Third Order System

Introduction

A PID controller gives a third order closed loop system, for example in Disturbance Rejection. You choose the gains so that the system behaves like a second order system.

How it works

Set the desired characteristic equation as a second order part times one extra pole $p$:

$$(s+p)(s^2+2\zeta {\omega }_{n}s+{\omega }_n^2)=s^3+(p+2\zeta {\omega }_n)s^2+(2p\zeta {\omega }_n+{\omega }_n^2)s+p{\omega }_n^2$$

Normally you want a dominant second order response. So put the third pole far to the left:

$$p>\zeta {\omega }_n\text{usually 5 to 10 times }\zeta {\omega }_n$$

Then the fast pole dies out quickly, and the response is almost the same as the second order part.

Finding the gains

Compare with the PID error equation $s^3+k_v{s}^2+k_{p}s+k_i=0$:

$$k_v=p+2\zeta {\omega }_n{k}_p=2p\zeta {\omega }_n+{\omega }_n^2{k}_i=p{\omega }_n^2$$

Choose $\zeta \ge 1$ and ${\omega }_n\le 0.5{\omega }_{res}$ as before.