What are common methods for tuning PID controllers?

PID tuning methods include: Ziegler-Nichols (closed-loop - find ultimate gain and period, or open-loop - measure delay and time constant from step response), Cohen-Coon (based on process reaction curve), IMC (Internal Model Control - based on process model), relay autotuning (inject relay, measure oscillation), and optimization-based methods (ITAE, ISE criteria). Each method has trade-offs: Ziegler-Nichols gives aggressive tuning (may need detuning), model-based methods require accurate identification. Modern systems use autotuning algorithms.