Reduction of high degree characteristic polynomials to 2nd degree for infinite range of gain applying gain margins and phase margins curve

The goal of this paper is to introduce an innovate analytic procedure to reduce the high degree of any characteristic equation to minor degrees applying the criterion of gain margin and phase margin curve (GMPMC) to design control systems. The philosophy of GMPMC is to develop quantitative results so in this case, the main characteristic of the GMPMC procedure is to develop and present system behaviors for infinite variations of gain and show its total characteristics in a curve which contains all their possible gain margins and phase margins, so all polynomial approaches to the minor degrees have exactly the same phase margin of the original one when gain changes in infinite intervals. Another important characteristic of this procedure allows us to approach the final response y(t) with the contribution of all its space state variables once one of them has been eliminated