γ-passive system and its phase property and synthesis

In this paper the concept of passivity is generalized from the viewpoint of the rate of energy dissipation, introducing a new supply rate with an index γ. A stabilizability result for nonlinear systems is obtained using the argument of the supply rate and the dissipative inequality. In the case of linear systems, it is shown that the index γ can be used as a measure of phase of transfer functions, which provides a phase curve shaping method. A stability result of a feedback system involving a γ-positive real system is also obtained and its effectiveness is shown by an example. Finally, a design method of γ-passive system is discussed using a bilinear transformation of nonlinear systems, called Cayley-transform