Stability analysis of systems with delay-dependent coefficients and commensurate delays

This paper develops a method of stability analysis of linear time-delay systems with commensurate delays and delay-dependent coefficients. The method is based on a D-decomposition formulation that consists of identifying the critical pairs of delay and frequency, and determining the corresponding crossing directions. The process of identifying the critical pairs consists of a magnitude condition and a phase condition. The magnitude condition utilizes the Orlando’s formula, and generates frequency curves within the delay interval of interest. Such frequency curves correspond to the delay-frequency pairs such that the decomposition equation has at least one solution on the unit circle. The delay interval of interest is divided into continuous frequency curve intervals (CFCIs). Under some nondegeneracy assumptions, the number of frequency curves remains constant within each CFCI, and the associated decomposition equation has one and only one solution on the unit circle at any point on a frequency curve. By traversing through the frequency curves, all the crossing points can be identified. The crossing direction is related to the sign of the lowest-order nonzero derivative of the phase angle with respect to the delay, which is a generalization of the existing literature even for the case with single delay. This conclusion allows one to determine the crossing direction by examining the phase angle vs delay diagram. An example is presented to illustrate how a stability analysis can be conducted if some nondegeneracy assumptions are violated.