## 6 Conclusions  
The present work provides the output regulation theory for the case of unmodeled reference/disturbance signals and non-modeled failures that influence the actuator's operation through the HGO theory and the UI observer. The estimation of faults and unmodeled signals has been carried out using HGO of different dimensions and UI observers, demonstrating that even for simpler faults in the mechanism, the proposed modification enhances the performance of the output regulation. The output regulation theory, which applies to cases of unmodeled reference and disturbance signals with faults in actuators, is an alternative. As a result, it has been possible to mitigate the tracking error further for nonlinear problems observed, not only for exogenous reference/perturbation systems but also for internal model disturbances. This opens up new opportunities for its application on various platforms, including robotics, process control, synchronization, and entertainment. Finally, a Furuta Pendulum system with a different trajectory to follow has been presented in real-time to demonstrate the advantages of the proposed approach. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Aspect**} & \multicolumn{2}{c}{**Low-Frequency**} \\ \cline{2-3}  & \multicolumn{2}{c}{**Signal**} \\ \cline{2-3}  & **Simulation** & **Experiment** \\ \hline Convergence time & \(\sim\)2 s & Visually similar \\ \hline Fault estimation & High accuracy & Real-time fault \\ accuracy & via UI observer & estimation is accurate \\ \hline Control signal & \multirow{2}{*}{Within bounds \(\beta=22\)} & Satisfies LMI \\ compliance & & constraints \\ \hline Remarks & Smooth tracking and & Clear improvement versus \\ stable control & & previous method \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Aspect**} & \multicolumn{2}{c}{**High-Frequency**} \\ \cline{2-3}  & \multicolumn{2}{c}{**Signal**} \\ \cline{2-3}  & **Simulation** & **Experiment** \\ \hline Reference & \(\Psi_{2}(t)\) & \(\Psi_{2}(t)\) \\ \hline Tracking & Less precise tracking & Better tracking, \\ performance & due to signal frequency & reduced oscillations \\ \hline HGO convergence & Converges with & Still effective, captures \\ some degradation & & signal trend \\ \hline \multirow{2}{*}{\(x_{2}\) stabilization} & Controlled, with some & Vertical link more stable \\  & disturbance & than previous method \\ \hline Convergence time & \(\sim\)3 s & Slightly longer, with \\  & & fewer oscillations \\ \hline Fault estimation & Acceptable, some & More precise than \\ accuracy & transient error & prior strategy \\ \hline Control signal & Within bounds & Remains compliant \\ compliance & despite demand & with constraints \\ \hline MSE & – & 29.72 (vs. 71.96 in [24]) \\ \hline Remarks & Limited by & Strong fault \\  & mechanical response & rejection and tracking \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between the simulation and experimental results for high-frequency signals. **Author Contributions:** Conceptualization, [PERSON], [PERSON] and [PERSON]; data curation, [PERSON]; formal analysis, [PERSON], [PERSON], [PERSON] and [PERSON]; funding acquisition, [PERSON]; investigation, [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON]; methodology, [PERSON]-M., [PERSON], [PERSON] and [PERSON]; project administration, [PERSON], [PERSON] and [PERSON]; resources, [PERSON], [PERSON] and [PERSON]; software, M.A.-M.; supervision, [PERSON], [PERSON], [PERSON] and [PERSON]; validation, [PERSON], [PERSON] and [PERSON]; visualization, [PERSON], [PERSON] and [PERSON]; writing--original draft, [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON]; writing--review and editing, [PERSON], [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript.