Distribution of Roots of Quasi-Polynomials of Neutral Type and Its Application-Part I: Determination of the Number of Roots and Hurwitz Stability Criteria

This article proposes several criteria for the distribution of roots of quasi-polynomials of neutral type with complex coefficients. Compared with Pontryagin's results, the derived criteria can be numerically implemented because the interval of the frequency for analyzing the behavior of the quasi-polynomial can be determined. Moreover, some Hurwitz stability criteria to judge whether all the roots of the quasi-polynomials are in the open left-half complex plane are provided. These Hurwitz stability criteria can be employed to analyze the stability of linear time-invariant systems with commensurate delays. It should be pointed out that on the one hand, the derived criteria are general since quasi-polynomials of retarded type and quasi-polynomials with real coefficients are their special cases. On the other hand, the conditions in Hurwitz stability criteria are all necessary and sufficient. Furthermore, as a special case, several criteria for the distribution of roots of the quasi-polynomials with real coefficients are presented. For the proposed criteria, this article provides some examples to illustrate the implementation and presents the detailed analysis and proofs.