You've got the basic idea of the AIV correct.
It is activated via a vacuum hose from a solenoid. When this solenoid allows intake vacuum to enter the small hose, the valve should activate.
There is a way to check the valve by sucking on the small vacuum hose, pinching it with pliers to keep the vacuum in it, and then blowing and sucking on the two 1" hose connectors (one at a time). Search on the forum for details -- or pm me if you cannot find the procedure. But I know that it's been posted.
If the valve fails the test, you need to replace it. If the valve passes, then it's not getting the correct control signals through the small vacuum line. This could be from faulty wiring, a bad solenoid or improper routing of the vacuum lines.