mathematics//dynamical systems//delay and lag

"Responds late" can hide different mechanisms. A **pure delay** reproduces a shifted signal, \(y(t)=u(t-\tau_d)\): a change in the input does not appear in the output before \(\tau_d\).

"Responds late" can hide different mechanisms. A pure delay reproduces a shifted signal, y(t)=u(t−τd)y(t)=u(t-\tau_d)y(t)=u(t−τd​): a change in the input does not appear in the output before τd\tau_dτd​.

In a delay equation,

x˙(t)=f(x(t),x(t−τd),u(t)),\dot x(t)=f\bigl(x(t),x(t-\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-\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.\tau_c\dot a=u-a,\qquad \tau_c>0.τc​a˙=u−a,τc​>0.

aaa is a filtered state and τc\tau_cτc​ a time constant. For a unit step from rest, a(t)=1−e−t/τca(t)=1-e^{-t/\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\tau_d−ωτd​ and unit gain; the lag has phase −arctan⁡(ωτc)-\arctan(\omega\tau_c)−arctan(ωτc​) and gain 1/1+(ωτc)21/\sqrt{1+(\omega\tau_c)^2}1/1+(ωτc​)2​. Different mathematical signatures. A drone motor can show lag, digital latency, or both.

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)+∫0t​H(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. Digital latency and hold also belong to discretization.