## 2 Takagi-Sugeno Fuzzy Modeling of DC-DC Converters
### T-S Fuzzy Model of a Buck-Boost Converter
On the other hand, \(\eta_{j}(\delta_{j})\) are the membership functions of the fuzzy sets \(M_{ji}\) and \(h_{i}(\delta(t))=\Pi_{j=1}^{n}\eta_{j}(\delta_{j})\) the weight contribution of the rule. The final outputs of the fuzzy systems are obtained as the weighted sum of all the local contributing submodels, leading to:  
\[ \begin{array}{l}\dot{x}(t) = \sum_{i=1}^{r}h_{i}(\delta(t))[A_{i}x(t)+B_{u_{i}}u(t)+B_{w_{i}}w( t)]\\ \\ \dot{z}(t) = \sum_{i=1}^{r}h_{i}(\delta(t))[C_{z_{i}}x(t)]\end{array}  \tag{4}\]  
where, \(0\leq h_{i}(\delta(t))\leq 1\), \(\sum_{i=1}^{r}h_{i}(\delta(t))=1\), for \(i=1,2,\ldots r\). From the incremental bilinear model (2), the T-S fuzzy model of the buck-boost converter can be obtained [20,21]. With the selection of the scheduling variables \(\delta(t)\), in most cases the state variables, the local linear models \((r=2^{j})\) necessary for the construction of the total fuzzy model are obtained from the extreme values of \(\delta(t)\). These extreme values will allow defining the rule base \((R_{i})\) of the fuzzy model, as well as the membership functions \(\eta_{j}(\delta_{j})\) necessary for the fuzzy weighting of the locally valid linear submodelsassociated to each \(R_{i}\) implication. Thus, the T-S fuzzy model approximation of the buck-boost converter can be constructed using the following steps:  
1. Find the system scheduling variables \(\delta(t)\) : \[\dot{\vec{x}}(t)=A(\delta(t))\vec{x}(t)+B_{u}(\delta(t))\vec{d}(t)\] (5)  
\[A(\delta(t))=A=\left[ \begin{array}{cc}0&\frac{D^{\prime}}{L}\\ -\frac{D^{\prime}}{C}&-\frac{1}{RC}\end{array} \right]\quad B_{u}(\delta(t))= \left[ \begin{array}{cc}\frac{V_{g}}{V_{g}^{\prime}D}-\frac{\delta_{2}(t)}{L} \\ \frac{D^{\prime}_{g}D}{D^{\prime}RC}+\frac{\delta_{1}(t)}{C}\end{array} \right]\] (6) being \(\delta(t)=[\delta_{1}(t)\quad\delta_{2}(t)]=\left[\widetilde{i_{L}}(t)\quad \widetilde{v}_{o}(t)\right]\). 2. Calculate \((r=2^{j})\) local linear models from extreme values of \(\delta(t)\). For the ordered pairs: \(\left(\delta_{1_{min}},\delta_{2_{min}}\right)\), \(\left(\delta_{1_{max}},\delta_{2_{min}}\right)\), \(\left(\delta_{1_{min}},\delta_{2_{max}}\right)\), \(\left(\delta_{1_{max}},\delta_{2_{max}}\right)\), \(\left(\delta_{1_{max}},\delta_{2_{max}}\right)\) is obtained: \[B_{u_{1}}=\left[ \begin{array}{cc}\frac{V_{g}}{V_{g}^{\prime}D}-\frac{\delta _{2_{min}}}{L}\\ \frac{V_{g}D}{(D^{\prime}R)C}+\frac{\delta_{1_{max}}}{C}\end{array} \right] \quad B_{u_{2}}=\left[ \begin{array}{cc}\frac{V_{g}}{V_{g}^{\prime}D}-\frac{ \delta_{2_{min}}}{L}\\ \frac{V_{g}D}{(D^{\prime}R)C}+\frac{\delta_{1_{max}}}{C}\end{array} \right]\] (7) \[B_{u_{3}}=\left[ \begin{array}{cc}\frac{V_{g}}{V_{g}^{\prime}D}-\frac{ \delta_{2_{max}}}{L}\\ \frac{V_{g}D}{(D^{\prime}R)C}+\frac{\delta_{1_{min}}}{C}\end{array} \right] \quad B_{u_{4}}=\left[ \begin{array}{cc}\frac{V_{g}}{V_{g}^{\prime}D}-\frac{ \delta_{2_{max}}}{L}\\ \frac{V_{g}D}{(D^{\prime}R)C}+\frac{\delta_{1_{max}}}{C}\end{array} \right]\]
3. Design the membership functions. From the extreme values of \(\delta(t)\), the membership functions are defined as follows: \[\eta_{small}(\delta_{1})=\frac{\delta_{1_{max}}-\delta_{1}}{\delta_{1_{max}}- \delta_{1_{min}}}\quad\eta_{big}(\delta_{1})=1-\eta_{small}(\delta_{1})\] (8) \[\eta_{small}(\delta_{2})=\frac{\delta_{2_{max}}-\delta_{2}}{\delta_{2_{max}}- \delta_{2_{min}}}\quad\eta_{big}(\delta_{2})=1-\eta_{small}(\delta_{2})\]
4. Build up the rule-base \(R_{i}\) of the T-S fuzzy model. Fuzzy model of the buck-boost converter is defined by the following four rules: \[ \begin{array}{l}\textbf{R}_{1}:\textbf{If }\delta_{1}\textit{ is small and }\delta_{2}\textit{ is small}\\ \textbf{R}_{2}:\textbf{If }\delta_{1}\textit{ is big and }\delta_{2}\textit{ is small}\\ \textbf{R}_{3}:\textbf{If }\delta_{1}\textit{ is small and }\delta_{2}\textit{ is big}\\ \textbf{R}_{4}:\textbf{If }\delta_{1}\textit{ is big and }\delta_{2}\textit{ is big}\\ \end{array}