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Characteristic Ratio Symmetric Polynomials and Their Root Characteristics

Young Chol Kim
International Journal of Control, Automation, and Systems, vol. 19, no. 5, pp.1890-1906, 2021

Abstract : For a real polynomial p(s)=a_ns^n+...+a_1s+a_0, its characteristic ratios are defined by \alpha_i :=a_i^2/a_{i-1}a_{i+1} for i =1, 2, ..., n-1, and the generalized time constant is defined by \tau := a_1/a_0. In contrast, every coefficient of the polynomial p(s) can be represented in terms of \alpha_i and \tau. We present a novel family of polynomials named characteristic ratio symmetric (CRS), where a polynomial p(s) is said to be CRS if \alpha_i = \alpha_{n-i} for 1 <= i <= (n-1) with any \tau. This paper deals with the relationships between the roots and {\alpha_i, t} of a CRS polynomial. It is shown that some of the roots of the CRS polynomial are on the circle of a specific radius \omega_c while the rest appear in fourtuples {\lambda_i, \omega_c^2 / \lambda_i, \lambda_i^*, \omega_c^2 / \lambda_i^*}. For CRS polynomials of the fifth or lower order, we derive that the damping ratio and natural frequency of every root of these polynomials can be uniquely represented in terms of only {\alpha_1, \alpha_2, \tau} or {\alpha_1, \tau} for less than third order. It is also shown that a special polynomial named K-polynomial is a CRS polynomial and the damping of an nth-order K-polynomial can be adjusted by just choosing a single parameter \alpha_1.

Keyword : Characteristic ratio, characteristic ratio assignment(CRA), characteristic ratio symmetric(CRS), damping ratios, K-polynomial, natural frequency, polynomials, roots.

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