Surf Wiki
Save to docs
general/control-theory

From Surf Wiki (app.surf) — the open knowledge base

Hautus lemma

Lemma in control theory

Summary

Lemma in control theory

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, gives equivalent conditions for certain properties of control systems.

A special case of this result appeared first in 1963 in a paper by Elmer G. Gilbert, and was later expanded to the current PBH test with contributions by Vasile M. Popov in 1966, Vitold Belevitch in 1968, 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 \mathbf{A}\in M_n(\Re) and a \mathbf{B}\in M_{n\times m}(\Re) the following are equivalent:

  1. The pair (\mathbf{A},\mathbf{B}) is controllable
  2. For all \lambda\in\mathbb{C} it holds that \operatorname{rank}[\lambda \mathbf{I}-\mathbf{A},\mathbf{B}]=n
  3. 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:

  1. The pair (\mathbf{A},\mathbf{B}) is stabilizable
  2. For all \lambda\in\mathbb{C} that are eigenvalues of \mathbf{A} and for which \Re(\lambda)\ge 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:

  1. The pair (\mathbf{A},\mathbf{C}) is observable.
  2. For all \lambda\in\mathbb{C} it holds that \operatorname{rank}[\lambda \mathbf{I}-\mathbf{A};\mathbf{C}]=n
  3. For all \lambda\in\mathbb{C} that are eigenvalues of \mathbf{A} it holds that \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 \mathbf{A}\in M_n(\Re) and a \mathbf{C}\in M_{m\times n}(\Re) the following are equivalent:

  1. The pair (\mathbf{A},\mathbf{C}) is detectable
  2. For all \lambda\in\mathbb{C} that are eigenvalues of \mathbf{A} and for which \Re(\lambda)\ge 0 it holds that \operatorname{rank}[\lambda \mathbf{I}-\mathbf{A};\mathbf{C}]=n

References

Notes

References

  1. (2018). "Linear Systems Theory". Princeton University Press.
  2. (2018). "Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas". Princeton University Press.
  3. Popov, Vasile Mihai. (1966). "Hiperstabilitatea sistemelor automate". Editura Academiei Republicii Socialiste România.
  4. Popov, V.M.. (1973). "Hyperstability of Control Systems". Springer-Verlag.
  5. Belevitch, V.. (1968). "Classical Network Theory". Holden–Day.
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

control-theory lemmas
Want to explore this topic further?

Ask Mako anything about Hautus lemma — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report