projsl

Scilab Function
Last update : 12/01/2004

projsl - linear system projection

### Calling Sequence

  `[slp]=projsl(sl,Q,M)`

### Parameters

- `sl,slp`: `syslin` lists
- `Q,M`: matrices (projection factorization)

### Description

`slp`= projected model of `sl` where `Q*M` is the full rank factorization of the projection.

If `(A,B,C,D)` is the representation of `sl`, the projected model is given by `(M*A*Q,M*B,C*Q,D)`.

Usually, the projection `Q*M` is obtained as the spectral projection of an appropriate auxiliary matrix `W` e.g. `W` = product of (weighted) gramians or product of Riccati equations.

### Examples

    rand('seed',0);sl=ssrand(2,2,5);[A,B,C,D]=abcd(sl);poles=spec(A)
    [Q,M]=pbig(A,0,'c');  //keeping unstable poles
    slred=projsl(sl,Q,M);spec(slred('A'))
    sl('D')=rand(2,2);  //making proper system
    trzeros(sl)  //zeros of sl
    wi=inv(sl);  //wi=inverse in state-space
    [q,m]=psmall(wi('A'),2,'d');  //keeping small zeros (poles of wi) i.e. abs(z)<2
    slred2=projsl(sl,q,m);
    trzeros(slred2)  //zeros of slred2 = small zeros of sl
    //  Example keeping second order modes
    A=diag([-1,-2,-3]);
    sl=syslin('c',A,rand(3,2),rand(2,3));[nk2,W]=hankelsv(sl)
    [Q,M]=pbig(W,nk2(2)-%eps,'c');    //keeping 2 eigenvalues of W
    slr=projsl(sl,Q,M);  //reduced model
    hankelsv(slr)

### See Also

`pbig`,

### Author

F. D.;