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