A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 75

Fig. 18 The system of a
parallel-double-inverted
pendulum

where

I 1 þ M 1 l 1 ; I 2 þ M 2 l 2 ; I 1 ¼ I 2 ¼ ð 21 Þ

M ¼ M 0 þ M 1 þ M 2 ð 22 Þ

a 1 ¼ M 1 l 1 ; a 2 ¼ M 2 l 2 ; a 3 ¼ M 1 l 1 g ; a 4 ¼ M 2 l 2 g ð 23 Þ

_

where x is the position of the cart, x is the velocity of the cart, u stands for the

u

angular velocity of the first pendulum with respect to the vertical line, is the

a

angular velocity of the first pendulum, is the angular position of the second

a

pendulum, represents the angular velocity of the second pendulum, M 1 is the mass

of the first pendulum, M 2 is the mass of second pendulum, M 0 is the mass of the

cart, l 1 denotes the length of the first pendulum with respect to its center, l 2 stands

for the length of the second pendulum with respect to its center, f r is the friction
coef cient of the cart with ground, I 1 first pendulum

with respect to its center, I 2 represents the inertia moment of the second pendulum
with respect to its center, C 1 is the angular frictional coef

is the inertia moment of the

cient of the rst pen-

dulum, C 2 stands for the angular frictional coef cient of the second pendulum, and

u is the control effort.
To obtain the state space representations of the dynamic equations, the state
space variables are defined as x ¼½x 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6

T . This vector includes the
position of the cart, the velocity of the cart, the angular position and velocity of the

first pendulum, the angular position and velocity of the second pendulum. After
linearization around the equilibrium point x a ¼½

T , the state space

x 1 ; 0 ; 0 ; 0 ; 0 ; 0

representation is obtained according to Eq. ( 25 ).