The best way to automate the up-down movement of your electric screen is an after market screen trigger.
The trigger can be activated by your projector or any switched AC outlet on the back of your audio amp or other equipment that you may have in your rack. You can purchase the aftermarket trigger from Stewart. It may be easier and cheaper to build your own trigger by purchasing a SPDT relay at your local electronics store. If you elect to purchase a relay, make sure that its rated at least 15 amps. The coil of the relay should be 12 volt DC. You will also need to purchase a 12 volt wall wart transformer. Plug the transformer into the switched ac outlet on the back of the equipment. The wire leads from the transformer are connected to the 12 volt coil side of the SPTD relay. The screen has 3 active leads, up, down and common, that operate the motor. When all is said and done, the end result is when you power up your system the screen will descend on it's own. When you shut down your system the screen will auto retract. You can buy the parts for about $15 USD.