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.
There exist multiple forms of the lemma: The Hautus lemma for controllability says that given a square matrix
(
)
the following are equivalent: The Hautus lemma for stabilizability says that given a square matrix
the following are equivalent: The Hautus lemma for observability says that given a square matrix
)
the following are equivalent: The Hautus lemma for detectability says that given a square matrix
the following are equivalent: