all λ∈ℂ that are eigenvalues of A and for which ℜ(λ)≥0 it holds that rank⁡[λI−A,B]=n

Hautus Lemma for observability

The Hautus lemma for observability says that given a square matrix A∈Mn(ℜ) and a C∈Mm×n(ℜ) the following are equivalent:

  1. The pair (A,C) is observable.
  2. For all λ∈ℂ it holds that rank⁡[λI−A;C]=n
  3. For all λ∈ℂ that are eigenvalues of A it holds that rank⁡[λI−A;C]=n

Hautus Lemma for detectability

The Hautus lemma for detectability says that given a square matrix A∈Mn(ℜ) and a C∈Mm×n(ℜ) the following are equivalent:

  1. The pair (A,C) is detectable
  2. For all λ∈ℂ that are eigenvalues of A and for which ℜ(λ)≥0 it holds that rank⁡[λI−A;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. ↑ 1.0 1.1 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. ↑ 5.0 5.1 Belevitch, V. (1968). Classical Network Theory. San Francisco: Holden–Day.
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