Demo on the interpretation of the norm of linear systems:

1. SISO (Single Input Single Output) system:

Let us consider the second order system:
Numerical application:
w0=2;
xi=0.1;
G=tf(w0*w0,[1 2*xi*w0 w0^2]);
The norm of :
norm(G,2)
ans =
2.2361e+00
sqrt(w0/4/xi)
ans =
2.2361e+00

1.1 Time-domain interpretation:

The impulse response of :
figure
impulse(G)
The square root of the numerical intregration of the impulse response squared:
[Y,T]=impulse(G);
Z=trapz(T,Y.^2);
sqrt(Z)
ans =
2.2360e+00
except the numerical integration error, we recognize the time-domain interpretation of the norm :

1.2 Frequency-domain interpretation:

The Bode response:
figure
bode(G)
The square root of the numerical integration of the frequency-domain response squared and divided by
[MAG,PHASE,W]=bode(G);
sqrt(trapz(W,MAG.^2)/pi)
ans =
2.2390e+00
except the numerical integration error, we recognize the frequency-domain interpretation of the norm :
since the frequency-domain response magnitude is an even function of .

1.3 Stochastic interpretation:

It can be also shown that the norm is also the standard deviation of the random signal corresponding to the steady-state of the output signal when the input signal is a normalized centered white noise (the PSD (Powed spectral density) of is : 1).
To simulate such a response one have to keep in mind that is not possible to simulate a white noise on a digital computer because the variance of a white noise is infinite. We use a discrete-time approximation: the signal is sampled with a sampling period and hold over seconds on the value of a centered normal random variable . Each sample is stochatically independant from the other. Let be the variance of the this normal random variable, then the PSD of such a signal is: So to approximate a unitary withe noise, one have to choose
The sampling period must be chosen such the half sampling frequency is very big with respect to the bandwidth of the system
dt=0.01;
T=[0:dt:1000]; % the stop-time is large to simulate the steady state
U=randn(size(T))/sqrt(dt);
Y=lsim(G,U,T);
figure
plot(T,U)
hold on
plot(T,Y); legend('u(t)','y(t)')
The standard deviation of the output in steady state is computed statistically with the Matlab function std:
std(Y(10000:end)) % it is assumed that the steady steady is reached after 100s
ans =
2.1542e+00
except the numerical error:.
Now we can consider multi-input multi-output (MIMO) systems.

2. MIMO systems

Let us consider a stable system with states outputs and inputs randomly generated by Matlab function rss
n=5;p=3;m=2;
G=rss(n,p,m);
The direct-feedtrough matrix of must be null otherwise its norm is infinite. It is obvious from the:
G.d=0*G.d;
The norm of :
norm(G,2)
ans =
2.1244e+00
From the time-domain definition of the norm of a MIMO system defined by the state-space matrices , , :
Let (the observability grammian) then is solution of the Lyapunov equation: and :
Qo=lyap(G.a',G.c'*G.c)
Qo = 5×5
3.0918e+00 -4.5208e-02 -1.0684e+00 -9.5959e-02 8.4245e-02 -4.5208e-02 8.3381e-01 2.9562e-01 -7.5605e-02 3.2785e-01 -1.0684e+00 2.9562e-01 1.1538e+00 6.6253e-02 2.1174e-01 -9.5959e-02 -7.5605e-02 6.6253e-02 2.4290e-01 -5.8832e-03 8.4245e-02 3.2785e-01 2.1174e-01 -5.8832e-03 5.1811e-01
sqrt(trace(G.b'*Qo*G.b))
ans =
2.1244e+00

2.1 Time-domain interpretation:

The impulse response of :
figure
impulse(G)
Computation of :
[Y,T]=impulse(G);
size(Y)
ans = 1×3
288 3 2
is the matrix of the impulse responses:
Y(i,j,k) is the value at the time T(i) of the response of the j-th output to an single impulse on the k-th input.
Z=zeros(length(Y),1);
for j=1:p,for k=1:m,Z=Z+Y(:,j,k).^2;end;end;
The square root of the numerical intregration of :
sqrt(trapz(T,Z))
ans =
2.1256e+00
except the numerical integration integration errors, we found .

2.2 Frequency-domain interpretation:

The Bode response:
figure
bode(G)
[MAG,PHASE,W]=bode(G);
size(MAG)
ans = 1×3
3 2 83
is the matrix of the frequency-domaine responses:
MAG(i,j,k) is the value at the time T(k) of the magnitude of transfer from the j-th input to the i-th output
Z=zeros(length(MAG),1);
for i=1:p,for j=1:m,Z=Z+squeeze(MAG(i,j,:).^2);end;end;
sqrt(trapz(W,Z)/pi)
ans =
2.1144e+00
except the numerical integration integration errors, we found again .

2.3 Stochastic interpretation:

It is the direct extension of the approach presented for SISO systems. One should keep in mind that the m normalized centered white noises on the m inputs of the system must be stochastically independent from each other. Then the results is:
.
In the following Matlab sequence, since is generated randomly:
dt=2*pi/(100*max(abs(eig(G))))
dt =
2.1218e-02
Tst=10/(min(abs(eig(G))))
Tst =
1.0888e+01
Tfin=Tst+10000*dt;
T=[0:dt:Tfin];
U=[];
for i=1:m, U=[U;randn(size(T))/sqrt(dt)];end;
Y=lsim(G,U,T);
sig2=0;
for i=1:p,sig2=sig2+var(Y(end-10000:end,i));end;
sqrt(sig2)
ans =
2.1699e+00