I'm trying to understand how the infinity norm of a transfer matrix is calculated. For example, assume a simple transfer matrix(adsbygoogle = window.adsbygoogle || []).push({});

[tex] G(s) = \begin{pmatrix} \frac{s}{s+1} & \frac{s}{s^2+s+1} \\ \frac{s-1}{s+2} & \frac{s-1}{s+1} \end{pmatrix} [/tex]

Now, I'm trying to compute the [tex] \mathcal{H}_{\infty} [/tex]-norm of G(s), that is

[tex] \| G \|_{\infty} := \sup_{\omega \in \mathbb{R}} \| G(i\omega) \| [/tex]

So I'm looking for a real valued frequency such that [tex] \| G(i\omega) \| [/tex] achieves its maximum value. The [tex] \mathcal{H}_{\infty} [/tex]-norm and the corresponding frequency can easily be found with MATLAB:

s=tf('s');

g11=s/(s+1);

g12=s/(s^2+s+1);

g21=(s-1)/(s+2);

g22=(s-1)/(s+1);

G=[g11,g12;g21,g22];

tol=1e-6;

[ninf,fpeak]=norm(G,inf,tol)

The result is: ninf=1.6973 (abs), fpeak=1.0651 rad/s. Now I want to reproduce this result by my own calculations:

[tex] \| G \|_{\infty} := \sup_{\omega \in \mathbb{R}} \| G(i\omega) \| = \sup_{\omega \in \mathbb{R}} \sqrt{ | \frac{i\omega}{i\omega +1} |^2 + | \frac{i\omega }{(i\omega)^2+i\omega +1} |^2 + | \frac{i\omega -1}{i\omega +2} |^2 + | \frac{i\omega -1}{i\omega +1} |^2 } [/tex]

If I compute this with MATLAB for the frequency w=fpeak=1.0651 (which should yield the infinity norm),

w=1.0651;

g11=(i*w)/((i*w)+1);

g12=(i*w)/((i*w)^2+(i*w)+1);

g21=((i*w)-1)/((i*w)+2);

g22=((i*w)-1)/((i*w)+1);

h_inf_norm=sqrt( abs(g11)^2+abs(g12)^2+abs(g21)^2+abs(g22)^2 )

This is what I get: h_inf_norm=1.1222, which is not what I expected (that would be 1.6973). Even more strangely, this value increases as the frequency get closer to zero (the singular value plot shows something totally different).

I was wandering where I'm making the big mistake?

Thanks in advance.

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# Determine Infinity Norm of a Transfer Matrix

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