Computing the Zeros of Characteristic Function of Time-Delayed Systems

In this brief, a simple algorithm based on the contour integration is presented for computing the zeros of characteristic function of time-delayed systems with invariant time constant. Three measures are adopted to ensure the effectiveness of our algorithm: First, by generating
${n}$
evenly spaced points in the specified region,
${n}$
modified integral functions based on these points can be introduced. Compared with the existing contour integration method, the highest degree of the integral functions to calculate is reduced from
$2{n}$
to
${n}$
+1 due to the above integral functions’ modification. Second, through a linear conformal mapping, the integral contour is mapped onto the unit circle at the origin, and the characteristic function to be solved is mapped correspondingly. Third, perform another round of calculation using the modified integral functions based on
$n$
newly solved zeros instead of
${n}$
evenly spaced points. Results show that our algorithm can solve a large number of zeros with satisfying accuracy.