## 2 Modeling of the Power Stage
### Control-Oriented Model  
This section derives a mathematical model for the hybrid power system of Figure 2, which is aimed at supporting the controller design. The first step is to analyze the power system when the activation signal of the MOSFETs is \(u=1\) (\(\overline{u}=0\)), thus \(M_{1}\) is ON and \(M_{2}\) is OFF, as illustrated in Figure 3.  
The resulting differential equations for this operation condition, for both the inductors and capacitors, are:  
\[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]  
\[\frac{di_{L2}}{dt}=\frac{v_{sco}+v_{ci}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{ L2}\cdot R_{L2}+(i_{dc}-i_{L2})\cdot R_{b}-v_{\text{loc}}}{L_{2}} \tag{2}\]  
\[\frac{dv_{ci}}{dt}=\frac{-i_{L2}}{C_{i}} \tag{3}\]  
\[\frac{dv_{sc}}{dt}=\frac{-i_{L1}-i_{L2}}{C_{sc}} \tag{4}\]  
The second step is to analyze the power system when the activation signal of the MOSFETs is \(u=0\) (\(\overline{u}=1\)), thus \(M_{1}\) is OFF and \(M_{2}\) is ON, as illustrated in Figure 4. Then, the resulting differential equations of this new condition are given in Equations (5)-(8). \[\frac{di_{L1}}{dt}=\frac{-v_{ci}-(i_{L1}+i_{L2})\cdot R_{on}-i_{L1}\cdot R_{L1 }}{L_{1}} \tag{5}\]  
\[\frac{di_{L2}}{dt}=\frac{-(i_{L1}+i_{L2})\cdot R_{on}-i_{L2}\cdot R_{L2}+(i_{ dc}-i_{L2})\cdot R_{b}-v_{\text{loc}}}{L_{2}} \tag{6}\]  
\[\frac{dv_{ci}}{dt}=\frac{i_{L1}}{C_{i}} \tag{7}\]  
\[\frac{dv_{sc}}{dt}=0 \tag{8}\]  
\[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]0 HESS operation for \(u=1\). \[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]1 HESS circuital interface for bidirectional power flow. The switched model of this power system is obtained by combining each pair of the previous differential equations using the activation signal (\(u\)) and its complement (\(\overline{u}\)) as follows:  
\[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]2  
\[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]3 \[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]4  
\[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]5  
\[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]6  
A linearized representation of the previous model is obtained to perform the adaptive controller design. This linearization process is based on the averaging method, in which all the switched differential Equations (9)-(12) are averaged within the switching period \(T=1/F\), where \(F\) is the switching frequency. Taking into account that the duty cycle is \(d=(1/T)\int_{0}^{T}u~{}dt\), the averaged differential equations have \(u\) replaced by \(d\). The linearized model is developed in small-signal conditions, where \(\bar{x}\), \(\bar{d}\), and \(\bar{y}\) are the small-signal variations of the state vector, input vector, and output, respectively. Then, the model states are defined as follows:  
\[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]7  
Based on Equation (13), the small-signal state vector is \(\bar{x}=[\bar{x}_{1}\ \bar{x}_{2}\ \bar{x}_{3}\ \bar{x}_{4}]^{T}\). Finally, the state-space representation of the small-signal model is given in Equation (14), where \(A_{m}\) and \(B_{m}\) are matrices evaluated in steady-state conditions around a given operation point. \[\frac{di_{L1}}{dt}=\frac{v_{sco}-(i_{L1}+i_{L2})\cdot(R_{on}+R_{sc})-i_{L1} \cdot R_{L1}}{L_{1}} \tag{1}\]8 HESS operation for \(u=0\). \[\dot{\vec{x}}=\underbrace{