# Finding the H infinity norm of a system

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I am running a system with ode45. So, I have an input array and an output. The definition of H infinity norm says that it is L2 norm of output by L2 norm of input. So I just used norm function on both arrays and divided them both. Although, the system must have the H infinity norm of 1 but it comes out to be less than 1. I have the code as:

clear all

%%

%paramters of leader vehicle

q=0; %position

v=0; %velocity

a=0; %acceleration

x_0=[q;

v;

a]; %initial conditions

t1=linspace(0,5,1000);

L1=4; %lenght of car(m)

dr=5; %gap b/w cars(m)

%parameters of second vehicle

q1=0; %position

v1=0; %velocity

a1=0; %acceleration

x_1=[q1;

v1;

a1]; %initial conditions

%initial conditions of the platoon

x=[x_0; x_1];

[t1,y1]=ode45(@code3,t1,x);

%% second step

%paramters of leader vehicle

q=0; %position

v=0; %velocity

a=1; %acceleration

x_0=[q;

v;

a]; %initial conditions

t2=linspace(5,10,1000);

%parameters of second vehicle

q1=0; %position

v1=0; %velocity

a1=0; %acceleration

x_1=[q1;

v1;

a1]; %initial conditions

%initial conditions of the platoon

x=[x_0; x_1];

[t2,y2]=ode45(@code3,t2,x);

%% third step

%paramters of leader vehicle

q=y2(end,1); %position

v=y2(end,2); %velocity

a=0; %acceleration

x_0=[q;

v;

a]; %initial conditions

t3=linspace(10,20,1000);

%parameters of second vehicle

q1=y2(end,4); %position

v1=y2(end,5); %velocity

a1=y2(end,6); %acceleration

x_1=[q1;

v1;

a1]; %initial conditions

%initial conditions of the platoon

x=[x_0; x_1];

[t3,y3]=ode45(@code3,t3,x);

%% concating

time=[t1;t2;t3];

%parameters of 1st

q_1=[y1(:,1);y2(:,1);y3(:,1)];

v_1=[y1(:,2);y2(:,2);y3(:,2)];

a_1=[y1(:,3);y2(:,3);y3(:,3)];

%parameters of 2nd

q_2=[y1(:,4);y2(:,4);y3(:,4)];

v_2=[y1(:,5);y2(:,5);y3(:,5)];

a_2=[y1(:,6);y2(:,6);y3(:,6)];

figure(1)

plot(time,q_1,time,q_2)

grid on

figure(2)

plot(time,v_1,time,v_2)

grid on

figure(3)

plot(time,a_1,time,a_2)

ylim([-2 2])

grid on

Nm=norm(a_2)/norm(a_1);

##### 0 Comments

### Answers (2)

Sam Chak
on 22 Aug 2022

Edited: Sam Chak
on 26 Aug 2022

Edit 1: Looking at your code, it seemed that what you computed was the H₂-norm, .

Edit 2: There are some algorithms provided in Kuster's thesis: "H-infinity Norm Calculation via a State Space Formulation".

##### 6 Comments

Torsten
on 23 Aug 2022

Edited: Torsten
on 25 Aug 2022

I don't know how it works here, but given a function u you get by a differential equation

du/dt = f(u,t)

and you want to calculate

sqrt( integral_{t=0}^{t=T} u(t)^2 dt )

you have to solve the additional differential equation

dU/dt = u^2, U(0) = 0

The L2-norm of u will then be

sqrt(U(end))

But when reading about the output of MATLAB's hinfnorm, this doesn't seem to be what you want:

Paul
on 25 Aug 2022

Edited: Paul
on 25 Aug 2022

Following up on this comment ... Can the equations not be writen in state space form only because it's impractical to do so, or because it's not possible to do so, i.e., the system is not LTI and causal?

If the latter, then I'm not sure how the inifity norm is defined? Is there a definition?

If the former, how do you know that the infinity norm of the system is 1?

As far as computing the 2-norm of the input and output goes, I think using norm() is a reasonable approximation as long as (actually, it would be norm*sqrt(sampling period), but the sqrt(sampling period) cancels when computing the ratio)

aa) the system is BIBO stable (actually, I'm not sure if the inifnity norm is even defined for an unstable system, can't recall off hand, but even if it is, it woudn't be too useful)

a) the input has a finite 2-norm

b) the input and ouput are sampled at a fixed rate, which ode45 doesn't do with default settings

c) the output has finite 2-norm, which I think basically means the (square of the?) ouput has to at least approach zero asymptotically.

d) the simualtion is simulated long enough s.t. one can use a finite number of samples of the input and output to estimate the 2-norm of each.

Assuming that the system is LTI, and assuming that its infnity norm is 1, and assuming that the calculation of the 2-norms of the input and output are reasonably approximated, we also must bear in mind that, in general, the infinity norm of the system is NOT the ratio of the 2-norms of the output and input. Rather the infinity norm is an upper bound on this ratio, i.e.,

2-norm(y) <= inf-norm(H)*2-norm(u).

So, the obtained result might be correct after all.

##### 2 Comments

Paul
on 26 Aug 2022

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