What Is PID Control?
PID control is a type of feedback control widely used to bring the output value $y(t)$ of a controlled system closer to the target value $r(t)$. It computes a control input $u(t)$ based on the error $e(t) = r(t) - y(t)$ between the system output and the target value, and applies it to the controlled system to achieve stabilization, improved target tracking, and disturbance rejection.
P Control (Proportional Control)
The control input $u(t)$ in P control is determined proportionally to the error $e(t)$.
$$ u(t) = K_P e(t) $$
- $K_P$: Proportional gain.
- Characteristics: Increasing $K_P$ speeds up the response to the error and reduces it. However, making $K_P$ too large can cause oscillatory responses and eventually make the control unstable. Also, since the control input decreases as the error decreases, a steady-state error (residual error in steady state) may remain between the output and the target value.
PI Control (Proportional-Integral Control)
The control input $u(t)$ in PI control is determined by the sum of a proportional term and an integral term.
$$ u(t) = K_P e(t) + K_I \int_0^t e(\tau)d\tau $$
- $K_I$: Integral gain.
- Characteristics: The integral term accounts for the accumulation of past errors and therefore eliminates steady-state error. Since it continues to produce a control input until the error reaches zero, it can ultimately achieve exact tracking of the target value. However, the integral action introduces a lag in the response, which can lead to overshoot (exceeding the target value) and hunting (oscillation around the target value) in response to rapid changes.
PID Control (Proportional-Integral-Derivative Control)
The control input $u(t)$ in PID control is determined by the sum of three components: a proportional term, an integral term, and a derivative term.
$$ u(t) = K_P e(t) + K_I \int_0^t e(\tau)d\tau + K_D \frac{de(t)}{dt} $$
- $K_D$: Derivative gain.
- Characteristics: The derivative term predicts the trend of the error by reflecting its rate of change into the control input. This enables fast responses to rapid changes in the error, suppresses overshoot and hunting, and reduces response lag.
PID control achieves a well-balanced control system that satisfies stability, target tracking, and responsiveness by operating based on the “present (P term),” “past (I term),” and “future (D term)” of the error. Proper tuning of the three parameters $K_P, K_I, K_D$ is essential for PID control to function effectively.
