A Method to Determine Oscillation Emergence Bifurcation in Time-Delayed LTI System with Single Lag

Table 1

General forms of the characteristic equation for time-delayed system.

Dim	General forms of the characteristic equation	𝑁𝑐
1	𝑠+𝑎1+𝑎2𝐷=0	2
2	𝑠^2+𝑎1𝑠+𝑎2+(𝑎3𝑠+𝑎4)𝐷+𝑎5𝐷^2=0	5
3	𝑠^3+𝑎1𝑠^2+𝑎2𝑠+𝑎3+(𝑎4𝑠^2+𝑎5𝑠+𝑎6)𝐷+(𝑎7𝑠+𝑎8)𝐷^2+𝑎9𝐷^3=0	9
4	𝑠^4+𝑎1𝑠^3+𝑎2𝑠^2+𝑎3𝑠+𝑎4+(𝑎5𝑠^3+𝑎6𝑠^2+𝑎7𝑠+𝑎8)𝐷+(𝑎9𝑠^2+𝑎10𝑠+𝑎11)𝐷^2+(𝑎12𝑠+𝑎13)𝐷^3+𝑎14𝐷^3=0	14
5	𝑠^5+𝑎1𝑠^4+𝑎2𝑠^3+𝑎3𝑠^2+𝑎4𝑠+𝑎5+(𝑎6𝑠^4+𝑎7𝑠^3+𝑎8𝑠^2+𝑎9𝑠+𝑎10)𝐷+(𝑎11𝑠^3+𝑎12𝑠^2+𝑎13𝑠+𝑎14)𝐷^2+(𝑎15𝑠^2+𝑐16𝑠+𝑎17)𝐷^3+(𝑎18𝑠+𝑎19)𝐷^4+𝑎20𝐷^5=0	20