Here’s how it works: unifi devices need to communicate with the controller over tcp/8080 to maintain their provisioned state. By default, the controller adopts the device with http://controller-ip:8080/inform, which means that if you ever change the controller IP, you’ll must adopt your devices again.
There are several other ways to adopt the device, most notably using the DHCP option 43 and using DNS. Of those, setting up DNS is generally easier. You’d provision the DNS to point at your controller and then update the inform address on all your devices (including the USG).
Now, there’s still a problem of keeping your controller IP and DNS address in sync. Unifi, generally, doesn’t do DNS names for its DHCP leases, and devices can’t use mDNS, so you’ll have to figure a solution for that. Or, you can just cut it short and make sure the controller has a static IP―not a static DHCP lease, but literally, a static address. It allows your controller to function autonomously from USG, as long as your devices don’t reach to it across VLANs.