1. Heated Bed Pad
2. Heated Mattress

Since the board is set up from the factory for bang-bang control for the heatbed and not PWM/PID control, I don't think there is any tuning you can do for the bed without reflashing the firmware with settings for PID/PWM. PWM/PID cannot be used with an external mechanical relay, only a FET or some solid-state relays. If using the on-board FET, it should be possible to use PWM/PID but only with reflashed firmware.

Heated Bed Pad

After the bed reaches temperature, is it stable within a few degrees C?

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If while heating up, there is massive temperature overshoot and/or a bad oscillation of temperature above and below the target temp, then yes, there is a problem that will need some tuning or redesign ( the bed thermistor may not be thermally connected to the bed well enough). Otherwise, don't worry about it.

• If you want to use PID for your heated bed, there is one more option you need to add to the M303 command, and that is E for extruder. The heated bed is extruder negative one, the first hotend is zero, the second hotend is one and so on.
• Switch to an entirely separate 120v AC heated bed. This is the option I’m currently investigating. Switch all my electronics out to 24v (ugh). But in the meantime, at least the stupid z-artifact is gone. PID Autotune your hotend and heated bed in RepRap Firmware.
• Feb 15, 2016  I was able to tune my extruder PID controller, however I cannot test my heat bed PID tuner. I can't find the correct G-Code to autotune the PID controller for the heat bed. I've tried M303 P1 S, M303 C8 S175, and M304 as indicated on reprap.org , however none of these return any values in the window and act like nothing was typed.
• Jan 16, 2018  In this video, I will show you how to do a heated bed PID calibration, to improving the way it will heat up. Prevent some overshoot of the temperature and make it faster to stabilize on its target.

Mar 16, 2016 Also tried PID tuning, nozzle temp are stable. So, in my opinion the next step could be the firmware with enabled pid tuning for the heated bed. The next days I try to print on blue tape. In use a heated bed from aluminium and print directly to this surface. Jun 09, 2016 I finally got it to work, you have to heat your bed withing 10 degrees of your target temperature then run the pid autotune, then it will work, as long as you have the pid autotune enabled in your marlin code and that other option below it commented out you are good to go, try that and let me know if it worked for you, if you haven't already figured it out.

Heated Mattress

I haven't had any issues with wild temperature swings. I have my bed set up so I can use it with 12v through the FET for lower temperatures or with a flip of a switch, powered by a 20V power supply through a relay.