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:$gamma+2\; Re\; [c^T\; (jomega\; I-A)^\{-1\}b]\; ge\; 0$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 Yakubovich and Popov.