## 3 Jacobian Matrix Derivation for MES
### Jacobian Matrix for DHN  
Referring to [11], the state variable of the DHN is set as \(\mathbf{x}_{\rm h}=[\mathbf{m};\,T_{\rm n}^{\rm S^{\prime}};\,T_{\rm n}^{\rm R}]\). Based on the proposed matrix-based DHN model, the mismatch vector of the DHN \(\mathbf{F}_{\rm h}\) is obtained via  
\[\mathbf{F}_{\rm h}=\left[ \begin{array}{c}\Delta\mathbf{\Phi}_{\rm n}=C_{\rm P}\mathbf{A} _{\rm h,n}\mathbf{m}\Big{(}\mathbf{T}_{\rm n}^{\rm f^{\prime}}-\mathbf{T}_{\rm sp}^{\rm f^ {\prime}}\Big{)}-\mathbf{\Phi}_{\rm n}\\ \Delta\mathbf{\Phi}_{\rm s}=C_{\rm P}\mathbf{A}_{\rm h,s}\mathbf{m}\Big{(}\mathbf{T}_{\rm sp}^ {\rm f^{\prime}}-\mathbf{T}_{\rm s}^{\rm f^{\prime}}\Big{)}-\mathbf{\Phi}_{\rm s}\\ \Delta\mathbf{H}_{\rm f^{\prime}}=B_{\rm h}\mathbf{K}_{\rm h}\mathbf{m}|\mathbf{m}|\\ \Delta\mathbf{T}_{\rm n}^{\rm g^{\prime}}=\Big{(}\mathbf{D}_{1,\rm m}^{\rm S}+\mathbf{D}_ {2,\rm n}^{\rm S}-\mathbf{D}_{3,\rm n}^{\rm S}\Big{)}\mathbf{T}_{\rm n}^{\rm g^{ \prime}}-\mathbf{D}_{3,\rm n}^{\rm S}\mathbf{T}_{\rm s}^{\rm g^{\prime}}\\ \Delta\mathbf{T}_{\rm n}^{\rm R^{\prime}}=\Big{(}\mathbf{D}_{1,\rm n}^{\rm R}-\mathbf{D}_{2,\rm n}^{\rm R}\Big{)}\mathbf{T}_{\rm n}^{\rm R^{\prime}}-\mathbf{D}_{2,\rm n}^{\rm R }\mathbf{T}_{\rm sp}^{\rm R^{\prime}}-\mathbf{D}_{3,\rm n}^{\rm R}\mathbf{T}_{\rm s}^{\rm R ^{\prime}}\end{array} \right] \tag{46}\]  
It is worthwhile noting that \(\hbar T_{\rm s}\) nodes are excluded from \(\Delta\mathbf{\Phi}_{\rm s}\); if the DHN topology is radial, \(\Delta\mathbf{h}_{\rm f}\) ought to be omitted. Similarly, the Jacobian matrix of the DHN \(\mathbf{J}_{\rm hh}\) is given by the partial derivatives of \(\mathbf{F}_{\rm h}\) with respect to \(\mathbf{x}_{\rm h}\):  
\[\mathbf{J}_{\rm hh}=\left[ \begin{array}{ccc}\frac{\partial\Delta(\mathbf{\Phi}_{h} )t_{\rm h}}{\partial\mathbf{m}}&\frac{\partial\Delta(\mathbf{\Phi}_{h}t_{\rm h})}{ \partial\mathbf{T}_{\rm n}^{\rm G^{\prime}}}&\mathbf{0}\\ \frac{\partial\Delta\mathbf{T}_{\rm n}^{\rm f^{\prime}}}{\partial\mathbf{m}}&\frac{ \partial\Delta\mathbf{T}_{\rm n}^{\rm f^{\prime}}}{\partial\mathbf{T}_{\rm n}^{\rm S}} &\mathbf{0}\\ \frac{\partial\Delta\mathbf{T}_{\rm n}^{\rm g^{\prime}}}{\partial\mathbf{m}}&\mathbf{0}&\frac {\partial\Delta\mathbf{T}_{\rm n}^{\rm g^{\prime}}}{\partial\mathbf{T}_{\rm n}^{\rm R }}\end{array} \right] \tag{47}\]  
The matrix-based formulations for submatrices in the first row of \(\mathbf{J}_{\rm hh}\) follow [11]. \(\partial\Delta\mathbf{T}_{\rm n}^{\rm f^{\prime}}/\partial\mathbf{T}_{\rm n}^{\rm S \prime}\) and \(\partial\Delta\mathbf{T}_{\rm n}^{\rm R^{\prime}}/\partial\mathbf{T}_{\rm n}^{\rm R^{ \prime}}\) are commonly computed via a flow chart or formulated recursively in existing DHN models. Instead of employing these inefficient methodologies, the matrix form of the two partial derivatives can be directly derived from the proposed thermal model as follows:  
\[\left\{ \begin{array}{c}\frac{\partial\Delta\mathbf{T}_{\rm n}^{\rm g^{\prime}}} {\partial\mathbf{T}_{\rm n}^{\rm S}}=\mathbf{D}_{1,\rm n}^{\rm S}+\mathbf{D}_{2,\rm n}^{ \rm S}-\mathbf{D}_{3,\rm n}^{\rm S}\\ \frac{\partial\Delta\mathbf{T}_{\rm n}^{\rm g^{\prime}}}{\partial\mathbf{T}_{\rm n}^{ \rm R^{\prime}}}=\mathbf{D}_{1,\rm n}^{\rm R}-\mathbf{D}_{2,\rm n}^{\rm R}\end{array} \right. \tag{48}\]  
As for \(\partial\Delta\mathbf{T}_{\rm n}^{\rm g^{\prime}}/\partial\mathbf{m}\) and \(\partial\Delta\mathbf{T}_{\rm n}^{\rm R^{\prime}}/\partial\mathbf{m}\), most existing models adopt a practical and convenient approach, which approximates them as zero [11]. However, studies have shown that this approximation can compromise the convergence of the EFC [33]. Hence, to achievethe complete Jacobian matrix of the DHN with gradient descent direction, \(\partial\Delta\Upsilon_{\text{n}}^{\prime}/\partial m\) and \(\partial\Delta\Upsilon_{\text{n}}^{R^{\prime}}/\partial m\) are derived in matrix form as  
\[\left\{