Page 95 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 83

FIGURE 3.1 Notation for: (i) the number of roots in each of the two half-planes separated by each
of the two slanted straight lines through the origin with opposite slope (n 1u , n 1l and n 2u , n 2l ), and (ii)
the number of roots in each of the two half-planes separated by the vertical axis (n 1 and n 2 ).

Proposition 2: If δ . 0 the fractional order system is unstable.
When δ # 0, the system may be stable, but the stability conditions also
depend on n, n 1 , and n 2 5 n 2 n 1 . By considering that, in the absence of
real roots, all of the above numbers are even, the following two stability con-
ditions can easily be proved. The first only requires the knowledge of n.

Proposition 3: If δ 52 i, i nonnegative, and n , i 1 4, the fractional order
system is stable.
Proof. Assume that the fractional order system is unstable. Since all roots
appear in conjugate pairs, the number of roots in the instability sector is 2 or
more. Therefore, in order for δ 52 i, the sector opposite to the instability
sector must contain at least i 1 2 roots, and the polynomial degree n, which
is greater than, or equal to, the sum of the roots in both sectors, must be
equal, at least, to i 1 4, contrary to the assumption that n , i 1 4.

Proposition 4: If δ 52 i, i nonnegative, n 2 5 i 1 j, j nonnegative, and
n , i 1 j 1 2, the fractional order system is stable.
Proof. It is enough to consider that, under the adopted assumptions,
n 2 n 2 , 2 so that no root may lie in the instability sector.
More general stability conditions require the acquisition of additional
information, which in some cases may be worthwhile. For instance, to deter-
mine whether some roots lie inside the instability sector S, the slope of the
π
two slanted lines (i.e., the angle ) can gradually be taken to zero. If the dif-
2q
ference between the numbers of roots in the LHP and RHP sectors decreases