Multivariate theory‐based passivity criteria for linear fractional networks

In traditional linear network theory, the positive‐real (PR) criteria are widely used to judge the passivity of elements and networks in the light of the fact that there exists an equivalent relationship between the passivity and the PR property of their immittance functions (matrices). However, the equivalence will no longer hold when the fractional elements are introduced into the network, and the PR criteria are not suitable in complex frequency domain anymore. On the other hand, the rapid development of fractional‐order circuits and systems and the corresponding study in fractional circuit analysis and designs put forward an urgent requirement for the passivity criterion, which can tackle linear fractional networks. Hence, in this paper, we propose new passivity criteria for linear fractional networks by aid of generalized Tellegen's theorem and multivariable PR theory. By using the proposed criteria, the passivity of linear fractional networks can be judged, and the steps of the proposed criterion are illustrated by examples.