Computing output passivity index of a simple system using getPassiveIndex

I am trying to compute the output passivity index of the system,

by calling

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

rho = -Inf

freq = NaN

The results in

rho = -Inf

freq = NaN

However, based on the definition of output passivity index here, I think the value of rho should be 0.2.

If we did

rho = 0.2;
Hc = feedback(H,0.2,+1)

Hc =
 
     s
  -------
  s^2 + 1
 
Continuous-time transfer function.

isPassive(Hc)

ans = logical
   1

we get a system that is just passive, and increasing rho any further would result in a closed loop system that is not passive.

Intuitively also, this makes sense. H is the transfer function from force to velocity of a (passive) spring-mass-dashpot system. 0.2 is the damping coefficient, and may be thought of as the surplus passivity that the system has.

I am wondering why getPassiveIndex doesn't return an output passivity index of 0.2, and if I am interpreting something incorrectly.

Answer 1

[Neelanshu]

Hi Siva,

This is a limitation of the approach used for computation of the "getPassiveIndex". The frequency-domain test used is only valid when "H" has no zero on the imaginary axis or at infinity.

Both conditions are not satisfied here resulting in rho = -Inf. You can add a small feedback to work around this issue and recover the appropriate rho as shown :

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

Unrecognized function or variable 'H'.

You may refer to the following MATLAB answer to know more about limitation of "getPassiveIndex" :

https://www.mathworks.com/matlabcentral/answers/1883402

Hope this helps.

[Siva]

Thank you, Neelanshu