A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 80

feedback controllers) of the Pareto front shown in Fig. 20 are the best possible
design points. Moreover, if any other set of decision variables is selected, the
corresponding values of the pair of those objective functions will locate a point
inferior to that Pareto front. Indeed, such inferior area in the space of two objectives
is top/right side of Fig. 20 . Thus, the Pareto optimum design method causes to

nd
important optimal design facts between these two objective functions. From
Fig. 20 , point B is the point which demonstrates such important optimal design
facts. This point could be the trade-off optimum choice when considering minimum
values of both sum of settling time and overshoot of the cart and sum of settling
time and overshoot of the first and second pendulums. The values of the design

variables obtained for three design points are illustrated in Table 20 . The control
effort, the angle of the first pendulum, the angle of the second pendulum, and the

position of the cart are illustrated in Figs. 21 , 22 , 23 and 24 . By regarding these

figures, it can be concluded that the point A has the best time response (overshoot
plus settling time) of the cart and the worst time responses (overshoot plus settling
time) of the pendulums while point C has the best time responses of pendulums and
the worst time response of the cart.

-3

x 10

10

Point A
Point B
Point C

8

6

4

2

0

-2

-4

-6

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Fig. 21 The control effort of the system of the parallel-double-inverted pendulum for design
points of the Pareto front of multi-objective hybrid of particle swarm optimization and the genetic
algorithm