Higher Order Systems

Similar to the second order systems, the response behavior of an nth-order System is determined by its n poles' locations in s-plane. Due to the increased number of possible pole locations, the analysis of a high order system can be difficult. However, it is still possible to qualitatively analyze the behavior of a high order system by looking only at its poles closest to the imaginary axis $j\omega$ of the s-plane (with small real part), as these poles dominate the system's behaviors, while poles far away from the imaginary axis (with large real part) have relatively little effect on the system's behavior.

Assume $p_{near}$ and $p_{far}$ are two of the n poles of a high order system, and $\vert Re[p_{near}]\vert \ll \vert Re[p_{far}]\vert$. When frequency $\omega$ changes along the imaginary axis, vector $j\omega-p_{near}$ will vary much more drastically than vector $j\omega-p_{far}$ (in terms of both magnitude and phase angle), i.e., the system's response is mostly determined by $p_{near}$ rather than $p_{far}$. In general, a high order system can be approximately analyzed as a second order system by looking at the two poles closest to the imaginary axis.