## 1 Introduction  
Continuous stirred tank reactors (CSTRs) are widely used in industrial chemical processes. These reactors maintain constant mixing, ensuring uniform reactant distribution and product removal. However, exothermic reactions within CSTRs can generate substantial heat, potentially leading to uncontrolled reactions and safety hazards. To prevent thermal runaway and overheating, effective temperature control through cooling mechanisms is essential for stabilizing the reaction process. Extensive research has been conducted on temperature control in CSTR processes, exploring various controller structures and tuning methods [1, 2]. While traditional fixed-gain proportional-integral-derivative (PID) controllers offer simplicity and cost-effectiveness [3, 4], their performance can deteriorate when operating conditions deviate from the design point or when applied to highly nonlinear systems like CSTRs. This limitation has driven the development of advanced PID controller variants [5]. Several approaches aim to address the nonlinearities inherent in control systems. These include gain scheduling, fractional-order techniques, and nonlinear PID control structures. Gain scheduling techniques enhance control performance by adjusting controller gains according to the system's current operating conditions, effectively linearizing the system around various operating points. [PERSON] and [PERSON] [6] presented a gain-scheduled composite nonlinear feedback controller for CSTRs, combining linear and nonlinear control laws. [PERSON] et al. [7] conducted a comparative study of 2-DOF (2-degree-of-freedom) PID and 2 DOF fractional-order PID (FOPID) controllers, reviewing their different forms. [PERSON] et al. [8] successfully applied a fractional-order PID controller to regulate CSTR process parameters within desired limits. The FOPID controller parameters were optimized using genetic algorithms and particle swarm optimization. Building upon these advancements, [PERSON] et al. [9] integrated the strengths of both fractional-order PID control and fuzzy logic. This hybrid approach utilizes fractional-order PID controllers as local controllers within distinct operating regions, enhancing control performance. A Takagi-Sugeno fuzzy gain scheduling mechanism then seamlessly transitions between the outputs of these local controllers, ensuring smooth and adaptive control across the entire operating range. Nonlinear proportional-integral-derivative (NPID) control strategies have been extensively investigated for systems exhibiting complex and time-varying dynamics, building upon [PERSON]'s pioneering research [10; 11]. NPID controllers offer significant advantages by allowing for customization through the adjustment of their nonlinear functions and parameters. This flexibility enables precise tuning of the control system to achieve optimal performance. Contributing to the several notable advancements in NPID control, [PERSON] [10; 11] pioneered a novel approach to nonlinear PID control. He introduced a class of controllers that incorporate a nonlinear gain element in cascade with a conventional linear fixed-gain PID controller. This framework explored three distinct nonlinear gain functions: the sigmoidal function, the hyperbolic function, and the piecewise-linear function. The stability of the closed-loop systems resulting from these configurations was rigorously analyzed using the Popov stability criterion. To improve the control performance, [PERSON] and [PERSON] [12] developed a dynamic NPID controller that adapts its parameters in real time based on the system's error. [PERSON] et al. [13] proposed an NPID controller that effectively combines linear and nonlinear elements, particularly by adjusting the integral gain using a Gaussian function. [PERSON] and [PERSON] [14] introduced a design method for a nonlinear variable-gain PID controller, where the gains are determined by a set of fuzzy rules. The optimal parameter values for this controller were determined using a genetic algorithm.