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
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.
The Kalman–Popov–Yakubovich lemma which was first formulated and proved in 1962 by Vladimir Andreevich Yakubovich[1] where it was stated that for the strict frequency inequality.
The case of nonstrict frequency inequality was published in 1963 by Rudolf E.
[2] In that paper the relation to solvability of the Lur’e equations was also established.
Both papers considered scalar-input systems.
The constraint on the control dimensionality was removed in 1964 by Gantmakher and Yakubovich[3] and independently by Vasile Mihai Popov.
[4] Extensive reviews of the topic can be found in [5] and in Chapter 3 of.