## 3 Continuous-Time Average State-Space Model
### Input Stage (AC-DC)  
Formulation: The input stage of the SST is the conversion of AC to DC signal, shown as architecture and equivalent diagram in Figures 2a and 2b, respectively. Equations (1) and (2) represent the input voltage; however, in Equation (2), the duty cycle is taken as an average over a switching time. Referred to Figure 2b, Equation (3) is formed by applying the KVL equation. Further, considering changes as a minute value in the DC signal, Equation (3) can be modified as Equation (4). Equation (5) is solved by subtracting Equation (3) from Equation (4). Thereafter, by applying Laplace transform to Equation (5), Equation (6) and its updated version in Equation (7) are obtained, which represent the open-loop transfer function of the input stage with the inductor current.  
Further, the transfer function of the input stage through DC-link capacitance current is presented in Equation (8). Similarly, Equations (9)-(11) are solved to reach Equation (12), which represents the ratio of the change in output DC link voltage to any change in input current.  
\[\mathrm{U_{s}}(t)=\mathrm{U_{dc}}\mathbb{D}=\mathrm{K_{m}}\mathrm{sin}(\omega t )\mathrm{U_{dc}} \tag{1}\]  
\[\mathrm{U_{s}}(t)=\mathrm{U_{dc}}\mathbb{D}_{1}(t)=\mathrm{U_{m}}\mathrm{sin}( \omega t) \tag{2}\]  
\[\mathrm{U_{m}}\mathrm{sin}(\omega t)-\mathrm{U_{dc}}\mathbb{D}_{1}(t)= \mathrm{L_{f}}\frac{\mathrm{d}(\mathrm{i_{s}}(t))}{\mathrm{dt}} \tag{3}\]  
\[\mathrm{U_{m}}\mathrm{sin}(\omega t)-\mathrm{U_{dc}}(\mathbb{D}_{1}+\triangle \mathbb{D}_{1})(t)=\mathrm{L_{f}}\frac{\mathrm{d}(\mathrm{i_{s}}+\triangle \mathrm{i_{s}}(t))}{\mathrm{dt}} \tag{4}\]  
\[-\triangle\mathbb{D}_{1}(t)\mathrm{U_{dc}}=\mathrm{L_{f}}\frac{\mathrm{d} \triangle\mathrm{i_{s}}}{\mathrm{dt}} \tag{5}\]  
\[-\triangle\mathbb{D}_{1}(s)\mathrm{U_{dc}}=\mathrm{s}\mathrm{L_{s}}\triangle \mathrm{i_{s}}(s) \tag{6}\]  
\[\frac{\triangle\mathrm{i_{s}}(s)}{\triangle\mathbb{D}_{1}(s)}=\frac{-\mathrm{ U_{dc}}}{\mathrm{s}\mathrm{L_{s}}}=\mathrm{G_{in}} \tag{7}\]  
\[\mathrm{C_{f}}\frac{\mathrm{d}\mathrm{U_{dc}}(t)}{\mathrm{dt}}=\mathbb{D}_{1}( \mathrm{i_{s}}(t)) \tag{8}\]  
\[\mathrm{C_{f}}\frac{\mathrm{d}(\mathrm{U_{dc}}(t)+\triangle\mathrm{U_{dc}}(t)) }{\mathrm{dt}}=\mathbb{D}_{1}((\mathrm{i_{s}}(t)+\triangle\mathrm{i_{s}}(t))) \tag{9}\]  
\[\frac{\mathrm{d}(\triangle\mathrm{U_{dc}}(t))}{\mathrm{dt}}=\frac{1}{\mathrm{C _{f}}}\mathbb{D}_{1}(\triangle\mathrm{i_{s}}(t)) \tag{10}\]  
\[\mathrm{s}\triangle\mathrm{U_{dc}}(s)=\frac{1}{\mathrm{C_{f}}}\mathbb{D}_{1}( \triangle\mathrm{i_{s}}(t)) \tag{11}\]  
\[\frac{\triangle\mathrm{U_{dc}}(s)}{\triangle\mathrm{i_{s}}(t)}=\frac{\mathbb{D} _{1}}{s\mathrm{C_{f}}}=\mathrm{G_{vi}} \tag{12}\]  
Figure 2: (**a**) AC-DC stage architecture, (**b**) Equivalent diagram of AC–DC stage.  
State-Space Model: Furthermore, the state-space model presented in Equations (13) and (14) denotes the state and output equation of the system where A, B, C, and D are the state, control, output, and transmission matrices, respectively. The overall transfer function of the input stage has been derived, and the state-space parameters are measured in the results section. Here \(x\) is the state variable, which is: inductor current (\(\mathrm{i_{ls}}\)) and capacitor voltage (\(\mathrm{U_{dc}}\)) and \(u\) is the input as \(\mathrm{U_{ref}}\).  
\[\dot{x}=\mathrm{A}x+\mathrm{B}u \tag{13}\]  
\[y=\mathrm{C}x+\mathrm{D}u \tag{14}\]  
Considering the input stage transfer function as derived in (7) and (12), two controllers have been designed using MATLAB SISOTOOL, as per Figure 3. Thus, corresponding to \(\mathrm{G_{vi}}\) and \(\mathrm{G_{in}}\), the PI controllers are \(G_{PI1}\) and \(G_{PI2}\), respectively.  
\[G_{PI1}=\frac{-3.82\times 10^{-6}s+6.5\times 10^{-6}}{s}\]  
\[G_{PI2}=\frac{-2.7\times 10^{-6}s+259.2\times 10^{-6}}{s}\]