Page 173 - Mathematical Techniques of Fractional Order Systems

Fractional Order Error Models With Parameter Constraints Chapter | 6 161

On the theoretical side, several new results have been recently published
on the FO controllers and observers in the book (Azar et al., 2017), where
several methods for FO control are treated in Part I: Fractional Order Control
Systems. Also, in Part II: Applications of Fractional Order Chaotic Systems
of Azar et al. (2017), numerous applications on control and synchronization
of chaotic systems are discussed. Other recent analytical developments worth
to mention are Merrikh-Bayat (2013), where the FO unstable pole-zero can-
cellation problem in linear feedback systems is presented; Ladaci and
Bensafia (2016), in which the indirect adaptive control using FO pole assign-
ment is discussed; Rapaic and Pisano (2014), where the case of variable deri-
vation order is analyzed through an estimation process; Liu et al. (2017), in
which an interesting adaptive fuzzy backstepping control technique of FO
nonlinear systems is presented; and Wei et al. (2017), where a discussion on
FO adaptive observers is presented, just to mention a few of them.
The next step in this sequence of studying FOEM is to study two EM
whose true and unknown parameters are coupled by a linear relationship, but
the AL for updating the unknown parameters are done using FOAL. This sit-
uation occurs when a FO plant is being controlled by an adaptive control
strategy of FO type (i.e., FOAL) and simultaneously, the AL for estimating
plant parameters are of FO type. This problem was already solved for the
case of the fractional order EM (FOEM) of type 1 (FOEM1) in Aguila-
Camacho and Duarte-Mermoud (2015, 2017). In this Chapter, the cases of
the so-called FOEM2 and FOEM3 when the true parameters of both FOEM
are coupled are treated in detail.
FOEM2 defines a special case of FOEM, where the whole state error vec-
tor is accessible to the designer. The analytical results presented in this chap-
ter show that it is possible to find coupled fractional AL, such that the
overall adaptive system is globally stable, when the FO is in the interval
αA 0; 1Š. Also, it is analytically proved that the mean value of the squared
ð
norm of the state error vector converges asymptotically to zero. The same
analysis is performed for the FOEM3, where the main difference with
FOEM2 lies in the fact that only one component of the state error vector is
available (output error), proving the same properties as in the FOEM2.
The presentation is organized as follows. After the introductory
Section 6.1, some preliminary results used in this study are presented in
Section 6.2. They form the basis for understanding the mathematical devel-
opments in analyzing the coupled FOEM2 (CFOEM2) and the CFOEM3 per-
formed in Sections 6.3 and 6.4, respectively. In Section 6.5 some simulation
examples on a CFOEM2 of dimension 2 are presented to illustrate the bene-
fit of using CFOEM theory under ideal conditions as well as in the presence
of noise. Finally, some general conclusions on this study are drawn in
Section 6.6.