mathematics//dynamical systems//delay and lag
Delay and lag are two different mechanisms by which a dynamic system responds late, the first shifting the input in time and the second filtering it, and telling them apart matters wherever phase lag decides the margins of a feedback loop. A **pure delay** reproduces a shifted signal, \(y(t)=u(t-\cc{\tau_d})\): a change in the input does not appear in the output before \(\cc{\tau_d}\). The dead time in copper, the time constant of a lag in violet.
Delay and lag are two different mechanisms by which a dynamic system responds late, the first shifting the input in time and the second filtering it, and telling them apart matters wherever phase lag decides the margins of a feedback loop. A pure delay reproduces a shifted signal, y(t)=u(t−τd)y(t)=u(t-\cc{\tau_d})y(t)=u(t−τd): a change in the input does not appear in the output before τd\cc{\tau_d}τd. The dead time in copper, the time constant of a lag in violet.
A hotel shower with three seconds of pipe between the tap and your back.
The water comes out cold, so you open the hot tap; nothing happens, so you open it more; then scalding water arrives, you close it hard, and ice follows. You are oscillating, because every correction answers a past that no longer exists; the cure, turn a little and wait, is what a controller facing a dead time is forced to do (loop delay).
- In a delay equation,
x˙(t)=f(x(t),x(t−τd),u(t)),\dot x(t)=f\bigl(x(t),x(t-\cc{\tau_d}),u(t)\bigr),x˙(t)=f(x(t),x(t−τd),u(t)),
the current value is usually not enough to continue the solution. A history function on [t0−τd,t0][t_0-\cc{\tau_d},t_0][t0−τd,t0] is needed.
A lag is not dead time. A first-order lag satisfies
τca˙=u−a,τc>0.\cd{\tau_c}\dot a=u-a,\qquad \cd{\tau_c}>0.τca˙=u−a,τc>0.
aaa is a filtered state and τc\cd{\tau_c}τc a time constant. For a unit step from rest, a(t)=1−e−t/τca(t)=1-e^{-t/\cd{\tau_c}}a(t)=1−e−t/τc: it responds from the start, even if it does not reach the new value immediately. Under a sinusoid of angular frequency ω\omegaω, the pure delay has phase −ωτd-\omega\cc{\tau_d}−ωτd and unit gain; the lag has phase −arctan(ωτc)-\arctan(\omega\cd{\tau_c})−arctan(ωτc) and gain 1/1+(ωτc)21/\sqrt{1+(\omega\cd{\tau_c})^2}1/1+(ωτc)2. Different mathematical signatures. A drone motor can show lag, digital latency, or both, and infrastructure latency is a pure delay in this sense: nothing arrives before it.
A transport delay is a volume over a flow. Material that travels through a pipe or along a belt reaches the sensor after the pipe volume divided by the flow, so at half the flow the dead time doubles; process plants schedule their controller gains on the flow for that reason (gain scheduling).
Convolution and memory. For a linear time-invariant system with no direct feedthrough,
y(t)=yfree(t)+∫0tH(t−τ)u(τ) dτ.y(t)=y_{\mathrm{free}}(t)+\int_0^t H(t-\tau)u(\tau)\,d\tau.y(t)=yfree(t)+∫0tH(t−τ)u(τ)dτ.
HHH is the impulse response or kernel; the integral is the convolution. Every past contribution generates a shifted, scaled response. HHH need not be positive, dimensionless or a probability.
An exponential memory admits a scalar sufficient state. Several exponentials may need several states. An ideal pure delay has, in general, no exact finite-dimensional LTI realization; a finite approximation does not remove that difference.
Reading data. A shift in cross-correlation can suggest a time scale, but can also reflect autocorrelation, mediation or a common input. It does not prove causality. Slow sampling, noise and uninformative experiments can make mechanisms indistinguishable.
Mediation through intermediate states (coupling) produces intermediate dynamics, not necessarily dead time. Phase lag matters in a feedback loop, where it decides the margins read in a frequency response. Digital latency and hold also belong to discretization.