# Standard Second Order System

Definition 1: Standard second order system

The standard second order system has a transfer function given by $H(s)=s_{2}+2ξωs+ω_{2}w_{2} $

where $ω>0$ and $ξ≥0$.

The quadratic formula tells us that the poles of H(s) are $s_{±}=−ξω±ωξ_{2}−1 $

There are hence three different forms of the solution based on where the poles lie.

Case 1: Two real distinct pole.

Case 4: Two complex conjugate poles with a zero real component: If ξ2 − 1 = 0, or simply if ξ = 0, then we have complex conjugate poles at s± = ±ωj These roots have zero real parts and hence the system may not be well behaved is there is resonance!