Stability analysis of double time-delay systems with PID controller based on the τ-decomposition method

This paper focuses on the issue related to the stabilization of double time-delay process with proportional-integral-derivative (PID) controllers. Inspired by the τ -decomposition method, the Rekasius substitution method is used to deal with the delay parameters in the system. Based on the expressions of the characteristic equations of the system, effective auxiliary characteristic equation is constructed, and revealed the constraint conditions of the relevant parameters that make the system stable. By solving for crossing frequency intervals, the complete crossing frequency set is obtained. Meanwhile, the double-delay range curve that makes the system stable are also constructed in the delay parameter plane. Then, this paper applied the proposed theory to a class of linear-time-invariant first and second order double time-delay systems with PID controller. The parameter constraints ensuring system stability were calculated and the feasibility of the theoretical method is verified by two examples.