How are control systems implemented in discrete time?

Digital controllers sample signals at fixed intervals and compute control output discretely. Continuous designs are discretized using: bilinear transformation (Tustin method, preserves frequency response), forward/backward difference (simpler but may cause instability), or zero-order hold equivalent (matches step response). Z-transform replaces Laplace transform: z = e^(sT) where T is sampling period. Sampling rate should be 10-20 times system bandwidth. Digital implementation includes quantization effects, computational delay, and anti-alias filtering.