The purpose of the charge controller is to fill the battery with charge as quickly as possible without causing any damage.
Letting the battery voltage rise to 13.5 volts, then holding it there using PWM is fine, but the battery saturates (fills with charge) slowly. Allowing the voltage to rise initially to 14.4 volts, then dropping back to 13.5 volts once the battery is full, speeds up the process. The battery should not be held at 14.4 volts indefinitely, nor should it be allowed to rise above 14.4 volts at any time.
So how do we know when the battery is fully charged? One method is to monitor current flow into the battery. During the saturation charge phase, the voltage is held at a constant 14.4 volts, but the current required to keep the voltage constant, gradually reduces. The reduction in current is not linear, but decays exponentially until a steady state current flow is reached.
We can’t measure current directly, but we can see it change by monitoring either PWM duty cycle change, or variations in the difference between voltage measurements taken when the FET is off and those when it is on.
A simpler, though less accurate approach is to use a timer. We can permit saturation charging at 14.4 volts for a number of minutes, then switch to float charging at 13.5 volts. However, the timer needs to be sophisticated enough to operate correctly regardless of external influences.
One such external influence is the grid-tied inverter. This device kicks in once the solar panel reaches about 14.2 volts, effectively preventing the battery ever going above this point. If the timer starts operating at 14.4 volts, it will fail to increment and the battery will be held at a high voltage for too long.
One approach would be to increment the timer slowly at voltages a little above 13.5 volts and more rapidly at voltages higher than this.