## 5 Conclusions  
The challenge to design MB and MF robust control strategies to achieve desired performance for different inputs in terms of transient behavior and steady-state/tracking error is addressed. System modeling using the approach of the BG technique was carried out and simulation results were analyzed. System components responsible for resistance/damping, inductance/inertia, and capacitance/stiffness were considered for the LLRE to develop a precise model.  
A model-based controller was then designed to address the robust control problem, which includes input and output disturbances, and un-modeled dynamics and uncertainties in the system, that cause poor control performance. The lead compensated \(H_{\infty}\) reference gain controller was proven to be a robust controller as it successfully attenuated the disturbances and, in the presence of uncertainty, provided the desired robust results. Before the integration of the lead compensator and the reference gain controller, the output was out of phase with the given sinusoidal time-varying input signal and there existed tracking error. This control synthesis provided robust trajectory tracking control of LLREs and addressed the challenges of tracking error faced by the previous model-based control approaches.  
The model-free control was implemented as an overall system approximation controller. An adaptive PD RBFNN controller was designed. The comparison between the performance parameters of the designed model-based and model-free control strategies is presented.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Reference Signal**} & \multicolumn{4}{c}{**Performance and Robustness**} \\ \cline{2-4}  & **Transient Behavior** & **Steady-State/Tracking Error** & **Robustness** \\ \hline \multicolumn{4}{c}{**Model-Based Control Strategy\(H_{\infty}\) Reference Gain Controller**} \\ \hline \multirow{3}{*}{Step} & Rise Time: 24 sPeak Value: & \multirow{3}{*}{Minimized to Zero} & \multirow{3}{*}{Achieved} \\  & 1 Settling Time: 35 s & & \\  & Rise Time: 3.2 sPeak Value: & & \\  & 1 Settling Time: 37 s & & \\  & Rise Time: 3.2 sFirst Peak Value: & & \\  & 2.6 Settling Time: 35 s & & \\ \hline \multicolumn{4}{c}{**Model-Based Robust Control DesignLead Compensated \(H_{\infty}\) Reference Gain Controller**} \\ \hline \multirow{3}{*}{Sinusoidal} & Rise Time: 1.6 First Peak Value: & \multirow{3}{*}{Minimized to Zero within 4 s} & \multirow{3}{*}{Achieved} \\  & 0.8 Settling Time: 4 s & & \\ \hline \multicolumn{4}{c}{**Model-Free Robust Control Design—Adaptive PD RBFNN Controller**} \\ \hline \multirow{3}{*}{Step} & Rise Time: 3.2 sSettling Time: 4.5 s & \multirow{3}{*}{Minimized to Zero} & \multirow{3}{*}{Achieved} \\  & Rise Time: 0.8 sSettling Time: 2 s & & \\ \cline{1-1} \cline{3-3} \cline{  
The tracking performances for the lead compensated \(H_{\infty}\) controller and the adaptive PD RBFNN controller are both effective; however, the response of the model-free controller is faster than that of the model-based controller. Future work will include improvements to the MB and MF control approaches for further optimization of the responses.  
Conceptualization, M.T.S. and S.Q.; methodology, M.T.S. and Z.K.; software, M.T.S.; validation, Z.K., A.M.M. and S.Q.; formal analysis, A.M.M.; investigation, M.T.S.; resources, J.Z.G. and Z.M.U.D.; data curation, A.M.M.; writing--original draft preparation, M.T.S.; writing--review and editing, Z.K.; visualization, A.M.M.; supervision, S.Q.; project administration, M.T.S., J.Z.G. and Z.M.U.D.; funding acquisition, J.Z.G. and Z.M.U.D. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Center of Robotics and Automation, Higher Education Commission (HEC), Pakistan under Grant Nos. RF-NCRA-037 and RF-NCAI-064.  
Not applicable.  
The authors declare no conflict of interest.