.TH cainv 1 "April 1993" "Scilab Group" "Scilab Function"
.so man1/sci.an 
.SH NAME
cainv -  Dual of abinv
.SH CALLING SEQUENCE
.nf
[Y,ns,nsg,nn,J,k,Z]=cainv(Sl,alfa)
.fi
.SH PARAMETERS
.TP 10
sl
: \fVsyslin\fR list containing the matrices \fV[A,B,C,D]\fR.
.TP
alpha 
: real number or vector (possibly complex, location of closed loop poles)
.TP
Y
: orthogonal matrix of size nx (dim of state space).
.TP
ns,nsg,nn
:
three integers 
.TP
J
: real matrix (output injection)
.TP
k
: integer (normal rank of \fVSl\fR)
.TP
Z
: non-singular linear system (\fVsyslin\fR list)
.SH DESCRIPTION
\fvcainv\fR finds a basis \fVY\fR and output injection \fVJ\fR
such that the abcd matrices of \fVss2ss(Sl,Y,[],J))\fR are: 
.nf
                  [*,*,*,*]             [*]
  (Y'A + J*C)*Y = [0,u,*,*]   B + J*D = [0]
                  [0,0,s,*]             [0]
                  [0,0,0,*]             [0]

            C*Y = [*,*,*,*]          D= [*]
                  [0,0,0,*]             [0]
.fi
where the modes of A22 (u) are unstable, modes of A33 (s) are stable,
pair (A44, C24) is observable, poles of A44 set to \fValfa\fR.
.LP
The \fVns\fR first columns of \fVY\fR span S= smallest (C,A) invariant
subspace which contains Im(B), \fVnsg\fR first columns of \fVY\fR
span Sg, \fVnn\fR first columns of \fVY\fR span \fVN=S+V\fR. 
(\fVns=0\fR iff B(ker(D))=0).
.LP
.nf
// 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
//i.e. H is such that:
// For any W which makes a column compression of [Yp(1:nsg,:);C]
// with Yp=Y' and [Y,ns,nsg,nn,J,k,Z]=cainv(syslin('c',A,G,C));
// [Yp(1:nsg,:);C]*W = [0 | *] one has
// H*W = [0 | *]  (with at least as many columns as above).
.fi
.SH SEE ALSO
abinv, dt_ility