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Construct an RMFD Representation from Impulse Response

pseries2rmfd.Rd

This (helper) function constructs a right matrix fraction description of a rational matrix from its impulse response. Of course, the impulse response must contain sufficently many lags. The constructed RMFD is in canonical form.

Usage

pseries2rmfd(obj, Hsize = NULL, mu = NULL, tol = sqrt(.Machine$double.eps))

Arguments

obj

pseries object or 3-D array with dimension \((m,n,l+1)\).

Hsize

integer vector c(f,p), number of block rows and block columns of the Hankel matrix which is used to construct the RMFD. If NULL a default choice is made.

mu

integer vector with the right-Kronecker indices. If NULL then the right-Kronecker indices are computed via a QR decomposition of the transpose of the Hankel matrix, see qr.

tol

tolerance parameter, used by qr.

Value

List with components

Xr

rmfd object which contains the RMFD resresentation of the rational matrix (in echelon form).

mu

integer vector with the right-Kronecker indices.

Details

There are a number of restrictions on the dimension \((m,n)\) and number of lags \(l\) of the impulse response, the number of block rows (\(f\)), block columns (\(p\)) of the Hankel matrix and the right-Kronecker indices \(\mu_i\):

We require that:

  • \(m>0\), \(p>0\), \(f>1\),

  • \(l \geq f+p-1\) and

  • \(\nu_i <f\).

If these restrictions are not satisfied an error is thrown.

Examples

# generate a random RMFD object
m = 3
n = 2
p = 1
q = 1
cc = test_polm(dim = c(n,n), degree = p, random = TRUE)
dd = test_polm(dim = c(m,n), degree = q, random = TRUE)
Xr = rmfd(cc,dd)

# compute impulse response of this matrix
Xi = pseries(Xr, lag.max = 2*(max(p,q)+1))

# reconstruct a matrix from this impulse response
out = pseries2rmfd(Xi)
out
#> $Xr
#> ( 3 x 2 ) right matrix fraction description d(z) c^(-1)(z) with degrees deg(c(z)) = p = 1, deg(d(z)) = q = 1
#> left factor d(z):
#>         z^0 [,1]        [,2]   z^1 [,1]       [,2]
#> [1,]  1.23437457  0.07761702 -0.7171007  0.4754158
#> [2,]  0.01355616 -1.16203156  1.1470586 -1.3252045
#> [3,] -0.69546910 -0.48787929 -3.4156439 -0.2642928
#> right factor c(z):
#>      z^0 [,1]  [,2]    z^1 [,1]        [,2]
#> [1,]        1     0  0.03204006  0.51470020
#> [2,]        0     1 -0.21188798 -0.07764691
#> 
#> $mu
#> [1] 1 1
#> 

# check that the lmfd object is a realization of the given impulse response
Xi1 = pseries(out$Xr, lag.max = 2*(max(p,q)+1))
all.equal(Xi, Xi1)
#> [1] TRUE