When a controller feels sluggish, it is tempting to increase gain until the plot moves faster. That can fail even with a perfectly known linear model. Some plants contain delay; others initially move the output the wrong way. Learn to tell these mechanisms apart before tuning.
Delay: the action arrives late
A pure delay T_d replaces u(t) with u(t−T_d). Its frequency factor e^(−sT_d) has magnitude one on s=jω but adds phase −ωT_d radians, so phase loss increases with frequency. Around a loop crossover this can erode phase margin even though the delayed plant's magnitude curve is unchanged by that pure delay.
A Padé approximation can turn a delay into a rational expression for some analyses, but it is an approximation with a frequency range and must be checked against the true delay. Do not present a Padé pole or zero as a physical component of the plant.
Right-half-plane zero: first the wrong direction
Consider the stable, strictly proper example G(s)=(1−s)/(1+s)². It has a zero at s=+1 and poles at s=−1. For a positive unit step its output is y(t)=1−(1+2t)e^(−t): it initially moves negative, then turns around and approaches +1. The first downward motion is called undershoot or inverse response.
A right-half-plane zero also adds a difficult phase relation for feedback design. Trying to make the loop much faster can demand more input or sacrifice robustness. This is a structural constraint of the chosen input-output path, not a mysterious flaw in one PID setting.
How to tell the two stories apart
With pure delay the output waits before responding; with a right-half-plane zero it may respond promptly but in the opposite direction. Real systems can contain both, along with sensor filtering and saturation. Record exactly which input and output define the transfer path before naming its zeros.
The current EIGENROOM motor Lab does not expose a configurable transport delay or a right-half-plane-zero plant. Its Bode view cannot reproduce the example in this article. Use the existing Nyquist/margins lecture to understand the phase consequence; a future verified example model is required for a faithful interactive experiment.
Delay and inverse response are distinct limits. Neither can be tuned away by assuming the plant will instantly obey a larger command.
CHECK YOUR UNDERSTANDING: What is the visible difference between pure transport delay and an inverse step response?
Pure delay postpones the first response; a right-half-plane zero can make the output move promptly in the wrong direction before it turns toward the target.