## 5 Formulation of the LMI Stability Criterion
### Additional LMI Requirements to Reduce the Controller Gain Magnitude  
It is crucial to highlight that the LMI-based problems developed in the previous theorems can result in a significantly large control gain \(\mathcal{K}\) of the proposed controller (25), making it impractical for real-world applications. As a result, reducing the controller's gain \(\mathcal{K}\) is our primary goal. To achieve this goal, since \(\mathcal{K}\) is computed via Relation (78), we introduce the subsequent conditions:  
\[\mathcal{R}^{\mathrm{T}}\mathcal{R}<\ell_{1}\mathcal{I}, \ell_{1}>0 \tag{94a}\] \[\mathcal{S}^{-1}<\ell_{2}\mathcal{I}, \ell_{2}>0 \tag{94b}\]  
Hence, we can reformulate the two constraints mentioned earlier, defined by expressions (94a) and (94b), as follows:  
\[\left[\begin{array}{cc}\ell_{1}\mathcal{I}&\star\\ \mathcal{R}&\mathcal{I}\end{array}\right]>0 \tag{95a}\] \[\left[\begin{array}{cc}\ell_{2}\mathcal{I}&\mathcal{I}\\ \star&\mathcal{S}\end{array}\right]>0 \tag{95b}\]  
In the intended LMIs (95a) and (95b), the two parameters \(\ell_{1}\) and \(\ell_{2}\) need to be minimized to effectively reduce the controller gain's size. It is essential to emphasize that we will incorporate the two additional LMIs, (95a) and (95b), into the previously expanded theorems to decrease the magnitude of the gain \(\mathcal{K}\). This endeavor aims to enhance the practicality and real-world applicability of the control approach.  
Note that the additional LMIs (95a) and (95b) do not change the feasibility feature of the different LMIs established in the previous theorems. Indeed, an unfeasible LMI-based problem without these two additional LMIs remains unfeasible by introducing these two LMIs.  
Our study focuses on the crucial knee joint and aims to improve the precise control of a lower-limb exoskeleton [60; 61]. To achieve this goal, we have implemented a state feedback controller specifically tailored to handle the challenges intrinsic to this field. Our study is unique from other studies [51; 54; 62] because we apply the LMI technique, which guarantees effective control and stability of the knee exoskeleton robot for rehabilitation. Proving that it can manage parametric uncertainties and disturbances more effectively, this invention greatly advances the body of knowledge already in existence and improves control performance.  
In this research, the parametric uncertainties (\(\Delta a\), \(\Delta f_{\forall v}\), \(\Delta f_{s}\)) in the knee-joint exoskeleton's nonlinear dynamics (5) were combined with the disturbing torque \(\delta_{t}\) to form the lumped disturbance \(\Delta_{t}\), simplifying the design process. The primary goal was to determine the maximum allowable lumped disturbance \(\tilde{\delta}_{\max}\). Individual parametric uncertainties in the robust controller design would significantly increase the complexity of the LMI formulation, particularly in the case of norm-bounded uncertainties [59], requiring additional subcases for analysis.  
* Case 1: All unknown parameters are assumed to have identical maximum values. While this simplifies the LMI computation, it imposes an unrealistic constraint, since parameters with varying levels of uncertainty are forced to adopt small values, ignoring the significance of higher uncertainties in some parameters.
* Case 2: Each parameter is assigned its own maximum uncertainty value. For the knee exoskeleton, this adds four additional parameters to the LMI design. With two parameters already fixed a priori in the first approach and one in the second and third approaches, this significantly increases the computational complexity. In the fourth approach, incorporating these four uncertainties results in an optimization problem involving eight parameters, including \(\ell_{1}\) and \(\ell_{2}\), for minimizing the gain \(\mathcal{K}\). This not only complicates the LMI minimization but may also influence the maximum/minimum values of \(\tilde{\delta}\) and \(\epsilon\).