Q: Do PESDs have an output relay to connect to my PLC or an alarm circuit?

A: PESDs do not have an output relay or auxiliary contacts. These devices are meant for verifying the presence and absence of voltage at the connected source only. They are passive devices and can be thought of simply as a wire that connects the internal LEDs to the line (although they are more complicated than that). Adding more devices to the circuit would decrease the reliability of the device.
They cannot close a contact, send a signal to a PLC, remain off until reset, and do not have a battery test function.