Generalizations of the Concept of the Positive Real Function

Two generalizations of the concept of the positive real function are made that are applicable to transfer functions whose poles outnumber their zeros by any amount. Some previously published results are briefly discussed. Then, the second generalization is developed still further. In particular, the properties of the zeros on the imaginary axis are established and are found to be analogous to the corresponding properties for positive real functions. In addition, several new tests for the generalized functions are developed, one of which serves as a new test for positive real functions wherein the Sturm sequence is replaced by a more easily calculated sequence. Reza's "double alternation" property for positive real functions is extended to the generalized functions. Finally, a property of the phase functions of the positive real functions is extended and the use of some common transformations is discussed which, in turn, leads to still another type of generalization.