Robust Stability Analysis is one of the most important and fundamental problems in analyzing linear systems. Motivated by Kharitonov's (1978) seminal theorem on robust stability for a class of polynomials, a lot of papers have tried to extend the result for such neat results. Some concentrated on the so-called dual problem for a diamond of polynomials. Some study the arbitrary D-domain in complex plane, e.g. the unit disc for discrete linear systems. Some consider the coefficients affinely dependent on parameter space, and even multi-affinely. In this paper, a series of historical results are unified by the non-linear mathematical programming approach with its Kuhn-Tucker conditions via value set analysis and zero exclusion principle. These results include: the famous Edge Theorem, the robust stability of diamond families, the robust stability of box polynomials for arbitrary D-domain, and the principal points concept for the robust stability of multi- affinely dependent coefficients for arbitrary D-domain This paper studies the stabilization of diamond plants with the fixed compensator whose numerator and denominator are chosen from even or odd polynomials. By the quadratic programming approach, it is proved that the necessary and sufficient conditions for the compensator to robustly stabilize the diamond plants is that it simultaneously stabilizes 16 vertex plants when both of the numerator and denominator are even or odd, and 8 vertex plants if the coefficients of the odd or even numerator and denominator of the fixed compensator positively and negatively interlaced. When numerator is even but denominator odd, or the reverse, it is suffices to check 32 specially selected edges.
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