After returning my first USB3 PCIe Card because it randomly stopped working and, just to experience the same problems with the card I got in exchange – thanks amazon – i finally found the root cause and permanent fix.
The Renesas uPD720201 USB 3.0 Host Controller seems to have broken S3 power saving implenented. Since you probably are not using your full size PCIe card while on battery, you can safely disable power saving features.
Edit your /etc/default/grub and add usbcore.autosuspend=-1 to your GRUB_CMDLINE_LINUX_DEFAULT. Afterwards run update_grub.
Turn your PC off and unplug from the wall outlet, this will bring your card out of S3 sleep.
Then boot your PC again – problem solved!