cainv

Dual of abinv

Calling Sequence

[X,dims,J,Y,k,Z]=cainv(Sl,alfa,beta,flag)

Arguments

: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) :

Description

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).

See Also

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