dt_ility
========

detectability test



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [k, [n [,U [,Sld ] ] ]]=dt_ility(Sl [,tol])




Arguments
~~~~~~~~~

:Sl linear system ( `syslin` list)
: :n dimension of unobservable subspace
: :k dimension of unstable, unobservable subspace ( `k<=n`).
: :U orthogonal matrix
: :Sld linear system ( `syslin` list)
: :tol threshold for controllability test.
:



Description
~~~~~~~~~~~

Detectability test for `sl`, a linear system in state-space
representation. `U` is a basis whose `k` first columns span the
unstable, unobservable subspace of `Sl` (intersection of unobservable
subspace of `(A,C)` and unstable subspace of `A`). Detectability means
`k=0`.

`Sld = (U'*A*U,U'*B,C*U,D)` displays the "detectable part" of
`Sl=(A,B,C,D)`, i.e.


::

    [*,*,*]
    U'*A*U = [0,*,*]
    [0,0,*]
    
    C*U = [0,0,*]


with `(A33,C3)` observable (dimension `nx-n`), `A22` stable (dimension
`n-k`) and `A11` unstable (dimension `k`).



Examples
~~~~~~~~


::

    A=[2,1,1;0,-2,1;0,0,3];
    C=[0,0,1];
    X=`rand`_(3,3);A=`inv`_(X)*A*X;C=C*X;
    W=`syslin`_('c',A,[],C);
    [k,n,U,W1]=dt_ility(W);
    W1("A")
    W1("C")




See Also
~~~~~~~~


+ `contr`_ controllability, controllable subspace, staircase
+ `st_ility`_ stabilizability test
+ `unobs`_ unobservable subspace
+ `stabil`_ stabilization


.. _unobs: unobs.html
.. _stabil: stabil.html
.. _contr: contr.html
.. _st_ility: st_ility.html