Fuzzy control optimized by a Multi-Objective Differential Evolution algorithm for vibration suppression of smart structures
Introduction
During the last years, there has been an increasing application of nature inspired and evolutionary approaches to many fields, and, especially when the task is optimization within complex domains of data or information. Differential Evolution (DE) is a stochastic, population-based algorithm that was proposed by Storn and Price [1], [2], [3]. Recent books for the DE can be found in [4], [5]. DE has the basic characteristics of the evolutionary algorithms. It focuses in the distance and the direction information of the other solutions. In the differential evolution algorithms [6], initially, a mutation is applied to generate a trial vector and, afterwards, a crossover operator is used to produce one offspring. The mutation step sizes are not sampled from an a priori known probability distribution function as in other evolutionary algorithms but they are influenced by differences between individuals of the current population.
In real world applications, optimization problems with more than one objectives are very common. In these Multi-Objective Optimization Problems, usually, there is no guarantee that a unique solution exists. The solution of Multi-Objective Optimization problems with the use of Evolutionary algorithms is a field that has been extensively studied during the last years. For a complete survey of the field of the Evolutionary Multi-Objective Optimization the reader is refereed to [7], [8].
In this paper, a Multi-Objective Differential Evolution (MODE) algorithm is used in order to optimize the parameters of a fuzzy control system that is used for the vibration control problem of a flexible structure (smart beam). The usage of MODE with a combination of continuous and discrete variables for the optimal design of the controller is presented. Beams are fundamental elements in many mechanisms and structures [9], [10], [11], [12], [13], [14]. Therefore, the modeling of the dynamic behavior as well as the control of beams is a significant model problem. A smart structure with bonded sensors and actuators as well as an associated control system, which enable the structure to respond to external excitations in such a way that it suppresses undesired effects, is considered. The choice of the control technique is important for the design of controllers which ensures the performance of the flexible structure under required conditions and at the same time can be easily applied.
This paper is based upon our previous contribution, Marinaki et al. [15], and includes the following additional research results: in [15] only one objective function was used while in the current paper the multi-objective approach is applied using three objective functions. In this paper, three new mutation operators, which are suitable for the multi-objective DE, have been developed while in paper [15] the classical mutation operators have been used for Differential Evolution. The results obtained are compared with the results of the fuzzy control system when its parameters are optimized by a multi-objective particle swarm optimization algorithm [16]. It should be mentioned here that the proposed method is quite general and can be used for the design of other smart structures like plates and shells. Results in this direction will be reported in the future.
The rest of the paper is organized as follows: In Section 2 the most important evolutionary multiobjective algorithms based on Differential Evolution are described. In Section 3, the finite element modeling of the smart beams is outlined. In Section 4, a detailed description of the proposed fuzzy control system optimized by MODE is given while in Section 5 the numerical results are presented. In the last section, the conclusions and some proposals for further research are given.
Section snippets
Evolutionary multiobjective optimization algorithms based on differential evolution
The solution of an optimization problem with two or more competitive objective functions satisfies the conflicting objective functions or a compromise between them. This problem is called multi-objective optimization. In multi-objective optimization, the solution is chosen from a set of solutions called Pareto front [8]. Given two vectors and where , , k is the number of the objective functions and are the values of the i-th
Models of smart beams and structures
A smart laminated composite beam with rectangular cross-section having length L, width b, and thickness h is considered (Fig. 1) where the control actuators (thickness ) and the sensors (thickness ) are piezoelectric patches symmetrically bonded on the top and the bottom surfaces of the host beam. Both piezoelectric layers are positioned with identical poling directions and can be used as sensors or actuators.
The mathematical formulation of the model is based on the Timoshenko beam theory
Fuzzy control and adaptively optimal fuzzy control
Fuzzy systems [37] can be applied in many fields and can solve different kinds of problems in various application domains. Fuzzy inference rules systematize existing experience and can be used for the mathematical formulation of nonlinear controllers. The feedback is based on fuzzy inference and may be nonlinear and complicated. Knowledge or experience on the controlled system is required for the application of this technique. Less knowledge of the logic and availability of more observations
Numerical results
The problem of a cantilever beam was studied in the present paper. The beam has a total length equal to 0.8 m and a square cross-section with dimensions . Material constants for the host beam are: Elastic modulus , density . A viscous damping coefficient equal to 0.01 is considered. The structure has been discretized with four finite elements resulting in a model with eight degrees of freedom.
A dynamic loading is used as disturbance that simulates a strong wind and
Conclusion
In this paper, a fuzzy control system optimized by Multi-Objective Differential Evolution (MODE) was used for the vibration control of beams with piezoelectric sensors and actuators. The parameters of the fuzzy controller were selected optimally by using the MODE algorithm. The results obtained for a sinusoidal loading pressure using the fuzzy controller system optimized by MODE are very promising. A comparison was performed using a Multi-Objective Particle Swarm Optimization algorithm. Future
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