Eigenvalue-Based Characterization and Test for Positive Realness of Scalar Transfer Functions

An eigenvalue-based characterization of positive realness of transfer functions of single-input single-output time-invariant linear systems is derived. Based on this characterization, we propose efficient computational procedures to determine if a given transfer function is positive real. The input for these eigenvalue-based tests is any given, not necessarily minimal, state-space representation of the linear system. Furthermore, the tests only involve standard matrix computations, such as computing eigenvalues of a matrix or a matrix pencil. Results of numerical experiments with these eigenvalue-based tests for positive realness are reported.