Finding the function of the supremum

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Aysel Alimirzayeva 2022-9-29

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回答： Osmar Tormena Júnior 2023-3-30

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.

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Answer 1

Torsten 2022-9-29

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编辑：Torsten 2022-9-29

在 MATLAB Online 中打开

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

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Walter Roberson 2022-10-1

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

Aysel Alimirzayeva 2022-10-2

@Walter RobersonThank you for your answer.

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Answer 2

Sam Chak 2022-9-30

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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

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Aysel Alimirzayeva 2022-9-30

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

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Answer 3

Osmar Tormena Júnior 2023-3-30

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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.