I am having weirdness from my battery charging. I had blamed hardware but I was wondering if linux could be responsible.
/sys/class/power_supply/C1ED/status shows one of
* Charging
* Discharging
* Unknown (= Discharging). This one worries me.

The battery is rarely charged, so I have no battery life. If it has a charge, the system elects to discharge the battery instead of using the plugged in mains supply. It has shut down over low battery with the power supply plugged in :-/.
If I unplug the battery, leave it a few seconds, and replace it, a charge starts. On suspend/hibernate, even if I have a charging light, the battery never fills. But if I switch off, I can charge it fully. And immediately I turn on, the system drains it again unless I take the battery out.

What control has linux over this? Anything I can check?

What kind of computer do you have?

Maybe you have a faulty DSDT file? I don't know much about it, but I know that it has something to do with the power management system. I know it's not exactly the same as yours, but in my case, I had to compile a kernel with a custom DSDT file in order for my laptop to recognize the battery power state. This has some info on the DSDT:
https://lesswatts.org/projects/acpi/overridingDSDT.php

You will need the program acpica from slackbuilds. Then copy the DSDT file from /sys/firmware/acpi/tables/ into your desktop and disassemble it with:
and recompile it to see if you get any errors. You can google the errors:
Once you get the errors fixed, save it and recompile it again. Then you'll have to compile a kernel and make that kernel use that DSDT file.

The DSDT is in the BIOS, and has just been updated to a 2011 version. It's a 2007/2008 laptop, and the dsdt worked for years. I nearly went at it when I bought the box, but it was just configuration. Windows uses the same dsdt, and the problem isn't serious enough for me to try a 'kill or cure' approach like recompiling the dsdt.

Solved this. The @#&$£! box _is_not_charging_ while off, on suspend, or hibernate. Hardware fault. Unplugging the battery and putting it back gives me a 'charging' light - but it's not charging. Linux isn't involved.