MATLAB: How is it guaranteed that ROBSTAB finds the worst-case stability margin over the complete frequency range without gridding

Robust Control Toolbox

How is it guaranteed that ROBSTAB finds the worst-case stability margin over the complete frequency range without grinding? Can you provide the principle of the underlying algorithm/method or a source? Can you also provide information on which approach is used to calculate the lower and upper bound on Mu?

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Before MATLAB R2016b, Mu calculations were done on frequency grids. While it was fine for complex uncertainty, this was prone to miss peaks and discontinuities in the presence of uncertain real parameters, often leading to over-optimistic stability margins.

Starting in MATLAB R2016b, functions like ROBSTAB, ROBGAIN, and WCGAIN use a modified algorithm that eliminates this issue. The main change is in the Mu upper bound computation, which seeks to certify entire intervals rather than just individual frequencies and to cover the entire frequency axis with such interval certificates. This is how we obtain a guarantee over the entire frequency range.

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MATLAB: Does robstab return stable perturbation

I agree that the values do not seem correct. In fact, strangely if you use the range [0.55, 1.5] with a nominal value of 1, then WCU will display the correct destabilizing value of tau (tau = 0). Furthermore, if you use a nominal value that is closer to the upper bound than the lower bound, then tau gets approximated to some value close to zero. I'm not sure why there is such an inconsistency in this function.

The issue seems to occur when the ureal parameter has an asymmetric range that is asymmetric past a certain degree, and occurs more often when the nominal value is closer to the lower bound than the upper bound.

I have notified the developers of this issue. Unfortunately, I suppose the only workaround for now is to use a symmetric range.

MATLAB: How is sensitivity output of wcgain and robstab functions determined

The sensitivity is determined via a finite difference approximation. For each block, the algorithm will perturb the amount of uncertainty by the fraction specified in wcgainOptions, recompute the mu upper bound, and take the ratio of variations.

For additional information, please see the source code rctutil.getCSensitivity for details. You can access this in MATLAB using the following:

>> edit  rctutil.getCSensitivity

It might be beneficial if you set breakpoints while running your scripts to see how the individual components are computed while determining sensitivity.

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MATLAB: Does robstab return stable perturbation

MATLAB: How is sensitivity output of wcgain and robstab functions determined

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