Properties of Positive Real Functions

Question

I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is PR/SPR, however I'm not sure how to approach the following example.

Say we are given two positive real transfer functions: $G_{1}(s)$ and $G_{2}(s)$

How can I tell if the sum $$ G_{1}(s) + G_{2}(s)$$ or the product $$ G_{1}(s) * G_{2}(s)$$

is PR/SPR?

Intuitively, I think the sum is SPR, but as for the product I am uncertain. Either way, is there a framework I can use to approach this or does anyone know of a proof I could look at?

Answer 1

At any given $s$ the two transfer functions give you two complex numbers $\rho_1+i\,\sigma_1$ and $\rho_2+i\,\sigma_2$, with the constraints that $\rho_1,\rho_2\geq0$.

For the summation you only have that it is strictly positive real iff $\rho_1$ and $\rho_2$ are not both simultaneously zero. For example $G_1(s)=\alpha\,G_2(s)$ with $\alpha>0$ would not satisfy this, or the transfer functions already need to be strictly positive real.

For the multiplication the real part would become $\rho_1\,\rho_2 - \sigma_1\,\sigma_2$. Therefore, the fact that $G_1(s)$ and $G_2(s)$ are positive real does not give you enough information about their imaginary parts to tell if the result is even positive real.