Key words: Fuzzy Control ；Differential Evolution ；Inverted Pendulum ；Optimization 1 Introduction In stabilization control of inverted pendulum systems, fuzzy control shows great potentials. However, it is difficult to determine the value of some parameters. Differential Evolution is a simple and efficient heuristic for global optimization over continuous spaces. This paper gives a better optimization algorithm for the fuzzy controller and it is verified. Theory developed by L. A. Zadeh, is a practical alternative for a variety of challenging control applications since it provides a simple and convenient method for constructing nonlinear controllers using heuristic information. Fuzzy control provides a user-friendly formalism for representing and implementing our ideas via the use of linguistic rules, so that we can achieve the expected performance easily. Usually, a block diagram of a fuzzy controller is shown in Fig.1 .
2 Fuzzy Control and Differential Evolution
2.1 Fuzzy logic systems and control Fuzzy control, based on the Fuzzy Logic
Fig.1 The block diagram of a fuzzy controller
(1) The rule-base contains a fuzzy logic quantification of the expert’s linguistic description of how to achieve the performance. (2) The inference mechanism emulates the expert’s decision making, interpreting and applying the knowledge to the plant. (3) The fuzzification converts controller
inputs into fuzzy information which the inference mechanism can easily use to activate and apply rules. (4) The defuzzification yields the results of the whole fuzzy controller, which is actually the inputs of the controller for the plant . Fuzzy control is also a kind of “intelligent control” technique. Although satisfactory 1
performance can be achieved with the fuzzy control method, its performance can still be improved, if some optimization algorithms are used to tune some of its parameters. Later on, we will provide a brief overview of the Differential Evolution algorithm which can be used to optimize the parameters of the fuzzy controllers. 2.2 Differential Evolution (DE) algorithm Differential evolution, proposed by Rainer Storn and Kenneth Price, is a heuristic approach for minimizing nonlinear and non-differential continuous functions based on population difference. Comparing with other Evolution Algorithms (EA), coded by real numbers, DE is robust, easy to use, and it converges faster, requires fewer control variables, and lends itself very well to parallel computations . Similar to Genetic Algorithm (GA), DE Algorithm also includes mutation, crossover, and selection, while the difference exists in the step of mutation. In DE, it is adding the difference of two individuals to a third one that realizes the mutation. The Benchmark DE (also called simple DE) can be described as follows : (1) Initializing: DE utilizes NP
The randomly chosen r1 , r2 , r3 are also chosen to be different from running indexes i . F is a real and constant factor 0, 2 . (3) Crossover: In order to increase the diversity of the perturbed parameter vectors, crossover is introduced. To this end, the trial vector:
vi ,G 1 (u1i ,G 1 , u2 i ,G 1 ,..., u Di ,G 1 ) if ( randb ( j ) CR )or j rnbr (i) if (randb( j ) CR )or j rnbr (i )
is formed, where
v ji,G 1 u...