Page 269 - Mathematical Techniques of Fractional Order Systems

258 Mathematical Techniques of Fractional Order Systems

For the considered second order (n 5 2) plant, (9.34) becomes,
s 5 λe 1 _ e ð9:35Þ
Here, λ is a constant and error e 5 θ r 2 θ, where θ r is the desired angular
position and θ is the current angular position of end points of the links in
radian. On differentiation (9.35) becomes,

_ s 5 λ_ e 1 € e ð9:36Þ
€
Putting € e 5 θ r 2 θ €
€ €
_ s 5 λ_ e 1 ðθ r 2 θÞ ð9:37Þ
The system output is forced to track this surface with the help of a reach-
ing law. The reaching law is to be designed in such a way that it guarantees
the stability of the closed loop system. Generally, three reaching laws are
widely used and have been reported in the literature namely exponential,
constant rate, and power rate laws (Liu and Wang, 2012). In the present
work, exponential law as given in (9.38) is utilized. According to this law,
_ s 52 E sgn sðÞ 2 ks; E . 0; k . 0 ð9:38Þ

where E and k are constants.
As SMC is a high gain switching controller, a fast oscillation phenomenon,
called chattering, occurs at the controller output due to the discontinuity in
sgn function in control action. To resolve this problem, normally saturation
function (9.39) is employed rather than sgn function (Liu and Wang, 2012).
8 11; s . Δ
> s
< ; jsj , Δ
sat sðÞ 5 Δ ð9:39Þ
>
21; s ,2 Δ
:
where Δ 0 , Δ , 1Þ forms the boundary layer.
ð
Now, from (9.38) and (9.39),
_ s 52 E sat sðÞ 2 ks; E . 0; k . 0 ð9:40Þ
Equating (9.38) and (9.40),
€ €
2 E sat sðÞ 2 ks 5 λ_ e 1 ðθ r 2 θÞ ð9:41Þ
€
Now substituting θ from the dynamics of manipulator (9.30),in (9.41) then,
€ 21 _
½
2 E sat sðÞ 2 ks 5 λ_ e 1 ðθ r 2 M θðފ τ 2 V θ;θ 2 G θðÞ Þ ð9:42Þ
This can also be expressed as,
_ €
h i
½
τ 5 V θ;θ 1 G θðÞ 1 M θðފ E satðsÞ 1 ks 1 λ_ e 1 θ r ð9:43Þ