A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 78

14 ; 900 K 3 15 ; 700 ð 34 Þ

3 ; 000 K 4 4 ; 000 ð 35 Þ

14 ; 000 K 5 12 ; 000 ð 36 Þ

;

K 6

; ð Þ

3 000 1 500 37

In this problem, the objective functions of the multi-objective optimization

algorithm are

F 1 = the sum of settling time and overshoot of the cart;
F 2 = the sum of settling time and overshoot of the

first pendulum + the sum of
settling time and overshoot of the second pendulum;

These objective functions have to be minimized simultaneously. The Pareto
front of the control of the system of the parallel-double-inverted pendulum obtained
via multi-objective hybrid of particle swarm optimization and the genetic algorithm
is shown in Fig. 20 . In Fig. 20 , points A and C stand for the best sum of settling
time and overshoot of the cart and the sum of settling time and overshoot of the rst

and second pendulums, respectively. It is clear from this figure that all the optimum

design points in the Pareto front are non-dominated and could be chosen by a
designer as optimum linear state feedback controllers. It is also clear that choosing a
better value for any objective function in a Pareto front would cause a worse value
for another objective. The corresponding decision variables (vector of linear state

2

1.9

A

1.8

1.7

B

1.6

1.5

C

1.4

1.3

0.42 0.425 0.43 0.435 0.44

Objective function 1 (F1)

Fig. 20 Pareto front of multi-objective hybrid of particle swarm optimization and the genetic
algorithm for the control of the system of the parallel-double-inverted pendulum