function [Q]= qdecmpc(A,B)
%
% Author: Nasser M. Abbasi, Northeastern University
%
% functionality:
%               -1
% contruct the P   , or Q , a constant nonsingular matrix which
% transforms  a noncontrollable state equation into a controllable
% one.
% The algorithm used is that of lemma 3.5 , page 74 of book
% computer aided analysis and design of Linear control systems
% by Jamshidi,Tarokh,Shafai
%
% arguments:
% A: IN, the A matrix of the state equation
% B: IN, the B matrix of the state equation
% Q: OUT, the Q transformation matrix 
%

if(nargin ~= 2)
  error('Number of input arguments must be 2, A,B matrices');
end

[row,col]= size(A);
if(row ~= col)
  error('Matrix A supplied is not a SQUARE matrix');
end

%
% first find the controllabilty matrix
%
rc= ltirc(A,B);
[original_row,original_col]= size(rc);

%
% t is the temp matrix we use to build Q from
%
t(1:original_row,1) = rc(1:original_row,1);

old_rc_col= 2;
new_rc_col= 1;

%
% Now loop over all columns of Rc, select those that are
% Linearly independent with each others, ignore the rest
%
for i=2:original_col
    temp= t;
    temp(1:original_row,new_rc_col+1)= rc(1:original_row,i);
    [row,col]= size(orth(temp));
    if (col==new_rc_col+1)
       t=temp;
       new_rc_col= new_rc_col+1;
    end
end

%
% now see if the matrix t build has rank same as A
% if not, we need to come up with arbitrary columns
% that are linearly independent with those allready
% extracted from Rc from above
%
[arow,acol]= size(A);
[row,col]= size(t);
if(col == acol)
  Q=t;   
else
  dif= acol-col;
  for i=1:dif
      temp=t;
      more=1;
      while (more==1)
            % 
            % The trick i use to find arbitrary column that is L.I. with
            % those allready in Q, is by random generation, using the
            % built in magic function, I keep looping, generating random
            % columns untill i find one that is L.I., then i use it
            %
            temp1= magic(original_row);
            temp(1:original_row,new_rc_col+i)= temp1(1:original_row,1:1);
            [row,col]= size(orth(temp));
            if(col == new_rc_col+i)
               more=0;
               t=temp;
            end
       end
   end
   Q=t;
end