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, named after Malo Hautus, can prove to be a powerful tool. This result appeared first in [1] and.[2] Today it can be found in most textbooks on control theory.

The main result

There exist multiple forms of the lemma.

Hautus Lemma for controllability

The Hautus lemma for controllability says that given a square matrix and a the following are equivalent:

  1. The pair is controllable
  2. For all it holds that
  3. For all that are eigenvalues of it holds that

Hautus Lemma for stabilizability

The Hautus lemma for stabilizability says that given a square matrix and a the following are equivalent:

  1. The pair is stabilizable
  2. For all that are eigenvalues of and for which it holds that

Hautus Lemma for observability

The Hautus lemma for observability says that given a square matrix and a the following are equivalent:

  1. The pair is observable
  2. For all it holds that
  3. For all that are eigenvalues of it holds that

Hautus Lemma for detectability

The Hautus lemma for detectability says that given a square matrix and a the following are equivalent:

  1. The pair is detectable
  2. For all that are eigenvalues of and for which it holds that

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.
  1. Belevitch, V. (1968). Classical Network Theory. San Francisco: Holden–Day.
  2. Popov, V. M. (1973). Hyperstability of Control Systems. Berlin: Springer-Verlag. p. 320.
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