Necessary conditions for positive realness Given a linear time-invariant (LTI) system with
$$
\begin{align}
\dot{x} &= A x + Bu \\ 
y &= C x + D u
\end{align}
$$
We know that the transfer function matrix $G(s) = C(s I - A)^{-1}B + D$ is positive real iff there exist matrices $P = P^T > 0, L, W$ such that
$$
\begin{align}
A^T P + P A &= -L L^T \\
P B - C^T &= -L W \\
D + D^T &= W^T W
\end{align} \tag{1}
$$
While we can find solutions to $(1)$ with numerical methods, I wonder if there are some "easy-to-check" neccessary conditions for positive realness of  LTI systems with multiple inputs and outputs (MIMO)?
I know that in the single input/output case a neccessary condition is for example that the relative degree of $G(s)$ is less than $2$ and that $G(s)$ must be minimum-phase, which is both very easy to check. Are there similar conditions for MIMO systems (except stability)?
Question: I am looking for a list of easy-to-check necessary conditions of positive realness of a transfer function matrix.
 A: Let us show one necessary condition. Suppose that all transfer functions (all elements of $G(s)$) are of relative degree at least one. Then $D=0$, thus $W=0$ and for the system to be SPR a necessary condition is that there exists a positive definite matrix $P$ such that $PB=C^\top$. Multiplying this equation by $B^\top$ yields $B^\top P B=B^\top C^\top$.
Ok, let us assume that the number of inputs is less or equal to the number of states and assume also that the matrix $B$ is of full rank. Then $B^\top P B$ is also positive definite. This assumption about the full rank is reasonable since if $B$ is not of full rank and $D=0$ then some of your input signals can be dropped away or combined together.
Let us now assume that all elements of $G(s)$ are of relative degree two or higher. Then the derivative of the output $y$ does not depend on $u$. Since $\dot{y} = CAx + CBu$, we obtain $B^\top C^\top=0$. Thus there does not exist $P$ satisfying the requirements.
As you can see, this is not a complete answer, and the idea can be further extended. For example, if any of the diagonal elements of $G(s)$ is of relative degree 2, then $B^\top C^\top$ cannot be positive definite. On the other hand, the full-rank of dimension conditions can be also relaxed given the structure of the matrix $B^\top C^\top$.
