Kalman–Yakubovich–Popov lemma

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 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 $\{x: x^T P x = 0\}$ 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.