Discrete-Time Control Systems

Core Idea

Discrete-time control systems operate on sampled signals x[k] = x(kT) and are analyzed using the z-transform, where the transfer function H(z) = Y(z)/U(z) describes the input-output relationship in the z-domain. The zero-order hold (ZOH) models the digital-to-analog conversion that holds each computed control value constant between samples, and the ZOH-equivalent pulse transfer function G(z) = (1 − z⁻¹)·Z{G(s)/s} captures both the continuous plant dynamics and the hold effect. Stability in the z-domain requires all closed-loop poles to lie strictly inside the unit circle |z| = 1, analogous to the left half-plane requirement in the s-domain. The mapping z = e^sT relates s-plane and z-plane pole locations: the imaginary axis in the s-plane maps to the unit circle, the left half-plane maps to the interior of the unit circle, and the negative real s-axis maps to the interval (0, 1) on the real z-axis. Jury's stability criterion or the bilinear transformation w = (z − 1)/(z + 1) followed by Routh-Hurwitz can assess z-domain stability without computing roots explicitly. Discrete-time root locus and Bode plot techniques mirror their continuous-time counterparts but use the unit circle and the z = e^jωT frequency mapping respectively, with the critical frequency being the Nyquist frequency ω_s/2 = π/T.

How It's Best Learned

Discretize a well-understood continuous-time system (e.g., a second-order plant with known pole locations) using the ZOH method at several sampling rates. Plot the z-plane pole locations alongside the original s-plane poles and verify the z = e^sT mapping. Design a discrete PID controller directly in the z-domain using root locus on the pulse transfer function, then simulate the closed-loop step response and compare with the continuous-time design to observe intersample ripple and latency effects.

Common Misconceptions

• The z-transform is not simply the Laplace transform with z substituted for s — the z-transform is defined for sequences x[k], not continuous signals, and the relationship z = e^sT means the z-plane wraps the s-plane vertically with period jω_s, causing aliasing of high-frequency dynamics.
• A continuous-time system with poles on the negative real axis does not map to z-plane poles on the negative real z-axis — it maps to the positive real interval (0, 1), while negative real z-axis poles correspond to oscillatory s-plane modes at half the sampling frequency.
• Increasing the sampling rate does not always improve discrete-time controller performance — excessively fast sampling can amplify quantization noise, increase computational load, and approach the numerical precision limits of the controller hardware without meaningful performance benefit beyond a certain point.

Explainer

You know from continuous-time control that a system's behavior in the s-domain is characterized by its transfer function G(s) = Y(s)/U(s), and that stability requires all closed-loop poles to lie in the left half of the s-plane (LHP). When a digital computer implements a controller, it reads sensor data at discrete moments — every T seconds — computes a control output, and holds that output constant until the next sample. This sampling and holding process fundamentally changes the mathematical framework: continuous signals become sequences x[k] = x(kT), and the Laplace transform gives way to the z-transform.

The z-transform is defined on discrete sequences: X(z) = Σ x[k] z^-k. The complex variable z is related to s by z = e^sT, which is the key mapping between the two domains. Every vertical line Re(s) = σ in the s-plane maps to a circle of radius e^σT in the z-plane. The imaginary axis (σ = 0) maps to the unit circle |z| = 1. The left half-plane (σ < 0, stable continuous-time poles) maps to the interior of the unit circle. The LHP stability criterion becomes: all closed-loop z-domain poles must lie strictly inside the unit circle. A pole at z = 0.8 corresponds to stable, decaying behavior; a pole at |z| = 1.1 corresponds to instability.

To obtain the discrete-time model of a continuous plant plus zero-order hold, you use the ZOH-equivalent pulse transfer function: G(z) = (1 − z^-1) · Z{G(s)/s}. The term (1 − z^-1) captures the ZOH, which holds the control input constant between samples. G(s)/s represents the continuous plant preceded by an integrator, because the ZOH introduces an integration-like effect on the held signal. The z-transform of that combined system, discretized at rate T, gives the discrete-time transfer function relating z-domain input U(z) to output Y(z). Once you have G(z), all the root-locus and frequency-response techniques from continuous-time design carry over — but applied to the z-plane, with the unit circle replacing the imaginary axis as the stability boundary.

Stability checking without computing roots explicitly uses either Jury's criterion — a tabular test analogous to Routh-Hurwitz — or the bilinear transformation w = (z − 1)/(z + 1), which maps the unit disk onto the LHP. After applying this substitution to the characteristic polynomial, you recover a polynomial in w and apply Routh-Hurwitz directly. The choice of sampling period T profoundly affects the design: too large and dynamics are aliased or poorly approximated (the ZOH hold introduces a lag of T/2, degrading phase margin); too small and quantization noise and computational limitations dominate. A common rule of thumb is to sample at 10–20 times the closed-loop bandwidth, but the appropriate rate depends on the specific plant dynamics and noise environment.