# Transfer function with cancellable zero pole and controllability

I have a transfer function (From Ogata's Modern Control Engineering)

$$\frac{s+2.5}{(s+2.5)(s-1)}$$

and the theory says the system has a pole zero cancellation and is uncontrollable.

They said that a state space rep of this transfer function has A and B matrix of:

$$A = \begin{bmatrix} 0 & 1 \\ 2.5 & -1.5 \end{bmatrix} \ \text{and} \ B = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$

Using rank(ctrb(A,B)) in MATLAB, the result is not equal to the dimension of the state space so it not controllable.

So I got curious and used tf2ss in MATLAB and got another state space rep:

$$A = \begin{bmatrix} -1.5 & 2.5 \\ 1 & 0 \end{bmatrix} \ \text{and} \ B = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$

Using rank(ctrb(A,B)) in MATLAB, I got a value equal to the dimension of the state space so it is controllable.

What have I misconceived? (and could someone teach me on how to make matrices in Markdown for the above?)

• Your A matrix has a typo, the 2 should be 2.5 and to learn about matrix formatting in MathJax take a look at this. – fibonatic Jul 22 '18 at 13:33
• Yea but the rank of the controllability matrix still remains 2 for the result I got from MATLAB.... – aldo Jul 22 '18 at 13:41

PS: One might argue that there still exists a similarity transform between those state space models. Only the transform should be rank deficient itself (not invertible) but still satisfy $\hat{A}\,T = T\,A$ and $\hat{C}\,T = C$.