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# cainv

Dual of abinv

### Syntax

[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

- abinv — AB invariant subspace
- dt_ility — detectability test
- ui_observer — unknown input observer

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