{title2=$u=ke,\quad k>0$}
{wiki}

<Proportional control> chooses a <control input> proportional to the current <tracking error>. For the ideal <controlled plant> and constant <reference signal> $r$, its <ordinary differential equation> is $\dot x=k(r-x)+w$. With $w=0$ and <initial condition> $x(0)=x_0$, the exact response is
$$
x(t)=r+(x_0-r)e^{-kt}.
$$
For $r=1$, $x_0=0$ and $k=2$, this gives $x(1)=1-e^{-2}\simeq0.864665$ and $x(2)=1-e^{-4}\simeq0.981684$. To verify the formula, substitute it into the <ordinary differential equation> and check the <initial condition>; its decaying <exponential function> explains the approach to the goal.

With a constant <control disturbance>, setting $\dot x=0$ gives the <equilibrium point> $x_*=r+w/k$ and <steady-state tracking error> $e_*=-w/k$. For $w=0.2$, doubling the <feedback gain> from $k=2$ to $k=4$ reduces the offset from $0.1$ to $0.05$. This improvement is a statement about the ideal model: it neglects <feedback delay> and <measurement noise>, so it does not establish that arbitrarily large <feedback gains> are safe. https://www.fbswiki.org/wiki/index.php/Introduction[Åström and Murray, Chapter 1: feedback benefits and drawbacks]{ref}

In the first diagram's switched <control disturbance> experiment, $k=2$ and $w=0.2$ from $t=2$ give, for $t\ge2$,
$$
x(t)=1.1+(1-e^{-4}-1.1)e^{-2(t-2)}.
$$
Thus $x(4)\simeq1.097833$ and $x(8)\simeq1.099999$. At the <equilibrium point>, the <controller> supplies $u=-0.2$ to cancel the <control disturbance>. <Proportional control> needs a nonzero <tracking error> to produce that persistent correction.


Proportional control

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