PID controller, tuning and creation
Without the integral term being used within the feedback it could be nonsense - any error will accumulate in the integral term and just appear illogical. However it could show something useful....

For instance, at the start of your graph the integral term develops a value because there is a small error between demand and actual. As you approach t=2500 it seems to be drifting down slightly as the actual psi drifts slightly up - this makes sense.

When demand changes about t=2500, the integral ramps up (as you'd expect) and keeps ramping because actual isn't attaining demand.

This tells me it is working, however at 3200 something goes wrong - I'd expect integral to keep ramping up but it appears to flat-line then disappear about 4000.

I'd be suspicious of what happens after about 3200.

Hope this helps.