arl2

Scilab Function
Last update : 12/01/2004

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')`

### Parameters

- `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);
    xbasc();
    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` , `imrep2ss` , `time_id` , `armax` , `frep2tf` ,