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# arl2

SISO model realization by L2 transfer approximation

### Syntax

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(); 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')

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