Properties of strictly positive real functions: Products and compositions

In this work necessary conditions on one side are sufficient conditions on another side given such that the product of strictly positive real (SPR) functions to be an SPR0 function. Also, sufficient conditions are given for the composition of SPR functions to be an SPR function. As a consequence, a result on robust SPR transfer functions, under non-linearly correlated perturbations in the coefficients, induced by the composition and product with SPR functions is presented. Based on these facts, we give an application on preservation of robust and simultaneous stabilisation of single-input single-output (SISO), linear time-invariant (LTI) systems. The applicability of the results is shown through two examples.