spantwo - sum and intersection of subspaces
Given two matrices A and B with same number of rows, returns a square matrix Xp (non singular but not necessarily orthogonal) such that :
[A1, 0] (dim-dimb rows)
Xp*[A,B]=[A2,B2] (dima+dimb-dim rows)
[0, B3] (dim-dima rows)
[0 , 0]
The first dima columns of inv(Xp) span range(A).
Columns dim-dimb+1 to dima of inv(Xp) span the intersection of range(A) and range(B).
The dim first columns of inv(Xp) span range(A)+range(B).
Columns dim-dimb+1 to dim of inv(Xp) span range(B).
Matrix [A1;A2] has full row rank (=rank(A)). Matrix [B2;B3] has full row rank (=rank(B)). Matrix [A2,B2] has full row rank (=rank(A inter B)). Matrix [A1,0;A2,B2;0,B3] has full row rank (=rank(A+B)).
A=[1,0,0,4;
5,6,7,8;
0,0,11,12;
0,0,0,16];
B=[1,2,0,0]';C=[4,0,0,1];
Sl=ss2ss(syslin('c',A,B,C),rand(A));
[no,X]=contr(Sl('A'),Sl('B'));CO=X(:,1:no); //Controllable part
[uo,Y]=unobs(Sl('A'),Sl('C'));UO=Y(:,1:uo); //Unobservable part
[Xp,dimc,dimu,dim]=spantwo(CO,UO); //Kalman decomposition
Slcan=ss2ss(Sl,inv(Xp));