## 4 Sufficient Conditions for Stability Control Rules  
The main objective of this section is to derive a sufficient condition for achieving the asymptotic stabilization of the wireless communication switching system using the multiple Lyapunov function method. Based on the system described by Equation (7), for a simpler analysis, define the augmented vector \(\mathbf{\Phi}(t)=\left[x_{i}(t)\quad 1\right]^{\mathrm{T}}\). The state equation of the system is then simplified to:  
\[\dot{\mathbf{\Phi}}(t)=\mathbf{L}_{i}\mathbf{\Phi}(t) \tag{27}\]  
where  
\[\mathbf{L}_{i}= \begin{bmatrix}A_{i}-B_{i}K_{i}\\ 0\end{bmatrix} = \begin{bmatrix}A_{i}\\ 0\end{bmatrix} - \begin{bmatrix}B_{i}\\ 0\end{bmatrix}\begin{bmatrix}K_{i}&0\end{bmatrix} =\mathbf{A}\mathbf{b}_{i}- \mathbf{B}\mathbf{b}_{i}\cdot\mathbf{K}_{ni} \tag{28}\]  
In the equation, \(\mathbf{A}\mathbf{b}_{i}= \begin{bmatrix}A_{i}\\ 0\end{bmatrix} \), \(\mathbf{B}\mathbf{b}_{i}= \begin{bmatrix}B_{i}\\ 0\end{bmatrix} \), \(\mathbf{K}_{ni}= \begin{bmatrix}K_{i}&0\end{bmatrix} \). **Lemma 1** (Schur Complement). : _For a given symmetric matrix \(\mathbf{S}= \begin{bmatrix}\mathbf{S}_{11}&\mathbf{S}_{12}\\ \mathbf{S}_{12}^{\mathrm{T}}&\mathbf{S}_{22}\end{bmatrix} \), the following three conditions are equivalent:_  
1. \(\mathbf{S}<0\)__
2. \(\mathbf{S}_{11}<0,\mathbf{S}_{22}-\mathbf{S}_{12}^{\mathrm{T}}\mathbf{S}_{11} ^{-1}\mathbf{S}_{12}<0\)__
3. \(\mathbf{S}_{22}<0,\mathbf{S}_{11}-\mathbf{S}_{12}^{\mathrm{T}}\mathbf{S}_{22} ^{-1}\mathbf{S}_{12}<0\)__  
_The following provides the Lyapunov stability theorem for the asymptotic stability of the system._  
**Theorem 1** (Lyapunov Asymptotic Stability Theorem). : _The fundamental idea is to analyze the stability of the system from the perspective of energy change. If the stored energy of the system decreases over time as the system evolves, then the system is stable; otherwise, the system is unstable. A scalar function \(V(x)\) can be used to represent the system's energy. Thus, if the scalar function is positive definite and_  
1. _If the time derivative of the scalar function_ \(\dot{V}(x)\) _is negative definite, then the origin is asymptotically stable;_
2. _If the scalar function_ \(\dot{V}(x)\) _is positive definite, then the origin is unstable._  
_The following provides a sufficient condition for the system described by Equation (7) to achieve asymptotic stability using the state feedback given by Equation (26)._  
**Theorem 2**. : _Assume that for \(i,j\in\{1,\cdots,M\}^{2}\), where \(M\) represents the number of wireless communication systems under different coding and modulation schemes, and \(\{A_{i},B_{i},C_{i},D_{i}\}\) is controllable. For such a closed-loop multi-parameter wireless communication system described as a switching system (27), if there exist \(M\) symmetric positive definite matrices \(\mathbf{X}_{i}\) such that the following inequalities hold:_  
\[\mathbf{Q}= \begin{pmatrix}-\mathbf{X}_{i}&(\mathbf{A}\mathbf{b}_{i}\mathbf{X} _{i}-\mathbf{B}\mathbf{b}_{i}\mathbf{K}_{ni}\mathbf{X}_{i})^{\mathrm{T}}\\ \mathbf{A}\mathbf{b}_{i}\mathbf{X}_{i}-\mathbf{B}\mathbf{b}_{i}\mathbf{K}_{ni }\mathbf{X}_{i}&-\mathbf{X}_{i}\end{pmatrix} <0 \tag{29}\]  
_then, at the switching moment, after switching from the \(i\)-th subsystem to the \(j\)-th subsystem, the multi-parameter wireless communication switching system is asymptotically stable._  
Proof. : Let the Lyapunov functions for the system at time t and at the switching moment \(t+\Delta t\) be respectively:  
\[V(\mathbf{\Phi}(t))=\mathbf{\Phi}^{T}(t)\mathbf{P}_{i}\mathbf{\Phi}(t)\text{, }V(\mathbf{\Phi}(t+\Delta t))=\mathbf{\Phi}^{T}(t+\Delta t)\mathbf{P}_{j} \mathbf{\Phi}(t+\Delta t)\text{, }\mathbf{P}_{i}=\mathbf{P}_{i}^{T}>0\text{, }\mathbf{P}_{j}=\mathbf{P}_{j}^{T}>0\text{,}\]The difference equation for the function is:  
\[