= Feedback delay
{title2=$\tau$}
A <feedback delay> means a <controller> reacts to an earlier <plant output>. An accurate measurement can still be stale: when the <plant output> has already crossed the <reference signal>, a <controller> using the earlier value may continue pushing it away from the target. In a pure constant-delay model, $y_m(t)=y(t-\tau)$ with $\tau\ge0$. This isolates timing from <measurement noise>; it does not describe every sampling or communication mechanism. https://stevefrank.org/control/controlBook.pdf#page=99[Steven A. Frank, Control Theory Tutorial, section 13.4: delayed feedback of an integrator]{ref}
For the ideal <scalar> <controlled plant> $\dot x=u$, a zero <reference signal>, no <control disturbance>, and <proportional control> with constant <feedback gain> $k>0$, the equations are
$$
y(t)=x(t),\qquad y_m(t)=x(t-\tau),\qquad u(t)=-k y_m(t),\qquad x'(t)=-k x(t-\tau).
$$
With $\tau=0$, this is an <ordinary differential equation> with <exponential decay> $x(t)=x(0)e^{-kt}$. With $\tau>0$, it is a <delay differential equation>, requiring an <initial history of delayed feedback>. The <delay stability of proportional integrator control> depends on the product $k\tau$: even this simple <controlled plant> loses <asymptotic stability> at $k\tau=\pi/2$.
\Image[/feedback-delay-stale-measurement.png]
{title=A stale <measured output> can command motion away from the target}
{description=Original schematic and exact toy trajectory for $k=1$, $\tau=1.2$, and <initial history of delayed feedback> $x(t)=1$ on $[-1.2,0]$. At $t=1.4$, the <plant output> is $x=-0.38$, but the <measured output> is $x(0.2)=0.8$, so $u=-0.8$ drives $x$ farther below zero. The shaded interval has actual and measured values of opposite signs. All plotted coordinates are <dimensionless quantities>; these are model predictions, not measured data.}
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