## 2 Development of Proposed Controller
### Integer and Fractional-Order PID Controllers  
PID is a feedback control algorithm extensively used in automation and engineering systems. It adjusts the system performance by calculating the difference between the desired set point and the actual value of the variable's process. The three parameters of PID are \(K_{p}\), \(K_{i}\), and \(K_{d}\), which determine its behaviour, and its controller block diagram is shown in Figure 1. The schematic depiction shows the set point denoted by \(R(s)\), the system's output response by \(Y(s)\), the error by \(E(s)\), and the controller signal by \(U(s)\). It is worth noting that all these representations are in the Laplace domain, represented by the variable \"s\". The proportional gain (\(K_{p}\)) influences the control action based on the current error. A higher \(K_{p}\) value results in a more aggressive response, leading the system towards the desired set point more quickly. However, setting \(K_{p}\) too high can cause overshoot and instability, making it crucial to choose an optimal value. The integral gain (\(K_{i}\)) is a crucial parameter in control systems that helps eliminate steady-state errors and correct long-term deviations between desired settings and actual performance. On the other hand, the derivative gain (\(K_{d}\)) is a parameter in control systems that considers the error signal's rate of change. Combining these three terms enables a PID controller to achieve stability, responsiveness, and precision in controlling various systems. The control signal of the PID controller is given as follows,  
\[U(s)=\bigg{(}K_{p}+\frac{K_{i}}{s}+K_{d}s\bigg{)}E(s). \tag{1}\]  
Fractional-order PID (FOPID) controllers are advanced versions of traditional PID controllers that utilize fractional-order computations to improve the control performance of complex systems [38]. In contrast to standard PID controllers that have three parameters (proportional, integral, and derivative gains), FOPID controllers incorporate two additional parameters (\(\lambda\) and \(\mu\)) to achieve a better control performance. The FOPID controller's \(\lambda\) parameter modulates the integral gain over time, representing a time-varying factor that can be adjusted based on system dynamics. This allows the controller to adjust the integral gain to meet changing system behaviour and control needs. The \(\mu\) parameter of the FOPID controller scales the integral gain, fine-tuning the magnitude of the integral response to system characteristics and control goals. Adjusting \(\mu\) accordingly, the FOPID controller can adapt the integral response to meet specific control requirements. The FOPID controller's block diagram is shown in Figure 1. The control signal of the FOPID controller is given as follows:  
\[U(s)=\bigg{(}K_{p}+\frac{K_{i}}{s^{\lambda}}+K_{d}s^{\mu}\bigg{)}E(s),\;0< \lambda,\mu<1. \tag{2}\]  
Achieving optimal control performance in FOPID controllers requires careful \(K_{p}\), \(K_{i}\), \(K_{d}\), \(\lambda\), and \(\mu\) tuning to balance responsiveness, stability, and robustness. Various optimization, heuristic, or model-based approaches can be employed to obtain appropriate parameter values for a specific control application. Integrating \(\lambda\) and \(\mu\) parameters and the standard gain elevates FOPID controllers' performance by enhancing control accu  
Figure 1: Block diagram of integer and fractional PID controllers.  
racy, noise rejection, and adaptability to nonlinear and time-varying dynamics. These controllers are becoming increasingly popular in fields such as process control, robotics, and power electronics, where precise and robust control is crucial for handling complex system behaviour.