## 2 Controller Presentation
This figure presents the response of the derivative terms under perturbations. While the classical \(D\) term amplifies noise and generates abrupt control actions, the \(D^{\text{FE}}\) term adaptively regulates its magnitude via the AE exponent, resulting in smoother and safer control signals. \[C_{P}(s)=k_{P}+\frac{k_{I}}{s}+k_{D}(s) \tag{2}\]0 Proposed adaptive _PI-D\({}^{\text{FE}}\)_ controller structure. This block diagram presents the architecture of the _PI-D\({}^{\text{FE}}\)_ controller, in which the adaptive exponent AE modulates the derivative action. The total control signal is composed of independent proportional, integral, and adaptive derivative components. A switching gain is included to enable or disable the derivative term as needed, providing flexibility in controller deployment. Equation (5) defines AE as follows:  
\[C_{P}(s)=k_{P}+\frac{k_{I}}{s}+k_{D}(s) \tag{2}\]1  
where \(\mu\) is the maximum limit predefined for AE; \(\sigma\) is a sensitivity factor (100 is the recommended value), which determines how reactive the algorithm will be to changes in \(Y(s)\); \(\epsilon\) is an indeterminate avoidance value (\(\epsilon\simeq 0\)); and \(\Lambda\) is an auxiliary gain (recommended values of 1 or 2) that can be added and is helpful in situations in which the gain must be modified indirectly from without altering the initial tuning. There are two possible scenarios:  
1. If the input is greater than the output, \(U(s)>Y(s)\), so the response of exponent AE is increasing and positive. For example, if \(Y(s)=0\), \(R(s)=1\), and \(\mu=\sigma=100\), this represents the condition \(Y(s)<R(s)\), where \(e(s)\) is positive and at its maximum. In this case, \(=100\), leading to a rapid response in \(D^{\AE}\) as its exponential increases. 2. If the output is greater than the input, \(Y(s)>U(s)\), so the response of AE is decreasing and negative. And, if \(Y(s)=2\), \(R(s)=1\), and \(\mu=\sigma=100\), this represents condition \(Y(s)>R(s)\), where \(e(s)\) is negative and at its maximum. In this case, is negative, leading to a rapid response in \(D^{\AE}\) as its exponential decreases. And results in a maximum negative decrease AE \(=-100\). \(\mu\) and \(\sigma\) can be adjusted based on the system's requirements, such as speed and noise rejection, as suggested by [PERSON] et al. (2013) for \(P\). These parameters are associated with a high crossover frequency of \(D\), which plays a critical role in achieving stability and optimal control system performance. The parameters \(\upmu\), \(\sigma\), and \(\epsilon\) directly influence the adaptive response of the exponent AE and, therefore, the behavior of the \(D^{\AE}\) term. Increasing \(\upmu\) allows a wider dynamic range for AE, making the controller more aggressive and responsive but potentially increasing overshoot. A higher \(\sigma\) makes the controller more sensitive to small changes in output, improving reactivity but also amplifying noise--excessively high \(\sigma\) can destabilize the system in noisy environments. Conversely, a lower \(\sigma\) smooths the response but reduces adaptability. The parameter \(\epsilon\) ensures numerical stability when the reference R(s) approaches zero. If \(\epsilon\) is too small, the system may become unstable; if too large, it weakens the effect of AE. Proper tuning of these parameters enables the controller to balance fast adaptation, robustness, and noise immunity effectively. The values presented in this article were carefully selected as ideal defaults for the initial presentation of the algorithm, providing a robust and balanced behavior across typical operating conditions. In an additional case, if \(Y(s)\ \simeq\ U(s)\), AE modifies its response, resulting in a null \(D^{\AE}\) response, even if noise variations in \(D\) occur.