## 5 Implementation of the Adaptive SMC and Detailed Circuital Simulations
### Analog Implementation of the Adaptive SMC  
The implementation of the control algorithm, using analog circuits, is presented in Figure 4. Such a circuit considers the implementation of the high-pass filter using opera \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Value** & **Nominal Condition** & **Energy Stored** \\ \hline \(L_{1}\) & 330 \(\mu\)H & 0 A & 0 Wh, 0 Ah \\ \hline \(L_{2}\) & 330 \(\mu\)H & 0 A & 0 Wh, 0 Ah \\ \hline \(C_{1}\) & 470 \(\mu\)F & 48 V & 0.15 mWh, 0.003 mAh \\ \hline \(C_{2}\) & 470 \(\mu\)F & 48 V & 0.15 mWh, 0.003 mAh \\ \hline \(\Delta\Psi\) & 0.3 A & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Circuital parameters of the application example. tional amplifiers, as discussed in [36]. Similarly, addition, subtraction, and multiplication by a constant value can be easily implemented with operational amplifiers, as reported in [36]. The multiplication and division operations can be implemented using analog integrated circuits like the AD734 described in [37]. Therefore, the calculation of the adaptive parameters \(k_{c}\) and \(k_{v}\), the calculation of the \(i_{R}\) signal, and the calculation of the switching function \(\Psi\) are easily implemented with analog circuits. Concerning the control law, the logical operations in Algorithm 2 can be implemented using two comparators and an S-R Flip-Flop, as depicted in Figure 4. When \(\Psi\geq+\Delta\Psi\), the signal \(Set=1\) activates the \(Set\) input of the Flip-Flop, which imposes \(u=1\) and \(\overline{u}=0\); similarly, when \(\Psi\leq-\Delta\Psi\), the signal \(Reset=1\) activates the \(Reset\) input of the Flip-Flop, which imposes \(u=0\) and \(\overline{u}=1\). Finally, Figure 4 shows a diagram of those control signals. The first simulation of this analog implementation, reported in Figure 5, evaluates the compensation of a load transient from steady state to battery charge with the maximum current derivative \(\left|\frac{d{i_{c}}}{dt}\right|=70\) A/ms. Figure 5a shows two fast current transients between 0 A and 2 A, and in both cases, the Zeta converter provides the fast current component, while the battery exhibits a slow change in the current, thus fulfilling the desired behavior of the system. In addition, the \(C_{2}\) voltage is constrained to the \(MO_{C2}\) limit for such a transient condition, which validates the design of the converter components. Moreover, since \(v_{b}=v_{R}\), then the average value of \(k_{c}\) is near 1 and the stability condition of the inductor currents is \(i_{L1}=i_{L2}\), which is confirmed in the simulation. It is observed that the switching function \(\Psi\) is permanently trapped inside the hysteresis band \([-\Delta\Psi,+\Delta\Psi]\); thus, the SMC is always inside the sliding mode, which ensures the global stability. Finally, the switching frequency is always below the maximum design limit of 120 kHz. Figure 5b presents a zoom of the previous simulation, which confirms that the Zeta converter provides the current transient, while the battery current changes with the frequency limit defined by the high-pass filter. In addition, the zoom of the inductor currents confirms the relation imposed by the first part of the switching function and confirms both currents' stable behavior. Finally, the zoom of the switching function \(\Psi\) shows the frequency reduction at the perturbation instant, which is caused by the fast change in the load current, where the SMC can impose three switching periods during the transient time. Figure 6 reports a second simulation, which evaluates the compensation of a load transient from steady state to battery discharge with the same maximum current derivative. Figure 6a shows the correct behavior of both the Zeta converter and battery currents, while \(C_{2}\) voltage and the inductor currents are regulated as expected. As in the previous case, the switching function is constrained to the hysteresis band, ensuring global stability.