Periodized characteristic equation and stability analysis of linear systems with delay

When the stability analysis for continuous linear time-invariant systems with pure time-delay is performed in the Laplace domain, then the answer is given by the roots of a transcendent characteristic equation. Instead of this original characteristic equation, the present paper investigates a periodized characteristic equation. The relations are based on a transcendent transformation, which maps the right half plane into the unit disc. As a second idea, the system dynamics are formulated by an integral equation. Then, applying the Fredholm theory on integral equations, this idea allows us to investigate an approximate periodized characteristic equations instead of the exact periodized characteristic equation. The paper provides constructive conditions under which the approximate and the exact equations possess the same number of unstable roots. The method is explained at the simple case with only one delay, but generalizations are drafted. The application of the method is shown by an example.