## 2 Controller Presentation  
\(PID\) contains three parts, as follows:  
\[C_{P}(s)=P+I+D \tag{1}\]  
The parallel structure of \(PID\) is selected as shown in Equation (2). \[C_{P}(s)=k_{P}+\frac{k_{I}}{s}+k_{D}(s) \tag{2}\]  
Now, a new concept of \(\AE\) is presented:  
\[C_{P}(t)=\frac{U(s)}{Y(s)}=k_{P}e(t)+\frac{k_{I}}{I}e(t)+\left[k_{D}\frac{de(t )}{dt}\right]^{\AE} \tag{3}\]  
\[C_{P}(s)=k_{P}+\frac{k_{I}}{s}+\left[k_{D}(s)\right]^{\AE} \tag{4}\]  
\(\AE\) dynamically adjusts the \(D\) system's behavior by accelerating or mitigating the response, depending on the desired control effect, as shown in Figure 1.  
For example, if the exponent is \(\AE=1\), we obtain classic \(PID\), and if \(\AE=0\), we obtain \(PI\). Extending this concept, \(\AE\) is assumed to take values within the range \((-\infty,\infty)\), including infinitesimally small values approaching zero, offering a continuum of control strategies between \(PI\) and \(PID\). This simple and robust idea considers an adaptive exponent () for \(D\), proposing that \(D\) be enhanced as a dynamic, adaptive parameter instead of a fixed constant. Based on the classical \(PID\) response (the Output signal amplitude system depicted in Figure 2), the proposed concept introduces an analysis of \(D\) across its full range of values \((-\infty,\infty)\). This analysis redefines \(D\) as a new, independent parameter with a \(D^{\text{E}}\) function, enabling the transformation of the system's response. By employing the proposed exponential function, \(D^{\text{E}}\), the system's future behavior can be effectively modified, as illustrated in Figures 2 and 3. Figure 2: _PID_ vs. _PI-D_\({}^{\rm E}\) responses. The figure highlights how the _PI-D_ controller modifies the output dynamics compared with a traditional PID. The use of the adaptive exponent \(\AE\) results in faster rise times, reduced overshoot, and shorter settling times, demonstrating improved robustness to setpoint changes. Figure 1: \(D^{\AE}\) modification of behavior response. This figure illustrates how the proposed \(D^{\AE}\) term dynamically adjusts the influence of the derivative component. Depending on the value of the exponent \(\AE\), the controller’s behavior can range from PI control (\(\AE=0\)) to classical PID control (\(\AE=1\)) or to even more aggressive responses. This adaptability enhances the system’s transient performance without relying on external filters. According to [PERSON] [4], maintaining the robustness of a \(PID\) design is essential; a positive \(D\) (\(D>0\)) can destabilize the system and provide excessive disturbance, while a negative signal (\(D<0\)) is negligible. To prevent instability, \(D\) can be deactivated. In the \(PI\)-\(D\) controller, \(D^{\mathcal{E}}\) remains continuously active, ready to respond to any disturbance or \(SP\) change. It dynamically adjusts the behavior of \(D\) in real time at any \(Y(s)\), as illustrated in Figures 4 and 5. Figure 4: \(PID\) and \(PI-D^{\mathcal{E}}\) responses with disturbance. The figure demonstrates the superior disturbance rejection capability of the \(PI-D^{\mathcal{E}}\) controller. Compared with the conventional PID, the \(PI-D^{\mathcal{E}}\) recovers equilibrium faster and exhibits reduced oscillatory behavior, ensuring improved robustness in non-ideal environments. Figure 3: \(D\) vs. \(D^{\mathcal{E}}\) responses. This plot shows how \(D\) either attenuates or intensifies the derivative action based on the system state. This adaptiveness prevents noise amplification and contributes to improved controller stability under disturbances or time-varying conditions. _PI-D_ employs an (AE) adaptive exponent within its structure. Unlike other _PID_-based controllers, this approach uses an adaptive exponent with an open-loop adaptive control mechanism, shown in Figure 6, allowing the behavior of \(D\) to be modified instantly using only the input-to-output relationship of C\({}_{\text{P}}\). Figure 5: \(D\) and \(D^{\text{FE}}\) responses with disturbance.