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.

  • \$\begingroup\$ Please provide an attempt at a solution \$\endgroup\$
    – Voltage Spike
    Mar 25 '21 at 16:20
  • \$\begingroup\$ 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. \$\endgroup\$ Mar 25 '21 at 16:27
  • \$\begingroup\$ 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 ! \$\endgroup\$
    – user215805
    Mar 25 '21 at 16:36
  • \$\begingroup\$ @MissMulan Cross posting is not appropriate meta.stackexchange.com/questions/64068/… \$\endgroup\$
    – Voltage Spike
    Mar 25 '21 at 17:20
  • \$\begingroup\$ Are you trying to derive \$|H(j\omega)|\$ from \$H(s)\$ of a 2nd order low pass filter? \$\endgroup\$
    – Andy aka
    Mar 25 '21 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 }}\$.

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