Gudermannian-based error-shaped PID controller tuned via THRO for nonlinear CSTR temperature control

Temperature regulation in nonlinear continuous stirred-tank reactors (CSTRs) is challenging due to the strong coupling between reaction kinetics and heat transfer, which leads to state-dependent gain variations and asymmetric transient behaviour. This study addresses the limitation of classical PID controllers under such nonlinear conditions by modifying the error-processing stage rather than the controller structure itself. A Gudermannian-based error-shaping mechanism is introduced into a standard PID architecture to obtain a smooth, bounded, and continuously differentiable transformation of the tracking error. This mapping attenuates large error magnitudes while preserving sensitivity near the setpoint, resulting in an error-dependent modulation of the proportional, integral, and derivative actions. The controller parameters are tuned using Tianji's Horse Racing Optimizer (THRO) within a closed-loop nonlinear simulation framework. Proposed gd-PID controller is evaluated on a nonlinear CSTR temperature model and compared with PI, PIDf, two-degree-of-freedom PID, and fractional-order PID controllers under identical conditions. Simulation results show that the gd-PID achieves a rise time of 0.586 min and a settling time of 1.852 min, while limiting overshoot to 0.2186%. The integral absolute error is reduced to 2.858, representing a reduction of more than 55% compared with conventional PID-based designs. Additional tests involving multi-step reference changes, feed-temperature disturbances, and measurement noise confirm that the proposed controller maintains stable and physically consistent behaviour across wide operating regions. The results indicate that Gudermannian-based error shaping, combined with THRO-based tuning, provides an effective and computationally simple approach for nonlinear CSTR temperature control.