Finding the function of the supremum

Hello.I have a function I want to calculate its supremum. The function is below.I don't know how to do this in Matlab.Please help me.

Answer 1

[Torsten]

sup-norm is a theoretical concept.

There is no such function in MATLAB to calculate it for a general function.

In the case above you could use

syms t
f = (1-exp(-t))*heaviside(t);
limit(abs(f),t,Inf)

ans = 

1

[Walter Roberson]

Note that limit() approaching infinity is not generally sufficient. For example if the function were multiplied by sin(t) then the maximum might occur before that.

[Aysel Alimirzayeva]

@TorstenThank you very much

[Aysel Alimirzayeva]

@Walter Roberson I didn't quite understand what you said. Could you please explain in more detail?

[Torsten]

@Aysel Alimirzayeva

Walter Roberson means that this way to calculate the inf-norm of a function cannot be generalized to other functions.

E.g. if u(t) = sin(t), then sup | u(t) | = 1, but lim (t->oo) | sin(t) | does not exist.

[Aysel Alimirzayeva]

@TorstenThanks for the detailed explanation.

[Walter Roberson]

or consider a Gaussian. The sup is at the peak but the limit at infinity is 0.

[Aysel Alimirzayeva]

@Walter RobersonThank you for your answer.

Answer 2

[Sam Chak]

Hi @Aysel Alimirzayeva

I'm not sure if the following is what you want. But this 1st-order transfer function (taking the Laplace transform)

produces the same step response as the given Continuous-Time Signal. So the following computes the ∞-Norm of the linear system, instead of the signal.

Also worth checking out these:

https://www.mathworks.com/help/ident/ref/lti.norm.html

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

https://www.mathworks.com/help/robust/gs/interpretation-of-h-infinity-norm.html

t = linspace(0, 10, 1001);
x = 1 - exp(-t);
G = tf(1, [1 1])

G =
 
    1
  -----
  s + 1
 
Continuous-time transfer function.

subplot(211)
plot(t, x), grid on, xlabel('t'), title('Continuous-Time Signal')
subplot(212)
step(G, 10), grid on, xlabel('t'), title('Step Response of Transfer Function')

[ninf,fpeak] = norm(G, inf)

ninf = 1

fpeak = 0

[Aysel Alimirzayeva]

@Sam Chak The information you gave me was very useful. Thank you very much.

Answer 3

[Osmar Tormena Júnior]

Following p-norm definition, the supremum may be defined as:

Which may be computed by

Bx = limit((int(abs(x)^p, dummy, -Inf, Inf))^(1/p), p, Inf);

for a suitable dummy integration variable.

Alas, the solution appears to be beyond the Toolbox analytical capability. It fails for common signals like constants, unit step, etc. Even when it doesn't fail, it does not return the final result — although it still works with isfinite() function to test if . Really, the only kind of signal this has worked out is a gaussian.