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Equation: $u(t) = K_p e(t)$ Role: The proportional element of a PID controller multiplies the error (difference between desired and actual output) by a gain factor ($K_p$), providing a control action that is directly proportional to the error.

Equation: $u(t) = K_i \int e(t)\,dt$ Role: The integral element of a PID controller accumulates the error over time and integrates it to compensate for past values, helping to eliminate steady-state error.

Equation: $u(t) = K_d \frac{d}{dt}e(t)$ Role: The derivative element of a PID controller predicts the future trend of the error by its rate of change, providing a damping force that reduces overshoot and improves system stability.

Equation: $K_p$ Role: The proportional gain determines the strength of the corrective action taken for a given error magnitude in the proportional element of the PID controller.

Equation: $K_i$ Role: The integral gain adjusts the weight of the accumulated error in the control response, affecting the speed at which the integral element corrects the steady-state error.

Equation: $K_d$ Role: The derivative gain modifies the impact of the rate of error change, influencing the ability of the derivative element to dampen system oscillations and enhance stability.

Equation: Direct Relation with $K_p$ Role: In disturbance rejection, the proportional element helps to quickly counteract disturbances but may not completely eliminate their effects.

Equation: Accumulation with $K_i \int e(t)\,dt$ Role: Integral action removes any bias or offset in the system by integrating the error over time, ensuring that the output converges to the desired reference value.

Equation: Prediction with $K_d \frac{d}{dt}e(t)$ Role: The derivative action can reduce the settling time by damping oscillations and minimizing overshoot, making the system respond more smoothly to changes.