Calculating Characteristic Roots of Multi-Delayed Systems with Accumulation Points via a Definite Integral Method

Multi-delayed systems, especially the neutral ones, have infinitely many and complex distributed characteristic roots that are crucial for system dynamics. The definite integral method, which determines the system stability by using only a definite integral, is extended in this paper for calculating all the characteristic roots in an arbitrarily given area on the complex plane of both retarded and neutral multi-delayed systems with constant discrete delays. Two simple algorithms are proposed for implementing the proposed method, by first calculating the distribution of the real parts of all the characteristic roots, then the imaginary parts by using an iteration method. The real part distribution can be used for the quick estimation of key characteristic roots such as the rightmost ones or the corresponding accumulation point(s), thus allowing adjusting the upper limit of the integral to further simplify the calculation procedure. Examples are given to show the feasibility and the efficiency of the proposed method through numerical analyses.