Page 94 - Mathematical Techniques of Fractional Order Systems

82 Mathematical Techniques of Fractional Order Systems

To this purpose, consider the “forbidden” RHP sector defined by (3.10)
(similar considerations hold, of course, for the opposite sector). If no roots
of AðwÞ are in the RHP, which may be checked by means of the standard
Routh test, then this sector, as well as the two sectors containing the points
of the RHP which do not belong to S, do not contain any root, either.
Therefore, the method in (Usher, 1957) can be adopted to determine the
number of roots inside any LHP sector symmetric with respect to the nega-
tive real semi-axis.
Also, since the number of the real roots in any interval of the real axis
can be found easily on the basis of the classic Sturm algorithm (see, e.g., the
lecture notes in Jia, 2016), whose computational complexity is not greater
2
than that of the Routh algorithm (i.e., Oðn Þ), only the root distribution of
the roots with a nonzero imaginary part need actually be determined.
Therefore, for notational simplicity and without loss of generality, the fol-
lowing assumption is made.

Assumption 2: The real polynomial AðwÞ has no real roots.

It follows that the degree n of AðwÞ is even because its complex roots
with nonzero imaginary part are in conjugate pairs (In other words, AðwÞ can
be though of as the even-degree factor containing all of the complex conju-
~
gate roots of an original polynomial AðwÞ 5 AðwÞA r ðwÞ, where A r ðwÞ is the
~
factor containing all of the real roots of AðwÞ. For simplicity, the following
analysis refers to AðwÞ only. The results, however, can easily be extended to
include the real roots).
By combining the information on the root distribution with respect to the
imaginary axis with the information on the root distributions with respect to
each of the two slanted straight lines with slope 6 π=2q (see again (3.10)),
some interesting results can be established straightaway. To state them in a
compact form, the following notation, illustrated in Fig. 3.1, is introduced:
1. n 1u , n 1l denote the number of roots above and, respectively, below the
slanted line through the origin with positive slope π=2q;
2. n 2u , n 2l denote the number of roots above and, respectively, below the
slanted line through the origin with negative slope 2π=2q; and
3. n 1 , n 2 denote the number of roots in the closed RHP and in the open
LHP, respectively;
4. the difference between the number of roots in S (unstable roots) and the
LHP sector (symmetric of S) is denoted by δ.
Clearly,
δ 5 n 2u 2 n 1u 5 n 1l 2 n 2l 5 n 2u 2 n 2l 5 n 1l 2 n 1u : ð3:18Þ

The following result is obvious.