how to find backup history of SSAS database
There is no dynamic management view(DMV) available to get this backup history details about SSAS database. You can get these details by using below method:


Implement this backup process with SSIS either using XMLA or SSIS task
Use to implement the logging and completion of backup task time to a log table.


Unlike a relational database backup file, an SSAS backup file can contain only one backup. In other words, you can’t create backup sets. If you want to retain multiple backups, you must create multiple files. Also, SSAS doesn’t provide options for differential backups or log file backups.

By default, two check boxes are selected in the Backup Database window -  Apply compression and Encrypt backup file. When the Apply compression option is selected, SSAS compresses the backup file, which saves disk space at the expense of higher CPU utilization during the backup operation. When the Encrypt backup file option is selected, SSAS accepts a user-supplied password, which it uses to encrypt the backup file. Note that, unlike the password for a SQL Server relational database backup set, this password prevents the reading of the backup data by other means. If you choose to encrypt SSAS backups, make sure that you keep a record of the password in a safe and separate location. More details are here, here and here.

Script for XMLA and using ascmd.exe is mentioned at this link.

Hope above helps.