Page 140 - Mathematical Techniques of Fractional Order Systems

128 Mathematical Techniques of Fractional Order Systems

20
k = 3
15
10 k = 2
5
k = 1
Im 0 k = 0
k = –1
–5
–10 k = –2

–15
k = –3
–20
–25 –20 –15 –10 –5 0 5 10 15 20 25
Re
FIGURE 4.7 Ranges of different branches of W k .

Denote the branch index of the Lambert W function whose range
^
includes the complex number 2 c by k i . This branch produces the trivial
λ i
c
2 λ i =λ i 52 c
2ce in (4.109) resulting in the root s 5 1=a which
λ i
value lW ^ k i
is invalid due to our initial assumption. Removing this branch from our
scope, for each eigenvalue λ i of matrix A, an infinite number of roots are
obtained via (4.109) as follows
no
2 λ i 2 c ^
s 5 lW k 2ce λ i=λ i ; kAZ 2 k i ð4:111Þ
ca
Thus it can be said that system (4.17) with the exponential weight func-
α
tion w αðÞ 5 ca , αA 0; 1Š, aAR . 0 , and cAR 2 0fg is stable, if and only if
½

2 λ i 2 c
Re lW k 2ce λ i=λ i , 0 ð4:112Þ
ca
^
no
holds for all iA 1; 2;?; ng and kAZ 2 k i . Note that due to the property
f
lW k zðÞ 5 lW 2k zðÞð Þ which is valid for zAC 2 R 3 0fg (Corless et al., 1996),
the root obtained from (4.111) for λ i 5 l and k 5 k 0 where lAC 2 R 3 0fg is
the complex conjugate of the root obtained from (4.111) with λ i 5 l and
k 52 k 0 . Thus among the complex eigenvalues, assessment of criterion
(4.112) for eigenvalues that are above the real axis suffices to decide on stabil-
ity of the system in practice. For the case c 5 1, a 5 1and the scalar matrix
A 52 4, the characteristic equation roots (4.111) are depicted in Fig. 4.8.
As it can be seen in Fig. 4.8, all of the roots are located on the open left
half plane, indicating that the system is stable in this case. This was expected