Understanding Proportional, Integral, and Derivative Controller Actions

Understanding Control System Controllers

Core Components of a Control Loop

In a control and regulation system, we can distinguish the following elements:

• Transducers and sensors
• Comparators and error detectors
• Control and regulation elements
• Final elements or actuators

The controller is the 'brain' of a control loop. It is responsible for comparing a physical variable with the desired value (setpoint), interpreting the error or deviation, and acting to minimize or eliminate this error. While the controlled variable is maintained at the expected value, the controller does not act on the final elements. If the control variable deviates from the setpoint, the controller modifies the signal acting on the final elements to correct this deviation, continuing until the controlled variable returns to the desired value.

Controller Actions: P, I, and D

Many controllers produce a control signal by processing the error signal (the difference between the reference signal and the output signal). This processing involves amplifying the error, calculating its time derivative, or integrating it over time, leading to specific control actions: Proportional (P), Derivative (D), and Integral (I). Regulators implement one of these actions, a combination of two (PI, PD), or all three (PID).

Proportional (P) Action

In proportional control, when a deviation occurs between the setpoint and the actual value, the final element is modified by an amount proportional to the error. If the error is small, the controller will cause a small change to the output. Conversely, if the error signal is large, it will produce a large change to the output. The transfer function of a purely proportional controller is: G(s) = Kp. This action alone can result in a permanent deviation called offset.

Integral (I) Action

With integral action, the output signal varies based on the magnitude of the deviation and the duration it persists. The controller output continues to change as long as a deviation exists. In practice, pure integral controllers are uncommon. They are usually combined with proportional action (PI controller) to respond effectively to the error. These two actions are complementary: the proportional action acts instantly, while the integral action acts over time. Through integral action, we can eliminate the permanent or residual deviation (offset), which is the principal drawback of purely proportional action. The transfer function of a PI controller is typically represented as: G(s) = Kp (1 + 1 / (Ti * s)).

Derivative (D) Action

In derivative or differential action, the controller output changes in proportion to the rate of change of the error. This action anticipates future error based on its current rate of change, providing an output proportional to the speed of the error signal. Like integral action, derivative action is not typically used alone. It is always linked with proportional action (PD controller) or with both proportional and integral action (PID controller). The transfer function of a PD controller is typically represented as: G(s) = Kp (1 + Td * s). It is a proactive or anticipatory action, providing control adjustments to potentially prevent overshoot before it fully develops.