# Energy-Feedback Load Simulation Algorithm Based on Fuzzy Control
## 2 Load Simulation Small-Signal Modeling
According to the concept of state-space average, the average switching period of the state variable is defined as:  
\[<x(t)>_{\mathrm{T_{S}}}=\frac{1}{T_{\mathrm{S}}}\int_{t}^{t+T_{\mathrm{S}}}x(\tau )\mathrm{d}\tau \tag{4}\]  
where \(T_{S}\) is the average switching period, and \(\tau\) is the response time. Inductance current and capacitance voltage vary less in boost circuit operating modes, and their rate of change is approximately constant. Firstly, the values of the state variables at the end of the on mode and off mode are solved and then obtained by combining with [PERSON]'s formula:  
\[ \begin{bmatrix}\frac{\mathrm{d}<i_{\mathrm{L}}>_{\mathrm{T_{S}}}}{\mathrm{d}t} \\ \frac{\mathrm{d}<i_{\mathrm{L}}>_{\mathrm{T_{S}}}}{\mathrm{d}t}\\ \end{bmatrix} = \begin{bmatrix}\frac{1}{t}\\ 0\end{bmatrix} <u_{\mathrm{IN}}>_{\mathrm{T_{S}}}+ \begin{bmatrix}0&\frac{-dt}{L} \\ \frac{dt}{C}&\frac{-1}{RC}\end{bmatrix}\begin{bmatrix}<i_{\mathrm{L}}>_{\mathrm{T_ {S}}}\\ <u_{\mathrm{C}}>_{\mathrm{T_{S}}}\end{bmatrix}  \tag{5}\]  
where \(dt=1-d\),\(d\) is boost driver duty cycle [16]. When the boost circuit operates stably, the average switching period of the state variable is constant, and thus Formula (5) can be solved to obtain:  
\[ \begin{bmatrix}I_{\mathrm{L}}\\ U_{\mathrm{C}}\end{bmatrix} =\frac{-LC}{\left(1-D\right)^{2}} \begin{bmatrix} \frac{-1}{RC}&\frac{1-D}{C}\\ \frac{-\left(1-D\right)}{L}&0\end{bmatrix}\begin{bmatrix}\frac{1}{t}\\ 0\end{bmatrix} U_{\mathrm{IN}} \tag{6}\]  
where \(I_{\mathrm{L}}\), \(U_{\mathrm{IN}}\), \(U_{\mathrm{C}}\), \(D\) are, respectively, the switching cycle average values of \(i_{\mathrm{L}}\), \(u_{\mathrm{IN}}\), \(u_{\mathrm{C}}\), \(d\) in the boost circuit at the static operating point. Since the boost converter is a nonlinear system, its state-space average Equation (5) is a nonlinear equation, and thus it needs to be solved by the perturbation method. The core idea of the perturbation method is to make small perturbations of \(i_{\mathrm{L}}\), \(u_{\mathrm{IN}}\), \(u_{\mathrm{C}}\), \(d\) around the static operating point, which is the process of slightly changing the average switching period of each parameter after perturbation, compared with the average switching period of the state variable at the static operating point, as shown in Equation (7). \[\left\{ \begin{array}{l}<i_{\mathrm{L}}>_{\mathrm{T_{S}}}=I_{\mathrm{L}}+ \widehat{i}_{\mathrm{L}}(t)\\ <u_{\mathrm{C}}>_{\mathrm{T_{S}}}=U_{\mathrm{C}}+\widehat{u}_{\mathrm{C}}(t) \\ <u_{\mathrm{IN}}>_{\mathrm{T_{S}}}=U_{\mathrm{IN}}+\widehat{u}_{\mathrm{IN}}(t) \\ d=D+\hat{d}\left(t\right)\end{array} \right. \tag{7}\]  
After substituting Equation (7) into the nonlinear Equation (5), it is directly connected with Equation (6) and solved by ignoring the second-order AC term:  
\[ \begin{bmatrix}\frac{\widehat{i}_{\mathrm{L}}(t)}{\mathrm{d}t}\\ \frac{d\widehat{u}_{\mathrm{L}}(t)}{\mathrm{d}t}\\ \end{bmatrix} = \begin{bmatrix}\frac{1}{t}\\ 0\end{bmatrix} \widehat{u}_{\mathrm{IN}}(t)+ \begin{bmatrix}\frac{U_{\mathrm{C}} }{t}\\ -\frac{I_{\mathrm{L}}}{\mathrm{C}}\end{bmatrix} \widehat{d}\left(t\right)+ \begin{bmatrix}0&\frac{-\left(1-D\right)}{L}\\ \frac{1-D}{C}&\frac{-1}{RC}\end{bmatrix}\begin{bmatrix}\widehat{i}_{\mathrm{L}}( t)\\ \widehat{u}_{\mathrm{C}}(t)\end{bmatrix}  \tag{8}\]  
Equation (8) is Laplace transformed with the initial condition of \(\widehat{i}_{\mathrm{L}}(t)=0\) into:  
\[\left\{ \begin{array}{l}s\widehat{i}_{\mathrm{L}}(s)=\frac{1}{t}\widehat{u}_ {\mathrm{IN}}(s)-\frac{\left(1-D\right)}{L}\widehat{u}_{\mathrm{C}}(s)+\frac{U_ {\mathrm{C}}}{L}\widehat{d}\left(s\right)\\ s\widehat{u}_{\mathrm{C}}(s)=\frac{1-D}{C}\widehat{i}_{\mathrm{L}}(s)-\frac{1 }{RC}\widehat{u}_{\mathrm{C}}(s)-\frac{\hat{h}}{C}\widehat{d}\left(s\right) \end{array} \right.