# Controllable realisation of $\frac{s^4+1}{4s^4+2s^3+2s+1}$ is both controllable and observable?

I am trying to find the controllable realization of the following transfer function:

$$H(s)=\frac{s^4+1}{4s^4+2s^3+2s+1}$$

I approach this by first using polynomial division to assure that $H(s)$ is strictly proper. This yields:

$$H(s)=1/4+\frac{-1/8\cdot s^3-1/8\cdot s+3/16}{s^4+1/2\cdot s^3+1/2\cdot s+1/4}$$

From this I can extract the controllable canonical form where the matrices A, B, C and D are:

$$A=\begin{bmatrix}0 & 1 & 0 & 0\\ 0 & 0 & 1& 0 \\ 0 & 0 & 0 & 1\\-1/4 & -1/2 & 0 & -1/2\end{bmatrix}$$

$$B=\begin{bmatrix}0\\0\\0\\1\end{bmatrix}$$ $$C=\begin{bmatrix}3/16 &-1/8 &0 &-1/8\end{bmatrix}$$ $$D=1/4$$

From this I am confused because when I construct the observability and controllability matrices I found that the system is both controllable and observable (both matrices have full rank). This seems like a contradiction. Can it be possible? What am I doing wrong?

• Controllability and obsrvabilty are not mutually exclusive, indeed having both is usually good – user28715 Oct 28 '17 at 16:02
• Ok thank you! Its just that I thought that in the canonical controllable realization the system cannot also be observable. But I guess this is false. How can I ensure that my system is both observable an controllable? – john melon Oct 28 '17 at 16:11
• You did the tests. You can try doing the same tests on a different canonical form and should get the same answer. The system doesn’t mind how we write it down. State space expressions can be realized by more than one way for a particular system. If the system is both, it should be both. If you can find a counter example, it would be interesting to see it – user28715 Oct 28 '17 at 16:23

Controllability is a property of $A,B$ and observability is a property of $A,C$ matrices.