## 4 Convergence Analysis  
**Theorem 2**. : _Considering the switched multi-area power system represented by Equation (7), which complies with Assumptions 1 and 2, with respect to \(\forall j\),1 and when \(j\
eq l\), employing the switching system model-free adaptive controller scheme (20)-(22), given positive scalars \(o_{1},o_{2}\), if there exist \(\eta_{i},\mu_{i},\rho_{i}\), and \(\lambda_{i}\) such that:_  
\[\aleph=\left( \begin{array}{cc}\Xi_{1}&\Xi_{2}\\ *&\Xi_{3}\end{array} \right)<0 \tag{23}\]  
_where \(\Xi_{1}=\left( \begin{array}{cc}(1-o_{1})Q_{i}(1-o_{1})-Q_{i}&(1-o_{1})Q_{i} o_{2}\\ *&o_{2}Q_{i}o_{2}-\Omega\end{array} \right)\), \(\Xi_{2}=\operatorname{diag}\left\{\hat{\delta}^{\frac{1}{2}},\hat{\delta}^{ \frac{1}{2}}\right\}\), \(\Xi_{3}=\operatorname{diag}\left\{-I,-I\right\}\). The tracking error \(e(k)\)of the switched multi-area power system (7) is bounded._  
Proof of Theorem 2.: If the conditions of the reset algorithm (17) are met, it becomes clear that the parameter \(\hat{\phi}_{a(k)}(k)\) is bounded. Let \(\hat{\phi}_{a(k)}(k)=\hat{\phi}_{a(k)}(k)-\phi_{a(k)}(k)\) represent the PDD estimation error. By subtracting \(\phi_{a(k)}(k)\) from both sides of the parameter estimation algorithm (20), we obtain:  
\[\hat{\phi}_{a(k)}(k)=\Bigg{(}1-\frac{\eta_{a(k)}\Delta u^{2}(k-1)}{\mu_{a(k)}+ \Delta u^{2}(k-1)}\Bigg{)}\hat{\phi}_{a(k)}(k-1)+\phi_{a(k)}(k-1)-\phi_{a(k)}(k) \tag{24}\]  
Obtaining the absolute value of Equation (24), we find:  
\[\left|\hat{\phi}_{a(k)}(k)\right|=\left|\Bigg{(}1-\frac{\eta_{a(k)}\Delta u^{ 2}(k-1)}{\mu_{a(k)}+\Delta u(k-1)}\Bigg{)}\right|\left|\hat{\phi}_{a(k)}(k-1) \right|+\left|\phi_{a(k)}(k-1)-\phi_{a(k)}(k)\right| \tag{25}\]  
Figure 1: Block diagram of SSMFAC. Since \(\left|\phi_{\alpha(k)}(k)\right|\leq d_{1}\) is bounded, there exists \(\left|\phi_{\alpha(k)}(k-1)-\phi_{\alpha(k)}(k)\right|\leq 2d_{1}\) Choosing \(\mu_{\alpha(k)}>0,0<\eta_{\alpha(k)}<1\), we can derive \(\eta_{\alpha(k)}\Delta u^{2}(k-1)<\Delta u^{2}(k-1)<\mu_{\alpha(k)}+\Delta u^{2 }(k-1)\). Consequently, there must exist \(\gamma\) constant y such that:  
\[0<\gamma<\left(\frac{\eta_{\alpha(k)}\Delta u^{2}(k-1)}{\mu_{\alpha(k)}+\Delta u ^{2}(k-1)}\right)<1 \tag{26}\]  
Substituting Equation (26) into Equation (25), we obtain:  
\[ \begin{split}&\left|\tilde{\phi}_{\alpha(k)}(k)\right|\leqslant(1- \gamma)\left|\tilde{\phi}_{\alpha(k)}(k-1)\right|+2d_{1}\\ &\leqslant(1-\gamma)^{2}\left|\tilde{\phi}_{\alpha(k)}(k-2) \right|+2d_{1}(1-\gamma)+2d_{1}\\ &\leqslant\cdots\\ &\leqslant(1-\gamma)^{k-1}\left|\tilde{\phi}_{\alpha(k)}(1) \right|+\frac{2d_{1}}{\gamma}\end{split}  \tag{27}\]  
Therefore, Equation (27) is bounded, and since \(\tilde{\phi}_{\alpha(k)}(k)\) is bounded, \(\hat{\phi}_{\alpha(k)}(k)\) is also bounded. The boundedness of tracking error. We define the system's tracking error as:  
\[e(k)=y_{d}(k)-y(k) \tag{28}\]  
By substituting Equations (19) and (13) into Equation (29), we obtain:  
\[ \begin{split} e(k+1)&=y_{d}(k+1)-y(k+1)\\ &=y_{d}(k+1)-y(k)-\phi_{\alpha(k)}(k)\Delta u(k)\\ &=e(k)-\frac{\rho_{\alpha(k)}\hat{\phi}_{\alpha(k)}(k)\phi_{ \alpha(k)}(k)}{\lambda_{\alpha(k)}+\left|\hat{\phi}_{\alpha(k)}(k)\right|^{2} }(e(k)-e(k_{r}))\\ &=(1-\Theta(k))e(k)+\Theta(k)e(k_{r})\end{split}  \tag{29}\]  
where \(\Theta(k)=\frac{\rho_{\alpha(k)}\hat{\phi}_{\alpha(k)}(k)\phi_{\alpha(k)}(k)} {\lambda_{\alpha(k)}+\left|\hat{\phi}_{\alpha(k)}(k)\right|^{2}}\)  
Next, consider the following Lyapunov function:  
\[V(k)=V_{\alpha(k)}(e(k))=e^{T}(k)Q_{\alpha(k)}e(k) \tag{30}\]  
For the \(i\) th subsystem:  
\[V_{i}(e(k))=e^{T}(k)Q_{i}e(k) \tag{31}\]  
Let \(\Delta V_{i}(k+1)=V_{i}(k+1)-V_{i}(k)\) to obtain  
\[ \begin{split}\Delta V_{i}(k+1)&=[(1-\Theta(k))e( k)+\Theta(k)e_{r}(k)]^{T}Q_{i}[(1-\Theta(k))e(k)+\Theta(k)e_{r}(k)]\\ &-e^{T}(k)Q_{i}e(k)\\ &=\ell^{T}(k)\Lambda\ell(k)\end{split}  \tag{32}\]  
where \(\ell(k)=[e(k)e(k_{r})],\Lambda=\left[ \begin{array}{cc}\
abla_{1}&\
abla_{2} \\ *&\
abla_{3}\end{array}