Fuzzy control optimized by a Multi-Objective Differential Evolution algorithm for vibration suppression of smart structures

Computers and Structures xxx (2014) xxx–xxx

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Fuzzy control optimized by a Multi-Objective Differential Evolution algorithm for vibration suppression of smart structures Magdalene Marinaki ⇑, Yannis Marinakis, Georgios E. Stavroulakis School of Production Engineering and Management, Technical University of Crete, Chania, Greece

a r t i c l e

i n f o

Article history: Accepted 29 September 2014 Available online xxxx Keywords: Multi-Objective Differential Evolution Multi-objective particle swarm optimization Active control of structures Fuzzy control Smart structures

a b s t r a c t Smart structures include elements of active, passive or hybrid control. The fuzzy control is considered which is a suitable tool for the systematic development of active control strategies. A new MultiObjective Differential Evolution algorithm for the calculation of the free parameters in active control systems is proposed and tested. In particular, the usage of MODE with a combination of continuous and discrete variables for the optimal design of the controller is proposed. Numerical applications on smart piezoelastic beams are presented. The results obtained are compared with the ones obtained with the fuzzy controller optimized by a MOPSO algorithm. Ó 2014 Published by Civil-Comp Ltd and Elsevier Ltd.

1. 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 [1–3]. 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]. ⇑ Corresponding author. E-mail addresses: [email protected] (M. Marinaki), [email protected] (Y. Marinakis), [email protected] (G.E. Stavroulakis).

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 [9–14]. 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.

http://dx.doi.org/10.1016/j.compstruc.2014.09.018 0045-7949/Ó 2014 Published by Civil-Comp Ltd and Elsevier Ltd.

Please cite this article in press as: Marinaki M et al. Fuzzy control optimized by a Multi-Objective Differential Evolution algorithm for vibration suppression of smart structures. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2014.09.018

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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.

2. 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 y1 and y2 where y1 ¼ ½y1;1 ; y1;2 ; . . . ; y1;k , y2 ¼ ½y2;1 ; y2;2 ; . . . ; y2;k  , k is the number of the objective functions and y1;i ; y2;i are the values of the i-th objective function for the two vectors, respectively. If each objective function is subject to minimization and y1;i 6 y2;i ; 8i 2 ½1; kÞ then it is considered that y1 dominates y2 . A vector y1 is called Pareto optimal if it is not dominated by any other feasible solution. For fairly complete reviews of Evolutionary Multiobjective Optimization algorithms the reader may consult [7,8,17,18]. Several methods have been proposed for the multi-objective optimization using the Differential Evolution algorithm [19]. Chang et al. [20] applied a differential evolution algorithm to fine tune the fuzzy automatic train operation for a typical mass transit system using the DE=Rand=1 version of differential evolution with an external archive. In this paper the selection mechanism is modified in order to enforce that the members of the new generation are nondominated not only regarding their objective values but also regarding a set of distance metric values (one assigned to each objective). Abbass et al. [21–23] developed the Pareto Differential Evolution (PDE). The initial population is initialized using a Gaussian distribution retaining only the nondominated solutions. Three parents are randomly selected (one as the main parent and also trial solution) and an offspring is generated which is placed in the population only if it dominates the main parent. This process continues until the population is completed, but if the number of nondominated solutions exceeds a certain threshold a distance metric is adopted to remove parents which are too close from each other. Abbass et al. [23] proposed a new version of the PDE called Self-adaptive Pareto Differential Evolution (SPDE) in which the crossover and mutation rates are self-adapted. Abbass [24] proposed, also, an algorithm called Memetic Pareto Artificial Neural Networks (MPANN) where the PDE uses a Back-Propagation algorithm to speed up the convergence. Kukkonen and Lampinen [25] proposed a version of Differential Evolution called Generalized Differential Evolution (GDE) to solve multi-objective optimization problems. In this algorithm, the original DE is modified by introducing Pareto Dominance as a selection criterion between the old population member and the trial vector. A second version of this algorithm is the GDE2 where a crowding distance measure was used to select the best solution when the old population vector and the trial vector are feasible and nondominated with respect to each other, in such a way that the vector located in the less crowded region will be part of the population of the next generation. Santana-Quintero and Coello Coello [26] proposed the -MyDE algorithm. This algorithm does not allow two solutions in the Pareto front with difference less than a threshold i in the objective function i.

Xue et al. [27,28] proposed the Multi-Objective Differential Evolution where if the trial member is dominated then it is replaced with one Pareto solution and if the trial vector is nondominated, then, it is inserted in the population. Iorio and Li [29] developed the Nondominated Sorting Differential Evolution (NSDE). This algorithm is a simple modification of the NSGA-II as in NSDE the genetic operators are replaced with the operators of the Differential Evolution. Robic and Filipic [30] proposed the Differential Evolution for Multi-objective Optimization (DEMO) which is based on the concept of Pareto Dominance. In this algorithm, when the trial vector dominates the parent vector, then, the trial vector replaces the parent, otherwise the trial vector is discarded. In order to truncate the population, DEMO applies a nondominated sorting mechanism. Parsopoulos et al. [31] proposed the Vector Evaluated Differential Evolution (VEDE) which is inspired by the Vector Evaluated Genetic Algorithm (VEGA) [32]. A number of M subpopulations are considered in a ring topology. Each population is evaluated using one of the objective functions of the problem, and there is an exchange of information among the populations through the migration of the best individuals. The new vector is introduced in the population if it dominates the main parent. The Pareto front is stored in an external archive. 3. 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 hA ) and the sensors (thickness hS ) 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 (which takes into account shear effects) and the linear theory of piezoelectricity. The numerical solution of the model is based on the development of superconvergent finite elements using the form of exact solution of the Timoshenko beam theory and Hamilton’s principle (see [10,33–36]). The linear constitutive equations of the two coupled fields, namely the elastic and the electric fields, read:

  T frg ¼ ½Q  feg  ½d fEg

ð1Þ

fDg ¼ ½d½Q feg þ ½nfEg

ð2Þ

where frg61 is the stress vector, feg61 is the strain vector, fDg31 is the electric displacement, fEg31 is the strength of applied electric field acting on the surface of the piezoelectric layer, ½Q 66 is the elastic stiffness matrix, ½d36 is the piezoelectric matrix and ½n33 is the permittivity matrix. Eq. (1) describes the inverse piezoelectric effect (which is exploited for the design of the actuator). Eq. (2) describes the direct piezoelectric effect (which is used for the sensor). Additional assumptions are used for the construction of

Fig. 1. Laminated beam with piezoelectric sensors, actuators and the schematic control system.

Please cite this article in press as: Marinaki M et al. Fuzzy control optimized by a Multi-Objective Differential Evolution algorithm for vibration suppression of smart structures. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2014.09.018

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the simplified model: (a) Sensor and actuator (S/A) layers are thin compared with the beam thickness. (b) The polarization direction of the S/A is the thickness direction (z axis). (c) The electric field loading of the S/A is uniform uni-axial in the x-direction. (d) Piezoelectric material is homogeneous, transverse isotropic and elastic. Therefore, the set of Eqs. (1) and (2) is reduced as follows:



rx sxz



 ¼

Q 11

0

0

Q 55



ex cxz



 d31  Ez 0

ð3Þ ð4Þ

The electric field intensity Ez can be expressed as

V hA

ð5Þ

where V is the applied voltage across the thickness direction of the actuator and hA is the thickness of the actuator layer. Since only strains produced by the host beam act on the sensor layer and no electric field is applied to it, the output charge from the sensor can be calculated using Eq. (4). The charge measured through the electrodes of the sensor is given by

8 ! Z 1< qðtÞ ¼ Dz dS 2 : Sef

þ

Dz dS

Sef

z¼2h

9 =

!

Z

z¼2hþhS

;

ð6Þ

where Sef is the effective surface of the electrode placed on the sensor layer. The current on the surface of the sensor is given by

iðtÞ ¼

dqðtÞ dt

ð7Þ

The current is converted into open-circuit sensor voltage output by

V S ¼ GS iðtÞ

ð8Þ

where GS is the gain of the current amplifier. Furthermore, it is supposed that the bending-torsion coupling and the axial vibration of the beam centerline are negligible and that the components of the displacement field fug of the beam are based on the Timoshenko beam theory which, in turn, means that the axial displacement is proportional to z and to the rotation wðx; tÞ of the beam cross section about the positive y-axis and that the transverse displacement is equal to the transverse displacement wðx; tÞ of the point of the centroidal axis ðy ¼ z ¼ 0Þ. The strain–displacement relationships read

ex ¼ z

@w ; @x

exz ¼ w þ

@w @x

ð9Þ

The kinetic energy of the beam with the layers can be expressed as

T¼

1 2

Z

_ T fugdV _ ¼ qfug

V

b 2

Z 0

L

Z

hþh S 2

2

_ þw _ 2 dzdx q½ðzwÞ

ð10Þ

2hhA

assuming that, for sake of simplicity, that the host beam and the piezoelectric patches have identical densities. The strain (potential) energy is given by

1 2

Z

fegT frgdV  2  2 # Z Z h " b L 2þhS @w @w ¼ dzdx Q 11 z þ Q 55 w þ 2 0 2hhA @x @x

U¼

V

ð11Þ

If the only loading consists of moments induced by the piezoelectric actuators and since the structure has no bending–twisting couple then the first variation of the work has the form

ð12Þ

where d is the first variation operator and M A is the moment per unit length induced by the actuator layer and is given by

MA ¼



Dz ¼ Q 11 d31 ex þ n33 Ez

Ez ¼

 Z L @w dx dW ¼ b MA d @x 0

Z

2h

h2hA

zrAx dz ¼

Z

2h 2hhA

zQ 11 d31 EAz dz



EAz ¼

VA hA

ð13Þ

Using Hamilton’s principle the equations of motion of the beam are derived. For the finite element discretization beam finite elements are used, with two degrees of freedom (d.o.fs) at each node: the transversal deflection wi and the rotation wi . They are gathered to form the degrees of freedom vector X i ¼ ½wi wi . After assembling the mass and stiffness matrices for all elements, it is obtained the equation of motion in the form

€ þ KX ¼ F m þ F e MX

ð14Þ

where M and K are the generalized mass and stiffness matrices, F e is the generalized control force vector produced by electromechanical coupling effects and F m is the external loading vector. It should be mentioned here that technical bending theories for plates can be constructed analogously. In order to take into account damping, _ we add a Rayleigh damping KX,

€ þ KX_ þ KX ¼ F m þ F e MX

ð15Þ

where K ¼ aM þ bK is the damping matrix, and a; b are suitable constants. They are taken to be equal to 0.01 in the numerical examples. The main objective is to design control laws for the smart beam bonded with piezoelectric S/A subjected to external induced vibrations. For this purpose, a two-input, single-output fuzzy control system optimized by MODE is applied in this paper. This configuration is suitable for a feedback control force based on the displacement and velocity at a given point (collocated configuration of the local controller). If needed, several independent and decentralized (local) fuzzy controllers, in general with different characteristics, can be installed at the various points of a larger structure. 4. Fuzzy control system optimized by differential evolution 4.1. 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 (experimental data) requires the use of some hybrid technique in the form of a neuro-fuzzy controller. In the area of smart systems the application of neuro-fuzzy control has been adopted by many authors. In particular, it seems to be suitable for the control of structures with complicated or nonlinear characteristics, like a fuzzy uncertainty modellization adopted for structural engineering in [38], the tuned mass damper systems for aseismic design [39,40] or the semi-active control using active friction devices or electrorheological fluid dampers [41]. In order to design a fuzzy control system (construct the control rules, state the membership functions, tune the parameters, etc.) various approaches have been proposed. In the last years, there is an increasing interest to optimize the fuzzy control systems with heuristic, metaheuristic, and nature inspired approaches. Mainly,

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genetic algorithms have been used for these purposes while Particle Swarm Optimization and Differential Evolution have not been used so extensively. For example in [42] a method for designing fuzzy logic controllers with symmetric partitioning of the universe of discourse is presented where both the rule base and membership functions of the input and output variables are designed optimally using a genetic algorithm. In [43] a genetic algorithm based optimal fuzzy controller design is proposed where the design procedure is accomplished by establishing an index function as the consequent part of the fuzzy control rule. An efficient genetic reinforcement learning algorithm for designing fuzzy controllers is proposed in [44]. In [45] the proposed algorithm combines the genetic algorithm (GA) and the least-squares estimate (LSE) method to construct the genetic algorithm-based neural fuzzy system for temperature control. An extensive analysis of the genetic fuzzy systems can be found in [46]. [47] presents a particle swarm optimization method for optimizing a fuzzy logic controller for a photovoltaic grid independent system consisting of a PV collector array, a storage battery, and loads (critical and non-critical loads). A learning approach based on Particle Swarm Optimization for the determination of the consequent parameters and premise parameters of a Takagi and Sugeno type fuzzy model is formulated and explained in [48]. [49] introduces the use of the adaptive particle swarm optimization (APSO) for adapting the weights of fuzzy neural networks. In [50] proposes a zero-order Takagi–Sugeno–Kang (TSK)-type fuzzy system learning using a two-phase swarm intelligence algorithm (TPSIA) where the first phase of TPSIA learns fuzzy system structure and parameters by on-line clustering-aided ant colony optimization and phase two aims to further optimize all of the free parameters in the fuzzy system using particle swarm optimization. In [51] a new technique for eliciting a fuzzy inference system (FIS) from data for nonlinear systems is proposed and the parameters of the refined fuzzy model are tuned by means of differential evolution. In [52] a method is presented for the automatic design of a hierarchical fuzzy logic controllers (HFLC) using differential evolution and the feasibility of the method is demonstrated by developing a two-stage HFLC for controlling a cart-pole with four state variables. A hybrid method to train Fuzzy Cognitive Maps, based on a two stage learning approach, first stage the nonlinear Hebbian learning algorithm and second stage the differential evolution algorithm, has been developed, tested and applied in three problems with different complexity levels in [53]. 4.2. Fuzzy control of smart structures In order to reduce the amplitudes of the displacement field of the cantilever beam system, a non-linear fuzzy controller [36] was constructed by using the Fuzzy Toolbox of Matlab. More specifically, a Mamdani-type Fuzzy Inference System, consisted of two inputs and one output, was developed. The system receives as _ while gives as inputs the displacement (u) and the velocity (u), output the increment of the control force (z). Triangular and trapezoidal shape membership functions were chosen both for the inputs and the output. In order to describe the present system-controller 15 rules were used. All rules have weights equal to 1 and use the AND-type logical operator. These rules are presented in the Table 1. The implication method was set to minimum (min), while the aggregation method was set to maximum (max). The defuzzified output value has been created by using the MOM (Mean of Maximum) defuzzification method. 4.3. Structural dynamics and fuzzy control The Houmbolt numerical integration method was chosen [36] in order to integrate the equations of motion (Eq. 15).

Table 1 Fuzzy inference system rules, e.g. if displacement is far up and velocity is up then control force is max. Velocity Up Null Down

Far Up

CloseUp

Max Med+ High+

Displacement Med+ Low Low+ Null Null Low+

Equilibrium

CloseDn

Far Dn

Null Low Med

High Med Min

According to this method, on the assumption that acceleration is constant within each time step, the Houmbolt factors are set to

c ¼ 0:5

b ¼ 0:25;

ð16Þ

The total integration time was chosen equal to 1 s, while the time step (Dt) equal to 0.001 s. The integration constants are

c1 ¼

1 bðDtÞ2

c

c5 ¼ ; b

;

c2 ¼

c 6 ¼ Dt



1 ; bDt

c

2b

1

c3 ¼

1 ; 2b

c4 ¼

c b Dt

; ð17Þ

In each step (t) of the numerical integration, the fuzzy controller provides a control force (z), according to the given input values _ Both the control force and the (displacement u and velocity u). external loads provide the next step’s (t þ Dt) values of displacement and velocity. 4.4. Fuzzy control optimized by Multi-Objective Differential Evolution for vibration reduction of smart structures Some parameters of the fuzzy control system described in Section 4.2 were optimized by Multi-Objective Differential Evolution. More precisely, the parameters that describe the triangular and trapezoidal membership functions, the weights of the rules, and the logical operator are considered as variables. Firstly, a number of individuals is randomly initialized, where an individual is a solution to the problem. In each individual, the first values correspond to the parameters of the membership functions and can take continuous values, the next 15 values correspond to the weights of the rules and can take continuous values in the range [0, 1] while the last 15 values correspond to the AND/OR – type logical operator and can take discrete values equal to 1 for AND and 2 for OR. It should be noted that in DE and in MODE implementations, usually, only continuous or discrete values exist. In our case, there is a combination of continuous and discrete values in each individual and in order to solve the problem a combination of the equations used for finding the new solution for the individual in DE and MODE for continuous and discrete problems is used. More precisely, the position of an individual is represented by a d-dimensional vector in problem space si ¼ ðsi1 ; si2 ; . . . ; sid Þ; i ¼ 1; 2; . . . ; N (N is the population size, d is set equal to the values’ parameters plus the values of weights plus the values of the logical type operator). The fitness functions that have to be minimized are the following three error functions:



err1 ¼ X  X

ð18Þ 2

m

where X is the L norm in R (m is the total number of the assumed d.o.fs), X ¼ ½X 1 ; . . . ; X m  is the nodal displacements and rotations array and X ¼ ½X 1 ; . . . ; X m  is the wished value of the nodal displacements and rotations array (in the application presented here, it is set equal to zero).



err2 ¼ V  V

ð19Þ

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where V is the L2 norm in Rm (m is the total number of the assumed d.o.fs), V ¼ ½V 1 ; . . . ; V m  is the nodal displacements’ and rotations’ velocities array and V ¼ ½V 1 ; . . . ; V m  is the wished value of the nodal displacements’ and rotations’ velocities array (here it is set equal to zero).



err3 ¼ G  G

5. The last mutation operator is the strategy that ensures that the parent is mutated using at least two difference vectors: nu X ui ðtÞ ¼ si1 ðtÞ þ bðsopt ðtÞ  si1 ðtÞÞ þ b ðsi2 ;l ðtÞ  si3 ;l ðtÞÞ

ð26Þ

l¼1

ð20Þ 2

m

where G is the L norm in R (m is the total number of the assumed d.o.fs), G ¼ ½G1 ; . . . ; Gm  is the nodal displacements’ and rotations’ accelerations array and G ¼ ½G1 ; . . . ; Gm  is the wished value of the nodal displacements’ and rotations’ accelerations array (here it is set equal to zero). In a single objective Differential Evolution, each individual is randomly placed in the d-dimensional space as a candidate solution. The mutation operator produces a trial vector for each individual of the current population by mutating a target vector with a weighted differential. This trial vector will, then, be used by the crossover operator to produce offspring. For each parent, si ðtÞ, the trial vector, ui ðtÞ, is generated as follows: a target vector, si1 ðtÞ, is selected from the population, such that i – i1 . Then, two individuals, si2 and si3 , are selected randomly from the population such that i; i1 ; i2 and i3 are all different. Using these individuals, the trial vector is calculated by perturbing the target vector as follows (in the continuous case)

ui ðtÞ ¼ si1 ðtÞ þ bðsi2 ðtÞ  si3 ðtÞÞ

ð21Þ

where b 2 ð0; 1Þ is the scale factor. The upper bound of b is usually the value 1 because as it has been proved if the b > 1 there is no improvement in the solutions [6,5] and the most usually utilized value is b ¼ 0:5. The target vector si1 in Eq. (21) is a random member of the population. The vectors si2 and si3 are selected usually at random. Except of this classic mutation operator, there is a number of different mutation operators that have been proposed [6,5]. In the following, these operators are presented. 1. In this operator, instead of a random target vector, the best vector is used and, thus, Eq. (21) is transformed as follows:

ui ðtÞ ¼ sopt ðtÞ þ bðsi2 ðtÞ  si3 ðtÞÞ

ð22Þ

2. In this mutation operator, instead of one difference vector a number (nu ) of difference vectors are used. This number is a parameter of the algorithm. The equation of the mutation operator is now the following: nu X ui ðtÞ ¼ si1 ðtÞ þ b ðsi2 ;l ðtÞ  si3 ;l ðtÞÞ

ð23Þ

l¼1

3. In this mutation operator, the only difference from the previous operator (Eq. (23)) is that instead of a random target vector the best vector is used: nu X ui ðtÞ ¼ sopt ðtÞ þ b ðsi2 ;l ðtÞ  si3 ;l ðtÞÞ

ð24Þ

l¼1

4. In this mutation operator, the best and the random target vectors are combined: nu X ui ðtÞ ¼ csopt ðtÞ þ ð1  cÞsi1 ðtÞ þ b ðsi2 ;l ðtÞ  si3 ;l ðtÞÞ

ð25Þ

l¼1

The parameter c shows the influence of the best and random target vectors and if its value is close to zero the algorithm has more exploration abilities, while if it is close to 1 the algorithm has more exploitation abilities [6]. Usually, in the beginning of the generations this value is set equal to 0 and as the generations increase this value is increased towards the value 1.

The Multi-Objective Differential Evolution (MODE) algorithm has to find the leader of each of the individuals. A number of different ways have been proposed [54]. The following strategy is used for confronting this issue: from the initial population of individuals an external archive is created using the nondominated solutions as in a number of MODE implementations [54]. In this archive, a number of classifications are performed based on each objective function. The feasible population is, then, evaluated by each objective function separately and the corresponding values are stored, while a global best solution arises for each one of the k objective functions. The initialization of the algorithm ends with the creation of the initial Pareto set with the nondominated solutions inside the initial population. The iterative procedure starts with the mutation operator which produces a trial vector for each individual of the current population by mutating a target vector with a weighted differential. An important issue is which individuals will be selected for the production of the trial vector, which individual will play the role of the parent and how the solutions of the Pareto front will affect the trial vectors. In this paper, three new mutation operators are applied that differ from the ones presented previously and lead to different equations for the trial vector than the ones presented previously (Eqs. (21)–(26)). Thus, the three different equations proposed, which are denoted as MODE1, MODE2 and MODE3, are, respectively, the following:

ui ðtÞ ¼ Paretoi1 ðtÞ þ bðsi2 ðtÞ  si3 ðtÞÞ

ð27Þ

ui ðtÞ ¼ si1 ðtÞ þ bðParetoi2 ðtÞ  Paretoi3 ðtÞÞ

ð28Þ

ui ðtÞ ¼ Paretoi1 ðtÞ þ bðParetoi2 ðtÞ  Paretoi3 ðtÞÞ

ð29Þ

where the solutions si are random solutions from the population while the solutions Paretoi are random solutions from the Pareto front. Thus, in the first version (MODE1), the target vector is selected randomly from the Pareto front while the other two vectors are selected from the population. In the second version (MODE2), the target vector is selected from the population and the other two vectors are selected randomly from the Pareto front. Finally, in the third version (MODE3), all vectors are selected randomly from the Pareto front. The reason that we use these three different operators is that, as it is mentioned previously, the most important part of Differential Evolution algorithm is the suitable calculation of the leader individuals from the external archive. If we use the classic mutation operators of a DE algorithm, we could not use the external archive in this phase and, thus, we would not use the nondominated solutions. In this phase three different versions of the external archive in combination with solutions of the population are selected in order to see which of them could give better Pareto front in our problem. In the first version, the random solution of the Pareto front is used as the target vector and the other two solutions which produce the weighted difference will be selected randomly from the population. As it was mentioned in the description of the Differential Evolution algorithm the target vector can be selected with a number of different ways. One of them is the selection of the best member. This is a very challenging issue in a Multi-objective Optimization problem, because as we have a number of different equivalent vectors (solutions) it is very difficult to find which one is the best or the leader. Thus, in this version, the leader is selected randomly from the Pareto front. In the second version, we use a variant where the target solution is selected randomly from the population and not from the Pareto front. The

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reason is that we would like to see the performance of the algorithm if the target solution was taken from the population. The other two solutions were selected from the Pareto front as we would like to, also, take advantage of the good solutions of the Pareto front. In the final version, only solutions from the Pareto front are used as we would like to develop a version that only the best solutions will be included in the mutation phase. As in the Pareto front there are solutions in different places of the solution space, the selection of the three solutions in this phase from the external archive will increase the exploration abilities of the algorithm. For the other two versions (MODE1 and MODE2) the exploration ability is increased as the one or two (depending on the version) solutions are in the Pareto front and the other two or one (depending on the version) solutions are from the population which give to the algorithm the possibility of searching for a better solution in promising regions (where the solution of the Pareto front is) and in possible non-promising regions (where the solution of the population may be). For the discrete values of the individuals, after the calculation of the target vector, the following equations are used in order to transform the continuous values calculated by Eq. (21) into discrete values

sigðui Þ ¼

ui ðtÞ ¼

1 1 þ expðui Þ



1; if rand3 < sigðui Þ 0;

if rand3 P sigðui Þ

ð30Þ

ð31Þ

After the completion of the mutation phase of the algorithm, a binomial crossover operator [6] or uniform crossover operator [5] is applied. In this crossover operator, the points are selected randomly for the trial vector and for the parent. Initially, a crossover operator number (Cr) is selected [5] that controls the fraction of parameters that are selected from the trial vector. The Cr value is compared with the output of a random number generator, randi ð0; 1Þ. If the random number is less or equal to the Cr, the corresponding value is inherited from the trial vector, otherwise it is selected from the parent, thus

s0i ðtÞ ¼



ui ðtÞ; if randi ð0; 1Þ 6 Cr si ðtÞ;

otherwise:

ð32Þ

Thus, the choice of the Cr is very significant because if the value is close or equal to 1, then, most of the values in the offspring are inherited from the trial vector (the mutant) but if the value is close to 0, then, the values are inherited from the parent [5]. After the crossover operator, the fitness function of the offspring s0i ðtÞ is calculated and if it is better than the fitness function of the parent, then, the trial vector is selected for the next generation, otherwise the parent survives for at least one more generation. The algorithm stops after a prespecified number of generations. In the next iterations, in order to insert an individual in the archive, there are two possibilities. Firstly, the individual is nondominated with respect to the contents of the archive and, secondly, it dominates any individual in the archive, but in this case all the dominated individuals have to be deleted from the archive. In order to keep computing time under control, the size of the archive has to be restricted in order its size not to be increased very quickly. Thus, a number, A, is used which is a parameter of the problem. When the number of individuals in the archive becomes greater than A, then, a classification of the individuals based on each objective function is performed and in the archive the A=k best individuals based on each classification remain. A pseudocode of the proposed Multi-Objective Differential Evolution algorithm is presented in Table 2.

5. 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 0:02 m  0:02 m. Material constants for the host beam are: Elastic modulus 73  109 N=m2 , density 2700 kg=m3 . 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 is a periodic sinusoidal loading pressure (sinð20tÞ) that influences the displacement of the second and the fourth (free end) finite elements. The purpose of the fuzzy controller is to reduce the oscillation. The controller is collocated, takes as input the displacement and the velocity of the free end and gives back the control force to be applied at the same point. The results obtained with the fuzzy controller optimized by MODE are compared with those of a fuzzy controller optimized by Multi-Objective Particle Swarm Optimization (MOPSO) algorithm. The parameters of the MODE1–MODE3 algorithms are selected after thorough testing and they are: the number of individuals is set equal to 30, the number of generations is equal to 20, the parameter b ¼ 0:7 and the parameter Cr ¼ 0:8 and the parameter A is set equal to the number of individuals. The whole algorithmic approach was implemented in MATLAB. The membership functions obtained by the fuzzy controller when optimized by MODE3 (Figs. 2 and 3) are presented in the following. In order to compare the proposed algorithm with another evolutionary optimization algorithm, a Multi-objective Particle Swarm Optimization (MOPSO) algorithm [16] is used. Particle Swarm Optimization (PSO) is a population-based swarm intelligence algorithm that was originally proposed by Kennedy and Eberhart as a simulation of the social behavior of social organisms such as bird flocking and fish schooling [55]. A comparison with a multiobjective version of the Particle Swarm Optimization is meaningful as the single objective Differential Evolution algorithm which is presented in [15] was compared with a single objective Particle Swarm Optimization algorithm [56,15]. The Multi-Objective Particle Swarm Optimization is presented and analyzed in [16]. In this algorithm we use the same objective functions as in the proposed algorithm and the same cantilever beam, thus, a comparison of these two algorithms can be performed. In [16] nine different versions of a Multi-Objective Particle Swarm Optimization (MOPSO) were used, where their main difference was in the use of the velocity equation of the particles. For more information

Table 2 Multi-Objective Differential Evolution. Initialization Selection of the number of individuals Generation of the initial population Evaluation of the population for each objective function Selection of the mutation operator Initialization of the Pareto set Main Phase Do while the maximum number of generations has not been reached: Do while the maximum number of parents has not been reached: Selection of the parent vector Creation of the trial vector by applying the mutation operator Evaluation of the individuals for each objective function if the offspring dominates the parent then Replace the parent with the offspring for the next generation endif enddo Update of the Pareto set Enddo Return Pareto front

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7

M. Marinaki et al. / Computers and Structures xxx (2014) xxx–xxx

Degree of membership

Input 1 MF FarL

1

CloseL

Far

Close

Equilibrium

R

R

0.8 0.6 0.4 0.2 0 −8

−6

−4

−2

0

2

4

6

8

−5

x 10

Displacement

Degree of membership

Input 2 MF Left

1

Null

Right

0.8 0.6 0.4 0.2 0 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1 −3

Velocity

x 10

Fig. 2. Membership functions for the inputs of the fuzzy controller optimized by MODE3.

Output MF Min

High− Med− Low− Null Low+ Med+ High+

Max

1

Degree of membership

0.8

0.6

0.4

0.2

0

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Control Force Fig. 3. Membership functions for the outputs of the fuzzy controller optimized by MODE3.

about the MOPSO versions please see [16]. The parameters of the MOPSO1–MOPSO9 algorithms used are: the number of swarms is set equal to 1, the number of particles is equal to 30, the number of generations is equal to 20, the parameters e1 and e2 are both set equal to 2.05 and the parameter A is set equal to the number of particles. The reason that the main parameters (number of individuals or particles and number of generations) of the two algorithms MODE and MOPSO are selected to be equal is that it is

desired to have the same number of function evaluations and to perform fair comparisons between the two algorithms. As it was very difficult to present and analyze the results of the three objective functions simultaneously, we test the algorithm initially for each combination of the two objective functions. In Figs. 4–6 the Pareto Fronts produced using the objective functions 1 and 2, the objective functions 1 and 3 and the objective functions 2 and 3 are presented, respectively. In these figures, the results of

Please cite this article in press as: Marinaki M et al. Fuzzy control optimized by a Multi-Objective Differential Evolution algorithm for vibration suppression of smart structures. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2014.09.018

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M. Marinaki et al. / Computers and Structures xxx (2014) xxx–xxx

three of the nine MOPSO variants of [16] are presented. The selection of these three variants was based on the produced Pareto fronts that each variant of MOPSO algorithm has given. In Fig. 4, the two algorithms that gave the most efficient Pareto fronts are MODE1 and MODE2. The range of values in MODE1 is better than the ranges of values in all other methods. However, the use of MODE2 algorithm produces a better Pareto front but with smaller range of values. These two methods gave, also, the smoother Pareto fronts. The Pareto front produced using MODE3 has the same characteristics as the Pareto front produced using MODE2 but with not so good values as the ones produced using MODE2. Pareto fronts produced using MOPSO1 and MOPSO2 have the same characteristics with Pareto front produced using MODE1 but with not so good values as the ones produced using MODE1. However, the Pareto front produced using MOPSO3 is better than the one produced using MOPSO1. Finally, the worst of the Pareto fronts is the one produced using MOPSO2 algorithm.

In Fig. 5, the Pareto fronts produced using the objective functions 1 and 3 are presented. As depicted in Fig. 5 the most efficient Pareto fronts are the ones produced using MODE1 and MODE2. However, the Pareto front produced by MODE1 is better than the one produced by MODE2, except that there is a small number of better solutions in the Pareto front produced by MODE2. In this figure the Pareto front produced by MODE3 is better than the one produced in Fig. 4, both in the values of the solutions and in the range of the values. The best Pareto front produced using the three MOPSO versions is the one produced using MOPSO3 algorithm. The Pareto front produced using MOPSO2 is better than the one produced in Fig. 4 as the range of the values is expanded in the whole solution space and the solutions are competitive with the solutions produced by the other algorithms. Finally, the Pareto front produced using MOPSO1 algorithm is a very efficient Pareto front with a very good expansion of the values in the solution space but with worst values than the ones produced using all the other algorithms.

−4

x 10

5 MODE1 MODE2

4.5

MODE3 MOPSO1 MOPSO2

objective function 2: err2

4

MOPSO3

3.5 3 2.5 2 1.5 1 0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 −4

objective function 1: err1

x 10

Fig. 4. Pareto fronts using objective functions 1 and 2 for all algorithms used in the comparisons.

0.13 MODE1 MODE2 MODE3 MOPSO1 MOPSO2 MOPSO3

0.12

objective function 3: err3

0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

objective function 1: err1

1 −4

x 10

Fig. 5. Pareto fronts using objective functions 1 and 3 for all algorithms used in the comparisons.

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M. Marinaki et al. / Computers and Structures xxx (2014) xxx–xxx

0.13 MODE1 MODE2

0.12

MODE3 MOPSO1

objective function 3: err3

0.11

MOPSO2 MOPSO3

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03

0

1

2

3

4

5

6 −4

objective function 2: err2

x 10

Fig. 6. Pareto fronts using objective functions 2 and 3 for all algorithms used in the comparisons.

Table 3 Nondominated solutions for each of the variants of the MODE and MOPSO algorithms using all three objective functions. MODE1 6 NDSa 1.81E05 3.81E06 6.34E06 4.95E06 1.38E05 1.21E05

MODE2 6 NDS 0.000163 0.000117 0.000127 0.000161 0.000121 0.000115

0.0429 0.0864 0.0466 0.0652 0.0785 0.0792

MOPSO1 6 NDS a 9.49E06 5.42E06 1.033E06 1.87E06 2.60E06 1.33E05

1.87E04 0.000121 6.97E05 6.87E05 8.24E05 0.000101

0.032459 0.033598 0.052880 0.049884 0.045905 0.045598

a

0.121 0.0511 0.0493 0.0457 0.0444 0.04408

1.01E05 1.33E06 1.87E06 2.60E06 1.33E05 9.78E06

7.86E05 0.000121 6.97E05 6.87E05 8.24E05 0.000101

0.049805 0.033598 0.052880 0.049884 0.045905 0.045598

5.42E06 7.40E06 2.60E06 1.33E05

0.000206 6.97E05 6.87E05 8.24E05 0.000101 1.96E04

0.043868 0.052880 0.049884 0.045905 0.045598 0.045838

0.033598 0.052880 0.049884 0.045905

5.42E06 1.33E06 1.87E06 2.60E06 3.00E06

7.11E05 0.000107 0.000137 7.57E05 7.06E05 0.000129

0.052 0.04404 0.04296 0.04807 0.05488 0.04391

1.33E06 1.87E06 2.60E06 1.33E05 9.78E06

6.97E05 6.87E05 8.24E05 0.000101 1.96E04

0.052880 0.049884 0.045905 0.045598 0.045838

0.001304 6.97E05 6.87E05 8.24E05 0.000101

0.040532 0.052880 0.049884 0.045905 0.045598

0.000121 6.97E05 6.87E05 8.24E05 0.000101

0.033598 0.052880 0.049884 0.045905 0.045598

MOPSO6 5 NDS 0.000121 7.34E05 8.24E05 0.000101

0.033598 0.058860 0.045905 0.045598

MOPSO8 5 NDS 0.000121 6.97E05 6.87E05 8.24E05

1.63E06 4.35E06 5.90E06 2.18E06 1.48E06 6.76E06 MOPSO3 5 NDS

MOPSO5 4 NDS

MOPSO7 4 NDS 5.42E06 1.33E06 1.87E06 2.60E06

7.61E05 0.000105 0.000169 0.000182 0.00022 0.000443

MOPSO2 6 NDS

MOPSO4 6 NDS 5.65E06 5.42E06 1.33E06 1.87E06 2.60E06 1.33E05

2.38E06 5.20E06 1.05E05 8.92E06 4.46E06 8.58E05

MODE3 6 NDS

6.34E05 1.33E06 1.87E06 2.60E06 1.33E05 MOPSO9 5 NDS

0.000121 6.97E05 6.87E05 8.24E05 9.03E05

0.033598 0.052880 0.049884 0.045905 0.045413

5.42E06 1.33E06 1.87E06 2.60E06 1.33E05

NDS = number of nondominated solutions.

In Fig. 6, the Pareto fronts produced using the objective functions 2 and 3 are presented. The produced Pareto fronts using the three versions of MODE algorithm have the same characteristics than the ones produced using the objective functions 1 and 3 and presented in Fig. 5. However, the results produced using MODE3 is slightly better and are competitive with the results produced using MODE1 algorithm. The only efficient Pareto front produced using a MOPSO version is the one produced using

MOPSO2 algorithm. The Pareto fronts produced using MOPSO1 and MOPSO3 algorithms are not very efficient and not competitive with the ones produced using the other four algorithms. As a concluding remark of the three figures is that the Pareto fronts produced using MODE1 algorithm are better than the ones produced using all the other algorithms. The MODE1 algorithm produces very efficient Pareto fronts with better solutions and the range and the expansion of the values in the solution space

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M. Marinaki et al. / Computers and Structures xxx (2014) xxx–xxx

objective function 3: err3

10

0.09 0.085 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 1.7 1.6 1.5

−4

x 10

1.4 1.3 1.2 1.1

objective function 2: err2

0.2

0.4

0.6

0.8

1.4

1.2

1

1.6

2

1.8 −5

x 10

objective function 1: err1

Fig. 7. Membership functions for the outputs of the fuzzy controller optimized by MODE3.

are better than the ones produced using the other algorithms. The MODE2 and MODE3 algorithms are very competitive between them and the results and Pareto fronts produced using these two algorithms are very efficient. The Pareto fronts produced using MOPSO algorithms are not very efficient in all cases. More precisely, although in Figs. 4 and 5 the Pareto fronts produced using MOPSO3 algorithm are very efficient and with competitive values with the ones produced using MODE versions, in Fig. 6 the produced Pareto front is not very efficient. Almost, the same holds for MOPSO1 algorithm with the exception that in the case of Fig. 5 the solutions that consist the Pareto front are not so good than the ones produced using MOPSO3 algorithm. The only algorithm that has not similar performance in all cases is MOPSO2 algorithm

−3

x 10

Velocity

Displacement

2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

Displacement before (blue) and after (red) fuzzy

0

0.2

0.4

0.6

0.8

1

5 4 3 2 1 0 −1 −2 −3 −4 −5

0.15

−5

Displacement before (blue) and after (red) fuzzy

0

0.2

−3

0.8

−0.2

1

2 0 −2

0

0.2

0.4

0.6

Time (sec.)

0.8

1

0

0

0.6

0.8

Acceleration before (blue) and after (red) fuzzy

0.5

−1.5

0.4

Velocity before (blue) and after (red) fuzzy

−1

−6

0.2

Time (sec.)

−0.5

−4

0

Time (sec.)

Acceleration

Velocity

Displacement

0.6

1

4

−8

0.4

1.5

6

0 −0.05

−0.15

x 10

8

0.1 0.05

−0.1

Time (sec.)

x 10

Acceleration before (blue) and after (red) fuzzy

Velocity before (blue) and after (red) fuzzy

Acceleration

−4

x 10

where while in the first case it performs worst than all the other algorithms, in the second case its results are competitive with the other MOPSO algorithms and in the last case the results are better than the other two MOPSO algorithms and competitive with the results of MODE algorithms. In Table 3, the nondominated solutions of the 3 variants of the proposed algorithm and of the 9 variants of the MOPSO algorithm are presented. The reason that we present the values of the Pareto fronts when three objective functions are used using a Table is that if these values were presented using figures, a very confusing figure would be presented and it would be very difficult to draw safe conclusions. An indicative Pareto front with three objective functions is presented in Fig. 7. As it can be seen from Table 3,

0.2

0.4

0.6

Time (sec.)

0.8

1

0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05

0

0.2

0.4

0.6

0.8

1

1

Time (sec.)

Fig. 8. Displacement, velocity and acceleration at the free end and at the other (second) finite element for the fuzzy controller optimized by MODE2.

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M. Marinaki et al. / Computers and Structures xxx (2014) xxx–xxx

the number of Pareto optimal solutions for the MODE algorithm is always equal to 6, while for the MOPSO algorithm varies between 4 and 6. The most important outcome is that all algorithms found almost the same solutions in the Pareto front which means that the proposed algorithm is suitable for the solution of the problem. In each iteration of the algorithm the external archive gives the nondominated solutions. The size of this archive was continuously changing during the iterations. The Pareto front of the nondominated solutions for the MODE1 is presented in Fig. 7. It should be noted that comparisons are performed based on the solutions of the Pareto Front as comparison based on the actual values of the displacement, velocities and acceleration are very difficult to be performed due to the fact that more than one objective functions are used. However in order to show the efficiency of the proposed method a solution from the Pareto Front is selected (Fig. 8). The displacement, the displacement velocity and the displacement acceleration for the fuzzy control optimized by MODE2 are presented in the graphs below (Figs. 8). The first three figures correspond to the displacement, the displacement velocity and the displacement acceleration at the free end and the other three figures correspond to the displacement, the displacement velocity and the displacement acceleration for the other (second) finite element. As it can be seen from these figures, the fuzzy controller optimized by MODE2 gives superior control results compared to the no control case. More precisely, the displacements, the displacement velocities are almost zero in the case of fuzzy controller optimized by MODE2 while the displacement accelerations have small oscillations but their values are near to zero as well.

6. Conclusion In this paper, a fuzzy control system optimized by MultiObjective 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 research is intended to be focused in the application of the proposed algorithm for the design of other smart structures like plates, shells and in the modification of the method in order to be used for the optimal design of other parts of the fuzzy system like the definition of the control rules. References [1] Storn R. On the usage of differential evolution for function optimization. In: Proceedings of the biennial conference of the north american fuzzy information processing society; 1996. p. 519–23. [2] Storn R. System design by constraint adaptation and differential evolution. IEEE Trans Evolution Comput 1999;3(1):22–34. [3] Storn R, Price K. Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 1997;11(4): 341–59. [4] Feoktistov V. Differential evolution – in search of solutions. NY: Springer; 2006. [5] Price KV, Storn RM, Lampinen JA. Differential evolution: a practical approach to global optimization. Berlin: Springer; 2005. [6] Engelbrecht AP. Computational intelligence: an introduction. England: John Wiley and Sons; 2007. [7] Abraham A, Jain L, Goldberg R. Evolutionary multiobjective optimization: theoretical advances and applications. London: Springer-Verlag; 2005. [8] Coello Coello CA, Van Veldhuizen DA, Lamont GB. Evolutionary algorithms for solving multi-objective problems. Springer; 2007. [9] Della CN, Shu D. Vibration of beams with piezoelectric inclusions. Int J Solids Struct 2007;44:2509–22. [10] Foutsitzi G, Marinova DG, Hadjigeorgiou E, Stavroulakis GE. Robust H2 vibration control of beams with piezoelectric sensors and actuators. In: International conference, physics and control, vol. 1; 2003. p. 157–62.

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Please cite this article in press as: Marinaki M et al. Fuzzy control optimized by a Multi-Objective Differential Evolution algorithm for vibration suppression of smart structures. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2014.09.018