Page 93 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 81

where the first addendum is a combination of modes proper to GðwÞ, the sec-
ond is a combination of modes proper to UðwÞ, and the third is a combina-
tion of modes proper to the “interaction” or “resonant” component
Y R ðwÞ9X I ðwÞ=IðwÞ. Some of the modes of Y R ðwÞ are not contained in both
GðwÞ and UðwÞ because the multiplicities of the roots of IðwÞ are greater
than the multiplicities of the same roots in AðwÞ and CðwÞ.

As already said, the possibility of decomposing the forced response into a
transient and a steady-state component depends on the system stability. The
next section deals with the problem of checking this fundamental property.

3.4 STABILITY CONDITIONS
Although Routh Hurwitz-like conditions have been derived to determine
how the roots of the characteristic pseudo-polynomial of a fractional order
system are distributed among the LHP and RHP half-planes of its principal
Riemann sheet or its sectors (Liang et al., 2017), no simple rules are as yet
available to establish directly from polynomial AðwÞ whether some of its
roots belong to given sectors of the w-plane (for arc angles different from π),
except for those given in (Ahmed et al., 2006) that deal with very special
cases. Indeed, conditions for all of the roots of a polynomial, or the eigenva-
lues of a matrix, to lie inside a minor LHP sector symmetric with respect to
the real axis were obtained in the 1970s (Gutman, 1979; Anderson et al.,
1974) from properties of Kronecker products of matrices (Graham, 1981)or
rational maps (Gutman and Jury, 1981). However, the same result, i.e., the
confinement of all the roots in the aforementioned minor LHP sector (no
roots in the corresponding major sector), had already been obtained well
before by means of Routh Hurwitz arguments (Usher, 1957; Lu ¨thi, 1942-
43) or could easily have been achieved based on generalizations of the
Routh Hurwitz criteria (Hurwitz, 1895; Routh, 1877) to polynomials with
complex coefficients (Frank, 1946; Billarz,1944). New formulations, exten-
sions, and improvements of similar algebraic conditions, including the analy-
sis of the critical cases and different tabular-form presentations, can be
found in Sivanandam and Sreekala (2012), Chen and Tsai (1993), Benidir
and Picinbono (1991), Agashe (1985), Hwang and Tripathi (1970) and, more
recently, in Bistritz (2013) where numerically very efficient variants are pre-
sented. A different approach has been followed in Kaminski et al. (2015)
where, for q . 1, a test based on regular chains for semi-algebraic sets (Chen
et al., 2013) has been suggested. Here, some simple conditions based on the
direct application of the Routh test to AðwÞ are suggested to check whether
some (not necessarily all) roots of AðwÞ lie in an RHP sector symmetric with
respect to the real axis.