# Research on Composite Control Strategy of Quasi-Z-Source DC-DC Converter for Fuel Cell Vehicles
## 2 Topology Operation Principle Analysis and Dynamic Modeling
### Dynamic Modeling
\tag{17}\]  
where the average values of the state variables are \(\overline{x}=\left[\tilde{l}_{1L1},\tilde{l}_{12},\overline{u}_{C1},\overline {u}_{C2},\overline{u}_{C3},\overline{u}_{C4},\overline{u}_{C5}\right]\) in a switching period. \(\overline{d}\) is the average duty cycle in a switching cycle. The average value of the input variable is \(\overline{u}=\overline{u}_{\text{in}}\). The average value of the output variable is \(\overline{y}=\overline{u}_{\text{o}}\). In the small-signal modeling method, each average variable is composed of a DC component and an AC small-signal component, as shown in Equation (18). \[\left\{ \begin{array}{l}\tilde{l}_{L1}=I_{L1}+\tilde{l}_{L1}\\ \tilde{l}_{L2}=I_{L2}+\tilde{l}_{L2}\\ \overline{u}_{C1}=U_{C1}+\overline{u}_{C1}\\ \overline{u}_{C2}=U_{C2}+\overline{u}_{C2}\\ \overline{u}_{C3}=U_{C3}+\overline{u}_{C3}\\ \overline{u}_{C4}=U_{C4}+\overline{u}_{C4}\\ \overline{u}_{C5}=U_{C5}+\overline{u}_{C5}\\ \overline{u}_{\text{in}}=U_{\text{in}}+\overline{u}_{\text{in}}\\ \overline{u}_{\text{O}}=U_{\text{O}}+\overline{u}_{\text{O}}\\ \overline{d}=D+\overline{d}\end{array} \right. \tag{18}\]  
where \(I_{L1}\), \(I_{L2}\), \(U_{C1}\), \(U_{C2}\), \(U_{C3}\), \(U_{C4}\), \(U_{C5}\), \(U_{\text{in}}\), \(U_{\text{O}}\) and \(D\) are DC steady-state components. \(\overline{u}_{L1}\),\(\overline{u}_{L2}\),\(\overline{u}_{C1}\),\(\overline{u}_{C2}\),\(\overline{u}_{C3}\),\(\overline{u}_{C4}\),\(\overline{u}_{C5}\),\(\overline{u}_{\text{in}}\),\(\overline{u}_{\text{O}}\) and \(\overline{d}\) are small-signal components. According to Equations (17) and (18) and ignoring the second order infinitesimal terms, the small-signal model of the converter can be written as Equation (19). \[\left\{ \begin{array}{l}\overline{x}=\overline{A}\overline{x}+\overline{B}_{1 }\overline{u}+\overline{B}_{2}\overline{d}\\ \overline{y}=\overline{C}\overline{x}+\overline{D}\overline{u}\end{array} \right. \tag{19}\]where the coefficient matrix \(\widetilde{A}\), \(\widetilde{B}_{1}\), \(\widetilde{B}_{2}\), \(\widetilde{C}\) and \(\widetilde{D}\) can be obtained as in Equation (20). \[\widetilde{A}=\left[ \begin{array}{cccccc}0&0&\frac{D-1}{L_{1}}&\frac{D}{L_{1} }&0&0&0\\ 0&0&\frac{D}{L_{2}}&\frac{D-1}{L_{2}}&0&0&0\\ \frac{1-D}{C_{1}}&-\frac{D}{C_{1}}&\frac{(r_{1}+r_{2})(D-1)}{r_{1}r_{1}r_{1}} &\frac{(r_{1}+r_{2})(D-1)}{C_{1}r_{1}}&\frac{D-1}{C_{1}r_{2}}&\frac{1-D}{C_{1}r _{1}}\\ -\frac{D}{C_{2}}&\frac{1-D}{C_{2}}&\frac{(r_{1}+r_{2})(D-1)}{C_{2}r_{1}r_{2}}& \frac{(r_{1}+r_{2})(D-1)}{C_{2}r_{1}}&\frac{1-D}{C_{2}r_{2}}&\frac{(r_{1}+r_{2} )(1-D)}{C_{2}r_{2}}\\ 0&0&\frac{D-1}{r_{2}}&\frac{D-1}{r_{2}}&\frac{D(r_{1}-r_{2})-r_{1}}{r_{2}r_{1}} &\frac{D(r_{2}-r_{1}+r_{1})}{r_{2}r_{1}}\\ 0&0&\frac{1-D}{C_{1}r_{2}}&\frac{1-D}{C_{2}r_{1}}&\frac{1-D}{C_{2}r_{2}}&\frac{D -R-r_{2}}{C_{2}R_{2}}&\frac{DR-R-r_{2}}{C_{2}R_{2}}\\ 0&0&\frac{(r_{1}+r_{2})(1-D)}{C_{1}r_{2}}&\frac{(r_{1}+r_{2})(1-D)}{C_{5}r_{1}r_ {2}}&\frac{D(r_{2}-r_{1})+r_{1}}{C_{5}r_{1}r_{2}}&\frac{DR-R-r_{2}}{C_{5}R_{2}} &\frac{DR_{1}-R_{1}r_{1}-R_{2}}{C_{5}R_{1}r_{2}}\\ \end{array} \right]\] \[\widetilde{B}_{2}=\left[