Finding the H infinity norm of a system

Question

Kashish Pilyal 2022-8-22

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1783300-finding-the-h-infinity-norm-of-a-system

评论： Paul 2022-8-26

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 个评论
显示 -2更早的评论隐藏 -2更早的评论

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

Sam Chak 2022-8-22

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1783300-finding-the-h-infinity-norm-of-a-system#answer_1030495

编辑：Sam Chak 2022-8-26

Hi @Kashish Pilyal

Perhaps try using hinfnorm() to compute the H∞ norm of a dynamic system.

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

.

Here is an excerpt from the article: Interpretation of H-Infinity Norm:

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

6 个评论
显示 4更早的评论隐藏 4更早的评论

Kashish Pilyal 2022-8-23

Hi @Torsten,

So, can you tell me how to find the L2 norm of a vector in my case? and how does it differ from norm(a_2)?

Torsten 2022-8-23

编辑：Torsten 2022-8-25

在 MATLAB Online 中打开

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:

https://de.mathworks.com/help/robust/ref/lti.hinfnorm.html

请先登录，再进行评论。

Answer 2

Paul 2022-8-25

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1783300-finding-the-h-infinity-norm-of-a-system#answer_1033570

编辑：Paul 2022-8-25

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 个评论
显示无隐藏无

Kashish Pilyal 2022-8-26

The system is non linear so I am unable to write it in state space form.

Paul 2022-8-26

How is the infinity norm defined for a nonlinear system? What is the basis for the claim that the inifinity norm of the system is one?

请先登录，再进行评论。

Finding the H infinity norm of a system

0 个评论
显示 -2更早的评论隐藏 -2更早的评论

回答（2 个）

6 个评论
显示 4更早的评论隐藏 4更早的评论

2 个评论
显示无隐藏无

另请参阅

类别

标签

产品

版本

Community Treasure Hunt

Finding the H infinity norm of a system

0 个评论 显示 -2更早的评论隐藏 -2更早的评论

回答（2 个）

6 个评论 显示 4更早的评论隐藏 4更早的评论

2 个评论 显示 无隐藏 无

另请参阅

类别

标签

产品

版本

Community Treasure Hunt

0 个评论
显示 -2更早的评论隐藏 -2更早的评论

6 个评论
显示 4更早的评论隐藏 4更早的评论

2 个评论
显示无隐藏无