MIMO state space <--> MIMO transfer function - How?

Question

Normally I use the controllability and observability canonical forms to transform a transfer function into a state space model.

I also find the poles, zeros and gain from a state space model to transform the transfer function into a transfer function.

But this is only in SISO-case. How would it be if I have a MIMO state space model and I want to transform that into a MIMO transfer function matrix?

I know that if the column length of the $B$-matrix is $2$ and the row length of $C$-matrix is $2$, then I will have a transfer function matrix of the dimension $2 \times 2$.

So can I still use the controllability and observability canonical forms to compute the MIMO transfer function matrix, into a MIMO state space model, just by using only one column of $B$-matrix and one row of the $C$-matrix at each time?

And if I want to transform a MIMO state space model into a MIMO transfer function, I need to find the poles, zeros and gain for each row and column from $C$ and $B$ matrix?

Are that correct?

Answer 1

For any continuous time state space model, so SISO, MISO, SIMO or MIMO you can always use the following formula to convert the state space model into a transfer function matrix

$$ G(s) = C (s\,I - A)^{-1} B + D. $$

Answer 2

If you realize a MIMO linear system channel-by-channel using canonical forms for each pair of 1 input variable to 1 output variable, the realization may be of larger order than the minimum. This subject is covered in Linear System Theory, 2/e - Rugh, Wilson J, Chapter 13, and many other books.