Page 96 - Mathematical Techniques of Fractional Order Systems

84 Mathematical Techniques of Fractional Order Systems

monotonically to zero as the sectors angle tends to zero, then the system is
stable. A similar procedure can be applied to detect roots with damping fac-
tor in a given range.
By simple adaptation to fractional order systems of the classic Mikhailov
stability criterion for integer order systems (Busłowicz, 2008; Mikhailov,
1938), the following graphically-based criterion also holds.

Proposition 5: The fractional order system is stable if and only if the phase
variation of the nth degree polynomial
B j π
AðρÞ9A ρ e 2q ð3:19Þ
π
as ρ varies from 0 to 1N is equal to n :
2q
B π
Δarg½Aðρފ 5 n : ð3:20Þ
2q
½0;NÞ
j π
Proof. Express A ρ e 2q in factored form as
π
j π n j
A ρ e 2q 5 a n L ρ e 2 p i ; ð3:21Þ
2q
i51
where the p i , i 5 1; 2; .. .; n, are the (possibly repeated) roots of AðwÞ. The
j π
phase variation of (3.21) as point ρe 2q moves along the slanted half-line
leaving the origin and making an angle π=ð2qÞ with the positive real axis is
the sum of the phase variations of its factors which can be regarded as vec-
π
j
tors applied at the p i s and pointing to ρe . Now, if a real root, say p k ,is
2q
outside the instability sector and thus negative, the initial phase of the corre-
j π
sponding factor (when ρ is equal to zero and the point ρe 2q coincides with
π
j
the origin) is zero and its final phase (when ρe 2q tends to infinity along the
aforementioned slanted half-line) is π=ð2qÞ.If a complex root, say p h , is out-
side the instability sector, consider the two factors associated with the pair of
%
conjugate poles p h and p . The sum of the initial phases of the two vectors
h
π
π
j
j
ρe 2 p h and ρe 2 p % is also zero, while the final sum of their phases is
2q
2q
h

FIGURE 3.2 Vector representation of the phase variation of the factors in (3.20) associated
with either real or complex roots outside or inside the RHP minor sector with central angle π=q
straddling the positive real axis.