## 7 Simulation Results
### Discussion  
It is noteworthy that the suggested controller uses feedback gains derived from four distinct methods to efficiently handle the control difficulty of the knee exoskeleton robot's position. Thus, as previously seen in Figures 3a-6a, the joint exoskeleton robot is controlled to the desired position \(\theta_{d}\), resulting in the convergence of the angular position error \(\phi\) to zero. Furthermore, as demonstrated in Figures 3b-6b, the angular velocity error \(\dot{\phi}\) approaches zero. In all four cases, it displays variations that are attributed to the perturbing torque's presence \(\Delta_{t}\). These outcomes show how resilient the controller is to the lumped disturbance. In Figures 3d-6d, the lumped disturbance applied to the robotic system is depicted by the torque load \(\Delta_{t}\). The lumped torque is represented as a signal varying between \(-\bar{\delta}\) and \(\bar{\delta}\), primarily implemented between 2 and 7 s. Accordingly, the simulation results are displayed in Figures 3-6 under the influence of the lumped disturbing torque \(\Delta_{t}\). It is clear that oscillations near the point of equilibrium are induced by the knee exoskeleton system. The regulated knee exoskeleton system returns to its desired state when the perturbing torque decreases. As indicated in Table 6, it is important to note that we assumed \(\bar{\delta}=0.8\) in the first example, \(\bar{\delta}=0.9\) in the second, and \(\bar{\delta}=0.8101\) in the third case. In the fourth case, \(\bar{\delta}=0.8422\). Simulation results demonstrate that the state feedback control law  
Figure 6: Simulation results obtained using the fourth designed method. Temporal variation of: (**a**) the angular position error \(\phi\), (**b**) the angular velocity error \(\dot{\phi}\), (**c**) the affine state-feedback controller \(u\), and (**d**) the lumped disturbance \(\Delta_{t}\). effectively counteracts the perturbing torque's effects, ensuring that the knee exoskeleton robot remains in the desired state. Figures 3c-6c illustrate the temporal evolution of the control law \(u\), showing that the knee exoskeleton stabilizes under the lumped disturbance with \(u\) approaching near-zero. As shown in Figure 7a,b, the four approaches demonstrate the controller's effectiveness but differ in stabilization times: approximately 1.8 s for the first method, under 2.8 s for the second, less than 1 s for the third, and about 1.4 s for the fourth. Initial control efforts (Figure 7c) also vary, with values of 37, 1.4, 22, and 2.2 for the first, second, third, and fourth methods, respectively. While all techniques effectively stabilize the system, differences in stabilization time and control effort highlight the need for further investigation under varied conditions, including different lumped disturbances. In our implementation, the computationally intensive step of solving the LMIs is performed offline during the design phase using MATLAB's LMI Toolbox, specifically the mincx function. This process was measured to take approximately 0.0528 seconds on a standard PC, ensuring an efficient offline design process. For real-time implementation, the resulting control law involves only matrix multiplications, which execute in less than 0.029 ms on lightweight embedded processors, demonstrating the suitability of the proposed approach for wearable exoskeletons. To further evaluate and thereby quantify the performance and efficacy of the adopted affine state-feedback control law in terms of the four different design approaches, we employ various metrics, namely the Root Mean Squared Error (RMSE), the Mean Squared  
Figure 7: Comparison of the four approaches: Temporal variation of (**a**) the angular position error \(\phi\), (**b**) the angular velocity error \(\dot{\phi}\), (**c**) the affine state-feedback controller \(u\), and (**d**) the lumped disturbance \(\Delta_{t}\). Error (MSE), the Mean Absolute Error (MAE), the Control Effort Indicator (CEI) and the Maximum Absolute Torque Indicator (MATI).