## 4 Results and Discussion
### Time Domain and Frequency Domain Analysis  
The proposed SST model is designed and analyzed in terms of time response, frequency response, and state-space characteristics. The time response analysis is conducted by the root locus technique, which determines the absolute stability and shows the location of poles or zeros in the s-plane. This technique also determines the physical performance of the model when the gain of the input to the plant is changed from 0 to \(\infty\). The technique is advantageous in controller design because it is organized and saves energy and time. This  
Figure 6: (a) DC-AC stage architecture (b) Equivalent diagram of DC–AC stage. study is limited to the linearized model, and both eigen analysis and frequency domain testing are applied to the linear model. In addition, time-domain analysis is studied. Input stage numerical study: To ensure the stable operation of the system, a controller is added through MATLAB SISIOTOOL, given in Equations (59) and (60) for \(\text{G}_{\text{in}}\) and \(\text{G}_{\text{vi}}\), respectively. The overall closed-loop transfer function of the input stage converter is given by Equation (61). \[\text{PI}_{\text{in}}=\frac{-\big{(}3.825\times 10^{-6}\text{s}+6.5\times 10^{-6} \big{)}}{\text{s}} \tag{59}\]  
\[\text{PI}_{\text{vi}}=\frac{-2.7\times 10^{-6}\text{s}+0.0002592}{\text{s}} \tag{60}\]  
\[\text{G}_{\text{input}}=\frac{2.9\times 10^{-6}\text{s}^{2}+1.717\times 10^{-6} \text{s}+1.76\times 10^{-8}}{7.65\times 10^{-6}\text{s}^{4}+6.5\times 10^{-6} \text{s}^{3}+6.69\times 10^{-6}\text{s}^{2}+1.717\times 10^{-6}\text{s}+1.76 \times 10^{-8}} \tag{61}\]  
\[\text{A}= \begin{bmatrix}-0.8497&-0.8745&0.2244&-0.0023\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ \end{bmatrix} \]  
\[\text{B}= \begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix} \]  
\[\text{C}= \begin{bmatrix}0&0.3791&0.2244&0.0023\\ \end{bmatrix} \]  
\[\text{D}= \begin{bmatrix}0\end{bmatrix} \]  
Input stage controllability and observability test: To test the system controllability and observability, the controllable matrix (\(\text{M}_{\text{c}}\)) and observable matrix (\(\text{M}_{0}\)) can be measured, which are expressed in Equation (62) and Equation (63), respectively, while their corresponding values are given below. Given the matrices \(M_{\text{c}}\) and \(M_{0}\), the ranks of both are 4, which signifies the system is controllable and observable because the ranks of matrices (\(\text{M}_{\text{c}}\) and \(\text{M}_{0}\)) are the same as 4. \[\text{M}_{\text{c}}= \begin{bmatrix}\text{B}\text{ AB A}^{2}\text{B A}^{3}\text{B}\end{bmatrix}  \tag{62}\]  
\[\text{M}_{0}= \begin{bmatrix}\text{C}\\ \text{CA}\\ \text{CA}^{2}\\ \text{CA}^{3}\end{bmatrix}  \tag{63}\]  
\[\text{M}_{\text{c}}= \begin{bmatrix}1&-0.8497&-0.1526&0.6482\\ 0&1&-0.8497&-0.1526\\ 0&0&1&-0.8497\\ 0&0&0&1\end{bmatrix} \]  
\[\text{M}_{0}= \begin{bmatrix}0&0.3791&0.2244&0.0023\\ 0.3791&0.2244&0.0023&0\\ -0.0977&-0.3292&-0.0851&-0.0009\\ -0.2462&0.0003&0.0210&0.0002\end{bmatrix} \]  
Isolation stage controllability and observability test: To test the system controllability and observability, the controllable and observable matrices as \(\text{M}_{\text{c}}\) and \(\text{M}_{\text{0}}\) respectively, and the values are given below. \[\text{M}_{0}=1\times 10^{7} \begin{bmatrix}0.0000&0&0\\ -0.0000&-0.0000&0.0000\\ -6.6707&1.6839&-4.5691\end{bmatrix} _{3\times 3}\]\[\text{M}_{\text{c}}=1\times 10^{18} \begin{bmatrix}0.0000&-0.0000&0.0053\\ 0.0000&-0.0000&1.3348\\ -0.0000&0.0000&1.5723\end{bmatrix} _{3\times 3}\]  
From the \(\text{M}_{\text{c}}\) and \(\text{M}_{0}\), the ranks of both are 3, which signifies the system is controllable and observable. To ensure the stability of the system and its closed-loop function, a feedback controller and a filter circuit must be added, which is expressed in Equation (64).