Why does getPassiveIndex incorrectly compute the output passivity index, rho?

Why does "getPassiveIndex" in Control System Toolbox MATLAB R2020a calculate a wrong value of the output passivity index?

Given the following syntax, rho should be 0.2

>> H = tf([1 0],[1 0.2 1]);
>> [rho,freq] = getPassiveIndex(H,'output')

rho = 

     -Inf

freq = 

      NaN

Why is rho calculated as "-Inf"?

Answer 1

[MathWorks Support Team]

This is a limitation of the approach used to compute the output passivity index. We use a frequency-domain test which is only valid when "H" has no zero on the imaginary axis or at infinity, see Theorem 11 in

https://www.researchgate.net/publication/347232443_Sector_bounds_in_stability_analysis_and_control_design

Both conditions are not satisfied here, which is why "getPassiveIndex" returns rho = -Inf. You can recover the expected result by adding a small feedthrough to work around these technical issues:

>> [rho,freq] = getPassiveIndex(H+1e-4,'output')

rho =

  2.0000e-01

freq =

   1

In general, systems where passivity fails to hold strictly tend to be numerically problematic.