Let system G(s) be: $$ G(s)=\sum_{i=0}^{10}\frac{(-1)^i}{(2i+1)^2}\frac{\omega_i}{s^2+2\zeta_i \omega_i s+\omega_i ^2}$$ $$\omega_i=\frac{(2i+1)\pi}{T}\, T=1\, \zeta_i=0.2$$ Its impulse response is an approximation (truncated Fourier series) of a slowly decaying triangular wave. Using the balanced truncation method, find an r-order approximation Gr(s) of G(s) such that the following error bound, is guaranteed: $$\lVert G-G_r\rVert_\infty<0.1\lVert G\rVert_H$$ and plot the bode plots (using matlab) of G and Gr and their impulse response on the time interval [0,5]
so far, i've managed to find the Hankel singular values of the system and also $\lVert G\rVert_H$ by using the matlab command of balreal(G) but i'm having an issue with turnicating the system to guarantee the demand of $\lVert G-G_r\rVert_\infty<0.1\lVert G\rVert_H$
thanks in advance for any advice