arl2 SISO model realization by L2 transfer approximation

## Syntax

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

## 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();
    plot2d([], [v';zeros(80,1)], 2, '051',' ',[1,-0.5,100,1.5])

    [d,n,e] = arl2(v, poly(1,'z','c'),1)
    plot2d([], ldiv(n,d,100), 2, '000')
    [d,n,e] = arl2(v,d,3)
    plot2d([], ldiv(n,d,100), 3, '000')
    [d,n,e] = arl2(v,d,8)
    plot2d([], ldiv(n,d,100), 5, '000')

    [d,n,e] = arl2(v,poly(1,'z','c'),4,'all')
    plot2d([], ldiv(n(1),d(1),100), 10, '000')

v = ones(1,20); plot2d([], [v';zeros(80,1)], 2, '051',' ',[1,-0.5,100,1.5]) [d,n,e] = arl2(v, poly(1,'z','c'),1) plot2d([], ldiv(n,d,100), 2, '000') [d,n,e] = arl2(v,d,3) plot2d([], ldiv(n,d,100), 3, '000') [d,n,e] = arl2(v,d,8) plot2d([], ldiv(n,d,100), 5, '000') [d,n,e] = arl2(v,poly(1,'z','c'),4,'all') plot2d([], ldiv(n(1),d(1),100), 10, '000')

## See also

- ldiv
- imrep2ss
- time_id
- armax
- frep2tf