With simple proportional control, if Kp is fixed, the only way that the control will increase is if the error increases. The addition of a derivative term to the controller, Kd, adds the ability of the controller to anticipate an error. This has the adverse effect of making a system sluggish and oscillatory since when the error signal changes sign, it may take a while for the integrator to unwind, leading to an overshoot in the opposite direction. If there is a persistent, steady error, the integrator builds over time, increasing the control signal and driving the error down. The addition of an integral term to the controller, Ki, helps to reduce the steady-state error. This is another way of saying that the response has reached a steady state. The steady-state error is the difference between the input and the output of a system as time goes to infinity. Increasing Kp also tends to reduce, but not eliminate, the steady-state error. This makes the controller react quicker for a given error in the closed-loop system, but it also means that it will likely overshoot more as well. Increasing the proportional gain, Kp, has the effect of proportionally increasing the control signal for the same level of error. u(t) is the controller output, and e(t) is the error value. The important parts of this equation are Kp, Ki, and Kd.
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