PIreglern

A PI-Regler, short for proportional-integral regulator, is a type of feedback controller used in control systems. It computes the control signal based on the current error between a desired setpoint and the measured process variable. The proportional term reacts to the present error, while the integral term accumulates past errors to eliminate steady-state offset.

In continuous time, the controller output u(t) is given by u(t) = Kp e(t) + Ki ∫0^t e(τ) dτ, where e(t) = setpoint − process_value, Kp is the proportional gain and Ki is the integral gain. In discrete-time implementations, with sampling period T, u[k] = Kp e[k] + Ki T ∑_{i=0}^k e[i]. Some implementations use Ki′ = Ki T to simplify notation. Many digital controllers include an anti-windup mechanism to prevent integrator saturation when actuator limits are reached.

Characteristics and limitations: The PI action reduces or eliminates steady-state error for many processes with steady input changes and introduces an integral action that can slow response and risk overshoot or oscillations if Ki is set too high. Tuning aims to balance speed of response and stability. Common tuning methods include Ziegler–Nichols, Cohen–Coon, and model-based approaches. In practice, integrator windup is addressed by anti-windup schemes such as back-calculation, clamping, or conditional integration.

Applications: PI regulators are widely used in process control systems, such as temperature, flow, pressure, and level control, as well as in motor control and basic robotics. They are preferred when systems display a significant steady-state error without integral action and when a simple, robust controller is desirable.

Relation to PID: A PI-Regler is a subset of PID controllers, omitting the derivative term. For some processes, a PI controller provides sufficient performance with lower sensitivity to noise than a full PID.