Page 263 - Mathematical Techniques of Fractional Order Systems

252 Mathematical Techniques of Fractional Order Systems

designed by Efe (2008) to improve the robustness of fuzzy SMC by the use
of an ANFIS. In this work, fractional order has been used rather than integer
order to improve the robustness of controller. The plant is used for study as
a two degree of freedom direct-drive robot arm. The obtained result from the
order controller is compared to its integer order counterpart, where it is
reported that the proposed controller displayed better tracking performance,
insensitivity towards disturbances, and a high degree of robustness.
To control two different plants, polar robot and coupled tank system, a
novel fractional order fuzzy SMC with proportional derivative surface
(FOFSMCPD) is proposed by Delavari et al. (2010a). The discount in signum
function is reduced in that work by using the FL when the controller reaches
the sliding surface. The parameters of the controller are optimized by using
genetic algorithm (GA) which was proposed by J. Holland and his collabora-
tors in the 1960s and 1970s. The performance index is considered as reaching
time and on root mean square value of tracking error. Extensive simulation
studies revealed that the FOFSMCPD controller outperforms on its integer
order counterpart for set point tracking capability and chattering reduction
task. In the present chapter, FOFSMCPD controller is considered for a two-
link rigid robotic manipulator control task. To show the efficacy of the
FOFSMCPD controller, its performance is compared with classical integer
order SMCPD (IOSMCPD) controller. The performances of the FOFSMCPD
controller are calculated in servo as well as regulatory mode with and without
model uncertainty in mass as well as length of both the links.
Followed by the introduction and detailed literature survey presented in
Section 9.1, the dynamic mathematical model of the plant is described in
Section 9.2. The design and implementation of IOSMCPD and FOFSMCPD
controllers are presented in Section 9.3 whereas optimization of controller gains
by GA is shown in Section 9.4. The detailed performance evaluation is pre-
sented in Section 9.5 and the conclusion of this chapter is given in Section 9.6.

9.2 DYNAMIC MODEL OF MANIPULATOR SYSTEM

In this section, the dynamic model of a two-link planar rigid robotic manipu-
lator system is presented (Craig, 2005). A manipulator diagram is shown in
Fig. 9.1 where l 1 and l 2 are considered as length of link-1 and link-2,
whereas m 1 and m 2 are taken as mass of link-1 and link-2, respectively. To
find the mathematical model of the two-link robotic manipulator system, it
has been assumed that masses of both the links are on the tip of respective
links. θ 1 and θ 2 are considered as the angular position of link-1 and link-2 in
radian whereas τ 1 and τ 2 are taken as input torques to the manipulator
system, respectively in N-m. A popular approach, Lagrangian-Euler formula
is used to find the mathematical model of the two-link manipulator system.
According to this consideration, the kinetic energy of the link will be,