Page 271 - Mathematical Techniques of Fractional Order Systems

260 Mathematical Techniques of Fractional Order Systems

€
Substituting θ from plant dynamic model (9.30) in (9.50), it becomes
_
α

2 E sat sðÞ 2 ks 5 λD e 1 D α21 € ½ 21 τ 2 V θ;θ 2 G θðÞ Þð9:51Þ
ðθ r 2 M θðފ
It can be further simplified as,
21 _ 12α α €
½ M θðފ τ 2 V θ;θ 2 G θðÞ 5 D ð E sat sðÞ 1 ks 1 λD eÞ 1 θ r ð9:52Þ
h i
_
€
α
τ 5 M θðފ D 12α ð E sat sðÞ 1 ks 1 λD eÞ 1 θ r 1 V θ;θ 1 G θðÞ ð9:53Þ
½
Eq. (9.53) explores the required torques as an FOFSMCPD controller out-
puts to control the angular positions of end-effector of manipulator system. In
the design of FOFSMCPD controller, E and k are the two positive gains of
exponential law which direct the manipulator output so as to track the surface,
whereas the saturation function reduces the chattering. Therefore, a satisfactory
compromise between robustness and chattering can be made after proper varia-
tion of these two variables. It is presented in Delavari et al. (2010a) that the con-
cept of an FL control (FLC) can be easily combined with SMC to preserve the
advantage of both the techniques. In the present chapter, an FLC is considered
to vary the gains E and k with varying surface to reduce chattering. A significant
reduction in chattering is obtained which will be explained and presented in
Sections 9.5 and 9.6. The detailed design of FLC is presented as follows.

9.5.1 Stability Analysis Using Lyapunov Stability Criteria
The main objective of SMC controller design is the overall guaranteed stabil-
ity of the feedback control system. According to the Lyapunov stability theo-
rem, the overall system will be stable and will reach the sliding surface if
the condition s_ s , 0 will be satisfied. For this, the Lyapunov function is con-
sidered as:

1
V 5 s 2 ð9:54Þ
2
_
V 5 s_ s ð9:55Þ
Substituting _ s from exponential law (9.40),
_
V 5 s 2 E sat sðÞ 2 ksÞ where; E . 0; k . 0 ð9:56Þ
ð
Here, (9.56) governs two cases. In the first case when the sliding surface
s . 0 whereas in the second case s , 0. After considering the first case:
Case I: When s . 0 then sat sðÞ . 0
Let, sat sðÞ 5 φ, where, φ . 0.
From Eq. (9.56)
_
ð
V 5 s 2 Eφ 2 ksÞ ð9:57Þ