Connecting the LC circuit to dc voltage source without transient

Question

Let's say I have following circuit

where the transistors are controlled via a software in the microcontroller.

I have been looking for a control algorithm how to achieve a state when both of the upper transistors are open (turned on) and this state has been achieved without transient in the LC circuit. It is worthwhile to say that both of the inductor current (\$i_L\$) and capacitor voltage (\$v_C\$) are measurable.

I have tried a naive approach when I open both of the transistors when the capacitor voltage (\$v_C\$) is equal to the source voltage (\$v_{bat}\$). Unfortunately this approach doesn't lead to success - the transient in the LC circuit occurs as can be seen in the attached graph.

Edit:

1. As far as the decrease of the capacitor voltage \$v_C\$ in the time interval \$<0, 0.18>\,\mathrm{s}\$. The voltage decrease is caused by transistor switching in the below given circuit

which is connected in parallel to the capacitor. The transistor is controlled also by the software in the microcontroller. The idea behind gradually reducing of the capacitor voltage was to balance the capacitor and battery voltage before opening (turning on) the upper transistors (I supposed that it should be sufficient for elimination of the transient).

2. As far as the nominal values of the passive elements

\$R = 19\,\mathrm{m\Omega}\$, \$L = 1\,\mathrm{mH}\$, \$C = 3.36\,\mathrm{mF}\$, \$R_b = 1.32\,\mathrm{\Omega}\$.

Answer 1

I have been looking for a control algorithm how to achieve a state when both of the upper transistors are open and this state has been achieved without transient in the LC circuit.

I don't believe there is a solution that solves this completely. You are storing energy in the inductor from the battery and then releasing that energy into the capacitor some short time later. You are bound to get a damped resonant oscillation. It's therefore a numbers-game where you attempt to get that ripple voltage down to an acceptable level.

Storing/transferring less energy but more frequently will reduce the capacitor ripple (as will increasing the size of the capacitor). Standard practice in this type of converter.