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 γ > 0, two n-vectors b, c and an n by 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:xTPx = 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 Yakubovich and Popov.

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