- What does the integral term (Ki) actually do that the proportional term (Kp) cannot?
- The proportional term responds only to the present error. If a constant disturbance (like a steady push) exists, the cart will settle at a position where the proportional force just balances the disturbance, resulting in a persistent steady-state error. The integral term sums all past errors over time. This accumulating action allows the controller to generate a corrective force even when the instantaneous error is small, eventually driving the steady-state error to zero.
- Why does increasing the derivative gain (Kd) sometimes make the system smoother?
- The derivative term acts on the rate of change of the error. When the cart is moving rapidly toward the set-point, a positive derivative term produces a braking force proportional to the velocity. This damping effect reduces overshoot and oscillation, leading to a smoother, more critically damped approach to the target. However, if Kd is set too high, it can over-damp the system, making it sluggish, or amplify high-frequency noise.
- How do the random velocity impulses relate to real-world control systems?
- These impulses model unpredictable external disturbances or internal system noise that a real controller must handle. Examples include wind gusts affecting a drone's position, varying load on a conveyor belt, or sensor noise. A well-tuned PID controller must reject these disturbances quickly and return the system to its set-point with minimal deviation, demonstrating its robustness.
- What are the main simplifications in this 1D model compared to a real system?
- This model assumes ideal conditions: the control output translates directly into acceleration without delay or saturation limits, the cart's mass and friction are often normalized or ignored, and the sensor provides perfect, instantaneous position feedback. Real systems have actuator limits, sensor lag, quantization, and more complex dynamics (like stiction or backlash), which introduce additional challenges for controller design and tuning.