Key Engineering Takeaways
- •Proportional gain (Kp) reduces rise time; Derivative gain (Kd) dampens overshoot; Integral gain (Ki) eliminates steady-state error.
- •Unconstrained integral accumulation when actuators saturate causes severe overshoot (Integral Windup). Always implement conditional clamping.
- •Raw numerical differentiation of noisy encoder or IMU data magnifies noise. Always apply a 1st-order low-pass filter to the derivative term.
Prerequisites
- • Basic differential calculus
The Continuous & Discrete PID Equation
The continuous time control output $u(t)$ is:
$$u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt}$$
In discrete digital firmware executing at fixed sampling time $\Delta t$:
$$u[k] = K_p e[k] + K_i \sum_{j=0}^k e[j]\Delta t + K_d \frac{e[k] - e[k-1]}{\Delta t}$$
Tags:#PID#Control Systems#Anti-Windup#Derivative Filter#Tuning#Ziegler-Nichols