= Closed-loop control
{title2=$e(t)=r(t)-y_m(t)$}
= Feedback control
{synonym}
<Feedback control> selects a <control input> using observations of the actual <controlled plant>. In a tracking loop, compare the <reference signal> $r$ with the <measured output> $y_m$ to form the <tracking error> $e=r-y_m$, then let the <controller> change $u$. The <control input> changes the <plant output>, which is measured again: that return path closes the loop. A <controller> can use earlier observations as well as the latest one. https://www.fbswiki.org/wiki/index.php/Feedback_Principles[Åström and Murray, Feedback Systems, Chapter 2: Feedback Principles]{ref}
For the ideal <controlled plant>, <proportional control> $u=2(1-x)$ initially commands $u(0)=2$. As $x$ approaches $1$, the command shrinks. When a positive <control disturbance> pushes $x$ above $1$, the <tracking error> becomes negative and the <control input> pushes back. The result is a bounded offset rather than the continuing drift of <open-loop control>. <Integral action> can remove that offset by remembering past <tracking errors>.
A control selected from the observed current state, rather than specified only as a function of the initial state and time. For $x_{t+1}=x_t+u_t$ and stage cost $x_t^2+2x_tu_t+7u_t^2$, the infinite-horizon feedback $u_t=-x_t/3$ gives $x_t=(2/3)^tx_0$ and total cost $2x_0^2$. The finite-horizon Bellman coefficients satisfy $\Pi_0=0$, $\Pi_s=6(1+\Pi_{s-1})/(7+\Pi_{s-1})$ and increase to two, proving optimality.
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