## 1 Introduction
### Parameter Stability Region Construction Method  
Once the system state matrix is obtained by linearizing operation according to its current operation mode, and the elements of the state matrix, especially the diagonal elements, are functions of the parameters of the controllers that dominate the dynamic characteristics. To ensure the rapid construction of PSR, a novel method is proposed. Based on GDT, the eigenvalue distribution range of the system state matrix containing controller parameter variables is rapidly estimated, substituting direct eigenvalue calculation. Then, the feasible range of the controller parameters is backward-deduced according to the small signal stability criterion; thus, the stability region of controller parameters is obtained.  
According to Lyapunov stability condition [13] for small signal stability, the system is stable when all eigenvalues of the state matrix are on the left side of the imaginary axis. When all disks of the state matrix, which encircle all eigenvalues, are on the left side of the imaginary axis, the corresponding system obviously satisfies the Lyapunov stability condition. Therefore, if the elements of the state matrix are functions of the controller parameters, the stability constraints of the parameters can be obtained according to the positions and distance between the Gerschgorin disk corresponding to the parameter-containing state matrix and the imaginary axis of the complex plane, that is, the corresponding stability region of the controller parameters can be backward-deduced via the intersection of the stability constraints of the parameters.  
Specifically, the following two stability conditions are required to construct the PSR according to GDT. Firstly, the centers of Gerschgorin disks of state matrix \(A\) are all located in the left side of the imaginary axis. Secondly, the distances between the centers of the Gerschgorin disks and the imaginary axis are larger than their corresponding radiuses of the Gerschgorin disks. Supposing that the matrix element is a function of the parameter \(x\), then the stability constraints are shown in the following  
\[\left\{\begin{array}{l}\mathrm{Re}(a_{ii}(x))<0\\ R_{i}(x)\leq|\mathrm{Re}(a_{ii}(x))|,\quad(i=1,2,\cdots,n)\end{array}\right. \tag{4}\]  
where Re means taking the real part, \(a_{ii}\) is the i-th diagonal element of matrix \(A\) and the corresponding center of the Gerschgorin disk, \(n\) is the dimension of the matrix.  
It is worth noting that this method to construct the PSR is relatively conservative because Equation (4) is a sufficient condition for stability rather than a necessary and  
Figure 1: Gerschgorin disk theorem schematic.  
sufficient condition. Therefore, an extension method of the PSR is introduced in the following to reduce the conservatism of the PSR.  
The boundary points of the PSR in the direction of the focused parameter extension are selected to re-form the eigenvector matrix, and the extended PSR can be obtained by repeating the formation process of the PSR mentioned above and then superposing all the PSRs. Obviously, the conservativeness of the PSR can be reduced by expanding the PSR several times, forming the extended PSR. As computational complexity also increases correspondingly, the conservatism and the computational complexity of the PSR need to be balanced for practical online application.