## 7 Simulation Results
### Results Obtained for Fixed Parameters \(\delta\) and \(\epsilon\)  
In this section, we consider that \(\epsilon\) and \(\delta\) are fixed _a priori_. The LMI stability condition, considering fixed parameters \(\epsilon\) and \(\delta\), is defined through Theorem 1. Relation (78), in which the matrices \(\mathcal{S}\) and \(\mathcal{R}\) are the solutions of the LMI problem in Theorem 1, describes the gain \(\mathcal{K}\) of the selected controller (25). After applying the fixed values \(\delta=0.8\) and \(\epsilon=1\) to the LMI problem that arises from this theorem, we arrive at the following results:  
\[\mathcal{S}=\left[ \begin{array}{cc}0.5632&-2.0302\\ -2.0302&8.7185\end{array} \right],\qquad\mathcal{R}=\left[ \begin{array}{cc} -0.9909&-16.2810\end{array} \right],\qquad\mathcal{M}_{1}=\left[ \begin{array}[ ]{cc}0.8667&0\\ 0&16.9965\end{array} \right],\]  
\[m_{2}=25.7225,\qquad\qquad\mathcal{M}_{3}=\left[ \begin{array}{cc}0.8695&0\\ 0&52.0745\end{array} \right],\qquad\qquad\text{and}\qquad\qquad\gamma=4.7447\]  
Hence, the gain \(\mathcal{K}\) is calculated as follows:  
\[\mathcal{K}=\left[ \begin{array}{cc}-52.8728&-14.1794\end{array} \right] \tag{116}\]  
Note that these previous numerical outcomes and therefore the feedback gain \(\mathcal{K}\) were obtained by solving the LMI (77) in Theorem 1 with _a priori_ selection of \(\epsilon\) and \(\delta\). Thus, to demonstrate (numerically) the feasibility of the LMI (77), it is essential to vary the two parameters \(\epsilon\) and \(\delta\). To streamline the complexity of the computation involved in solving the LMI problem, specific values of \(\epsilon\) and \(\delta\) are selected, allowing the LMI problem to be either feasible or unfeasible. The chosen values for \(\epsilon\) include \(10^{2}\), \(10^{1}\), \(1.0\), \(10^{-1}\), \(10^{-2}\), and \(10^{-3}\), while for \(\delta\), the values are \(1.64\), \(1.63\), \(1.5\), \(1.0\), \(0.7\), \(0.62\), and \(0.61\). Thus, we solve the LMI (77) in Theorem 1 for each possible combination of these two parameters \(\epsilon\) and \(\delta\). In the following, only the resulting gain \(\mathcal{K}\) will be presented. Table 2 illustrates the obtained numerical outcomes of the gain \(\mathcal{K}\) by resolving the LMI (77) in Theorem 1 for these possible various values of \(\epsilon\) and \(\delta\) and without considering the two LMIs (95a) and (95b) for reducing the magnitude of the gain \(\mathcal{K}\). In this table, the symbol (!) represents an unfeasible LMI. Notably, the LMI was found to be feasible for values of \(\delta\) between \(0.62\) and \(1.63\). Thus, the LMI problem is unfeasible for all \(\delta\leq 0.61\) and for all \(\delta\geq 1.64\). Moreover, notice from Table 2 that as the parameter \(\epsilon\) decreases and the parameter \(\delta\) increases, the feedback gain \(\mathcal{K}\) increases substantially, and for the two values \(\epsilon=10^{-2}\) and \(\epsilon=10^{-3}\), the gain \(\mathcal{K}\) becomes extremely large and is not practicable. For these values of the parameter \(\epsilon\), the gain \(\mathcal{K}\) cannot be considered in the controller for reliable control of the joint exoskeleton robot.