arl2
====

SISO model realization by L2 transfer approximation



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


::

    h=arl2(y,den0,n [,imp])
    h=arl2(y,den0,n [,imp],'all')
    [den,num,err]=arl2(y,den0,n [,imp])
    [den,num,err]=arl2(y,den0,n [,imp],'all')




Arguments
~~~~~~~~~

:y real vector or polynomial in `z^-1`, it contains the coefficients
  of the Fourier's series of the rational system to approximate (the
  impulse response)
: :den0 a polynomial which gives an initial guess of the solution, it
  may be `poly(1,'z','c')`
: :n integer, the degree of approximating transfer function (degree of
  den)
: :imp integer in `(0,1,2)` (verbose mode)
: :h transfer function `num/den` or transfer matrix (column vector)
  when flag `'all'` is given.
: :den polynomial or vector of polynomials, contains the
  denominator(s) of the solution(s)
: :num polynomial or vector of polynomials, contains the numerator(s)
  of the solution(s)
: :err real constant or vector , the l2-error achieved for each
  solutions
:



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

`[den,num,err]=arl2(y,den0,n [,imp])` finds a pair of polynomials
`num` and `den` such that the transfer function `num/den` is stable
and its impulse response approximates (with a minimal l2 norm) the
vector `y` assumed to be completed by an infinite number of zeros.

If `y(z) = y(1)(1/z)+y(2)(1/z^2)+ ...+ y(ny)(1/z^ny)`

then l2-norm of `num/den - y(z)` is `err`.

`n` is the degree of the polynomial `den`.

The `num/den` transfer function is a L2 approximant of the Fourier's
series of the rational system.

Various intermediate results are printed according to `imp`.

`[den,num,err]=arl2(y,den0,n [,imp],'all')` returns in the vectors of
polynomials `num` and `den` a set of local optimums for the problem.
The solutions are sorted with increasing errors `err`. In this case
`den0` is already assumed to be `poly(1,'z','c')`



Examples
~~~~~~~~


::

    v=`ones`_(1,20);
    `clf`_();
    `plot2d1`_('enn',0,[v';`zeros`_(80,1)],2,'051',' ',[1,-0.5,100,1.5])
    
    [d,n,e]=arl2(v,`poly`_(1,'z','c'),1)
    `plot2d1`_('enn',0,`ldiv`_(n,d,100),2,'000')
    [d,n,e]=arl2(v,d,3)
    `plot2d1`_('enn',0,`ldiv`_(n,d,100),3,'000')
    [d,n,e]=arl2(v,d,8)
    `plot2d1`_('enn',0,`ldiv`_(n,d,100),5,'000')
    
    [d,n,e]=arl2(v,`poly`_(1,'z','c'),4,'all')
    `plot2d1`_('enn',0,`ldiv`_(n(1),d(1),100),10,'000')




See Also
~~~~~~~~


+ `ldiv`_ polynomial matrix long division
+ `imrep2ss`_ state-space realization of an impulse response
+ `time_id`_ SISO least square identification
+ `armax`_ armax identification
+ `frep2tf`_ transfer function realization from frequency response


.. _ldiv: ldiv.html
.. _imrep2ss: imrep2ss.html
.. _frep2tf: frep2tf.html
.. _time_id: time_id.html
.. _armax: armax.html