Finding the exact value of the maximum allowable upper bound of the delay parameters in the multiple-delay LTI systems

This paper deals with the problem of determining the exact value of the maximum allowable upper bound (MAUB) of delay of an LTI system in the presence of the delay parameters. The delay parameter is supposed to be time-invariant, and this paper proposes a systematic method to exactly solve the mentioned problem. For this purpose, a modified system matrix is defined based on the delay system’s model that involves an extra complex variable on the boundary of the unit circle. It will be shown that the characteristic polynomial of the modified system can equivalently assess the system’s stability. The characteristic polynomial of the modified system is a quadratic function of the imaginary part of the extra complex variable which enables us to rewrite the polynomial based on only the real part of the extra variable. This property is exploited to develop an explicit formula to find the MAUB based on the critical numbers (complex numbers that force the modified characteristic polynomial to have pure imaginary roots) and their related pure imaginary roots. The method is tested for some sample delay systems and the results demonstrate a tighter bound for the MAUB of the systems.