The direct method for stability analysis of time delayed LTI systems

A general class of linear time invariant systems with time delay is studied. Recently, they attracted considerable interest in the systems and controls community. The complexity arises due to the exponential type transcendental terms in their characteristic equation. The transcendentiality brings infinitely many characteristic roots, which are cumbersome to elaborate as evident from the literature. A number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay. This study offers an exact, structured and robust methodology to bring a closure to the question at hand. Ultimately we present an explicit analytical expression in terms of the system parameters, which not only reveals the stability regions (pockets) in the domain of time delay, but it also declares the number of unstable characteristic roots at any given pocket. The method starts with the determination of all possible purely imaginary (resonant) characteristic roots for any positive time delay. To achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation. It is proven that the number of such resonant roots for a given dynamics is finite. Each one of these roots is created by infinitely many time delays, which are periodically distributed. An interesting property is also claimed next, that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency. These two findings facilitate a simple and practical stability method, which is the highlight of the work.