Controllability and observability of a transfer function

Question

I have the following transfer function:

$$G(s)=\dfrac{1}{(s+1)^{10} (s+0.8)^{10}}$$

There is no pole-zero cancellation, but still, when I convert this into the state-space form using tf2ss command, the state space model is not minimal. Why is it so?

Answer 1

is this what you are looking for?

expand((s+0.8)^10) = s^10 + 8*s^9 + (144*s^8)/5 + (1536*s^7)/25 + (10752*s^6)/125 
  + (258048*s^5)/3125 + (172032*s^4)/3125 + (393216*s^3)/15625 
  + (589824*s^2)/78125 + (524288*s)/390625 + 1048576/9765625

expand((s+10)^10) = s^10 + 100*s^9 + 4500*s^8 + 120000*s^7 + 2100000*s^6 
 + 25200000*s^5 + 210000000*s^4 + 1200000000*s^3 
 + 4500000000*s^2 + 10000000000*s + 10000000000


[A B C D]=tf2ss([1 8 144/5 1536/25 10752/125 258048/3125 172032/3125 
      393216/15625 589824/78125 524288/390625 1048576/9765625],
          [1 100 4500 120000 2100000 25200000 210000000 1200000000 
        4500000000 10000000000 10000000000])

rank([C;C*A;C*A^2;C*A^3;C*A^4;C*A^5;C*A^6;C*A^7;C*A^8;C*A^9]) 

Answer 2

This is not an answer, you but might note that you can write the system as: $$\hat{h}(s) = 1 - \frac{2}{(s+1)} + \frac{9}{5 (s+1)^2} - \frac{24}{25 (s+1)^3} + \frac{42}{125 (s+1)^4} - \frac{252}{3125 (s+1)^5} + \frac{42}{3125 (s+1)^6} - \frac{24}{15625 (s+1)^7} + \frac{9}{78125 (s+1)^8} - \frac{2}{390625 (s+1)^9} + \frac{1}{9765625 (s+1)^{10}},$$ from which you can read off a minimal realization pretty quickly.