## 6 Comparison with the Classical PD and PID Controllers
### Comparison with Existing Works  
In the field of exoskeleton control, several advanced control strategies have been proposed to address the inherent challenges of these systems. For example, [PERSON] et al. (2018) designed and simulated a PID controller for a lower-limb exoskeleton, demonstrating good performance in handling linear and mildly nonlinear systems. However, as the complexity of the system increases with significant nonlinearities and disturbances, classical PID approaches face limitations in maintaining stability and achieving high trajectory tracking accuracy [63]. Similarly, [PERSON] et al. (2017) proposed a PID controller for lower-limb rehabilitation, but their findings revealed difficulties in preserving system stability under complex dynamic disturbances [64]. [PERSON] et al. (2021) further improved PID control by implementing an adaptive PID controller using particle swarm optimization, showing better performance in varying conditions. Despite these improvements, the adaptive PID approach remains sensitive to system variations and external disturbances, limiting its robustness in practical applications [17].  
On the other hand, affine Proportional-Derivative (PD) controllers have been employed to improve the robustness and stability of knee exoskeletons [60], and state-feedback controllers have shown potential in effectively regulating knee-joint movements by accounting for frictional forces and system dynamics [30]. Adaptive methods, such as Model Reference Adaptive Control (MRAC) and Fuzzy PID control, have been explored to dynamically adjust to changing system parameters [27; 52; 54]. Recent advancements also include composite control strategies that combine linear and nonlinear control laws, offering robust stabilization in the presence of disturbances [31] and adaptive controllers like RISE controllers designed to handle high levels of nonlinearity in exoskeletons [70].  
While these existing strategies represent significant advances, they often struggle to cope with the full complexity of nonlinear, uncertain exoskeleton systems. Our approach, based on LMI, offers a distinct advantage by providing a robust mathematical framework for addressing the challenges of system stability and performance. LMI-based methods allow for systematic derivation of stability conditions, which can be solved efficiently using optimization algorithms. This makes our approach particularly suited for exoskeleton systems where nonlinearities, parameter uncertainties, and external disturbances must be robustly managed. Additionally, the use of quadratic Lyapunov functions within the LMI framework ensures system stability under various operating conditions [33; 34].  
Thus, our LMI-based approach offers significant added value in terms of stability, robustness, and overall performance in complex practical applications, positioning it as a promising solution for future experimental implementations in knee exoskeleton systems. The ability to systematically address stability and robustness issues, combined with the flexibility offered by LMI techniques, distinguishes our work from existing methods and provides a solid foundation for real-world applications.