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In this chapter, a hybrid of particle swarm optimization and the genetic algorithm
developed previously by authors (Mahmoodabadi et al.
2013
) is described and used
to design the parameters of the linear state feedback controller for the system of a
parallel-double-inverted pendulum. In elaboration, the used operators of the hybrid
algorithm include mutation, crossover of the genetic algorithm and particle swarm
optimization formula. The classical crossover and the multiple-crossover are two
parts of the crossover operator. A fuzzy probability is used to choose the particle
swarm optimization and genetic algorithm operators at each iteration for each
particle or chromosome. The optimization algorithm is based on the non-dominated
sorting concept. Moreover, a leader selection method based upon particles density
and a dynamic elimination method which con
nes the numbers of non-dominated
solutions are utilized to present a high convergence and uniform spread. Single and
multi-objective problems are utilized to assess the capabilities of the optimization
algorithm. By using the same benchmarks, the results of simulation are contrasted
to the results of other optimization algorithms. The structure of this chapter is as
follows. Section
2
presents the genetic algorithm and its details including the
crossover operator and the mutation operator. Particle swarm optimization and its
details involving inertia weight and learning factors are provided in Sect.
3
. Sec-
tion
4
states the mutation probabilities at each iteration which is based on fuzzy
rules. Section
5
includes the architecture, the pseudo code, the parameter settings,
and the flow chart of the single-objective and multi-objective hybrid optimization
algorithms. Furthermore, the test functions and the evolutionary trajectory for the
algorithms are provided in Sect.
5
. State feedback control for linear systems is
presented in Sect.
6
. Section
7
presents the state space representation, the block
diagram, and the Pareto front of optimal state feedback control of a parallel-double-
inverted pendulum. Finally, conclusions are presented in Sect.
8
.
2 Genetic Algorithm
The genetic algorithm inspired from Darwin
is theory is a stochastic algorithm based
upon the survival fittest introduced in 1975 (Holland
1975
).
Genetic algorithms offer several attractive features, as follows:
'
An easy-to-understand approach that can be applied to a wide range of problems
with little or no modi
cation. Other approaches have required substantial
alteration to be successfully used in applications. For example, the dynamic
programming was applied to select the number, location and power of the lamps
along a hallway in such a way that the electrical power needed to produce the
required illuminance will be minimized. In this method, signi
cant alternation is
needed since the choice of the location and power of a lamp affect the decisions
made about previous lamps (Gero and Radford
1978
).
Genetic algorithm codes are publicly available which reduces set-up time.
The inherent capability to work with complex simulation programs. Simulation
does not need to be simpli
ed to accommodate optimization.
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