Page 628 - Mathematical Techniques of Fractional Order Systems

Enhanced Fractional Order Chapter | 20 599

by means of fuzzy logic models is proposed that guarantees the feedback
control system stability and that is able to attenuate the effects of additive
noises and estimation errors on the tracking performance to any prescribed
error level via the sliding mode robust tracking design technique (Bourouba
and Ladaci, 2017). However, the important problem of sliding mode techni-
ques from the control perspective is the discontinuity of the control signal
required to obtain robustness (Khettab et al., 2017b; Bensafia et al., 2017).
This destructive phenomenon, so-called chattering may affect control accu-
racy or incur an unwanted wear of a mechanical component.
Various solutions to reduce the chattering have been studied in the litera-
ture (Ge and Ou, 2008; Lin and Balas, 2011; Ho et al., 2009). Comparing
with a similar previous work, an improved synchronization technique is pro-
posed here for a robust sliding mode control of nonlinear systems with frac-
tional order dynamics that is able to eliminate the chattering phenomena for
uncertain systems with unknown parameters’ variation. The main contribu-
tion of this work consists in combining a saturation function with the sliding
mode controller in order to improve the control signal quality by eliminating
the undesirable chattering. The Gru ¨nwald Letnikov numerical approxima-
tion method is used for fractional order differential equation resolution
with improved performance result. Stability analysis is performed for the
proposed control strategy using Lyapunov theory and numerical simulation
results on the synchronization of two fractional order chaotic systems
illustrate the effectiveness of the proposed fractional fuzzy adaptive synchro-
nization strategy (Khettab et al., 2015; Khettab et al., 2017b,c; Boulkroune
et al., 2016a,b)
This chapter is organized as follows: Section 20.2 presents a basic and
brief review on the state of the art for the addressed problem and for frac-
tional calculus, fractional derivatives, and its relation to the approximation
solution. In section 20.3 is an introduction to the basics and a description of
the T S fuzzy systems. Section 20.4 and 20.5 generally propose adaptive
fuzzy robust H N control of uncertain fractional order systems in the pres-
ence of uncertainty and its stability analysis. In section 20.6, application of
the proposed method on fractional order expression chaotic systems is inves-
tigated. Finally, the simulation results and conclusion will be presented in
section 20.7.

20.2 BASIC DEFINITION AND PRELIMINARIES FOR
FRACTIONAL CALCULUS

The fractional calculus is relatively an old topic dating back more than 300
years. It is a generalization of integration and differentiation to noninteger
α
order fundamental operators, denoted by D , where a and t are the limits of
t
a
the operator.