9. PI controller with anti-windup function

The aim of this section is to give the fundamental reason why the vast majority of controllers used in industry are proportional-integral with anti-windup, or "PI with anti-windup" controllers.

It is often possible to obtain a good approximation of the relationship between the measurement y and the control u by means of a first-order nonlinear system, stable in the open loop (not necessarily asymptotically, which includes the pure integrator) and for which we know the sign of the gain, which we'll take to be strictly positive. This means that for the block diagram in figure 1 we have y ≥ x...