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, 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 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