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Quadratic Strictly Positive Real Factors of Stable Transfer Functions

Primarily human-written textHuman understanding: all partsmath.OC — Optimization and Controlcs.SC — Symbolic Computation

Contributed by Jinlin Wang, Zhenyu Sun ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We characterize factorizations G=G0∘HG=G_0\circ H in which GG and G0G_0 are stable, minimum-phase, biproper real-rational functions with simple poles and zeros and positive gains at infinity, and HH is a quadratic strictly positive real function. Classical rational decomposition identifies the quadratic right-factor classes with real Möbius involutions fixing GG. 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 GG 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.

Provenance statement

The text is primarily human-written. AI assistance was used for proof review, symbolic cross-checks, prose editing, and LaTeX preparation. Exact symbolic checks used SymPy. All parts have been checked and understood by a human.

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Version history

  1. v1 · 18 pagesSubmitted by Zhenyu SunInitial depositCurrentNov 12, 2026