# Infinity norm of system matrix

1. Oct 7, 2009

### azizz

1. The problem statement, all variables and given/known data

This problem is related to the system theory, using the H-infinity frame work to determine the maximum gain of a multivarible system. The system is described as

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

with s a complex variable, known as the laplace operator. I want to compute the H-infinity norm of G(s), and if I'm not mistaking I can do this with

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

In words: I'm looking a real valued frequency such that the two-norm of G(iw) achieves its maximum value. So that can be done with

$$\| G(i\omega) \| = \sqrt{ \left| \frac{i\omega}{i\omega+1} \right|^2 + \left| \frac{i\omega}{(i\omega)^2+i\omega+1} \right|^2 + \left| \frac{i\omega-1}{i\omega+2} \right|^2 + \left| \frac{i\omega-1}{i\omega+1} \right|^2}$$

Actually, my question if is this is ok? Because I tried to compute this with matlab and that did not give the correct result (see below).

2. Solutions according to matlab

First I tried to compute the infinity norm of G(s):

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.

Then I wanted to compare this result to the result I get with the formule I gave above. We now know that the frequency that gives the infinity norm is equal to fpeak, thus filling this in gives;

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 )

However, what I get is: h_inf_norm=1.1222, and not ninf=1.6973... Why is that?
All hints/tips are welcome :-) Thanks in advance.