There are many ways to build a DC to AC inverter.
For instance, I earn my living lately by selling inverters (each set includes a charger too).
Based on the components available now in the local market (during the actual war), I made it rather simple (not SMPS) after adding extra functions:
(1) The output Vrms is regulated if the battery voltage > 12V (or > 24V if the model is for two batteries). This is done by decreasing the output duty cycle properly.
(2) A Soft start for the output is generated during the 1st second (by increasing the output duty cycle drom 5% to 80%, I keep 20% for the dead time between the half cycles).
(3) The current overload monitoring is instantaneous though an idle delay is added to pass the inrush current required by some appliances (820ms).
The inverter transformer is also used for charging when the mains power is on.
Obviously the charger regulates the charging current (set to about 20A) and limits the maximum battery voltage (to about 14.5V, after which the battery takes the current it needs, down to its leakage one, till its voltage decreases below 13.5V). 2 triacs at the transformer primary (connected to 2 different terminals, only one triac is active at a time) are used to regulate the charging current.
The main electronic components are the power MOSFET (here we have 75NF75N) and the MCU (here we have AT89C2051).
Fortunately, I was able to have a 8051 assembler (for DOS) that can generate the object file for the MCU after I write the appropriate assembly program.