Kalman–Yakubovich–Popov lemma

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The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number \gamma > 0, two n-vectors B, C and an n x n Hurwitz matrix A, if the pair (A,B) is completely controllable, then a symmetric matrix P and a vector Q satisfying

A^T P + P A = -Q Q^T\,
 P B-C = \sqrt{\gamma}Q\,

exist if and only if


\gamma+2 Re[C^T (j\omega I-A)^{-1}B]\ge 0

Moreover, the set \{x: x^T P x = 0\} is the unobservable subspace for the pair (A,B).

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 Kalman, who brought together results by Vladimir Andreevich Yakubovich and Vasile Mihai Popov.