The asymptotic analysis of multiple imaginary characteristic roots for LTI delayed systems based on Puiseux–Newton diagram

The paper proposes a novel procedure for the asymptotic expansions of root loci around multiple imaginary roots of an exponential polynomial, which is necessary for the stability analysis of the LTI systems with commensurate delays. With the LTI delay systems given as exponential polynomials (also called quasi-polynomial), we seek to characterise the asymptotic behaviours of the characteristic roots of such systems in an algebraic way and determine whether the imaginary roots cross from one half-plane into another or only touch the imaginary axis. According to the Weierstrass preparation theorem, the quasi-polynomial equation is equivalent to an algebraic equation in the neighbourhood of a singular point. Furthermore, our result gives an explicit expression of the coefficients of the algebraic equation in infinite power series of delay parameter, and the determinations of such power series coefficients refer to the computation of residues of memorphic functions. Subsequently, the classic Puiseux–Newton diagram algorithm can be used to calculate the algebraic expansions of the reduced equation directly. Thus, the asymptotic behaviours of root loci around singular points of the quasi-polynomial equation are obtained. Some illustrative simulations are given to check the validity of the proposed method on asymptotic analysis with a powerful software.