Preserving positive realness through discretization

The importance of positive real transfer functions relies on the fact that they are associated with positive linear systems. Those systems possess the property that their input-output product time-integral, which is a measure of the total enerty, is nonnegative. Such a property can be also formulated in the discrete context. It is shown that a discrete positive real transfer function is obtained from a positive real continuous one of relative order zero being strictly stable poles via discretization by a sampler and zero-order hold device provided that the direct input-output transmission gain is sufficiently large. It is also proved that a discrete positive real transfer function may be obtained from a stable continuous one of relative order zero and high direct input-output gain which posses simple complex conjugate critically stable poles even in the case that this one is not positive real. For that purpose, the use of an appropriate phase-lag or phase lead compensating network for the continuous transfer function may be required to ensure positive realness of the discrete transfer function.