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# 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).```