Page 270 - Mathematical Techniques of Fractional Order Systems

Design of Fractional Order Fuzzy Sliding Mode Controller Chapter | 9 259

Eq. (9.43) gives the required torques for a two-link manipulator system
by an IOSMCPD controller to control the angular positions of end-effector
of manipulator links. In this design, two gains, E and k, came from exponen-
tial law which uses the manipulator output to track the desired trajectory
whereas saturation function is used to reduce the chattering. Therefore,
proper variation in E and k can make a suitable compromise between robust-
ness and chattering reduction.

9.5 FOFSMCPD CONTROLLER DESIGN

A generalized integer order sliding surface for second order plant is given in
(9.35) which can be expressed as,

s 5 λe 1 _ e ð9:44Þ
It was shown by Efe (2008) and Delavari et al. (2010a) that by inclusion
of the fractional order differentiator and integrator instead of the integer
order in SMC, the robustness of the controller can be significantly increased.
In this context, fractional order calculus has been introduced in the design of
SMC to achieve an enhanced level of robustness and the design procedure
for the fractional SMC implementation is given below.
After introducing fractional order operator in (9.44) which has been
considered from (Efe, 2008; Delavari et al., 2010a,b)
α
s 5 λe 1 D e; 0 , α , 1 ð9:45Þ
Eq. (9.45) can be written as,
s 5 λe 1 D α21 _ e ð9:46Þ
On further differentiation (9.46) becomes,

_ s 5 λ_ e 1 D α21 € e ð9:47Þ
On introducing the fractional order differentiator for the first term in
(9.47) it becomes,
α α21
_ s 5 λD e 1 D € e ð9:48Þ
In FOFSMCPD controller design, the exponential law (9.40) is utilized
so that the system output can be forced to follow this surface. In this regard,
comparing (9.40) and (9.48) it becomes,
α α21
2 E sat sðÞ 2 ks 5 λD e 1 D ð€ eÞ ð9:49Þ
€
€
Putting € e 5 θ r 2 θ (9.49) becomes
α α21 € €
2 E sat sðÞ 2 ks 5 λD e 1 D ðθ r 2 θÞ ð9:50Þ