Page 146 - Mathematical Techniques of Fractional Order Systems

134 Mathematical Techniques of Fractional Order Systems

Masoud, 2009; Mihailo and Aleksandar, 2009; Si-Ammour et al., 2009;
Liqiong and Shouming, 2010; Lianglin et al., 2011; Boukal et al., 2014a,b,
2015a, 2017a,c).
The main goal of the feedback controller design is to stabilize
unstable systems (Boyd et al., 1994; Ahmad et al., 2004; Amato et al., 2007;
Ahn et al., 2007; Amato et al., 2009, 2010; Boukal et al., 2015b, 2016b,
2017b) or to improve the stability in the presence of transient phenomena
which do not die out quickly. From a practical point of view, it is very
important to be able to analyze the stability of dynamic FOS. The fundamen-
tals of the rigorous mathematical theory of stability were laid down
in the works of the prominent Russian mathematician A.M. Lyapunov
100 years ago.
Recently Trigeassou et al. (2011) proposed the application of Lyapunov’s
method to linear and nonlinear fractional derivative equations (FDEs) by
choosing a specific Lyapunov function candidate. Their approach is based on
the concept of diffusive representation of the fractional integration operator.
In this work, the fractional order time-varying delay systems stability
analysis is studied with the help of the diffusive representation of the frac-
tional order derivative and Lyapunov theory. Moreover, new sufficient
delay-independent and delay-dependent stability criteria are derived in a lin-
ear matrix inequality (LMI) formulation, which can be easily solved. In addi-
tion, based on this result, sufficient conditions of the existence of stabilizing
pseudo-state feedback controllers for an unstable plant can be checked. The
controller gain matrix can be computed by solving one of the obtained matri-
ces inequalities. An unstable FOS-TVD (time-varying delay) and
stable FOS-TVD for a bounded delay are studied as numerical examples to
illustrate the effectiveness of both the proposed stability analysis method and
the controller synthesis.
The chapter will be organized as follows. In Section 5.2, some prelimi-
naries on the fractional order derivative and some useful inequalities which
will be used later are provided. In Section 5.3, the problem that motivated
the present work is described. Section 5.4 is dedicated to the stability analy-
sis of FOS in presence of time-varying delays by using a Lyapunov function
candidate based on the concept of diffusive representation of the fractional
integration operator. Using the stability conditions formulated in terms of
LMIs in Section 5.4, pseudo-state-feedback controllers are derived in
Section 5.5. Numerical simulation results demonstrate the effectiveness of
the proposed method are shown in Section 5.6. Finally, concluding remarks
on the presented results and open problems are given in Section 5.7.
Notation: In the sequel of the chapter, the following notations are used. R n
and R n 3 m to denote the n dimensional Euclidean space and the set of all
T
n 3 m real matrices, respectively; A denotes the transpose of a matrix A;
T
matrix A is symmetric positive definite if and only if A 5 A and A . 0.