Numerical integration and differentation in the PID algorithm

Introduction

PID is elaborated algorithm to use in numerous ways. Though it might difficult to set up the parameters correctly, it makes the robot move more smoothly and thus more rapid. The P is for Proportional, I is the Integral and D is Differential. We use PID in line following.

Aim

Proportion, Integration and Derivation.

Plotting, Error function.

Robot

Almost any robot will do, but we will use Asimov 2/ Verne robot.

Sensors

A light sensor is used.

Theory

If applying only proportional coding, the steering is affected by the amount of difference in reflectance from the average. The integral term is a sum (or integral) of the previous (historical) errors, and differential is used to forecast the next error based on the previous errors.

Term	Math	Meaning
Error	$\epsilon$	Sensor value - target value
Integral	$I$	Integral + error
Derivative	$D$	Error - previous error

Steering is $S=K_{p}\epsilon +K_{i}I+K_{d}D$.

We use three different parameters $K_{p}$, $K_{i}$ and $K_{d}$. Furthermore, the integral needs to forget the very old data, and thus we need an $\alpha$ term (which can be multiplied together to $K_{i}$ already). Of course, the parameters depend on the line and robot.

References

See [PID Controller For Lego Mindstorms Robots]

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