11.1 The Theory

The drive system of a robot is control system. In order for this system to behave predictably, it should be a closed-loop control system. This means some feedback should be used to adjust the output to the motors. This section discusses the theory of a well-known method of closed loop control: PID (proportional integral and differential).

For any control, there is a set point. The set point is what is the feedback should read. Let us use $s(t)$ to represent the set point (reference) at time $t$. The actual feedback, $f(t)$, however, if very unlikely to be at $s(t)$. As a result, there is an error term, which is simply the difference of the two, $e(t)=s(t)-f(t)$.

Based on the error term, a control system adjusts the output to the system.

In continuous terms, the output of a control system, $k(t)$ is expressed as follows:

$$k(t)=K_{p}e(t)+K_{i}\int_0^t{e(x)dx}+K_{d}\frac{de(t)}{dt}$$ (11.1)