## 5 Formulation of the LMI Stability Criterion  
### Development of the Stability Condition  
In this section, we focus on formulating an LMI requirement to determine the gain \(\mathcal{K}\) of the controller (25)-(26), aiming to regulate and stabilize the knee-joint exoskeleton robot at the intended state. Thus, we propose to achieve this aim by applying the Lyapunov approach. Let us examine the subsequent Lyapunov function:  
\[\mathcal{V}(\mathcal{X})=\ \ \mathcal{X}^{\text{T}}\mathcal{P}\mathcal{X}\qquad \text{such that}\qquad\mathcal{Z}(\mathcal{X})>0 \tag{29}\]Therefore, its time derivation is expressed as follows:  
\[\dot{\mathcal{V}}(\mathcal{X})=\ \ 2\mathcal{X}^{\mathrm{T}}\mathcal{P}\dot{ \mathcal{X}}\qquad\text{such that}\qquad\mathcal{Z}(\mathcal{X})>0 \tag{30}\]  
Using the proposed control law (28), the established expression (30) becomes as follows:  
\[\dot{\mathcal{V}}(\mathcal{X})=2\mathcal{X}^{\mathrm{T}}\mathcal{P}(\mathcal{ A}+\mathcal{BK})\mathcal{X}+2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{D}_{1} \Phi_{1}(\mathcal{X})+2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{D}_{2} \Phi_{2}(\mathcal{X})+2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{B}\Delta_{t} \tag{31}\]  
To further enhance the control outcomes, let us adopt the following constraint:  
\[\dot{\mathcal{V}}(\mathcal{X})+\zeta\mathcal{V}(\mathcal{X})<0\qquad\text{ such that}\qquad\mathcal{Z}(\mathcal{X})>0 \tag{32}\]  
here, a fixed positive scalar \(\zeta>0\) is indicated. Consequently, the stability condition for the closed-loop exoskeleton system is as follows:  
\[2\mathcal{X}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{BK}) \mathcal{X}+\zeta\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{X}+2\mathcal{X}^{ \mathrm{T}}\mathcal{P}\mathcal{D}_{1}\Phi_{1}(\mathcal{X})+2\mathcal{X}^{ \mathrm{T}}\mathcal{P}\mathcal{D}_{2}\Phi_{2}(\mathcal{X})\] \[+2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{B}\Delta_{t}<0 \qquad\text{such that}\qquad\mathcal{Z}(\mathcal{X})>0 \tag{33}\]  
We primarily rely on condition (33) to ensure the stability of the regulated exoskeleton robot. Therefore, the subsequent steps will establish specific LMI constraints on the gain \(\mathcal{K}\). According to the Young inequality (Lemma 1), and using \(\Phi_{1}(\mathcal{X})=\text{sgn}(\dot{\phi})=\pm 1\) for \(\dot{\phi}\
eq 0\) leading to \(\Phi_{1}(\mathcal{X})^{2}=1\), it is simple to demonstrate that in the inequality condition (31), we obtain:  
\[2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{D}_{1}\Phi_{1}(\mathcal{X})\leq \mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{M}_{1}\mathcal{P}\mathcal{X}+ \mathcal{D}_{1}^{\mathrm{T}}\mathcal{M}_{1}^{-1}\mathcal{D}_{1} \tag{34}\]  
with \(\mathcal{M}_{1}=\mathcal{M}_{1}^{\mathrm{T}}>0\), and where the scalar function \(\Phi_{1}(\mathcal{X})\) is defined by Expression (23a). Moreover, recall that the function \(\Phi_{2}(\mathcal{X})\) is a scalar function defined by Expression (23b). Therefore, applying the Young inequality (Lemma 1) to the term in the condition (33) containing the function \(\Phi_{2}(\mathcal{X})\), we can demonstrate the subsequent equation:  
\[2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{D}_{2}\Phi_{2}(\mathcal{X})\leq \mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{D}_{2}m_{2}\mathcal{D}_{2}^{ \mathrm{T}}\mathcal{P}\mathcal{X}+m_{2}^{-1}\Phi_{2}(\mathcal{X})^{2}\quad \text{with}m_{2}>0. \tag{35}\]  
Let us examine Expression (23b) of the nonlinear function \(\Phi_{2}(\mathcal{X})\).