The left and right matrix-fraction description (MFD) of a multivariable system, often also called the linear-fraction transformation (LFT), provides an alternative form of model description to the transfer-function matrix that can be considered as a natural generalization of the classical single-input, single-output (SISO) system transfer-function description, namely g(s) = n(s) d-1(s) = d-1(s) n(s).
Assume that G(s) is a strictly proper, or proper, q × p matrix, then G(s) can be expressed as
where DR(s) is a p × p diagonal matrix containing the least common denominators of each of the p columns of G(s), and NR(s) is a q × p matrix containing the resulting numerator terms. This is known as a right matrix-fraction form of G(s), since the matrix of denominator terms is on the right.
The transfer-function matrix G(s) can equally be expressed as
where DL(s) is a q × q diagonal matrix containing the least common denominators of each of the q rows of G(s), and NL(s) is a q × p matrix containing the corresponding numerator terms. This is known as a left matrix-fraction form of G(s), since the matrix of denominator terms is on the left.
MFDs are not in general unique, as there exist many left and right matrix-fraction descriptions corresponding to a given transfer-function matrix G(s). For example, for the transfer-function matrix model
a left MFD of G(s) can easily be written as
However, a simpler left MFD of G(s) can also be determined as
where the numerator and denominator matrices of this latter model are left coprime (see Section 4.3).
