E The Kalman-Yakubovich Lemma

Lemma E.1 (Kalman-Yakubovich) Consider a controllable linear time-invariant system \[ \dot{x} = A x + b u \\ y = c^T x. \] The transfer function \[ h(p) = c^T (p I - A)^{-1} b \] is strictly positive real (SPR) if and only if there exist positive definite matrices \(P\) and \(Q\) such that \[ A^T P + P A = - Q \\ Pb = c. \]