Cockatoo Predator

Hi there, I live on Sydney's north shore at Willougbhy, and have quite a few Cockatoos who feed in my garden in the afternoons.  Over the past week, there has been a bird of prey around making them really nervous.  I have only seen it in flight.  It is quite dark with a signifcant barred (longish) tail.  It does chase the flock of Cockatoos in flight.  My first thought was Peregrine, but it seems to have darker underparts - and while the Black Falcon looks better in colour it does not have a barred tail.  Any idea what would be trying to eat the Cockatoos this close into town?

Thanks so much 

Regards