How to prevent circuit from shutting down when motor drain too much power
Depending on how long the power rail drops too low:


A Schottky diode from the 12V supply feeding a suitably large capacitor powering the PLC will keep it alive as long as the capacitor voltage is adequate.
A Schottky diode fed backup battery will give you a longer holdup time. 


If the 12V supply is well regulated then a battery of 3 x LiIon cells in series will float at 4V each (or slightly less with a Schottky diode drop supplying the battery). 
You could use 8 x NiMh with a Schottky diode feed
((12-0.3)) / 8 = 1.46 V/cell).
Lead acid batteries do not last very long when floated at 12V so are unsuited to this task unless extra effort is made to keep them charged properly. 

You could operate a buck-boost converter from the input to boost the input during the low voltage periods. This would probably be less easy to implement than a battery based solution. 


It is possible that the outage is caused by the power sag waveforms rather than the voltage itself. It may be that adding a noise filter or just reasonable capacitor filtering at the PLC 12V feed point will be adequate.