 Closed-loop control

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. ^[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 $x_t^2+2x_t{u}_t+7u_t^2$, the infinite-horizon feedback $u_t=-x_t/3$ gives $x_t=(2/3{)}^t{x}_0$ 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.