You can check ADS connection in SAP via multiple different methods. Make sure before checking below ADS RFC should working fine in SM59.
1. SA38 – FP_CHECK_DESTINATION_SERVICE (With tick mark).
It should give your reply with tick mark.
2. SA38 – FP_PDF_TEST_00
3. SA38 – FP_TEST_00
If there are any other error like SOAP/exception or any, you need to check the Java server Node or check server Trace. if restart of server node works & ADS start working. 1 cause of this, it means you are running out of memory in system. check the memory at OS level.