## 4 Results and Discussion
### Time Domain and Frequency Domain Analysis
The corresponding state-space parameters \((\text{A},\text{B},\text{C},\text{and}\text{D})\) followed by the test matrix (controllable and observable matrix) of the isolation stage are shown below, which is also controllable and observable because the rank of the matrix is the same as that of the test matrix. Further, the closed-loop transfer function of the proposed DAB converter is expressed in Equation (65) and after the assigned value, it is presented in Equation (66). \[\text{G}_{\text{filter}}=\frac{0.001362\text{s}+1}{4.916\times 10^{6}\text{s}^{ 2}+0.04229\text{s}} \tag{64}\]  
The corresponding A, B, C, and D matrix are:  
\[\text{A}=1\times 10^{3} \begin{bmatrix}-8.851&0\\ 0.0010&0\end{bmatrix} _{2\times 2}\text{B}= \begin{bmatrix}1\\ 0\end{bmatrix} \text{C}=1\times 10^{5} \begin{bmatrix}0.0028&2.0300\end{bmatrix}  \text{D}=[0]\]  
\[\text{M}_{\text{c}}=1\times 10^{3} \begin{bmatrix}0.0010&-8.5851\\ 0&0.0010\end{bmatrix} \text{M}_{\text{o}}=1\times 10^{6} \begin{bmatrix}0.0003&0.2 030\\ -2.1707&0\end{bmatrix} \]  
\[\text{G}_{\text{closed-loop}}=\text{G}_{\text{converter}}\times\text{G}_{ \text{filter}} \tag{65}\]  
\[\text{G}_{\text{closed-loop}}=\frac{61.29\text{s}^{3}+5.697\times 10^{6}\text{s}^{ 2}+8.275\times 10^{12}\text{s}+6.072\times 10^{15}}{4.926\times 10^{-6}\text{s}^{ 5}+0.497\text{s}^{4}+9.25\times 10^{14}\text{s}^{3}+7.914\times 10^{8}\text{s}^{ 2}+2.649\times 10^{11}\text{s}} \tag{66}\]  
Output stage controllability and observability test: This sub-section is similar to Sections 3.1 and 3.2. As in Equations (62) and (63), \(\text{M}_{\text{c}}\) and \(\text{M}_{\text{0}}\) are controllable and observable matrices, and the values are given below. The ranks of these matrices are 3, which signifies the system is controllable and observable, which subsequently means that the output stage is controllable and observable. \[\text{M}_{\text{c}}=\mathbf{1}\times\mathbf{10}^{\mathbf{10}} \begin{bmatrix} \mathbf{0}&-\mathbf{0.000}&2.950\\ \mathbf{0}&\mathbf{0}&-\mathbf{0.000}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix} \]  
\[\text{M}_{\text{0}}=\mathbf{1}\times\mathbf{10}^{\mathbf{22}} \begin{bmatrix} \mathbf{0}&\mathbf{0}&\mathbf{0}\\ -\mathbf{0.000}&\mathbf{0}&\mathbf{0}\\ -\mathbf{0.0001}&-7.4418&\mathbf{0}\end{bmatrix} \]  
The stability criterion is shown in Table 1. Further, the performance index is measured through frequency domain analysis and presented in Table 2 for the input, isolation, and output stages. The frequency and time domain analysis collectively offers a better understanding of the SST behavior. The designer can recognize the critical points and stability margins to ensure reliable operation and robust performance. For instance, in the b \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **System** \\ \hline GM = +ve, PM = +ve & Stable \\ GM = -ve, PM = +ve & Unstable \\ GM = +ve, PM = -ve & Unstable \\ \hline \end{tabular}
\end{table} 
Table 1: Frequency response indices criteria. response of SST, on account of the diverse frequency ranges, identifies the resonance points to ensure the system's stability. Similarly, the root locus plot is exhibited here, which can visualize the pole's movement in the system, which explains how a control parameter can change the system's stability. The Bode diagram reflects the variation in magnitude and phase with respect to frequency level at a log scale. Various parameters, such as GM, PM, \(\omega\)cg, and \(\omega\)cp, can be measured from the Bode plot. Thus, system stability margins can be determined. The Nyquist plot reflects the relationship between the imaginary part of the closed-loop transfer function and the radial frequency (\(\omega\)) in the real axis with the variation in '\(\omega\)' from zero to infinite in the polar coordinate. This plot determines the absolute and relative stability even if the system has delays in practical assumption. The plot also provides information such as GM, PM, BW, \(f_{peak}\), and \(g_{peak}\). The Nichols plot is the extension of the polar plot.