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276 Mathematical Techniques of Fractional Order Systems

TABLE 9.9 IAE Values and Objective Functions for Mass Uncertainty in
Both Links
IAE Link-1 IAE Link-2 Objective function
IOSMCPD 0.0014 0.0011 0.0027
FOFSMCPD 0.0011 0.0007 0.0021

functions where it can be easily detected that a 22.22% improvement is
observed by the FOFSMCPD controller over the IOSMCPD controller.
Enhancements of 21.42% and 36.66% are also obtained by the FOFSMCPD
over the IOSMCPD controller in IAE values for link-1 and link-2, respec-
tively. On the basis of uncertainty analysis, it can be concluded that the
FOFSMCPD controller outperforms the IOSMCPD controller.

9.9 CONCLUSION

In this chapter, integer order sliding mode proportional and derivative
(IOSMCPD) controller and fractional order fuzzy sliding mode proportional
and derivative (FOFSMCPD) controller are used to control a nonlinear,
MIMO, coupled complex system, two-link robotic manipulator. The efficacy
of controllers is tested for trajectory tracking task, disturbance rejection, and
uncertainty analysis. In a classical sliding mode controller (SMC), chattering
is a major problem. This problem is effectively handled by a combination of
FL-based intelligent technique and boundary layer technique. Exponential law
is used to design the SMC controllers which give the Lyapunov based stability
of overall system. The performance index was taken as the weighted sum of
integral of absolute error and chatter. The gains of the controllers are tuned by
GA. On the basis of the obtained simulated results, it can be concluded that
the FOFSMCPD controller outperforms the IOSMCPD controller in all the
aspects of performances carried out for evaluation of controllers. As a future
extension of this research work, performance of the proposed controller needs
to be validated on a real-time hardware system, as well as other variants of
the SMC like higher order SMC, Terminal mode SMC may be tried out.

REFERENCES
A ˚ stro ¨m, K.J., Wittenmark, B., 2008. Adaptive Control. Dover Publications, Mineola, New York.
Azar, A.T., 2010a. Fuzzy Systems. IN-TECH, Vienna, Austria, ISBN 978-953-7619-92-3.
Azar, A.T., 2010b. Adaptive neuro-fuzzy systems. In: Azar, A.T. (Ed.), Fuzzy Systems.
IN-TECH, Vienna, Austria, ISBN 978-953-7619-92-3.
Azar, A.T., 2012. Overview of type-2 fuzzy logic systems. Int. J. Fuzzy System Applicat.
(IJFSA) 2 (4), 1 28.