## 7 Simulation Results
### Results Obtained for Fixed Parameters \(\delta\) and \(\epsilon\)
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\delta\backslash\epsilon\) & **100** & **10** & **1** & **10\({}^{-1}\)** & **10\({}^{-2}\)** & **10\({}^{-3}\)** \\ \hline \(1.64\) & (!) & (!) & (!) & (!) & (!) & (!) \\ \(1.63\) & \(\begin{bmatrix}-10.08\\ -3.50\\ -14.19\\ -4.47\end{bmatrix}\) & \(\begin{bmatrix}-31.64\\ -9.49\\ -22.68\\ -7.50\end{bmatrix}\) & \(\begin{bmatrix}-351.95\\ -53.26\\ -22.68\\ -7.50\end{bmatrix}\) & \(\begin{bmatrix}-351.95\\ -53.26\\ -22.68\\ -7.50\end{bmatrix}\) & \(\begin{bmatrix}-351.95\\ -53.26\\ -146.01\\ -28.84\end{bmatrix}\) & \(\begin{bmatrix}-351.95\\ -749.10\\ -93.00\end{bmatrix}\) & \(10^{4}\times\begin{bmatrix}-4.50\\ -0.41\\ -0.07\end{bmatrix}\) & \(10^{5}\times\begin{bmatrix}-4.45\\ -0.41\\ -0.41\end{bmatrix}\) \\ \(1.00\) & \(\begin{bmatrix}-16.42\\ -5.23\end{bmatrix}\) & \(\begin{bmatrix}-16.72\\ -5.34\\ -5.34\end{bmatrix}\) & \(\begin{bmatrix}-54.76\\ -14.50\\ -43.78\end{bmatrix}\) & \(10^{3}\times\begin{bmatrix}-3.43\\ -0.31\\ -0.31\end{bmatrix}\) & \(10^{4}\times\begin{bmatrix}-3.18\\ -0.27\end{bmatrix}\) \\ \(0.70\) & \(\begin{bmatrix}-16.40\\ -5.22\end{bmatrix}\) & \(\begin{bmatrix}-16.70\\ -5.33\end{bmatrix}\) & \(\begin{bmatrix}-15.62\\ -43.64\end{bmatrix}\) & \(10^{3}\times\begin{bmatrix}-3.42\\ -0.31\end{bmatrix}\) & \(10^{4}\times\begin{bmatrix}-3.17\\ -0.27\end{bmatrix}\) \\ \(0.62\) & \(\begin{bmatrix}-14.24\\ -4.78\end{bmatrix}\) & \(\begin{bmatrix}-22.88\\ -7.55\end{bmatrix}\) & \(\begin{bmatrix}-125.62\\ -25.32\end{bmatrix}\) & \(\begin{bmatrix}-739.74\\ -91.96\end{bmatrix}\) & \(10^{4}\times\begin{bmatrix}-1.00\\ -0.07\end{bmatrix}\) & \(10^{5}\times\begin{bmatrix}-1.23\\ -0.05\end{bmatrix}\) \\ \(0.61\) & (!) & (!) & (!) & (!) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical results, \(\mathcal{K}^{\rm T}\), of the LMI (77) in Theorem 1 for different values of \(\epsilon\) and \(\delta\) and without the two LMIs (95a) and (95b) for reducing the size of \(\mathcal{K}\). To enhance the achieved numerical outcomes presented in Table 2 while ensuring a feasible controller, it is imperative to diminish the feedback gain \(\mathcal{K}\). To accomplish this objective, we introduce the two LMIs denoted by (95a) and (95b) and subsequently minimize the two parameters \(\ell_{1}\) and \(\ell_{2}\). In Table 3, we showcase the numerical outcomes for the feedback matrix gain \(\mathcal{K}\) with identical parameter values for \(\epsilon\) and \(\bar{\delta}\) as those used in Table 2, in addition to considering the two supplementary LMIs (95a) and (95b). The data in Table 3 clearly demonstrate that the magnitudes of the gains \(\mathcal{K}\) are notably smaller than those in Table 2. For certain combinations of the parameters \(\epsilon\) and \(\bar{\delta}\), a substantial reduction in the size of \(\mathcal{K}\) is observed. Nevertheless, for some other cases, there is only a marginal reduction in the individual elements of the gain matrix \(\mathcal{K}\). Consequently, based on the findings in Table 3, the most favorable value for the parameter \(\epsilon\) that yields satisfactory results, and consequently, a reasonable magnitude (in the order of \(10^{2}\)) for the feedback gain \(\mathcal{K}\), is \(\epsilon=0.1\). We provide the graphical results shown in Figure 3 using the gain \(\mathcal{K}\) described by Equation (116) within the suggested affine state-feedback controller (25). It is evident that the exoskeleton robot traveled exactly to \(\theta_{d}=40^{\circ}\), the desired location. Moreover, once the knee exoskeleton robot is stable at the optimal position \(\theta_{d}\), the control effort \(u\) decreases to the desired control. Figure 3: \(Cont.\)