## 7 Simulation Results  
This section contains the presentation of some numerical and simulation results that serve as a testament to verify that the established LMI conditions are effective in maintaining stability in the exoskeleton robot for actuated knee-joint rehabilitation. The parameter values of the robot, governed by the dynamics in (5), are comprehensively detailed in Table 1.  
Moreover, Figure 2 presents the simulated dynamic model of the knee rehabilitation exoskeleton utilized in this study. This model was implemented and simulated in MATLAB/Simulink ([[https://www.mathworks.com/](https://www.mathworks.com/)]([https://www.mathworks.com/](https://www.mathworks.com/))) to evaluate the effectiveness of the derived LMI conditions in achieving robust position control. The diagram highlights the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(\
u_{\star}\) & 0.323 & kg\(\cdot\)m\({}^{2}\)/rad \\ \(m_{\star}\) & 0.000105 & kg \\ \(l_{\star}\) & 0.25 & m \\ \(f_{\circ}^{\star}\) & 0.35 & N\(\cdot\)m\(\cdot\)s/rad \\ \(f_{s}^{\star}\) & 0.059 & N\(\cdot\)m\(\cdot\)A \\ \(g\) & 9.8 & N \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter values for the adopted knee-joint rehabilitation exoskeleton robot [54; 60].  
interactions between various system components, including the viscous and solid friction parameters, and terms accounting for external disturbances.  
The numerical studies in this part are intended to confirm the effectiveness and validity of the suggested design strategy for the gain \(\mathcal{K}\) in the robust state-feedback controller (25). Our primary goal is to show that the LMI-based minimization/optimization issue, which we constructed in the various theorems in the preceding sections, is practically feasible. In the following simulations, we fix the candidate Lyapunov function's decay rate \(\zeta\) to \(\zeta=2\).  
Moreover, the desired position \(\theta_{d}\) to which the exoskeleton robot is controlled and hence stabilized is selected to be \(\theta_{d}=40^{\circ}\). Furthermore, in all the simulation results, and for comparison purposes, we will select the following initial condition:  
\[\mathcal{X}_{0}=\left[\begin{array}{c}\phi\\ \dot{\phi}\end{array}\right]=\left[\begin{array}{c}40^{\circ}\\ 0^{\circ}\end{array}\right] \tag{114}\]  
Additionally, we incorporate the lumped disturbance \(\Delta_{t}\) in the form of a sinusoidal input, described as follows:  
\[\Delta_{t}=\delta\times\sin(\omega_{f}\times t) \tag{115}\]  
Here, \(\omega_{f}\) denotes the signal's pulse. The setting in the following is \(\omega_{f}=5\pi\). Moreover, this perturbation input will only be active from 2 to 7 s. After this period of time, \(\Delta_{t}=0\).  
**Remark 5**.: _In this study, the lumped disturbance \(\Delta_{t}\) is modeled as a sinusoidal input (115) with a reference position \(\theta_{d}=40^{\circ}\) and an initial condition (114) to validate the LMI conditions for maintaining stability during knee-joint rehabilitation. To test robustness under varied conditions, alternative disturbance models, such as random torque variations (\(\Delta_{t}=\delta\times(2\times\text{rand}-1)\)) or constant torque (\(\Delta_{t}=\tilde{\delta}\)), can be considered. Additionally, varying the desired position (\(\theta_{d}\) between \(0^{\circ}\) and \(90^{\circ}\)) and initial conditions allow a broader assessment of the controller's performance and robustness._  
Figure 2: Simulated dynamic model of the knee-joint rehabilitation exoskeleton robot implemented in MATLAB/Simulink.