Charging 24V battery from 12V-24V converter

You have a constant-voltage (CV) power supply, and you need constant-current (CC) power supply. You will want to add an external control loop to command the right voltage so the current is maintained. Luckily, it is not that hard -- your load will not change rapidly, and it can accept short overcurrent intervals.

You will want to add a microcontroller (arduino if you are a beginner, or something more rugged if you can do it) with a way to measure current and voltage going into the battery -- most likely this means you need a shunt and a voltage divider. This microcontroller must control the power supply in whatever way it needs to be controlled.

During bulk charge, the current is only limited by your power supply -- so increase voltage slowly until the current reaches required fraction (80%?) of power supply's capacity. As the battery is charging, change the voltage in small steps to maintain the current. When the battery voltage reaches threshold, you switch to holding mode and decrease the voltage to maintain a much smaller current.

As for charging the series of batteries, the method in the paper will work fine if all batteries are similar, and if they are always discharged simultaneously. If you buy batteries used, and each battery in series is from a different source, OR if you sometimes connect load to only one battery (as opposed to all four in series), then your chain will become unbalanced. If you are worried about that, add more voltage inputs to your MCU to measure the voltage between the batteries. If you detect disbalance, limit the charging current to 0.1C until it disappears.

I think batteries usually follow a CC/CV or constant current/constant voltage system to charge up batteries. The LiPoly batteries need such a mechanism. So, this can be achieved by a system that sends out PWMs that control the current and voltage in a more controlled manner. Ofcourse lead acid charging will be different. But please read up on that as I believe you are bypassing that.

And yes over-charging is definetily an issue in your case. You can ofcourse have a dedicated IC or even a combination of op-amp/comparators to do the job for you. When the voltage in the battery reaches a specific value then have the comparator/opamp circuitry turn off or isolate it via a MOSFET(again check the current rating and heat dissipation here).

What you are doing is blindly charging the battery which can very well end up in a pile of ash.