## 4 Design Procedure and Application Example
### Summary of the Design and Control Processes  
The summary of the design process for the parameters of both the power stage and SMC is synthesized in Algorithm 1. Such a process starts with the design of inductor \(L_{2}\) to fulfill the maximum load current derivative. To ensure at least two switching periods within the transient, a constant \(k_{L}\leq 1/2\) is used to fulfill inequality (46), where smaller \(k_{L}\) produces higher number of periods per transient, but very small \(L_{2}\) values will require very small \(\Delta\Psi\) limits, which could introduce implementation problems due to circuit sensitivity. Therefore, \(k_{L}\) must be selected to ensure that \(\Delta\Psi\) magnitude enables the correct operation of the comparators used to implement the control law (53). Next, \(L_{1}\) is set equal to \(L_{1}\) to provide a balance in the current ripple of both inductors. The capacitor \(C_{2}\) is designed to limit the voltage deviation up to \(MO_{C2}\), and \(C_{1}\) is designed to balance the energy stored in both capacitors. Then, the steady-state values of both the current (\(k_{c,ss}\)) and voltage (\(k_{\gamma,ss}\)) constants are calculated from the nominal values of the application, and both constants are used to calculate the hysteresis band value \(\Delta\Psi\). Finally, depending on the application, the high-pass filter is parameterized in agreement with the safe frequency range defined for the battery transients. ```
0:\(v_{b}\), \(v_{R}\), \(MO_{C2}\), \(t_{s,C2}\), \(\max\left(\left|\frac{d_{i}}{dt}\right|\right)\), \(\Delta i_{o}\), \(F_{safe}\), \(F_{sw}\)
1:\(L_{2}\leftarrow\) near commercial value of \(\left\{k_{L}\cdot\frac{v_{b}}{\max\left(\left|\frac{d_{i}}{dt}\right|\right) \right\}\right\}\) where \(k_{L}\leq\frac{1}{2}\)
2:\(L_{1}\gets L_{2}\)
3:\(C_{2}\leftarrow\) near commercial value of \(\left\{\frac{\Delta i_{c}}{2\cdot\pi\cdot F_{safe}\cdot(v_{R}\cdot MO_{C2})}\right\}\)
4:\(C_{1}\leftarrow\) near commercial value of \(\left\{C_{2}\cdot\left(\frac{v_{R}}{v_{b}}\right)^{2}\right\}\)
5:\(k_{c,ss}\leftarrow\left\{\frac{v_{b}}{\sqrt{k_{L}}}\right\}\)
6:\(k_{c,ss}\leftarrow\left\{-\frac{3.9\cdot C_{2}}{k_{c,ss}\cdot t_{s,C2}}\right\}\)
7:\(\Delta\Psi\leftarrow\max(\left|\Delta i_{L2}\right|,\left|k_{v,ss}\cdot\Delta v _{C2}\right|))\) calculating \(\Delta i_{L2}\) from (22) and \(\Delta v_{C2}\) from (23)
8:\(G_{hp}\leftarrow\left\{\frac{s}{s+2\cdot\pi\cdot F_{safe}}\right\}\)
9:return\(L_{1}\), \(L_{2}\), \(C_{1}\), \(C_{2}\), \(\Delta\Psi\), \(G_{hp}\)
```  
**Algorithm 1** Design of both the power stage and SMC parameters  
Implementing the adaptive controller requires processing, in an infinite loop, a series of calculations based on measurements and fixed parameters. Such a control process is synthesized in Algorithm 2. The algorithm requires the desired reference for the voltage in \(C_{2}\) and the value of the two inductor currents, both the battery and \(C_{2}\) voltages, and the load current. The next step is to detect the high-frequency transients on the load current using the high-pass filter. Then, both the current (\(k_{c}\)) and voltage (\(k_{v}\)) gains are adapted for the actual operation condition, and with those values, the switching function \(\Psi\) for the operation conditions is calculated. Using the values of \(\Psi\) and \(\Delta\Psi\), the control law of the SMC (53) is executed, which produces the binary values for controlling the MOSFETs (\(u\) and \(\overline{u}\)). Finally, the loop starts again by measuring the physical variables.