Stabilizing sets of the low-order controllers for the systems with multiple time delays

The low-order stabilization of the systems with multiple time delays is more difficult compared with that of the rational system and the system with single time delay. For an arbitrarily-given LTI (linear time-invariant) multiple time-delay plant, the stabilization problem of the Proportional (P) and Proportional-Integral (PI) is solved in this paper. Based on the generalized Nyquist criterion applicable to the quasipolynomial, analytical algorithms to determine the entire stabilizing sets of the P and PI controllers are presented, which avoids the delay approximation. The control gains chosen in the resulting stabilizing set can ensure the stability of the closed-loop system. The results demonstrate that, the characterizations of the stabilizing P and PI controllers are in the closed form and quasi-closed form, respectively. The proposed method is applicable to stable, unstable or non-minimum phase system with multiple time delays. Numerical example is provided to illustrate the validity of the proposed algorithms.