A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 74

n n

deterministic nature. Furthermore, A 2 R is the system or dynamic matrix, B 2

n m is the input matrix and B v is the disturbance matrix. Measurements are made
on this system which can be either the states themselves or linear combinations of
them:

R

y t ðÞ ¼ Cx t ðÞþ w ð t Þ ð 16 Þ

r r n is the output matrix. The vector

where y ðÞ2 R is the output vector and C 2 R

w stands for the measurement disturbance.

In order to establish linear state feedback around the above system, a linear
feedback law can be applied as follows:

ð t Þ

u t ðÞ ¼ Kx t ðÞþ r ð Þ ð Þ

t 17

m n

Þ
denotes the reference input vector of the system having dimensions the same as the
input vector ut

In this formula, K 2 R stands for a feedback matrix (or a gain matrix). r ð t

. The resulting feedback system is a full state feedback system due
to measuring all of the states. To design the state feedback controller with an
optimal control input and minimum error, the hybrid optimization algorithm is
applied and the optimal Pareto front of the controller is shown in the following
section.

ðÞ

7 Pareto Optimal State Feedback Control of a Parallel-
Double-Inverted Pendulum

The model of a parallel-double-inverted pendulum system is presented in this
section. The work deals with the stabilization control of a system which is a
complicated nonlinear and unstable high-order system. Figure 18 illustrates the
mechanical structure of the inverted pendulum. According to the

figure, the cart is
moving on a track with two pendulums hinged and balanced upward by means of a
DC motor. In addition, the cart has to track a (varying) reference position. This
system includes two pendulums and one cart. The pendulums are attached to the
cart. While the cart is moving, the system has to be controlled in such a way that
pendulums are placed in desired angels. The dynamic equations of the system are as
follows:

I 1 u þ C 1 u a 3 sin u þ a 1 x cos u ¼ 0 ð 18 Þ

I 2 a þ C 2 a a 4 sin a þ a 2 x cos a ¼ 0 ð 19 Þ

2 sin 2 sin

€ þ

f r _

þ u u u u þ a a a a ¼ ð Þ

M x x a 1 cos a 2 cos u 20