Digital Control Systems

Digital implementation

A digital controller samples measurements, computes a control command and updates an actuator through a hold or power stage. Its realized dynamics include sampling, computation delay, quantization, saturation and numerical precision.

Difference equation

For C(z) = (b0 + b1z⁻¹)/(1 + a1z⁻¹), the zero-initial-condition recursion is u[k] = −a1u[k−1] + b0e[k] + b1e[k−1]. Initial controller memory must be specified when analyzing a nonzero-state response. A discrete integrator can use I[k] = I[k−1] + Ki T e[k] with backward rectangular integration, or another stated approximation. The sample period belongs in the integral gain. Derivative estimates amplify measurement noise and usually need filtering.

Worked example

Let C(z) = (0.2 + 0.1z⁻¹)/(1 − 0.8z⁻¹). Then u[k] = 0.8u[k−1] + 0.2e[k] + 0.1e[k−1]. For e[k] = 1 for k ≥ 0, with e[−1] = u[−1] = 0: u[0] = 0.2; u[1] = 0.8(0.2) + 0.2 + 0.1 = 0.46; u[2] = 0.8(0.46) + 0.3 = 0.668. The controller pole is 0.8, and its steady output for constant error 1 is C(1) = 0.3/0.2 = 1.5. This is controller stability alone; feedback stability also depends on the plant.

Practical checks

Use deterministic update timing and account for the measured worst-case computation delay. Scale signals consistently and check overflow, rounding and finite-word-length effects. Apply output limits with anti-windup where integral action is used. Test startup, reset and sensor-failure behavior against the equipment requirements.

Quick check

1. Can computation delay change stability? Yes.
2. Does a stable digital controller guarantee a stable loop? No.
3. Why retain previous errors and outputs? The recurrence requires those state values.