A proportional–integral–derivative controller (PID controller or three-term controller) is a feedback-based control loop mechanism commonly used to manage machines and processes that require continuous control and automatic adjustment. It is typically used in industrial control systems and various other applications where constant control through modulation is necessary without human intervention. The PID controller automatically compares the desired target value (setpoint or SP) with the actual value of the system (process variable or PV). The difference between these two values is called the error value, denoted as ${\displaystyle e(t)}$.

It then applies corrective actions automatically to bring the PV to the same value as the SP using three methods: The proportional (P) component responds to the current error value by producing an output that is directly proportional to the magnitude of the error. This provides immediate correction based on how far the system is from the desired setpoint. The integral (I) component, in turn, considers the cumulative sum of past errors to address any residual steady-state errors that persist over time, eliminating lingering discrepancies. Lastly, the derivative (D) component predicts future error by assessing the rate of change of the error, which helps to mitigate overshoot and enhance system stability, particularly when the system undergoes rapid changes. The PID output signal can directly control actuators through voltage, current, or other modulation methods, depending on the application. The PID controller reduces the likelihood of human error and improves automation.

A common example is a vehicle’s cruise control system. For instance, when a vehicle encounters a hill, its speed will decrease if the engine power output is kept constant. The PID controller adjusts the engine's power output to restore the vehicle to its desired speed, doing so efficiently with minimal delay and overshoot.

The theoretical foundation of PID controllers dates back to the early 1920s with the development of automatic steering systems for ships. This concept was later adopted for automatic process control in manufacturing, first appearing in pneumatic actuators and evolving into electronic controllers. PID controllers are widely used in numerous applications requiring accurate, stable, and optimized automatic control, such as temperature regulation, motor speed control, and industrial process management.

The distinguishing feature of the PID controller is the ability to use the three control terms of proportional, integral and derivative influence on the controller output to apply accurate and optimal control. The block diagram on the right shows the principles of how these terms are generated and applied. It shows a PID controller, which continuously calculates an error value ${\displaystyle e(t)}$ as the difference between a desired setpoint ${\displaystyle {\text{SP}}=r(t)}$ and a measured process variable ${\displaystyle {\text{PV}}=y(t)}$: ${\displaystyle e(t)=r(t)-y(t)}$, and applies a correction based on proportional, integral, and derivative terms. The controller attempts to minimize the error over time by adjustment of a control variable ${\displaystyle u(t)}$, such as the opening of a control valve, to a new value determined by a weighted sum of the control terms. The PID controller directly generates a continuous control signal based on error, without discrete modulation.

In this model:

Tuning – The balance of these effects is achieved by loop tuning to produce the optimal control function. The tuning constants are shown below as "K" and must be derived for each control application, as they depend on the response characteristics of the physical system, external to the controller. These are dependent on the behavior of the measuring sensor, the final control element (such as a control valve), any control signal delays, and the process itself. Approximate values of constants can usually be initially entered knowing the type of application, but they are normally refined, or tuned, by introducing a setpoint change and observing the system response.

Control action – The mathematical model and practical loop above both use a direct control action for all the terms, which means an increasing positive error results in an increasing positive control output correction. This is because the "error" term is not the deviation from the setpoint (actual-desired) but is in fact the correction needed (desired-actual). The system is called reverse acting if it is necessary to apply negative corrective action. For instance, if the valve in the flow loop was 100–0% valve opening for 0–100% control output, meaning that the controller action has to be reversed. Some process control schemes and final control elements require this reverse action. An example would be a valve for cooling water, where the fail-safe mode, in the case of signal loss, would be 100% opening of the valve; therefore 0% controller output needs to cause 100% valve opening.