Transfer function of higher order

Dear all,

I need to input a transfer function of the form shown below:

    G(s) = 1/{((s+1)^10)*((s+3)^10)}

I have two questions:

1)- I am using conv command in a for loop to generate (s+1)^10, the most interesting thing that i observe that the roots of the polynomial that i obtain is NOT 1 with algebraic multiplicity of 10. Even if you go for (s+1)^3 or (s+1)^4, the roots won't be three times or four times 1 but somewhere near to 1 (not exactly 1). WHY IT IS SO?

2)- Surprisingly, the system G, when converted to ss form using tf2ss, the state-space realization is not minimal. BUT how on earth this is possible? there is no zero in G, where the pole zero cancellation is occurring so that the G is not controllable or observable?

Thanks in advance.

[Azzi Abdelmalek]

when matlab resolves roots of your polynom, it uses a numerical method, that means there will be some errors calculations. so, where is the problem? i checked that the system is not controllable, i think it's the representation tf2ss is not adequate.

Answer 1

[Azzi Abdelmalek]

% to generate use poly without loop instead conv:

N=1   % numerator 
D=poly([ones(1,10)*(-1) ones(1,10)*(-3)])  % -1 and -3 are your poles
[A,b,c,d]=tfn2ss(N,D)   % my function to find another ss representation (see below)
%then check your controllabilité

% the function ft2re

   function [F,h,c,d]=tfn2ss(N,D)
   if nargin==1;D=N;N=1;end
   D1=D(2:end);n=length(D1);m=length(N);
   if m==n;N1=N;else N1=[zeros(1,n-m) N];end
   F=[-D1' [eye(n-1) ; zeros(1,n-1)]];h=N1'; c=[1 zeros(1,n-1)];d=0;

 % for more details about this function

http://www.mathworks.com/matlabcentral/fileexchange/37697-transfert-function-to-state-space-representation

[M.Mohsin Siraj]

Thanks for your answer. But still the state-space realization is not minimal. The observablity matrix loses rank. Which i couldn't understand why.

[Azzi Abdelmalek]

you are right, the controllabilite is ok; but not the obs. the problem is numeric

Answer 2

[Azzi Abdelmalek]

%we now both that your system is observable and controllable , then the problem is numeric, i suggest that you multiply your denominator by 0.0001

 D=D*0.0001

% after when you calculate your observator or controller, you have to take into account that your output signal is multiplied by (1/0.0001)

% i advise you also to use my function tfn2ss, i tried it and it works. it did'nt work with tf2ss (it just works for controllability)

[M.Mohsin Siraj]

Thanks for your answer again. But D=D*0.0001 will make the system unstable. Basically, what i am trying to do is to convert my state-space realization to balanced realization. For that i need a minimal realization such that i can transform it to balanced state. So i start with a transfer function and want to get a balanced state-space realization. I first convert transfer function to ordinary state-space and then transform the ordinary state-space to balanced states. I hope it clears the situation i am facing. Thanks

[Azzi Abdelmalek]

what I suggested is not to replace D with 0.0001 * D when you simulate your system. It's just to calculate your observator, once the calcus finished, you change just the loi of your observator or controller.

[M.Mohsin Siraj]

Thanks, i got your point. But i am not designing an observer here... I want to have a balanced realization of a stable transfer function. Is such kind of problems very common in MATLAB? I mean i haven't worked on higher order systems. So for example, above 30 or 40 order does it start making problems?