cainv - Dual of abinv
cainv finds a bases (X,Y) (of state space and output space resp.) and output injection matrix J such that the matrices of Sl in bases (X,Y) are displayed as:
[A11,*,*,*,*,*] [*] [0,A22,*,*,*,*] [*] X'*(A+J*C)*X = [0,0,A33,*,*,*] X'*(B+J*D) = [*] [0,0,0,A44,*,*]  [0,0,0,0,A55,*]  [0,0,0,0,0,A66]  Y*C*X = [0,0,C13,*,*,*] Y*D = [*] [0,0,0,0,0,C26] 
The partition of X is defined by the vector dims=[nd1,nu1,dimS,dimSg,dimN] and the partition of Y is determined by k.
Eigenvalues of A11 (nd1 x nd1) are unstable. Eigenvalues of A22 (nu1-nd1 x nu1-nd1) are stable.
The pair (A33, C13) (dimS-nu1 x dimS-nu1, k x dimS-nu1) is observable, and eigenvalues of A33 are set to alfa.
Matrix A44 (dimSg-dimS x dimSg-dimS) is unstable. Matrix A55 (dimN-dimSg,dimN-dimSg) is stable
The pair (A66,C26) (nx-dimN x nx-dimN) is observable, and eigenvalues of A66 set to beta.
The dimS first columns of X span S= smallest (C,A) invariant subspace which contains Im(B), dimSg first columns of X span Sg the maximal "complementary detectability subspace" of Sl
The dimN first columns of X span the maximal "complementary observability subspace" of Sl. (dimS=0 if B(ker(D))=0).
If flag='st' is given, a five blocks partition of the matrices is returned and dims has four components. If flag='pp' is given a four blocks partition is returned (see abinv).
This function can be used to calculate an unknown input observer:
// DDEP: dot(x)=A x + Bu + Gd // y= Cx (observation) // z= Hx (z=variable to be estimated, d=disturbance) // Find: dot(w) = Fw + Ey + Ru such that // zhat = Mw + Ny // z-Hx goes to zero at infinity // Solution exists iff Ker H contains Sg(A,C,G) inter KerC (assuming detectability) //i.e. H is such that: // For any W which makes a column compression of [Xp(1:dimSg,:);C] // with Xp=X' and [X,dims,J,Y,k,Z]=cainv(syslin('c',A,G,C)); // [Xp(1:dimSg,:);C]*W = [0 | *] one has // H*W = [0 | *] (with at least as many aero columns as above).