Passivity analysis of a system and its approximation

We consider the following problem: given two mathematical system models, one of which could represent a physical system accurately and the other could be an approximation of the system, what passivity properties of the system can be inferred from studying only the approximate model. Our results show that an excess of passivity (whether in the form of input strictly passive, output strictly passive or very strictly passive) in the approximate model guarantees a certain passivity index for the system, provided that the norm of the error between the two models is sufficiently small in a suitably defined sense. Further, we consider QSR dissipative systems and show that QSR dissipativity has a similar robustness property, even though the supply rates for the system and its approximation may be different.