Write the standard PID control equation and briefly explain each term.

Write the standard PID control equation and briefly explain each term.

• A. u(t) = Kp e(t) + Ki e(t) + Kd de/dt; e is the error, Kp is proportional gain, Ki integral gain, Kd derivative gain.
• B. u(t) = Kp e(t) + Ki ∫ e(τ)dτ + Kd de/dt; e is the error, Kp is proportional gain, Ki integral gain, Kd derivative gain.
• C. u(t) = Kp e(t) - Ki ∫ e(τ)dτ + Kd de/dt; e is the error, Kp is proportional gain, Ki integral gain, Kd derivative gain.
• D. u(t) = e(t).

Answer: (B)

The standard PID control law combines three actions: respond to the present error, correct accumulated past error, and anticipate future error. The equation is u(t) = Kp e(t) + Ki ∫0^t e(τ) dτ + Kd de/dt, where e(t) is the difference between the setpoint and the process variable. The proportional term Kp e(t) gives a correction proportional to the current error, driving the system toward the setpoint quickly. The integral term Ki ∫0^t e(τ) dτ accumulates past errors, removing any steady-state offset by boosting the control action as long as there is persistent error. The derivative term Kd de/dt uses the rate of change of the error to dampen the response and reduce overshoot, improving stability and settling time. This form uses the integral of the error over time and the derivative of the error with respect to time, with all three terms added together. The other presented forms either misplace the integral (as a simple e(t) term), change the sign, or omit the integral and/or derivative actions, so they don’t match the standard PID structure.

The standard PID control law combines three actions: respond to the present error, correct accumulated past error, and anticipate future error. The equation is u(t) = Kp e(t) + Ki ∫0^t e(τ) dτ + Kd de/dt, where e(t) is the difference between the setpoint and the process variable.

The proportional term Kp e(t) gives a correction proportional to the current error, driving the system toward the setpoint quickly. The integral term Ki ∫0^t e(τ) dτ accumulates past errors, removing any steady-state offset by boosting the control action as long as there is persistent error. The derivative term Kd de/dt uses the rate of change of the error to dampen the response and reduce overshoot, improving stability and settling time.

This form uses the integral of the error over time and the derivative of the error with respect to time, with all three terms added together. The other presented forms either misplace the integral (as a simple e(t) term), change the sign, or omit the integral and/or derivative actions, so they don’t match the standard PID structure.