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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.

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  • $\begingroup$ At least, if relative degrees of all the transfer functions is 2 or bigger, then it is not SPR. $\endgroup$ – Arastas May 15 at 9:22
  • $\begingroup$ @Arastas Interesting, how is that? Do you have a reference for that or is it somehow obvious? $\endgroup$ – SampleTime May 15 at 18:14
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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 $B^\top C^\top$.

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  • $\begingroup$ That is interesting, but I don't really understand how $B$ with not full rank can be changed to have full rank (except if controls are redundant of course)? How would that work if the number of inputs is smaller than the number of states (which is usually the case)? $\endgroup$ – SampleTime May 18 at 11:27
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    $\begingroup$ It does not matter if the number of inputs is smaller than of the states. E.g. you have $2$ inputs and five states, $B$ is $5\times 2$, and you assume that it is of the rang $2$. If $B$ is of the rang $1$, then your system can be reduced to a system with one input. $\endgroup$ – Arastas May 18 at 16:57

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