systems working on the application of air are called pneumatics.controllers in which air plays a role in deriving the various control action is called pneumatic controllers. ofcourse the air used is dry air suitable for instrumentation operation.
You can see my explanation of "Air Logic Controls" used by some to replace Electric Relays or PLC's especially in Hazardous,Explosive atmospheres in Ch. 19 of a training book I use to teach Fluid Power here.
Look at the Basic book for an explanation of different Logic Elements and in the Advanced book at Section 2 for several circuits using Air Logic. Air Logics is in Section 2 since that book is laid out alphabetically.