## 1 Introduction
### Stability Margin Evaluation Indexes
For the matrix \(\mathbf{A}\), there are the following relations  
\[\mathbf{A}\cdot[x_{1},\cdots x_{n}]=[x_{1},\cdots x_{n}]\cdot\mathbf{D} \tag{8}\]  
i.e.,  
\[\mathbf{A}=[x_{1},\cdots x_{n}]\cdot\mathbf{D}\cdot[x_{1},\cdots x_{n}]^{-1} \tag{9}\]  
The matrix \(\mathbf{A}\) is multiplied by \(\mathbf{P}^{-1}\) on the left and then by \(\mathbf{P}\) on the right. \[\mathbf{P}^{-1}\mathbf{A}\mathbf{P}=\mathbf{P}^{-1}[x_{1},\cdots x_{n}]\cdot\mathbf{D}\cdot[x_{1},\cdots x_{n}]^{-1}\mathbf{P}=\mathbf{B} \tag{10}\]  
Let \(\mathbf{P}1=[x_{1},\cdots x_{n}]^{-1}\mathbf{P}\), which means the matrix \(\mathbf{B}\) can also be expressed as \(\mathbf{B}=\mathbf{P}1^{-1}*\mathbf{D}*\mathbf{P}1\), and \(\mathbf{B}\) is named after the transition matrix. The right eigenvector matrix \(\mathbf{X}\) is equal to \(\mathbf{P}\) at the initial set of parameters (\(a_{0}\), \(b_{0}\)), and, currently, \(\mathbf{P}1\) is a unit diagonal matrix. The eigenvalues of \(\mathbf{B}\) are equal to \(\mathbf{A}\), thus \(\mathbf{B}\) is exactly the eigenvalue diagonal matrix of \(\mathbf{A}\). It is worth noting that, on the one hand, \(\mathbf{A}\) is stable in the neighborhood of the initial set of parameters, which means its eigenvalues \(\lambda_{1}\cdots\lambda_{n}\) are all non-negative, but the diagonal elements of \(\mathbf{A}\) may still exist non-negative elements; thus, the preconditions for applying GDT to the PSR construction cannot be satisfied. However, matrix \(\mathbf{B}\) at the initial parameters is the same as the eigenvalue diagonal matrix \(\mathbf{D}\) of matrix \(\mathbf{A}\), the Gerschgorin disks of \(\mathbf{B}\) are distributed as a set of points, the radius of each disk is \(0\) and the center of each disk is the eigenvalue. In the neighborhood of the initial parameter set, the eigenvalues of \(\mathbf{A}\) vary, and \(\mathbf{B}\) is approximately equal to the diagonal matrix of the eigenvalues of \(\mathbf{A}\). As long as system \(\mathbf{A}\) is stable, the eigenvalues of \(\mathbf{A}\) are all negative; in other words, the diagonal elements of the \(\mathbf{B}\) matrix are negative, satisfying the precondition for the application of GDT to structure PSR. On the other hand, it can be further proved that the eigenvalues of \(\mathbf{A}\) are located in the disks of \(\mathbf{B}\) in the neighborhood of the initial parameter set; thus, the disks of \(\mathbf{B}\) can be used to constrain the eigenvalues of \(\mathbf{A}\). Therefore, when the disks of \(\mathbf{B}\) are located on the left side of the imaginary axis, the eigenvalues of \(\mathbf{A}\) are also located on the left side of the imaginary axis. In consequence, the PSR can be constructed via the method in Section 2 by replacing state matrix \(\mathbf{A}\) with the corresponding system transition matrix \(\mathbf{B}\) containing parameter variables. Although there is one more eigenvalue calculation used to apply system transition matrix \(\mathbf{B}\) for the PSR construction compared with the state matrix, the unsatisfied precondition that the diagonal elements of the initial system state matrix \(\mathbf{A}\) must be negative is eliminated, and the application scope of the PSR construct method is expanded. The superiority of the computational efficiency of the proposed method, compared with the point-wise eigenvalue calculation to construct PSR, increases its potential for online application to prevent and control SSO issues. Case Study of the Construction of the Sub/Super-Synchronous Oscillation-Oriented Dominant Controller Parameters Stability Region