= Actuator lag
\b[Actuator lag] is delayed adjustment of an <actuator>'s actual output to its commanded value. A simple dynamic model is
$$
\tau\dot u+u=K u_{\rm cmd},\qquad \tau>0,
$$
where $u_{\rm cmd}$ is the command, $u$ the delivered quantity, $K$ the steady-state conversion and $\tau$ the <time constant>. A first-order actuator model with an estimated time constant appears in MathWorks's aircraft-parameter estimation example.<Estimate Model Parameter Values in Simulink Design Optimization documentation>[model parameters and actuator initial state]{ref}
For a step command of amplitude $U$ from zero initial output,
$$
u(t)=KU(1-e^{-t/\tau}).
$$
The response reaches about63.2percent of its final value at one time constant and95.0percent at three time constants. In an original example with $K=1$, $U=10$ output units and $\tau=0.20\,\mathrm s$, the delivered value at0.20seconds is6.32, and its95percent rise takes about0.599seconds. The command itself has already reached10 at time zero.
The same model's <frequency response> is $K/(1+i\omega\tau)$. At $\omega\tau=1$, its magnitude is $|K|/\sqrt2$ and its <phase lag> is45degrees for positive $K$. Thus an actuator that is accurate for a constant command can still attenuate and phase-shift a rapidly changing command.
A direct identification experiment records the command and the actual output on a shared time base, applies a small step around an operating point, and estimates the final change and time to63.2percent. This is an explanatory procedure for the model, not a claim about a particular measured actuator. Sensor filtering, delay and an unknown initial state can alter the fitted response and must be included when relevant.
In a <control system>, treating $u=u_{\rm cmd}$ discards this state. The example in <unmodeled dynamics> shows how doing so changes closed-loop motion from a single exponential to a damped oscillation. Physical <actuator saturation> and rate limits are additional effects; the linear first-order lag equation does not represent them automatically.
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