Passivity criteria for controlled affine resistive components

Passivity criteria for controlled affine resistive (CAR) components are presented. First, the definitions of passive component and non-dynamic component are recalled, as formulated in the frame of the behavioral approach (i.e., based on the concept of forwards concatenation set of an admissible process at a given time). Then, three lemmas, one theorem, and four corollaries on quadratic functions are stated and proven, which provide the mathematical tools needed. On this basis, passivity criteria for CAR components are derived, under both the standard (i.e., homogeneous, hybrid, and transmission) matrix representations and the general matrix representation. Finally, by resorting to a kind of canonic representation, two general results on the passivity of CAR components are obtained.