- PID Controller Definition: A PID controller is a crucial device in control systems that adjusts the control action based on proportional, integral, and derivative terms of the error.
- Control Parameters: Proportional (Kp), integral (Ki), and derivative (Kd) components each uniquely influence the response and stability of control systems.
- Proportional Control: This mode adjusts the output proportionally to the error, which is the difference between desired and actual performance.
- Integral and Derivative Actions: Integral control focuses on the accumulation of past errors, whereas derivative control predicts future errors, helping to refine the control process.
- Applications and Limitations: PID controllers are versatile and widely used in modern industrial applications, though they face challenges in noisy environments and optimal control scenarios.
PID control, or proportional-integral-derivative control, is a feedback method in a control system. It calculates the control signal from the present error, accumulated past error and rate of change of error. The three gains are tuned for the plant and the required response.
PID is one controller family, and its suitability depends on the plant dynamics and design goals. Nicolas Minorsky published an analysis of automatic steering in 1922. In continuous parallel form, the actuating signal is the sum of a proportional error term, an integral of error and a derivative of error. The formula is:
The Laplace transform of the actuating signal incorporating PID control is
A controller can use one or two of the three actions by setting unused gains to zero. Common forms include P, I, PI and PD as well as PID. The derivative term reacts to the rate of change of error, but it also amplifies measurement noise, so practical controllers often filter it. The integral term accumulates error and can remove steady-state offset in a suitable stable loop, although actuator limits can cause integral windup. Early controllers used mechanical or pneumatic parts such as springs, levers and masses. Electronic and digital implementations now run in devices such as industrial PLCs (programmable logic controllers). The gains are written as Kp, Kd and Ki. Their effects on a closed loop control system depend on the plant and tuning. The table gives common qualitative effects on rise time, settling time, overshoot and steady state error, not universal results.
| Control Response | Rise time | Settling time | Overshoot | Steady state error |
| Kp | decrease | small change | increase | decrease |
| Kd | small change | decrease | decrease | no change |
| Ki | decrease | increase | increase | eliminate |
PID control combines proportional, derivative and integral actions. Each action uses a different feature of the error signal.
Proportional Control: The proportional action makes the actuating signal proportional to the current error. That error is the difference between the reference input and the measured feedback signal.
Derivative Control: In the PD form, the actuating signal combines a proportional error term with the derivative of error. The derivative term responds to how quickly the error is changing. The PD control law is:
Integral Control: In PI control, the actuating signal combines a proportional error term with the accumulated integral of error. The PI control law is:
A PID controller must be tuned for the plant and operating range. Poor tuning can produce slow response or oscillation, while actuator saturation can cause integral windup.
Basic PID control uses measured error without an explicit process model. Strong nonlinear behaviour, long dead time, actuator constraints or changing plant dynamics can require added logic or another control method. Derivative action is sensitive to high-frequency measurement noise, which is why practical PID designs often use a first-order derivative filter.
