Kalman–Yakubovich–Popov lemma
The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair
is completely controllable, then a symmetric matrix P and a vector Q satisfying
exist if and only if
Moreover, the set is the unobservable subspace for the pair
.
The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain.
It was derived in 1962 by Rudolf E. Kalman,[1] who brought together results by Vladimir Andreevich Yakubovich and Vasile Mihai Popov.
Multivariable Kalman–Yakubovich–Popov lemma
Given with
for all
and
controllable, the following are equivalent:
- for all
- there exists a matrix
such that
and
The corresponding equivalence for strict inequalities holds even if is not controllable. [2]