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cainv — Dual of abinv
[X,dims,J,Y,k,Z]=cainv(Sl,alfa,beta,flag)
| Sl | : syslin list containing the matrices [A,B,C,D]. |
| alfa | : real number or vector (possibly complex, location of closed loop poles) |
| beta | : real number or vector (possibly complex, location of closed loop poles) |
| flag | : (optional) character string 'ge' (default) or 'st' or 'pp' |
| X | : orthogonal matrix of size nx (dim of state space). |
| dims | : integer row vector dims=[nd1,nu1,dimS,dimSg,dimN] (5 entries, nondecreasing order).If flag='st', (resp. 'pp'), dims has 4 (resp. 3) components. |
| J | : real matrix (output injection) |
| Y | : orthogonal matrix of size ny (dim of output space). |
| k | : integer (normal rank of Sl) |
| Z | : non-singular linear system (syslin list) |
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]
[0,0,0,0,A55,*] [0]
[0,0,0,0,0,A66] [0]
Y*C*X = [0,0,C13,*,*,*] Y*D = [*]
[0,0,0,0,0,C26] [0]
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).
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