Page 277 - Mathematical Techniques of Fractional Order Systems

266 Mathematical Techniques of Fractional Order Systems

4. Objective function of the individuals will be evaluated and after compar-
ing the objective function value, a new population will be generated by
using cross-over and mutation.
5. At last, results will be decoded to find the solution of problem.

9.6.2 Tuning of Controller for Trajectory Tracking Task
All the simulations presented in the current chapter have been performed in
MATLAB/SIMULINK (R2012a) on a personal computer having Intel coret
i5 processor which is working at 3.33 GHz, 4 GB RAM with a 32-bit operat-
ing system. For the ODE solver task, fourth-order Runge Kutta method was
used where the sampling time is considered as 1 ms. Initially optimized gains
remain unaltered throughout all the further studies like disturbance rejection
and uncertainty analysis. In the present work, the torque restrictions for both
links have been considered as [ 20, 20] N-m. The reference trajectories (θ r1
and θ r2 ) for link-1 and link-2 have been shown in Eqs. (9.64) and (9.65),
respectively, which are expressed as,
θ r1 5 sin ωtf g ð9:64Þ

θ r2 5 cos ωtf g ð9:65Þ
where ω is termed as the angular frequency of desired trajectory.
The objective function in the current chapter is taken as:

IAE 5 IAE 1 1 IAE 2 ð9:66Þ
ð
Chatter 5 ½ jj jjŠdt ð9:67Þ
s 1 1 s 2
Here, IAE 1 and IAE 2 represent the integral of absolute errors for link-1
and link-2, respectively. The chatter is defined as the integration of sum of
absolute values of surfaces for link-1 and link-2, respectively, as described in
Eq. (9.67) where s 1 and s 2 represents the surface of link-1and link-2, respec-
tively. In the present work, the objective function has been taken as the
weighted sum of integral of absolute error (IAE) and the chatter which is
defined in Eq. (9.68).

Objective function 5 ðW 1 IAEÞ 1 ðW 2 chatterÞ ð9:68Þ
The aggregate IAE is defined as the sum of the IAE values of both the
links, i.e., IAE 5 IAE 1 1 IAE 2 . The weightage parameters W 1 and W 2 are
chosen as 0.999 and 0.001. These numbers were chosen to give equal weigh-
tage to both the terms, IAE and chatter, during the optimization. Table 9.4
shows the optimized gains of IOSMCPD and FOFSMCPD controllers where
Fig. 9.10 depicts the OBF versus iteration curve for IOSMCPD and
FOFSMCPD controllers. The obtained IAE values and objective function
values are depicted graphically in Fig. 9.11. It can be easily observed from