Effector saturation
= Effector saturation
<Effector saturation> is a capacity limit beyond which a larger command cannot produce a larger corrective action. In the <homeostatic temperature model>, take $u=\min(4x,1/2)$ during the sustained unit disturbance. Before reaching the cap, $x=(1-e^{-5t})/5$. The cap is reached at $t_c=\log(8/3)/5$, when $x=1/8$. Thereafter
$$
x(t)=\frac12-\frac38e^{-(t-t_c)},\qquad u(t)=\frac12.
$$
The lasting displacement is now $1/2$, compared with $1/5$ without the cap. The <negative feedback> command still asks for more cooling, but the action cannot supply it. This illustrates a capacity limit without predicting a human <temperature> limit. Increasing <feedback gain> cannot remove a binding cap; <integral action> cannot overcome inadequate physical capacity either.