## 5 Formulation of the LMI Stability Criterion
### Design of the LMI Condition
#### 5.2.2 LMI Design with Fixed Parameter \(\bar{\delta}\) and Unknown Parameter \(\epsilon\)  
The parameter \(\epsilon\) is treated as an unknown quantity to be reduced in the second approach for the LMI criteria, whereas the parameter \(\bar{\delta}\) remains constant. Now, let us reevaluate the inequality of the matrix (77). As can be seen from (77), the parameter \(\epsilon\) is related to the parameter \(\gamma\) in various terms of expression, making it a non-LMI condition with regard to both \(\epsilon\) and \(\gamma\).  
To convert the matrix inequality (77) into an LMI, we introduce a new parameter \(\eta=\frac{\gamma}{\epsilon^{2}}\). This implies that \(\epsilon=\sqrt{\frac{\gamma}{\eta}}\). Recall that \(\gamma>0\) and \(\eta>0\). Accordingly, in order to minimize the parameter \(\epsilon\), we need to minimize \(\gamma\) and minimize \(\eta\), or alternatively, we should minimize the quantity \(\left(\gamma+\eta^{-1}\right)\). To minimize this previous quantity, let us consider the following condition:  
\[\gamma+\eta^{-1}<\vartheta_{1} \tag{79}\]  
where \(\vartheta_{1}>0\).  
Hence, we conclude that to minimize \(\epsilon\), we should minimize the parameter \(\vartheta_{1}\). It is clear that inequality (79) can be reformulated as follows:  
\[\vartheta_{1}-\gamma-\eta^{-1}>0 \tag{80}\]  
We can determine the following LMI condition by applying the expression (80) and the Schur complement (Lemma 2):  
\[\left[\begin{array}{cc}\theta_{1}-\gamma&1\\ \star&\eta\end{array}\right]>0 \tag{81}\]  
with \(\gamma>0\).  
Thus, we can formulate the following theorem based on LMI conditions.  
**Theorem 2**.: _let us pose that, for a fixed scalar \(\zeta\geq 0\), and for some parameter \(\bar{\delta}\geq 0\) fixed a priori, there exist matrices \(\mathcal{S}>1\), \(\mathcal{R}\), \(\mathcal{M}_{1}>0\), and \(\mathcal{M}_{3}>0\), as well as scalars \(m_{2}>0\), \(\gamma>0\), \(\vartheta_{1}>0\), and \(\eta>0\), so that the subsequent LMI-based minimization issue is feasible:_  
\[\begin{split}&\text{min}~{}~{}\vartheta_{1}\\ &\text{s.t.}\\ &\left[\begin{array}{ccc}\Xi_{11}&\mathcal{S}&\mathcal{O}&\mu_{2} \mathcal{S}\mathcal{C}_{1}^{\mathrm{T}}&\mathcal{O}\\ \star&-\gamma\mathcal{I}&\mathcal{O}&\mathcal{O}&\mathcal{O}\\ \star&\star&\eta\mathcal{D}_{1}\mathcal{D}_{1}^{\mathrm{T}}-\mathcal{M}_{1}& \eta\mathcal{D}_{1}&\eta\mathcal{D}_{1}\mathcal{B}^{\mathrm{T}}\\ \star&\star&\star&\Xi_{44}&\eta\mathcal{B}^{\mathrm{T}}\\ \star&\star&\star&\star&\eta\mathcal{B}\mathcal{B}^{\mathrm{T}}-\left(\sigma \left(\frac{1}{\bar{\delta}^{2}}-1\right)^{2}+\frac{1}{\bar{\delta}^{2}} \right)\mathcal{M}_{3}\end{array}\right]<0\\ &\left[\begin{array}{cc}\vartheta_{1}-\gamma&1\\ \star&\eta\end{array}\right]>0\end{split} \tag{82b}\]_where the quantity \(\Xi_{11}\) is defined by expression (70), and \(\Xi_{\mathbf{44}}=\eta\mathcal{I}-\left(\sigma\Big{(}\frac{1}{\mu_{3}}-1\Big{)}^ {2}+\frac{1}{\mu_{3}}\right)m_{2}\)._  
_Consequently, the exoskeleton robot's nonlinear dynamics (5) are stabilized to the intended position by the suggested controller (25). The control law's gain \(\mathcal{K}\) is computed using relation (78). Moreover, the following formula may be used to obtain the minimal value, \(\epsilon_{min}\), of the parameter \(\epsilon\):_  
\[\epsilon_{min}=\sqrt{\frac{\gamma}{\eta}} \tag{83}\]  
Two LMI conditions, represented by Equations (82b) and (82c), were found in the preceding theorem. These expressions include the unknown variables \(\mathcal{S}\), \(\mathcal{R}\), \(\mathcal{M}_{1}\), \(m_{2}\), \(\mathcal{M}_{3}\), \(\gamma\), \(\eta\), and \(\phi_{1}\).