Novel Properties of Spectrum for Stability Test of Linear Delay Differential Systems

This paper reveals novel properties of spectrum of linear delay differential systems, and concerns the stability of a general class of time delay systems. The stability issue can be well characterized by the distribution of spectrum of the associated characteristic equation. Benefit properties on finite and bounded are formulated. And the finiteness condition and distribution boundary are obtained in a practical way. Based on the argument principle, an analytical formula for computing the number of unstable zeros (with positive real part) is deduced, which is determined by definite integral. Moreover, a procedure put no restriction on system except for the delay engendering purely imaginary roots is proposed to investigate the stability of linear time delay systems. At last, typical examples are given to illustrate that the method carried out is reliable and efficient.