PLC to Control 3Khz-6Khz Single Phase Inverter

Which PLC would you like? Allen Bradley (Rockwell), Siemens, Mitsubishi etc. most modern PLC's will give PID functionality.

You haven't stated what the inverter is for or how you intend to interface the PLC to it. I would guess similar to an inverter motor drive with several digital controls and one or two analogue speed (in this case frequency) signals.