From Stabilizing Regions to Certified Controllers: Closing the Selection Gap in Unified PID/PI Analysis for Time-Delay Plants

Abstract

A recent unified treatment of PID tuning for time-delay plants (An, Tang, Sun, Zhang and Chen, Automatica, 2026) combines the D-partition method with a boundary gradient vector (BGV) to orient the boundaries of stabilizing, relative-stability and stability-margin regions. That method answers a feasibility question, namely where admissible gains lie, and it leaves a manual interior-point test to fix the unstable-pole count in each cell, with the choice of a single controller left to the user. This note makes three contributions. First, the one operation the BGV leaves manual, the absolute unstable-pole count, is available analytically: exactly for delay-free designs through a companion-matrix or Routh count, and through an argument-principle (Mikhailov) evaluation for retarded-type delay loops. Labelling every cell with its analytic count removes the interior-point test and decides the whole partition. Second, we add the step the BGV framework cannot reach, a time-domain selection rule that returns one certified controller: among monotone step responses we choose the minimum-settling-time PI gains, characterized by a tangency condition, with monotonicity guaranteed by external positivity (a nonnegative closed-loop impulse response). Third, we flag a neutral-type pitfall that the unified analysis never delimits: an ideal PID with derivative action on a first-order-plus-dead-time (FOPTD) plant is of neutral type, with a root chain on the imaginary axis when k Kd = T. We reproduce the authors' delay-free benchmark exactly, recovering both admissible Kp intervals, and demonstrate the full pipeline on a FOPTD plant, delivering a certified monotone, fast-settling PI controller that the region-only method can neither locate nor justify; the selected gains match an independent closed-form tangency rule to within one percent. All claims are validated numerically.