## 2 Takagi-Sugeno Fuzzy Modeling of DC-DC Converters
### T-S Fuzzy Model of a Buck-Boost Converter  
Figure 1 shows the well-known circuit of a buck-boost converter, which is capable to step-up/down the output voltage \(v_{o}(t)\) from the input one \(V_{g}\). \(R\) represents the converter nominal load, while \(L\) and \(C\) stand for the inductance and capacitance values, respectively. Besides, source \(i_{o}(t)\) represents the load current disturbance. The state variables considered for the analysis are the inductor current \(i_{L}(t)\) and the capacitor voltage \(v_{o}(t)\). The binary signal \(u(t)\) turn the MOSFET ON when \(u=1\) and OFF when \(u=0\) at a constant switching frequency \(\frac{1}{T_{s}}\), such as shown in Figure 1. The step-up/down operation of the converter in steady state depends of the ratio \(\frac{T_{\rm out}}{T_{s}}\), which represents the duty cycle \(d\) of the converter. It is further assumed that the converter operates  
Figure 1: Schematic circuit of a buck-boost converter. in continuous conduction mode (CCM), and without parasitic elements. The following expression shows the state-space bilinear model of the buck-boost converter [23]:  
\[ \begin{array}{l}\dot{\tilde{x}}(t)=A\tilde{x}(t)+B_{w}\tilde{w}(t)+B_{u}( \tilde{x})\tilde{u}(t)\\ \tilde{z}(t)=C_{z}\tilde{x}(t)+D_{zw}\tilde{w}(t)+D_{zu}\tilde{u}(t)\end{array}  \tag{1}\]  
where \(\tilde{x}(t)\), \(\tilde{u}(t)\), \(\tilde{w}(t)\) are the incremental averaged values of the state vector, input vector, and disturbance inputs around equilibrium values \(X\), \(U\) and \(W\), respectively. Besides, \(\tilde{z}(t)\) represents the controlled output \(\widetilde{v}_{o}(t)\). Thus, based on the circuit of Figure 1, the expression (1) can be written as :  
\[ \begin{array}{l}\tilde{x}(t)=\left[\begin{array}{c}\widetilde{l}_{L}\\ \widetilde{v}_{C}\end{array}\right]\quad X=\left[\begin{array}{c}\frac{V_{r} D}{KD_{w}^{2}}\\ -\frac{1}{D^{\prime}}\end{array}\right]\quad\tilde{u}(t)=\left[\tilde{d}(t) \right]\quad\widetilde{w}(t)=\left[\widetilde{i}_{o}(t)\right]\quad\tilde{z}(t )=\left[\widetilde{v}_{C}(t)\right]\\ \\ A=\left[\begin{array}{cc}0&\frac{D^{\prime}}{KD_{u}^{\prime}}\\ -\frac{D^{\prime}}{C}&-\frac{1}{RC}\end{array}\right]\quad B_{u}(\tilde{x})= \left[\begin{array}{c}\frac{V_{r}}{KD_{w}^{\prime}}-\frac{\widetilde{v}_{C}( t)}{\widetilde{v}_{c}}\\ \frac{V_{r}D}{(D^{\prime}R)C}+\frac{\widetilde{h}_{o}(t)}{C}\end{array}\right] \quad B_{w}=\left[\begin{array}{c}0\\ -\frac{1}{C}\end{array}\right]\\ \\ C_{z}=\left[\begin{array}{cc}0&1\end{array}\right]\quad D_{zw}=[0]\quad D_{zu }=[0]\end{array}  \tag{2}\]  
where \(D^{\prime}=1-D\) is the complementary steady-state duty cycle. In order to ensure zero steady state error in the output voltage \(v_{o}(t)\), a new state variable \(x_{3}(t)=\int\!\left(v_{o}(t)-V_{ref}\right)\) has been introduced in the model (2). This integral function forces \(v_{C}(t)\to V_{ref}\) when \(t\to\infty\), where \(V_{ref}\) is the voltage reference. Takagi-Sugeno Fuzzy representation allows to describe the dynamics of a non-linear system by means of a set of local linear models based on fuzzy rules, which are smoothly connected by membership functions. The rule set of a T-S fuzzy model is written as:  
\[ \begin{array}{l}R_{i}:If\ \delta_{1}\ \ \textit{is}\ \ M_{i1}\ \ \textit{and}\ \ \textit{\ldots}\ \textit{and}\ \ \delta_{j}\ \ \textit{is}\ \ M_{ji}\ \ \textit{then}\\ \\ \left\{\begin{array}{l}\dot{x}(t)=A_{i}x(t)+B_{u_{i}}u(t)+B_{w_{i}}w(t)\quad i =1,2,\ldots r\\ \dot{z}(t)=C_{z}x(t)\end{array}\right.\end{array}  \tag{3}\]  
where \(r\) is the number of the submodels, \(A_{i}\) are system matrices of the i-th linear submodel, \(B_{u_{i}}\) are input matrices, \(B_{w_{i}}\) are disturbance inputs matrices, \(C_{z_{i}}\) are controlled output matrices, \(x(t)\) is the global state-space vector, \(u(t)\) is the input vector, \(w(t)\) is the disturbance input vector, \(z(t)\) is the controlled output vector, \(M_{ji}\) are the fuzzy sets, and \(\delta_{j}\) are the scheduling vector or premise variables [13].