= Open-loop control
{title2=$u(t)=u(t;x_0)$}
An <open-loop control> is prescribed as a function of time, possibly after calculation from an initial state and a dynamical model. It is executed without using observations of the actual current state to change the input. A simulated <feedback control> law can generate an open-loop waveform, but the resulting laboratory control does not automatically correct errors in the actual state or model.
For the ideal <controlled plant> with $w=0$, plan the <control input>
$$
u(t)=\begin{cases}1,&0\le t<1,\\0,&t\ge1.\end{cases}
$$
<Integration> gives $x(t)=\min(t,1)$, so the planned command reaches the <reference signal> $r=1$ at $t=1$. If an unplanned constant <control disturbance> $w=0.2$ begins at $t=2$, the same <open-loop control> gives $x(t)=1+0.2(t-2)$ for $t\ge2$: $x(4)=1.4$ and $x(8)=2.2$. Knowing the nominal response did not provide a way to correct the actual <plant output>.
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