How Can I Find the H-infinity Norm of a Transfer Function?

In summary, the H-infinity norm of a transfer function can be determined by evaluating the maximum singular value of the transfer function over all frequencies. This involves computing the transfer function's frequency response, typically by substituting complex variables into the function and analyzing it using techniques such as the Nyquist criterion or numerical methods. The H-infinity norm is particularly useful in control theory for assessing system stability and performance.
  • #1
billtodd
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Homework Statement
below
Relevant Equations
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I want to find ##\|G\|_\infty##, the solution in the pic uses the Bode plot. But to tell you the truth I am worse at drawing it.
So basically what I thought is I want to find: ##|G(i\omega)|=1/\sqrt{(25−\omega^2)^2+9\omega^2}##.
So basically I much prefer to find the minimum value of what is in the sqrt from highschool calculus methods (differentiating and equating to zero).
I found something of the sort of 24.21 in the denominator, which is close to 25 but far from 15.
Am I correct in my reasoning?
 

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  • #2
never mind I get the max is at 1/14.30908802. I had some really bad error.
 
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FAQ: How Can I Find the H-infinity Norm of a Transfer Function?

What is the H-infinity norm of a transfer function?

The H-infinity norm of a transfer function is a measure of the maximum gain of the system over all frequencies. It represents the worst-case amplification of the input signal to the output signal and is defined as the supremum of the transfer function's magnitude over the entire frequency spectrum.

How do I calculate the H-infinity norm for a given transfer function?

To calculate the H-infinity norm of a transfer function, you can use various methods such as frequency response analysis, numerical optimization, or specialized software tools. One common approach is to evaluate the transfer function's magnitude along the imaginary axis (substituting s = jω) and find the maximum value of this magnitude across all frequencies ω.

What tools or software can I use to find the H-infinity norm?

Several software tools can help you compute the H-infinity norm, including MATLAB (using the 'norm' function), Python with libraries like SciPy, and control system design software like Control System Toolbox or Simulink. These tools often provide built-in functions that streamline the calculation process.

Are there any specific properties of the H-infinity norm I should be aware of?

Yes, the H-infinity norm has several important properties. It is always non-negative, it is zero if and only if the transfer function is zero, and it is subadditive. Additionally, the H-infinity norm is invariant under state-space realizations, meaning that different representations of the same system will yield the same norm value.

Can the H-infinity norm be used for stability analysis?

Yes, the H-infinity norm is often used in control theory for stability analysis and robust control design. A smaller H-infinity norm indicates better performance and robustness of the system against disturbances and uncertainties. Control strategies can be designed to minimize the H-infinity norm, thereby enhancing system stability and performance.

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