# How to calculate the $\mathcal{H}_2$ norm of a second order transfer function?

I'm having trouble to calculate the $$\\mathcal{H}_2\$$ norm of a second order transfer function

$$H(s) = \frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2}$$ where $$\\xi>0\$$ and $$\\omega_n>0\$$. I know that the $$\\mathcal{H}_2\$$ norm is given by $$||H_2|| = \bigg\{\int_{-\infty}^{\infty}|H(j\omega)|^2d\omega\bigg\}^{1/2}$$ and that the magnitude of the frequency response is given by $$|H(j\omega)| = \frac{1}{\sqrt{\bigg(\dfrac{2\xi\omega}{\omega_n}\bigg)^2+\bigg(1-\dfrac{\omega^2}{\omega_n^2}\bigg)^2}}$$ Can someone help me with this? Is there another away to calculate it? Thanks a lot.

• Please provide an attempt at a solution Mar 25, 2021 at 16:20
• That's the only way I know. The H2 norm is a scalar positive number and it is the Hilbert's length of vector H. Mar 25, 2021 at 16:27
• You can use parseval's relation and try to integrate in time domain , it should be easy for $$\xi>=1$$but for$$0<\xi<1$$ it would be messy ! Mar 25, 2021 at 16:36
• @MissMulan Cross posting is not appropriate meta.stackexchange.com/questions/64068/… Mar 25, 2021 at 17:20
• Are you trying to derive $|H(j\omega)|$ from $H(s)$ of a 2nd order low pass filter? Mar 25, 2021 at 17:55

From a state-space representation, the $$\H_2\$$ norm can be computed as $$\\sqrt{\text{Trace}\left(b q b^T\right)}\$$ or $$\\sqrt{\text{Trace}\left(c p c^T\right)}\$$, where $$\b\$$ and $$\c\$$ are the input and output matrices, and $$\q\$$ and $$\p\$$ are the observability and controllability gramians.
I have done the calculations below using Mathematica and the result is $$\\frac{1}{2}\sqrt{\frac{\omega _n}{\zeta }}\$$.