math.OC — Optimization and Control

Quadratic Strictly Positive Real Factors of Stable Transfer Functions

Contributed by Jinlin Wang, Zhenyu Sun

We characterize factorizations $G=G_0\circ H$ in which $G$ and $G_0$ are stable, minimum-phase, biproper real-rational functions with simple poles and zeros and positive gains at infinity, and $H$ is a quadratic strictly positive real function. Classical rational decomposition identifies the quadratic right-factor classes with real Möbius involutions fixing $G$. Within each class, we parameterize all normalized representatives and eliminate the scaling parameter and the frequency variable. The resulting sign conditions involve univariate polynomials of degree at most four. They determine all feasible denominator intervals and all factorizations in the class; for real-algebraic input, they give an exact decision procedure. We also give a scaling that maximizes the smaller of the inner positive-real margin and the outer spectral margin for a fixed denominator. For quadratic input, existence is equivalent to strict positive realness of $G$ and to an explicit coefficient inequality. Examples exhibit a disconnected feasible parameter set, separate algebraic and analytic obstructions, and a quartic input with exactly three admissible classes.