Page 91 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 79

the presence of resonance phenomena where an input frequency coincides
with a natural frequency of the system. For the sake of simplicity, in this
chapter the following assumption is made. (Indications on the extension of
the following procedure to the general case are given in Remark 1 at the end
of this section.)

Assumption 1: Polynomials AðwÞ,BðwÞ,CðwÞ, and DðwÞ have no common
factors.
Therefore the (strictly-proper) representations (3.4) and (3.9) are irreduc-
ible and no cancellation occurs in (3.9); in particular, resonance phenomena
are not possible. Under Assumption 1, (3.9) can uniquely be decomposed as
X A ðwÞ X C ðwÞ
YðwÞ 5 1 ð3:11Þ
AðwÞ CðwÞ
where X A ðwÞ and X C ðwÞ are the solutions of the polynomial Diophantine
equation (Ferrante et al., 2000; Kuˇ cera, 1993)
X A ðwÞCðwÞ 1 X C ðwÞAðwÞ 5 BðwÞDðwÞ ð3:12Þ
with deg½X A ðwފ , deg½Aðwފ 5 n, deg½X C ðwފ , deg½Cðwފ 5 k and, by the
strict properness of GðwÞ and UðwÞ, deg½BðwÞDðwފ , deg½Aðwފ 1 deg½Cðwފ.
In this case, in fact, equation (3.12) is equivalent to a set of n 1 k linear
equations in the n 1 k unknown coefficients x i and y i of polynomials:

X A ðwÞ 5 x A;n21 w n21 1 x A;n22 w n22 1 ... 1 x A;1 w 1 x A;0 ; ð3:13Þ

X C ðwÞ 5 x C;n21 w k21 1 x C;n22 w k22 1 ... 1 x C;1 w 1 x C;0 ; ð3:14Þ
obtained by equating the coefficients of the equal powers of w on both sides
of (3.12). By properly ordering the unknowns, this set can be written in a
matrix form where the coefficient matrix is nonsingular (Antsaklis and
Michel, 2006; Henrion, 1998) (being the Sylvester matrix associated with the
polynomials AðwÞ and CðwÞ that are co-prime by Assumption 1). It follows
from Cramer’s rule (Brunetti, 2014) that the aforementioned set of equations
admits one, and only one, solution. For clarity of exposition, this result is
restated next in the form of a proposition.

If AðwÞ and CðwÞ are co-prime and if
Proposition 1:
deg½BðwÞDðwފ , deg½Aðwފ 1 deg½Cðwފ; there is a unique pair of polyno-
mials X A ðwÞ and X C ðwÞ with deg½X A ðwފ , deg½Aðwފ and deg½X C ðwފ ,
deg½Cðwފ that solves the polynomial Diophantine equation (3.12).
The two addenda in (3.11) will be denoted by

X A ðwÞ X C ðwÞ
Y Σ ðwÞ 5 ; Y U ðwÞ 5 ð3:15Þ
AðwÞ CðwÞ