## 1 Introduction  
Pressure and level process plants are widely utilized in various industrial applications, including chemical, petrochemical, and food processing [1, 2]. Process plants commonly experience dynamic set point changes and disturbances significantly affecting control performance [3]. Commonly, PID (proportional-integral-derivative) controllers are used to control these processes in an efficient operation of these factories, which is critical to maintaining product quality and reducing operating costs. Traditional PID controllers sometimes respond promptly and effectively, resulting in slow response times, overshoots, and poor noise rejection [4]. Further, systems that exhibit non-minimal phase responses, high-order dynamics, and extensive transmission delays often limit the PID controllers' performance [5]. The complex-order controller is a fractional order designed to overcome the limitations of regular PID controllers. Complex-order PI (proportional-integral)-PD (proportional-derivative) controllers have fractional derivative and integral terms, allowing the controller to operate with high dynamics and longer transit delays [6]. A PID controller is a standard feedback control system seen in industrial environments, which is essential for the controlling system output depending on the input. The system performance varies with a proportional component proportional to the deviation between target performance and actual performance [5]. The integral component continuously corrects for steady-state errors by integrating accumulated errors over time. Abrupt fluctuations in the error rate are predicted and compensated for by the derivative component. PID controllers are widely used in most industries, such as temperature control, motor speed control, pump flow control, satellite communications [7], and aircraft altitude control [8]. Recently, PID has been used in robotics, robotic systems, and computer numerical control systems due to its advantages, including simple tuning, maintaining system stability, and minimizing overshoot [9]. It offers greater accuracy than manual adjustment methods and the flexibility to adjust the system output to various operating conditions [10]. PID controllers can be implemented in digital or analogue systems. Still, digital PID controllers are generally more accurate and more accessible to implement, quickly responding to input changes and adjusting system outputs accordingly [11]. In the literature, an algorithmic optimization technique based on the flower pollination algorithm [12] tunes a PI-PD cascade controller. It evaluated its performance based on settling time, overshoot, and undershoot. Similarly, researchers in [13] achieved the settling time, overshoot, and phase margin using their cascaded fractional-order PI-PD approach for the inverted pendulum. In [14], the researchers used arithmetic and trigonometric operators to obtain the fractional-order predictive PI controller parameters using the integer absolute time error (ITAE) as the objective function to achieve effective set-point tracking and disturbance rejection performance. In another study, hybrid probabilistic fractal search and pattern search techniques were used to tune a cascaded PI-PD controller and evaluated its performance based on the settling time, integer squared error (ISE), integer time squared error (ITSE), integer absolute error (IAE), and ITAE [15]. In [16], [PERSON] et al. used particle swarm optimization (PSO) and genetic algorithms to tune PID, I-PD, and fractional order PI-PD controllers in a DC-DC converter, where the performance criteria are the settling time and overshoot. In addition, modified PI-PD controllers are tuned using an automated tuning method, and their performance has been assessed based on the settling time, overshoot, and IAE, with comparisons made against conventional PID and modified PID controllers [17].