Stability Analysis and Parameter Design for Stabilizing One Delay Point in Two-Delay Differential Equations

This paper studies stability of a general quasi-polynomial equation with two rationally independent delays, in order to reveal exhaustive stability distribution in delay space and stabilize system at certain unstable delay. First, the quasipolynomial equation is transformed into a combination problem of vector triangles in complex plane. Range of frequency spectra could be obtained with help of length condition and sweeping frequency. And method to declare potential stability switching curves is inferred from the phase condition. Second, the proposed tendency of imaginary roots and its invariant properties lead to the revelation of exact stability map. Last, two strategies based on improving stable domains to cover delay points are presented to realize stabilization of system. Simulations are presented to demonstrate the feasibilty of our methods.