## 4 Simulation Results
### Inverted Pendulum on a Cart  
The _IPC_ model was obtained according to previous studies [14; 15; 16], with \(M=2.4\) kg, \(m=0.23\) kg, \(L=0.36\) m, and \(g=9.81\) m/s\({}^{2}\). The state space equations are as follows:  
\[\dot{x_{1}}=x_{2} \tag{17}\]  
\[\dot{x_{2}}=\frac{(2.4+0.23)9.81 Sinx_{1}-0.23\cdot 0.36\ x_{2}{}^{2}(Sinx_{1} Cosx_{1})-uCosx_{1}}{2.4\cdot 0.36+0.23\cdot 0.36\ \left(Sinx_{1}\right)^{2}} \tag{18}\]  
\[\dot{x_{3}}=x_{4} \tag{19}\]  
\[\dot{x_{4}}=\frac{u+0.23\cdot 0.36\ x_{2}{}^{2}(Sinx_{1})-0.23\cdot 9.81 Cosx_{1}Sinx_{1}}{2.4+0.23\left(Sinx_{1}\right)^{2}} \tag{20}\]  
Equilibrium and position control are summed into one control signal; therefore, position control is affected by equilibrium control. To gain a better understanding, let us consider a first _IPC_ (Figure 18), where its equilibrium is controlled by the _PI-D\({}^{\text{E}}\)_ controller and its position is controlled by the _PID_ controller (affected by the _PI-D\({}^{\text{E}}\)_ equilibrium controller). Considering a second _IPC_ (Figure 19), its equilibrium is controlled by a conventional _PID_ controller, and its position is controlled by another _PID_ controller (affected by the _PID_ equilibrium controller). Figure 19: _IPC#2_ control. _PID_ equilibrium control + _PID_ position control. Furthermore, the traditional derivative term is defined as Equation (21):  
\[D(t)=k_{D}\frac{de(t)}{dt} \tag{21}\]  
where \(k_{D}\) is a fixed gain, and e(t) is the control error. In contrast, the proposed term modifies the derivative behavior through an adaptive exponent as Equation (22):  
\[D^{\mathcal{E}}(t)=\left(k_{D}\frac{de(t)}{dt}\right)^{\mathcal{E}} \tag{22}\]  
where  
\[\mathcal{E}(t)=\Lambda\left(\mu-\frac{\sigma Y(s)}{R(s)+\epsilon}\right) \tag{23}\]  
This formulation allows the gain applied to the derivative to change dynamically in real time, based on the system's input-output ratio \(Y(s)/R(s)\) Equation (23), modulated by the sensitivity \(\sigma\), a maximum bound \(\mu\), and a robustness constant \(\epsilon\). The behavior of \(D^{\mathcal{E}}\) becomes nonlinear and adaptive:  
If \(\mathcal{E}=1\), the system behaves identically to the classical PID. If \(\mathcal{E}<1\), the derivative contribution is softened, reducing noise amplification. If \(\mathcal{E}>1\), the derivative contribution is enhanced, accelerating system response. If \(\mathcal{E}\to 0\), the derivative term vanishes, effectively converting the controller to a PI type in real-time under certain conditions. To illustrate this, we have also included Figures 3, 5, 18 and 19 comparing the behavior and structure of \(PI\)-\(D^{\mathcal{E}}\) and \(PID\), especially under disturbances, to highlight how the adaptive exponent enhances robustness and mitigates derivative kick and noise amplification. The following comparisons were employed: the equilibrium control comparison is \(PI\)-\(D^{\mathcal{E}}\) vs. \(PID\), and the position control comparison is the affected \(PID\) (indirectly affected by the \(PI\)-\(D^{\mathcal{E}}\) equilibrium controller) vs. the affected \(PID\) (affected by the \(PID\) equilibrium controller). Tables 5 and 6 show the parameters of the controllers for balance and position, respectively. The deactivation of \(I\) is due to the fact that the authors of the original test [14; 15; 16] conducted their simulations in this way, and we aimed to respect this condition by offering a comparison with the originally proposed \(PID\) controller. This does not preclude the fact that the value of the \(AE\) algorithm can be further explored by comparing it with \(PID\) controllers with all three activated terms (as with the first-order system). Figure 20 shows an angle control comparison between \(PID\) and \(PI\)-\(D^{AE}\). Figures 21-23 compare \(D\) and \(D^{AE}\) behavior. Typically, \(D\) is equal to \(k_{D}\)-\(de(t)\). However, a distinction is made. \(D\) and \(D^{AE}\) will be considered \(D\) control signals. \(k_  
Figure 24 shows a comparison of the position control response between two PID configurations.