## Appendix A  
If \(\mathbf{A_{\mathrm{L}}}+\mathbf{B_{\mathrm{L}}}\mathbf{K_{\mathrm{x}}}\) is Hurwitz stable, which means all the eigenvalues of the matrix have negative real parts \(\mathrm{Re}\lambda_{i}(\mathbf{A_{\mathrm{L}}}+\mathbf{B_{\mathrm{L}}}\mathbf{K_{\mathrm{ x}}})<0\). The congruence of the matrix \(\mathbf{A_{\mathrm{L}}}+\mathbf{B_{\mathrm{L}}}\mathbf{K_{\mathrm{x}}}\) by transformation matrix \(\mathbf{T_{1}}\) does not change the eigenvalues  
\[\mathbf{T_{1}}^{-1}(\mathbf{A_{\mathrm{L}}}+\mathbf{B_{\mathrm{L}}}\mathbf{K_{\mathrm{x}}}) \mathbf{T_{1}}=\mathbf{\Lambda_{\mathrm{L}}}+\big{(}\mathbf{\hat{B}_{\mathrm{L}}}\mathbf{\hat {K}_{\mathrm{x}}}\big{)} \tag{1}\]  
where \(\mathbf{T_{1}}^{-1}\) is the inverse matrix of \(\mathbf{T_{1}}\). Since \(\mathbf{\Lambda_{\mathrm{L}}}+\big{(}\mathbf{\hat{B}_{\mathrm{L}}}\mathbf{\hat{K}_{\mathrm{ x}}}\big{)}\) is the diagonal form of the matrix \(\mathbf{A_{\mathrm{L}}}+\mathbf{B_{\mathrm{L}}}\mathbf{K_{\mathrm{x}}}\), it leads to \(\mathrm{Re}\lambda_{i}\big{(}\mathbf{\Lambda_{\mathrm{L}}}+\big{(}\mathbf{\hat{B}_{ \mathrm{L}}}\mathbf{\hat{K}_{\mathrm{x}}}\big{)}\big{)}<0\). Following the concrete forms of matrices \(\mathbf{A_{\mathrm{L}}}\) and \(\mathbf{B_{\mathrm{L}}}\) in Lemma 1, another form of the real parts of these eigenvalues can be rewritten as  
\[\mathrm{Re}[\lambda_{i}(\mathbf{A})\pm\lambda_{i}(|\mathbf{\Delta A}|_{\mathrm{bound} })\ +\lambda_{i}\big{(}\mathbf{B\hat{K}_{\mathrm{x}}}\big{)}\pm\lambda_{i}\big{(}|\mathbf{ \Delta B}|_{\mathrm{bound}}\mathbf{\hat{K}_{\mathrm{x}}}\big{)}\big{]}<0 \tag{2}\]  
According to Assumption 1, even if the maximum value condition of the above formula is chosen, it still holds that  
\[ \begin{array}{l}\max\mathrm{Re}[\lambda_{i}(\mathbf{A})\pm\lambda_{i}(|\mathbf{ \Delta A}|_{\mathrm{bound}})+\lambda_{i}\big{(}\mathbf{B\hat{K}_{\mathrm{x}}}\big{)} \pm\lambda_{i}\big{(}|\mathbf{\Delta B}|_{\mathrm{bound}}\mathbf{\hat{K}_{\mathrm{x}}} \big{)}]\\ =-|\mathrm{Re}\lambda_{i}(\mathbf{A})|+|\mathrm{Re}\lambda_{i}(|\mathbf{\Delta A}|_{ \mathrm{bound}})|-\big{|}\mathrm{Re}\lambda_{i}(\mathbf{B\hat{K}_{\mathrm{x}}}) \big{|}+\big{|}\mathrm{Re}\lambda_{i}\big{(}|\mathbf{\Delta B}|_{\mathrm{bound}} \mathbf{\hat{K}_{\mathrm{x}}}\big{)}\big{|}<0\end{array}  \tag{3}\]  
Similarly, to feedback the pair matrix \(\big{(}\mathbf{A}+\mathbf{\Delta A}\quad\mathbf{B}+\mathbf{\Delta B}\big{)}\) with \(\mathbf{K_{\mathrm{x}}}\) and then congruence it with another transformation matrix \(\mathbf{T_{2}}\), then the eigenvalues of the transformation result can be obtained as  
\[ \begin{array}{l}\mathrm{Re}\lambda_{i}(\mathbf{A}+\mathbf{\Delta A}+(\mathbf{B}+\mathbf{ \Delta B})\mathbf{K_{\mathrm{x}}})\\ =\mathrm{Re}[\lambda_{i}(\mathbf{A})+\rho_{\mathrm{A}}\times\lambda_{i}(|\mathbf{ \Delta A}|_{\mathrm{bound}})\ +\lambda_{i}(\mathbf{B}\mathbf{K_{\mathrm{x}}})+\rho_{\mathrm{B}}\times\lambda_{i}\big{(} |\mathbf{\Delta B}|_{\mathrm{bound}}\mathbf{\hat{K}_{\mathrm{x}}}\big{)}]\end{array}  \tag{4}\]where \(\rho_{\rm A}\) and \(\rho_{\rm B}\) are random variables as time independent factors(see Section 3.1). Since the two variables \(\rho_{\rm A}\in[-1,+1]\) and \(\rho_{\rm B}\in[-1,+1]\), the following inequality can be obtained as  
\[ \begin{array}{l}\mathrm{Re}[\lambda_{i}(\mathbf{A})+\rho_{\rm A}\times\lambda_{i} (\big{|}\mathbf{\Delta}\mathbf{A}|_{\rm bound})+\lambda_{i}(\mathbf{B}\mathbf{K}_{\rm x})+\rho_ {\rm B}\times\lambda_{i}\big{(}|\mathbf{\Delta}\mathbf{B}|_{\rm bound}\hat{\mathbf{K}}_{ \rm x})\big{]}\\ \mathbf{<}-|\mathrm{Re}\lambda_{i}(\mathbf{A})|+|\mathrm{Re}\lambda_{i}(|\mathbf{\Delta} \mathbf{A}|_{\rm bound})|-|\mathrm{Re}\lambda_{i}(\mathbf{B}\hat{\mathbf{K}}_{\rm x})|+| \mathrm{Re}\lambda_{i}(|\mathbf{\Delta}\mathbf{B}|_{\rm bound}\hat{\mathbf{K}}_{\rm x}) \big{|}\end{array}  \tag{12}\]  
From the Formulas (11)-(12), it can be deduced that  
\[