PID (Proportional-Integral-Derivative) control is a feedback control mechanism widely used in industrial processes, robotics, and automation systems to regulate a process variable (PV) to a desired setpoint (SP). The PID controller continuously calculates an output control signal based on the error between the SP and PV, applying proportional, integral, and derivative actions. Here's how PID control works:

1. Proportional (P) Action:

   • The proportional action generates a control output proportional to the current error between the setpoint (SP) and the process variable (PV).
   • The control output is calculated using a proportional gain (Kp), which determines the proportional relationship between the error and the control output.
   • The P action responds to the present error and is responsible for reducing the steady-state error and improving the responsiveness of the control system.
   • Mathematically, the proportional term is given by: $P=K_p\times e(t)$, where $e(t)$ is the error at time t.

2. Integral (I) Action:

   • The integral action calculates the cumulative sum of past errors over time and generates a control output based on this integral of the error.
   • The integral term accounts for accumulated errors and eliminates steady-state error by continuously adjusting the control output until the error is minimized.
   • The integral action helps to eliminate any bias or offset in the control system and improves the stability of the system.
   • Mathematically, the integral term is given by: $I=K_i\times \int e(t)dt$, where $K_i$ is the integral gain.

3. Derivative (D) Action:

   • The derivative action anticipates future trends by measuring the rate of change of the error and generates a control output based on the derivative of the error.
   • The derivative term helps to dampen oscillations, reduce overshoot, and improve the transient response of the control system.
   • It provides corrective action based on the rate of change of the error, helping to stabilize the system.
   • Mathematically, the derivative term is given by: $D=K_d\times \frac{de(t)}{dt}$, where $K_d$ is the derivative gain.

4. Overall Control Output:

   • The overall control output of the PID controller is the sum of the proportional, integral, and derivative terms: $u(t)=P+I+D$
   • The control output is applied to the actuator or control device, which adjusts the system to bring the PV closer to the SP.

5. Tuning:

   • The performance of a PID controller depends on the proper selection of the proportional, integral, and derivative gains (Kp, Ki, Kd), known as tuning.
   • Tuning involves adjusting these gains to achieve desired control performance, such as stability, responsiveness, and disturbance rejection, while avoiding oscillations and overshoot.

In summary, PID control continuously calculates a control output based on the present error (proportional action), the cumulative sum of past errors (integral action), and the rate of change of the error (derivative action), aiming to regulate the process variable to the desired setpoint.