Revisiting Chien-Hrones-Reswick Method for an Analytical Solution

Abstract

This study presents an analytical method for tuning PI controllers in first-order with time delay (FOTD) systems, leveraging the Lambert W function. The Lambert W function enables exact pole placement, yielding direct analytical expressions for PI gains. The proposed approach identifies a critical condition that achieves a step response without overshoot and with minimum settling time, while also providing explicit tuning rules for systems where controlled overshoot is specified. The method demonstrates strong agreement with established empirical Chien-Hrones-Reswick tuning rules for both non-overshooting and overshooting cases, bridging the gap between theoretical analysis and empirical results.