If the screen door goes to the outside, I would strongly recommend against this as some of our wild life could then get INTO the lanai and your house. If you are really set on this an it does go to the outside, there are some units that only open for your pet by using an RFID tag on the pet which allows the door to open for the pet to pass.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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