Sampled-Data Loops, Pulse Transfer Functions & the ZOH

A pulse transfer function is not obtained by simply writing $s=z$. It is the input-sequence to output-sequence map of a specified continuous plant plus sampler/hold interface, normally under zero initial conditions. Change the hold, sample phase, delay, or measured instant and the discrete model changes.

The hold creates a continuous waveform from $u[k]$. The sampler observes only $y(kT_s)$; intersample extrema are not visible in the sequence alone.

Ideal sampler and sequence #

Sampling a continuous signal $x(t)$ at period $T_s$ gives sequence

$$x[k]=x(kT_s).$$

The impulse-train representation

$$x_s(t)=\sum_{k=-\infty}^{\infty}x[k]\delta(t-kT_s)$$

is useful for spectral analysis, but an ADC output is a sequence of numbers, not physically an impulse train. Keep the mathematical model distinct from aperture, quantization, encoding, and transport.

The unilateral z-transform for causal analysis is

$$X(z)=\sum_{k=0}^{\infty}x[k]z^{-k},$$

with initial-condition terms when transforming differences. The bilateral transform and its region of convergence are needed for general signal/stability questions.

Zero-order hold #

The ZOH applies

$$u(t)=u[k],\qquad kT_s\le t<(k+1)T_s.$$

Its Laplace-domain reconstruction factor is

$$H_{ZOH}(s)=\frac{1-e^{-sT_s}}{s}.$$

For continuous plant $G(s)$, the sample-to-sample pulse transfer under an ideal sampler and ZOH can be written

$$G_d(z)=(1-z^{-1})\mathcal Z\left\{\left.\mathcal L^{-1}\!\left[\frac{G(s)}{s}\right]\right|_{t=kT_s}\right\}.$$

The expression says: take the continuous step response, sample it, then difference adjacent samples because each held input step is the difference of cumulative steps.

Worked first-order plant #

For

$$G(s)=\frac{K}{\tau s+1},$$

define $a=e^{-T_s/\tau}$. The exact ZOH update is

$$y[k+1]=ay[k]+K(1-a)u[k].$$

With zero initial state,

$$G_d(z)=\frac{Y(z)}{U(z)}=\frac{K(1-a)z^{-1}}{1-az^{-1}}=\frac{K(1-a)}{z-a}.$$

The one-sample delay in the $z^{-1}$ form is causal: $u[k]$ affects $y[k+1]$. The discrete pole $a$ lies inside the unit circle for $\tau>0$. As $T_s$ grows, $a$ moves toward zero, but this does not mean coarse sampling improves the actual continuous loop—delay, information loss, and intersample behavior worsen.

Closed-loop pulse transfer #

If controller and plant pulse models share a consistent sample convention, a simple unity-feedback sequence model has

$$\frac{Y(z)}{R(z)}=\frac{D(z)G_d(z)}{1+D(z)G_d(z)}.$$

This algebra does not commute the sampler through arbitrary continuous blocks. In general, sampling destroys the ability to combine continuous transfer functions on both sides as if every block were discrete. Derive the equivalent at the sampler boundary or use the exact state-space update.

Fractional delay and sample phase #

A delay $L=mT_s+\delta$, with integer $m$ and $0<\delta<T_s$, contains $m$ exact factors $z^{-m}$ plus fractional intersample timing. Rounding $L/T_s$ changes phase and can overstate margin. Options include:

• augmenting an exact sampled state model with the delayed input pieces;
• fractional-delay filters such as a stated Thiran approximation;
• smaller base period or multirate lifting;
• explicitly time-stamped variable-delay models.

The modified z-transform historically tracks response at an offset inside the sample interval. Modern lifted or state-space models make the same concern explicit: the sample phase is part of the output definition.

Frequency response on the unit circle #

For a stable discrete LTI model, frequency response is evaluated at

$$z=e^{j\Omega},\qquad \Omega=\omega T_s,$$

with unique physical frequencies up to Nyquist $\omega_N=\pi/T_s$. As $\omega$ increases from 0 to $\omega_N$, $z$ traces the upper unit semicircle. Frequencies separated by integer multiples of $2\pi/T_s$ map to the same point.

The bilinear transform

$$z=\frac{1+sT_s/2}{1-sT_s/2}$$

maps the imaginary axis to the unit circle without aliasing but warps frequency:

$$\Omega=2\tan^{-1}\!\left(\frac{\omega T_s}{2}\right).$$

Prewarping can match one selected frequency exactly, not the entire axis.

Intersample behavior #

The discrete model can be perfect at $kT_s$ while the continuous output violates a bound between samples. Fast flexible modes, lightly damped plants, delayed sensing, and aggressive control make this especially important. Reconstruct the continuous response using the held input and continuous plant, then inspect peak state/output and actuator waveform throughout each interval.

Zeros introduced by ZOH discretization also matter. For higher relative degree and coarse sampling, sampling zeros can be near or outside the unit circle, restricting direct inversion and deadbeat design even when the continuous plant is minimum phase.

Lifted and multirate models #

When sensors and actuators operate at different rates, collect a block of fast samples into a vector and describe one slow-period update. This lifting converts a periodically time-varying multirate system into a higher-dimensional time-invariant discrete model. It preserves within-period ordering that a single-rate approximation can hide.

State the base clock, rate ratios, phase offsets, and which values are held or interpolated. A multirate diagram without phase semantics is underspecified.

Modeling checklist #

1. Identify continuous plant, sensor, actuator, analog filters, and delays.
2. Define sample and actuation instants on one timeline.
3. State the hold/reconstruction waveform.
4. Derive exact $A_d,B_d$ or the pulse transfer for that interface.
5. Preserve integer and fractional delay.
6. Check frequency response only up to Nyquist and account for warping/aliasing.
7. Inspect sampling zeros and controllability/observability.
8. Reconstruct intersample behavior before accepting the sequence design.

The pulse transfer function is a powerful boundary model. Its power depends on keeping the boundary honest.