The root-locus method of synthesis of stable polynomials by adjustment of all coefficients

Abstract

The problem of synthesis of an asymptotically stable polynomial on the basis of the initial unstable polynomial is solved. For the purpose of its solution, the notion of the extended (complete) root locus of a polynomial is introduced, which enables one to observe the dynamics of all its coefficients simultaneously, to isolate the root-locus trajectories, along which values of each coefficient change, to establish their interrelation, which provides a way of using these trajectories as “conductors” for the movement of roots in the desired domains. Values of the coefficients that ensure the stability of a polynomial are chosen from the stability intervals found on the stated trajectories as the nearest values to the values of appropriate coefficients of the unstable polynomial or by any other criterion, for example, the criterion of provision of the required stability reserve. The sphere of application of the root locus, which is conventionally used for synthesis of characteristic polynomials through the variation of only one parameter (coefficient) of the polynomial, is extended for the synthesis of polynomials by way of changing all coefficients and with many changing coefficients. Examples of application of the developed algorithm are considered for the synthesis of stable polynomials with constant and interval coefficients.