Integral feedback ($\dot{z}=r-y$)

Integral feedback makes a control signal depend on the accumulated tracking error. With reference signal $r(t)$ and measured output $y(t)$, introduceAt any finite equilibrium of this controller, $\dot{z}=0$ forces zero tracking error. This explains why a stable integral-control loop can reject a constant control disturbance despite an imperfect model.

For the scalar controlled plant $\dot{y}=-ay+u+d$, with constants $a,k_I>0$ and constant $r,d$, equilibrium has $y=r$ and $u=ar-d$. Perturbations satisfy ${\lambda }^2+a\lambda +k_I=0$, whose roots have negative real parts, so this particular closed loop is stable. The conclusion depends on the plant and gain: adding integration does not guarantee stability theory for an arbitrary plant. Actuator limits can also let the accumulated error grow while the requested input cannot be delivered.