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 and. Today it can be found in most textbooks on control theory.
Contents
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
- The pair
( A , B ) is controllable - For all
λ ∈ C it holds thatrank [ λ I − A , B ] = n - For all
λ ∈ C that are eigenvalues ofA it holds thatrank [ λ I − A , B ] = n
Hautus Lemma for stabilizability
The Hautus lemma for stabilizability says that given a square matrix
- The pair
( A , B ) is stabilizable - For all
λ ∈ C that are eigenvalues ofA and for whichℜ ( λ ) ≥ 0 it holds thatrank [ λ I − A , B ] = n