Discrete-Time Control
Digital controllers act at discrete sample instants, requiring difference equations, the z-transform, and attention to sampling effects.
Control by a Computer
Nearly all controllers today run on digital computers, which read sensors, compute, and update actuators at discrete instants rather than continuously. Discrete-time control is the theory of such sampled systems, where signals exist only at sample times and dynamics are described by difference equations instead of differential equations.
The sampling process
- An analog-to-digital converter samples the continuous output at a fixed sample period T.
- The controller computes the next command from the sampled values.
- A digital-to-analog converter with a hold, usually a zero-order hold, applies that command until the next sample.
- The choice of sample period T shapes both performance and stability.
The z-domain
Just as the Laplace transform serves continuous systems, the z-transform serves discrete ones. It turns difference equations into algebraic equations in the variable z, yielding discrete transfer functions and pole-zero descriptions. The stability region changes accordingly: discrete poles must lie inside the unit circle rather than in the left half-plane.
Discretization
A controller may be designed in continuous time and then converted to discrete form, using methods such as the zero-order-hold equivalent, the bilinear (Tustin) transform, or matched pole-zero mapping. Alternatively the plant is discretized first and the controller designed directly in the z-domain. Each path has trade-offs in accuracy and fidelity of the frequency response.
Sampling effects
Sampling introduces effects absent in continuous control. Too slow a sample rate degrades performance and can destabilize a loop, since the controller sees the plant only intermittently and reacts with delay. The zero-order hold itself adds phase lag of roughly half a sample period. And frequencies above half the sample rate alias into lower ones, corrupting measurements unless removed by an anti-aliasing filter.
Rule of thumb
A common guideline is to sample at least ten to twenty times faster than the closed-loop bandwidth, giving the digital controller enough resolution to behave like its continuous counterpart. Fast loops, such as real-time plasma position control, demand correspondingly high sample rates and low-latency computation.