A novel hyperbolic tangent-augmented controller framework for temperature control in jacketed continuous stirred tank reactors.

Accurate temperature regulation of jacketed continuous stirred tank reactors (CSTRs) remains a challenging task due to strong nonlinearities, tight coupling between mass and energy balances, and sensitivity to disturbances and operating-point variations. In this study, a novel augmented proportional–integral–derivative (PID) controller incorporating a hyperbolic tangent nonlinearity (APID-T) is proposed for robust temperature control of an exothermic CSTR. The controller structure extends the classical PID framework by embedding a bounded nonlinear term that enhances transient shaping and robustness while preserving simplicity and practical implementability. The tuning of the APID-T parameters is formulated as a constrained nonlinear optimization problem, where a composite objective function combining normalized overshoot and integral squared error is minimized. To solve this problem efficiently, the recently developed Schrödinger optimizer (SRA) is employed, exploiting its balanced exploration–exploitation mechanism. A detailed nonlinear dynamic model of the jacketed CSTR is considered, and stability characteristics around the nominal operating point are examined to ensure meaningful closed-loop operation. The proposed SRA-based APID-T design is extensively evaluated through comparative simulations against several state-of-the-art metaheuristic optimizers and alternative controller structures, including PI, PID with filter, two-degree-of-freedom PID, and fractional-order PID controllers. Performance is assessed using statistical indicators, convergence behavior, and time-domain response metrics under identical optimization settings. In addition, widely used error performance criteria, including the integral squared error, integral time absolute error, and integral time squared error, are computed to provide a comprehensive quantitative assessment of the tracking performance. The results demonstrate that the SRA-tuned APID-T controller consistently achieves lower objective-function values, faster convergence, reduced settling time, and significantly smaller overshoot compared with the competing approaches. Furthermore, frequency-domain analysis based on the Bode characteristics of the linearized open-loop system confirms favorable stability margins, supporting the robustness of the proposed control structure. Additional stability and robustness evaluations are conducted under practical non-ideal conditions, including feed-temperature disturbances, measurement noise, and multiple setpoint variations, where the controller maintains stable and accurate temperature regulation across the considered operating scenarios.