It depends on the battery type used. But there are many ways to do so.
Some batteries can report their own errors. Many smart lithium types do.
For lead acid you will notice performance issues when the inverter is under load.
With a Digital Multi Meter you can measure the voltage of each battery in the bank at rest. Also if there is voltage sag in a particular one in the bank under load. With a DC clamp meter you can check current drawn under load.
This data can also be seen in the VRM data as well. The type of testing done is usually based on what you are looking for though.
For the end user who has no technical knowledge they usually only notice when the batteries go flat before they are expecting them to during grid outage. The inverter switches off randomly or during heavier loads. Or they notice that their batteries take too long too charge and feel their Grid bill has gone up.
There are many different ways to do this. For the most simple one:
https://www.victronenergy.com/panel-systems-remote-monitoring/battery-alarm
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