How can the Robust Performance margin be computed using Mu Analysis "mussv" in R2019a?

I am using the Robust Control Toolbox for Robust Stability and Performance Analysis. For this purpose, the commands ROBSTAB and ROBGAIN can be used.

On your homepage you are giving an example of how ROBSTAB is connected with mu-analysis:

https://www.mathworks.com/help/releases/R2020a/robust/examples/robust-stability-robust-performance-and-mu-analysis.html

In the example, you show how the robust stability margins can be computed alternatively using the MUSSV function.

*Now my question is how can the robust performance margin be computed using MUSSV? *

I searched through the documentation but could not find any example. In an older document, I found how this was done. However, when following this procedure the computed robust performance margins are completely different from the ones computed by ROBGAIN. Out of curiosity, I also checked the former ROBUSTPERF command. And here the results match very well. What is ROBGAIN doing differently?

Answer 1

[MathWorks Support Team]

The ROBGAIN and ROBUSTPERF margins M differ in the following way:

1) A ROBUSTPERF margin M means that the gain stays below 1/M for uncertainty level up to M in normalized units.

2) A ROBGAIN margin M means that the gain stays below the specified value G for uncertainty level up to M in normalized units.

ROBUSTPERF uses the "performance" block approach:

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To check that the gain from u to y is robustly less than gamma, you add a "performance" block (dynamic uncertainty bounded by 1/gamma) and check the robust stability against this augmented uncertainty structure. This is rooted in the Small Gain Theorem:

M(s) |oo < gamma <=> LFT(DELTA,M) stable for all ||DELTA|<1/gamma

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Meanwhile, ROBGAIN solves a "skew-mu" problem where the performance block is "fixed" and the other blocks are allowed to vary. See. e.g.,

https://w3.onera.fr/smac/smart

for details.

While this is handled via the same MUSSV machinery, skew-mu is not a documented usage of the MUSSV function and therefore not something a user can try directly.