Page 175 - Mathematical Techniques of Fractional Order Systems

Fractional Order Error Models With Parameter Constraints Chapter | 6 163

Some active researchers in the field have stated that the Caputo definition
does not allow to take into account initial conditions properly, and for that
reason they state that it should not be used to model fractional systems.
However, still there is no consensus about this topic, since there are other
works where real systems have been modeled using fractional differential
equations with the Caputo’s definition (see for instance Freed and Diethelm,
2007; Tejado et al., 2014). Thus, it can be stated that there are systems
described using this kind of fractional differential equations, and for that rea-
son as control engineers it is worth to study this kind of systems, as well as
proposing control strategies for them. In the sequel, Caputo’s FOD will be
used in this work.

6.2.2 Stability of Fractional Order Systems
The known stability methods for integer order systems (IOS) are not
directly applicable to the stability analysis of fractional order systems
(FOS). However, many of them have been generalized to the FOS case.
The conditions under which linear time-invariant FOS are stable were stud-
ied by Matignon (1994). However, in the case of fractional order adaptive
systems (FOAS) this analysis is not valid, since they are nonlinear and
time-varying.
Lyapunov’s direct method provides an effective way of analyzing the sta-
bility of nonlinear and time-varying IOS. The FO extension of Lyapunov
direct method was proposed by Li et al. (2010). This method allows conclud-
ing asymptotic stability and Mittag Leffler stability for FOS. However, it
doesn’t address the frequent case of adaptive schemes where the FOD of the
Lyapunov function is only negative semidefinite, and conclusions about sta-
bility or uniform stability can be drawn. In that case, Theorem 1 is useful,
and it will be used in this work.

Theorem 1: (Duarte-Mermoud et al., 2015). Let x 5 0 be an equilibrium
point for the nonautonomous FOS (6.3).

C a
D xðtÞ 5 fðxðtÞ; tÞ; αAð0; 1Þ ð6:3Þ
t 0
It is assumed that there exists a continuous function VðxðtÞ; tÞ such that
(Duarte-Mermoud et al., 2015, Thm. 3 & Rmk. 1),
VðxðtÞ; tÞ is positive definite, and
C β
D t 0 VðxðtÞ; tÞ; with βAð0; 1Š, is negative semidefinite.
Then the origin of system (6.3) is Lyapunov stable.
Furthermore, if VðxðtÞ; tÞ is decrescent, then the origin of system (6.3) is
Lyapunov uniformly stable.