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# 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')```