Automatic pH control is not that easy as is shown by the fact that automatic titrators, the equipment that you want to build, are relatively expensive.
Besides titration, automatic titrators usually have also a pH stat function, which is for maintaining a certain fixed pH.
In first place, note that to have a good control, typically you must know which type of system you are working with.
Normally, to maintain a constant pH only one reagent is added, either acid or base.
This is because chemical systems typically become more acidic or more alkaline when they proceed, but do not oscillate.
When you add both reagents, they usually start to compete: a bit of acid is added, it passes the set point, base is added, it passes the set point in the other direction, etc.
This happens especially when the response of the system is slow and the mixing is not good (shaker plate is not a good idea, use a stirrer instead).
The way to solve this is by adjusting very well the concentrations of the reagents, the delay times, addition rates, etc., and this is different in every system.
You also should know in which part of the pH curve you are working to set the concentration of the reagents and the rate of addition.
The pH curve has three very different zones, a vertical part and two horizontal parts.
Depending on where you are, the conditions must be different.
Note also that the inflection point becomes less obvious when weak acids or bases are involved.
Finally, you might need to integrate also a temperature compensation since, depending on the $\Delta\epsilon$ you have set, the pH change with temperature will be to high.
In short, a PID might work in some specific system, but for a general purpose system you will probably need some more sophisticated control like a PLC.