## 5 Conclusions
\[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]  
The difference between the Lyapunov functions \(V_{i}(k+1)\) and \(V_{i}(k)\) at the \(k+1\) time and \(k\) time of power grids is defined in Equation (14). \[\Delta V_{i}(k)=V_{i}(k+1)-V_{i}(k)=\frac{1}{2}\left\|\tilde{ \mathbf{W}}_{a,i}(k+1)\right\|_{F}^{2}-\frac{1}{2}\left\|\tilde{\mathbf{W}}_{ a,i}(k)\right\|_{F}^{2}\] \[+\frac{1}{2}\left\|\tilde{\mathbf{W}}_{c,i}(k+1)\right\|_{F}^{2} -\frac{1}{2}\left\|\tilde{\mathbf{W}}_{c,i}(k)\right\|_{F}^{2}+\frac{1}{2} \left\|\mathbf{x}_{i}(k+1)\right\|^{2}-\frac{1}{2}\left\|\mathbf{x}_{i}(k) \right\|^{2} \tag{14}\]  
The terms in Equation (14) are defined as follows. \[\Delta V_{i,1}(k)=\frac{1}{2}\left\|\tilde{\mathbf{W}}_{a,i}(k+1 )\right\|_{F}^{2}-\frac{1}{2}\left\|\tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2 },\Delta V_{i,2}(k)=\frac{1}{2}\left\|\tilde{\mathbf{W}}_{c,i}(k+1)\right\|_{ F}^{2}-\frac{1}{2}\left\|\tilde{\mathbf{W}}_{c,i}(k)\right\|_{F}^{2}\] \[\Delta V_{i,3}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k+1)\right\|^ {2}-\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2},\Delta V_{i}(k)\ =\Delta V_{i,1}(k)+\Delta V_{i,2}(k)+\Delta V_{i,3}(k) \tag{15}\]  
According to Equation (12), the relationships between \(\tilde{\mathbf{W}}_{i}(k+1)\) and \(\tilde{\mathbf{W}}_{i}(k)\) are shown as follows. \[\tilde{\mathbf{W}}_{a,i}(k+1)=\mathbf{W}_{a,i}(k+1)-\mathbf{W}_{ a,i}^{*}(k+1)=\tilde{\mathbf{W}}_{a,i}(k)-\beta\Pi_{\Omega}\biggl{(}\frac{ \partial E_{a,i}(k)}{\partial\tilde{\mathbf{W}}_{a,i}(k)}\biggr{)}\] \[\tilde{\mathbf{W}}_{c,i}(k+1)=\mathbf{W}_{c,i}(k+1)-\mathbf{W}_{ c,i}^{*}(k+1)=\tilde{\mathbf{W}}_{c,i}(k)-\gamma\Pi_{\Omega}\biggl{(}\frac{ \partial E_{c,i}(k)}{\partial\tilde{\mathbf{W}}_{c,i}(k)}\biggr{)} \tag{16}\]  
where the gradient of the target learning function \(E_{a,i}(k)\) to \(\tilde{\mathbf{W}}_{a,i}(k)\) and the gradient of \(\tilde{\mathbf{W}}_{a,i}(k)\) to \(\tilde{\mathbf{W}}_{c,i}(k)\) can be expressed as follows. \[\frac{\partial E_{a,i}(k)}{\partial\tilde{\mathbf{W}}_{a,i}(k)} =u_{lim}\mathbf{1}^{\text{T}}[\mathbf{x}_{i}(k+1)-\mathbf{L}_{ i}\mathbf{x}_{i}(k)]\Bigl{[}1-\tanh^{2}(\tilde{\mathbf{W}}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k) )\Bigr{]}w_{a,i}^{\text{T}}\] \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]0  
In order to simplify the update rules of \(\tilde{\mathbf{W}}_{a,i}\) and \(\tilde{\mathbf{W}}_{c,i}\), this paper defines the following parameters. \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]1 \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]2According to Equations (A6) and (A7), Equation (A6) can be re-expressed as follows. \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]3 \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]4 (A8)  
The first term \(\Delta V_{i,1}(k)\) is taken from Equation (A5), and then Equation (A8) is substituted into the calculation of \(\Delta V_{i,1}(k)\). Thus, \(\Delta V_{i,1}(k)\) is re-expressed as Equation (A9).