From multivariable system transfer function matrix to state space representation

Question

I have the transfer function matrix $H(s) = \begin{bmatrix} {1\over s+1} & {2\over s+2} \\ {-2\over s^2+3s+2} & {2s\over s+1} \\ \end{bmatrix}$

And I want to calculate two equivalent state space representations one of dimension $4$ and the other of dimension $3$. How can I find a state space representation of dimension $3$? Which is the minimum dimension with which I can represent a linear time invariant system?

Answer 1

If $\hat{y}=\hat{H}(s) \hat{u}$, then a 4 dimensional system can be easily created by realising ${1 \over s+1}, {1 \over s+2}, {1 \over s+1}$ (that is, two separate copies of the first) and feeding the output of the first into (another) ${1 \over s+2}$. In particular, $\hat{x}_{11} = {1 \over s+1} \hat{u}_1$, $\hat{x}_{12} = {1 \over s+2} \hat{u}_2$, $\hat{x}_{21} = {1 \over s+2} \hat{x}_{11}$, $\hat{x}_{22} = {1 \over s+1} \hat{u}_2$, and $\hat{y}_1 = \hat{x}_{11} - 2 \hat{x}_{12}$, $\hat{y}_2 = 2 \hat{x}_{21} - 2 \hat{x}_{22} + 2 \hat{u}_2$.

The corresponding $A,B,C,D$ realisation is (Matlab/Octave notation):

A = [-1 0 0 0 ; 0 -2 0 0 ; 1 0 -2 0 ; 0 0 0 -1];
B = [1 0 ; 0 1 ; 0 0 ; 0 1];
C = [1 2 0 0 ; 0 0 -2 -2];
D = [0 0 ; 0 2];

# the following return 4:
rank([A*B B])
rank ([C ; C*A])

It is straight forward to check that $(A,B)$ is cc. and $(A,C)$ is co., hence the realisation is minimal. It follows that there is no 3 dimensional system that realises $\hat{H}(s)$.