For a given scenario in the context of control system, I'm trying to invesigate how the $H_\infty$ norm can be calculated for a transfer function as follows:
$$G(s)= \frac{w_n^2}{s^2 +2\zeta w_ns +w_n^2 } $$
$$\left \| G \right \| _\infty = \max \limits _{\omega} |G(j\omega)|$$
$$\left \| G \right \| _\infty = \sup \limits _{\omega} |G(j\omega)|$$
For measurment of infinity norm of this transfer function, how do I determine the value of 'w' will the value of |G(jω)| be maximum?
The straightforward way is to explicitly write $|(j\omega)^2+2\xi\omega_n(j\omega) +\omega_n^2|^2$ as a function of $\omega$ and find where this function has its smallest value by taking its derivative with respect to $\omega$.
A bit simpler is to open any textbook on linear systems and check what is the frequency response of a second-order system.