Constructor for Left Matrix Fraction Descriptions (LMFDs)

A Left Matrix Fraction Description (LMFD) of a rational matrix, \(x(z)\) say, is a pair \((a(z),b(z))\) of polynomial matrices, such that \(x(z) = a^{-1}(z) b(z)\). The polynomial matrix \(a(z)\) must be square and invertible.

Usage

lmfd(a, b)

Arguments

a, b

polm objects, or objects which may be coerced to a polm object, via x = polm(x). Either of the two arguments may be omitted.

Value

An object of class lmfd.

Details

Suppose that \(x(z)=a^{-1}(z) b(z)\) is an \((m,n)\)-dimensional matrix and that \(a(z)\) and \(b(z)\) have degrees \(p\) and \(q\) respectively. The corresponding lmfd object stores the coefficients of the polynomials \(a(z), b(z)\) in an \((m,m(p+1)+n(q+1))\) dimensional (real or complex valued) matrix together with an attribute order = c(m,n,p,q) and a class attribute c("lmfd", "ratm").

For a valid LMFD we require \(m>0\) and \(p\geq 0\).

See also

Useful methods and functions for the lmfd class are:

• test_lmfd generates random rational matrices in LMFD form.

• checks: is.lmfd, is.miniphase, is.stable and is.coprime.

• generic S3 methods: dim, str and print.

• arithmetics: Ops.ratm.

• matrix operations: bind.

• extract the factors \(a(z)\) and \(b(z)\) with $.lmfd.

• zeroes, pseries, zvalues, ...

Examples

### (1 x 1) rational matrix x(z) = (1+z+z^2)^(-1) (3+2z+z^2)
lmfd(c(1,1,1), c(3,2,1)) %>% print(format = 'c')
#> ( 1 x 1 ) left matrix fraction description a^(-1)(z) b(z) with degrees (p = 2, q = 2)
#> left factor a(z):
#>              [,1]
#> [1,]  1 + z + z^2 
#> right factor b(z):
#>               [,1]
#> [1,]  3 + 2z + z^2 

### (1 x 1) rational matrix x(z) = (3+2z+z^2)
lmfd(b = c(3,2,1)) %>% print(format = 'c')
#> ( 1 x 1 ) left matrix fraction description a^(-1)(z) b(z) with degrees (p = 0, q = 2)
#> left factor a(z):
#>       [,1]
#> [1,]     1 
#> right factor b(z):
#>               [,1]
#> [1,]  3 + 2z + z^2 

### (1 x 1) rational matrix x(z) = (1+z+z^2)^(-1)
lmfd(c(1,1,1)) %>% print(format = 'c')
#> ( 1 x 1 ) left matrix fraction description a^(-1)(z) b(z) with degrees (p = 2, q = 0)
#> left factor a(z):
#>              [,1]
#> [1,]  1 + z + z^2 
#> right factor b(z):
#>       [,1]
#> [1,]     1 

### (2 x 3) rational matrix with degrees p=1, q=1
x = lmfd(array(rnorm(2*2*2), dim = c(2,2,2)), 
         array(rnorm(2*3*2), dim = c(2,3,2)))
is.lmfd(x)
#> [1] TRUE
dim(x)
#> m n p q 
#> 2 3 1 1 
str(x)
#> ( 2 x 3 ) left matrix fraction description with degrees (p = 1, q = 1)
print(x, digits = 2)
#> ( 2 x 3 ) left matrix fraction description a^(-1)(z) b(z) with degrees (p = 1, q = 1)
#> left factor a(z):
#>      z^0 [,1]  [,2] z^1 [,1]  [,2]
#> [1,]     0.72  0.43     1.12 -0.15
#> [2,]    -0.93  1.65     0.48  0.31
#> right factor b(z):
#>      z^0 [,1]  [,2]  [,3] z^1 [,1]  [,2]  [,3]
#> [1,]     1.54  1.04 -0.03    -0.71  2.25  1.81
#> [2,]    -1.80 -0.22 -1.34     0.29  0.35  1.39

if (FALSE) {
### the following calls to lmfd() throw an error 
lmfd() # no arguments!
lmfd(a = test_polm(dim = c(2,3), degree = 1))  # a(z) must be square 
lmfd(a = test_polm(dim = c(2,2), degree = -1)) # a(z) must have degree >= 0
}