# Determine Infinity Norm of a Transfer Matrix

1. Oct 5, 2009

### azizz

I'm trying to understand how the infinity norm of a transfer matrix is calculated. For example, assume a simple transfer matrix

$$G(s) = \begin{pmatrix} \frac{s}{s+1} & \frac{s}{s^2+s+1} \\ \frac{s-1}{s+2} & \frac{s-1}{s+1} \end{pmatrix}$$

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

$$\| G \|_{\infty} := \sup_{\omega \in \mathbb{R}} \| G(i\omega) \|$$

So I'm looking for a real valued frequency such that $$\| G(i\omega) \|$$ achieves its maximum value. The $$\mathcal{H}_{\infty}$$-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:

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

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?