Page 150 - Mathematical Techniques of Fractional Order Systems

138 Mathematical Techniques of Fractional Order Systems

which can be rewritten as
K
X
Mðς; τÞ 5 pðςÞ 1 q i ðςÞ ð5:16Þ
i51
in the case when τ 5 0.
The advantage of this substitution is to transform the system domain
from the multisheeted Riemann surface into the complex plane. In fact, the
stability region of the original system is not given by the left half-plane, but
by the stability condition given in Matignon (1998)as
π
ArgðςÞ $ α ð5:17Þ
2
where ςAC.
By using this transformation, it is easier to remark that the imaginary
axis in the s-domain is mapped into the following lines
π
ArgðςÞ 56 α ð5:18Þ
2
in the new ς-domain.
The frequency-domain methods provide necessary and sufficient condi-
tions to analyze the stability of a system with or without delays. Despite the
advantages of the frequency-domain analysis, the Eq. (5.13) causes some dif-
ficulties in order to resolve it, this is due to the presence of the exponential
type transcendental terms and the noninteger orders. In addition, the presence
of time-varying delays or uncertainties make the solution more complicated,
which proves that the frequency-domain analysis has some limitations.
In this work, only the time-domain methods are considered, which allows
us to obtain a sufficient condition for the stability of the system (5.12) and
its most general cases. The main objective of the next section, is to improve
the stability analysis of this kind of system by using the Lyapunov theory,
where the Lyapunov Krasovskii functional and the diffusive representation
n
of the fractional integral operator I will be used.

5.4 STABILITY ANALYSIS IN PRESENCE OF TIME-VARYING
DELAYS
In the time domain, the stability analysis of time-varying delay systems are
based mainly on two known theorems, the Lyapunov Krasovskii stability
theorem and the Razumikhin theorem. The key idea of these theorems
consists of choosing the appropriate candidate Lyapunov function or
Lyapunov Krasovskii functional to get sufficient conditions for analyzing
the system (5.21) stability. These conditions are divided into two basic cate-
gories. One of them is independent of the time-delay length, and it is called