Enhanced PID controller tuning for nonlinear continuous stirred-tank heaters using a modified Newton-Raphson optimizer with random opposition and Lévy-flight learning.

Accurate temperature regulation in continuous stirred-tank heater (CSTH) systems is vital in chemical and thermal process industries, where deviations can cause energy inefficiencies, product quality degradation, or even safety hazards. However, CSTH systems pose a formidable control challenge due to inherent nonlinearities, parameter uncertainties, and susceptibility to external disturbances. Conventional proportional-integral-derivative (PID) tuning methods often struggle to handle these complexities, resulting in sluggish responses or instability. This study introduces a modified Newton-Raphson-based optimization (mNRBO), for optimal PID tuning tailored to nonlinear CSTH environments. The mNRBO framework integrates two key innovations: random opposition learning, to enhance population diversity and prevent premature convergence, and Lévy-flight-based guided learning, to improve global exploration and escape local optima. These mechanisms are systematically embedded into the Newton-Raphson-based optimizer (NRBO) to achieve a robust exploration-exploitation balance. A CSTH dynamic model is formulated using mass and energy conservation principles, and a multi-objective cost function evaluates rise time, settling time, overshoot, and steady-state error under realistic process constraints. Simulation studies compare mNRBO with NRBO, hippopotamus optimization, golden eagle optimizer, and slime mould algorithm. Results show that mNRBO achieves the lowest cost function value 53.29, smooth convergence with standard deviation 0.90, and superior closed-loop performance with rise time 62.05 s, settling time 206.88 s, overshoot 1.41%, and steady-state error 0.006%. These findings confirm that mNRBO delivers high-precision, disturbance-resilient control and is a promising solution for industrial thermal processes requiring reliability, efficiency, and precision.