Calculation of the rightmost characteristic root of retarded time-delay systems via Lambert W function

Generally, it is not easy to analyze the stability of time-delay systems, especially when the systems are of high order or they have multiple delays. For retarded time-delay systems, the stability can be determined by the rightmost characteristic root. This paper presents a case study on the calculation of the rightmost root. Three practical time-delay systems are discussed. The first system is an oscillator with delayed state feedback, the second one is a delayed neural network based on the FitzHugh–Nagumo model for neural cells, and the third one is a car model of suspension with a delayed sky-hook damper. By using the Lambert W function, the rightmost root becomes a root of a function associated with the principal branch of the Lambert W function. Then the rightmost root is located by using Newton–Raphson's scheme or Halley's accelerating scheme. Some suggestions for successful application of the proposed method are given.