[Ian, Lee]

NOTE: this should have been posted on June 21 2023 as this is the date that it was writen, but it was in my drafts for some reason. I have added latex since then

In the H2 system the gain for the flat FOM (F2) is 1/F1_gain, where F1_gain is the gain of the BNS FOM (F1). This works fine for the H2 norm because that is made up of the composite RMS from both FOMs so as one FOM's gain increases and the others decreases they cancel out. However, this is not the case in the H_infinity case. The H_infinity norm is the maximum value for the frequency response from the Zinf input to the Flat FOM output this value must be less than or equal to the gamma that we are using. Currently we are searching for a gamma around 1. The equation for the frequency response along the  Zinf input to the Flat FOM output is:

\[ \left | \frac{K(\omega)P(\omega)}{1-K(\omega)P(\omega)} F_{\text{flat}}(\omega)\right |\leq \gamma \]

where \( K(\omega)\) is the controller response, \(P(\omega)\) is the plant response and \(F_{\text{flat}}(\omega)\) is the Flat FOM's response. If the environmental noise is large like at low frequencies then \( K(\omega)P(\omega) \) must be large to control it. Thus at low frequencies the above equation becomes,

\[ \lim_{KP \to \infty }\left | \frac{K(\omega)P(\omega)}{1-K(\omega)P(\omega)} F_{\text{flat}}(\omega)\right | \approx\left |F_{\text{flat}}\right | \leq \gamma \]

Thus if we use the F2_gain=1/F1_gain then as F1_gain gets very small F2_gain gets very large. When F2_gain is very large, the gamma we are searching for (around 1) is much less than the actual H_infinty norm which is on the order of the F2_gain and the solver fails. This is what was causing the problems with the RMS plot. 

We switched the F2 gain to be all ones and acquired attached plot.

Note: when this PDF is printed the second color bar has the same colors as the first color bar.

Improvements for code and plots for paper:

1. Fix the color bar problem (noted above) for printing.
2. Make the color bar discreet
3. decrease the number of F1_gain points so that the plot is not so crowded 
4. Maybe add an H2 boundary line. See how it looks
5. change the F1_gain color bar. maybe connect like F1_gain points with a subtle line.
6. Add an example of the model maker (getSPOFF) to Buzz
7. make the code transpose the model right before solving in the H_infinity case