Invoke-ProcessASDatabase fails with "Cannot connect to server"
You have two options here, which I can think of.

You can stick with the current setup of your Automation Account and add to firewall of the Azure Analysis Services instance IPs of the Azure Automation.

There is official links to this list, which you can download from here:


Azure IP Ranges and Service Tags – Public Cloud


Second option is to go with  Hybrid Runbook Worker. You can set up a worker designated to run your PowerShell scripts and then you have to white-list only IP of this particular worker. You can even use a machine from the onpremise infrastructure.


Automate resources in your datacenter or cloud by using Hybrid
Runbook Worker