Through and Beyond the Disturbance

The initial condition is h2(tl) = h and we should integrate to any t less than t2, so we integrate tot. 

When this is done we have an expression for h(t) during the course of the disturbance. We must next integrate after the disturbance is complete, that is, from t2 out to any additional time t. Here we have the following equation to work with because qp is zero  

The initial condition for this period is h3[t2] = h2[t2] Therefore, we must evaluate the constant of integration very carefully. 

Finally, we know that if the disturbance is positive, then the inlet flow increases and the tank level should rise. If there were no control it would rise and stay at a new higher level. With control it should rise and then fall back to the control or design level. Tracking the change in level versus time we should see a sawtooth that looks like this  

---