PID Controllers

$u(t) = K_p\, e(t) + K_i \int_0^t e(\tau)\, d\tau + K_d\, \frac{de(t)}{dt}$

• P — reacts to the present error; alone it leaves a steady-state error.
• I — removes steady-state error; too much causes overshoot and oscillation.
• D — reacts to the rate of change, adding damping; sensitive to noise (filter it).

float pid_update(PID *c, float setpoint, float measured, float dt) {
    float error = setpoint - measured;
    c->integral += error * dt;
    if (c->integral > c->i_max) c->integral = c->i_max;      // anti-windup clamp
    if (c->integral < -c->i_max) c->integral = -c->i_max;
    float derivative = (error - c->prev_error) / dt;
    c->prev_error = error;
    return c->kp * error + c->ki * c->integral + c->kd * derivative;
}

Tuning: manual (raise $K_p$ until it oscillates, back off, then add $K_i$ and $K_d$), Ziegler–Nichols or auto-tune. Implement anti-windup whenever the actuator can saturate.