Aerospace Science and Technology 11 (2007) 174–182
www.elsevier.com/locate/aescte
Optimal positioning of piezoelectric actuators on a smart finusing bio-inspired algorithms
Ali Reza Mehrabian ∗,1, Aghil Yousefi-Koma 2
Advanced Dynamic and Control Systems Lab., School of Mechanical Engineering, College of Engineering, University of Tehran,P.O. Box 14875-347, Tehran, Iran
Received 29 May 2006; received in revised form 6 October 2006; accepted 8 January 2007
All air vehicles create a type of vortex, which is an energetic-swirling mass of air, called a trailing vortex off each wingtipwhen the plane is in motion. These trailing vortices can bethought of as small tornados that grow larger as they extend be-hind the plane. A vortex generator is really nothing more thana miniature wing-like device designed specifically to create avortex. Even though a vortex creates drag, it can also provideadvantages that outweigh its negative impact. One such advan-tage is the ability of a vortex to speed up the flow of air over awing and allow a plane to reach a higher angle of attack than itwould be able to otherwise. For high-performance twin-tail air-craft (HPTTA) such as the F/A-18 and F-15, buffet induction
tail vibrations occur when unsteady pressures associated withseparated flow, or vortices, excite the vibration modes of thevertical-fin-structural assemblies (see Fig. 1) [1,2]. At high an-gles of attack, flow separates at the leading edge of the wings,and vortices are generated at different locations such as thewing fuselage interface or the leading edge extensions. Thisphenomenon, along with the aeroelastic coupling of the tailstructural assembly, results in vibrations that can shorten thefatigue life of the empennage assembly and limit the flight en-velope due to the large amplitude of the fin vibrations. Thisis a significant problem particularly for the F/A-18 aircraft,which requires frequent inspection to prevent catastrophic fail-ure. There have been numerous studies on monitoring and con-trolling the buffet loads on both scaled model and actual aircraftby the use of active vibration control [3–5].
There are essentially two major techniques to control the tailbuffet problem: flow control or structural control. The flow con-trol methods aim into modifying the vertical flow-field aroundthe vertical tails to reduce the buffet loads. Passive flow control[6] and active flow control [7] methods have been proposed, butonly the passive methods received the most attention [5]. Active
A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182 175
vibration control can be achieved either by active rudder control[8] or integrated smart structures on the tail [9].
The physical model considered in this study is a simplescaled model of the vertical tail fin of an F/A-18 fighter jet,which approximately replicated the first two natural frequen-cies of the full-scale vertical fin [10]. This model included aflexible aluminum fin, with a thickness of 1 mm, fixed at thebase. A total of 24 piezo-ceramic actuators are bonded ontoboth sides (12 on each side) of the aluminum plate. Two ac-celerometers are used to monitor the dynamic response of thefin tip. Due to the integration of actuators and sensors with thehost structure, it is usually very cumbersome, if not impossible,to develop a mathematical model for a complex smart struc-ture. Thus, FEM is used to predict the structural response. Forinstance an electromechanical coupling effect of piezoelectricmaterials is employed to establish a FEM model of a flexibleplate with piezoelectric sensors and actuators [10].
This paper presents a new technique for optimal positioningof actuators on a smart structure. Neural networks are employedto find an optimal 3-dimensional surface for the Frequency Re-sponse Function (FRF) peak data obtained from a finite elementmodel of a flexible aircraft fin. This is performed on a completeset, where the FRF was measured at 48 points on the smart fin.Neural networks are shown to be a suitable algorithm for ob-taining a proper surface fit. Finally, the position of a single pairof piezoelectric actuator on the fin is optimized using weightingfactors on each of the modal surfaces. These weighting fac-tors were dependent on the desired control authority over eachmode to determine the optimum actuator location. In this studya novel numerical optimization algorithms called invasive weedoptimization (IWO) algorithm, which is inspired from coloniz-ing behavior of weeds, is employed to find the optimal positionof piezoelectric actuators.
2. Fin geometry and finite element modeling
Finite element methods are often used to model and predictthe dynamic response of a structure with integrated sensors andactuators. An excellent early example of this is the study byRahmoune et al. [11], where the electromechanical couplingeffect of piezoelectric materials was used to establish a finiteelement model of a flexible plate with bonded piezoelectric
Fig. 2. Experimental setup of the flexible fin with piezoelectric actuators (Na-tional Research Council Canada (NRC)).
Table 1Material properties of the structure components of the smart fin
Fig. 3. FEM model of the fin in experimental configuration [12].
sensors and actuators. In the present paper, PATRAN is usedto develop the FE model of the flexible fin, while NASTRANis employed as the FEM solver with thermal load analogy forpiezoelectric actuators. The conventional assumptions associ-ated with FE modeling of smart structures such as the assump-tion of a perfect bond and neglect of the glue stiffness and massapply to the model in this study [12,13]. The finite elementmodel was based on the experimental apparatus, which con-sisted of a flexible aluminum fin with a thickness of 1 mm fixedat the fin root shown in Fig. 2. The material properties of thestructure components are given in Table 1. Shell and solid ele-ments were employed in PATRAN for the aluminum plate andPZT, respectively. The FE model equivalent to the experimentalsetup is presented in Fig. 3. Natural frequencies and dampingratios of the integrated smart structure are presented in Table 2.Structural damping obtained from experiment was also incorpo-rated into the FEM model. The effects of the gluing and wiringof actuators resulted in higher natural frequencies recorded for
176 A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182
Table 2Modal frequencies and damping ratios of the smart fin
Mode Frequency [Hz] Damping ratio fromexperimentExperiment FEM Error [%]
Fig. 4. The fin mode shapes: (a) First mode (First bending), (b) Second mode(First torsional), (c) Third mode (Second bending).
Fig. 5. FEM of the smart fin showing actuator size [12].
the experimental apparatus as compared with the FE model.The first three dynamic mode shapes are shown graphically inFig. 4. For both the experimental and finite element models,five accelerometers were used to measure the dynamic responseof the integrated smart structure. These were positioned so thatthey could measure relatively high responses from the first threedynamic modes of the smart fin. Fig. 5 shows position of twoaccelerometers (numbers 1 and 3) employed in this study.
For the actuator position optimization, the same finite ele-ment model with the 48 possible actuator locations (6 rows of8 actuators) as shown in Fig. 5 was used. For each vibrationmode, the resulting FRF at the accelerometer was measuredonce for each of the 48 possible actuator positions (on each sideof the fin).
Each 25×25 mm actuator was excited within a range of 2 Hzaround each of the first three fundamental frequencies of theintegrated fin, corresponding to the first mode (first bending),second mode (first torsional), and third mode (second bending).Although the finite element model for the configuration opti-mization had 47 passive actuators for each FRF measurement,the compromise between stiffness and mass contributed by thepassive actuators, in fact resulted in less than 10% differencein fundamental frequency for all three modes as compared witha single actuator on the fin (17.5 Hz vs. 17.0 Hz, 47 Hz vs.
Fig. 6. Sample FRF of the fin tip acceleration.
49.1 Hz, and 73 Hz vs. 80.4 Hz). A typical example of the FRFof the fin is shown in Fig. 6 [12,13]. Phase angles were irrel-evant for the optimization of actuator configuration, but wereconsidered in the development of an active control system forthe fin by Yousefi-Koma et al. [9].
3. Surface fitting by neural networks
In order to optimize the position of a single pair of piezo-electric actuators continuously within the actuator test area, acontinuous fitness function describing the actuation authorityof a pair of actuators anywhere within the test area is required.This fitness function may be developed using a genetic algo-rithm designed to generate the best three-dimensional polyno-mial surface fit of the piezoelectric actuator FRF peak valueswithin the test area, and to a limited degree by extrapolation,outside of the test area as well [12,13]. Next, a second geneticalgorithm can be used to perform the optimization of actuatorposition. The main difficulty in the proposed method is tuning alarge number of coefficients for the three-dimensional polyno-mial surface to fit the FRF peak data, which makes the processof data fitting complicated and time consuming. For instance, a9th order polynomial needs 55 coefficients to be optimized tofit the FRF peak data, while there is no guarantee that a polyno-mial is the best fit for the data.
It is very well known that a multi-layer perceptron (MLP)neural-network (NN) can be used as a general function approx-imator that can approximate any function with a finite numberof discontinuities, arbitrary well, given sufficient neurons in thehidden layer [14]. Therefore, a MLP NN is employed to gener-ate three-dimensional surface fit for the FRF peak data.
3.1. MLP NN architecture and training algorithm
A NN consists of a series of layers starting with an inputlayer (also called ‘input vector’), ending with an output layerand having a number of ‘hidden’ layers in between. Each layerconsists of a series of linear or nonlinear nodes. The output ofeach node in a nonlinear layer is a nonlinear transformation ofthe weighted sum of the outputs of the nodes of the previouslayer. The weights that connect outputs of one layer to the nextare the parameters that characterize the system model. Fig. 7illustrates and MLP NN which consists of a single layer ofS perceptron neurons connected to R inputs through a set ofweights wi,j that is the strength of the connection from the j thinput to the ith neuron. The MLP shown in Fig. 7 consists of
A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182 177
Fig. 7. Architecture of a two-layer perceptron neural-network with a single hid-den layer with tan-sigmoid activation function and a linear output layer.
two layers and one input vector. The input transfer function ofthe first layer (also called ‘hidden layer’) is ‘tan-sigmoid’ andtransfer function for the second layer is ‘pure-linear’. The hid-den layer performs nonlinear summation of the input signalswhile the second layer only applies a linear summation on theoutput signal of the first layer.
System modeling using NN involves ‘training’ the neuralnetworks using a set of historical data. Training is an off-lineoptimization procedure that modifies the weights of the neuralnetwork in order to minimize the error (in some norm) be-tween the network output and the desired output value over theentire data set. A single data set is usually processed by the op-timization procedure for many times before an acceptable fitis obtained between NN output prediction and the actual out-put measurement. Each cycle over the data set is termed anepoch. The trained NN is then used as a model for predic-tion/estimation [15].
3.2. Improving generalization
One of the problems that arise in connection learning by NNis over or under-fitting of the provided training examples. NN,like other flexible nonlinear estimation methods such as kernelregression and smoothing splines, can suffer from either under-fitting or over-fitting. A network that is not sufficiently complexcan fail to detect fully the signal in a complicated data set, lead-ing to under-fitting. Under-fitting can also occur when gradient-based learning methods is used. In this case, for a NN providedwith sufficient number of layers and weights, the optimization(learning) algorithm may fail to find the global minimum of NNweights and stuck in a local minimum far from the global min-ima resulting in large errors on the training data set.
A network that is too complex may fit the noise, not just thesignal, leading to over-fitting. Over-fitting is especially danger-ous because it can easily lead to predictions that are far beyondthe range of the training data with many of the common typesof NN. Over-fitting can also produce wild predictions in MLPeven with noise-free data.
There are different approaches to overcome generalizationdifficulties. The best way to avoid over-fitting is to use lots oftraining data. For noise-free data, five times as many trainingcases as weights may be sufficient to prevent over-fitting. Butthe number of weights for fear of under-fitting cannot be re-duced arbitrarily.
Given a fixed amount of training data, there are different ap-proaches to avoiding under-fitting and over-fitting, and hencegetting good generalization:
Table 3Neural network properties for three-dimensional FRF peak data fitting for ac-celerometer 1
Note the all neural networks used are two-layer perceptron with tan-sigmoidactivation function in the hidden layer and linear function in the output layer.
a Levenberg–Marquardt (LM) training algorithm.b Mean squared error (MSE).c Bayesian regularization (BR) training algorithm.d Sum squared error (SSE).
Table 4Neural network properties for three-dimensional FRF peak data fitting for ac-celerometer 3
Note the all neural networks used are two-layer perceptron with tan-sigmoidactivation function in the hidden layer and linear function in the output layer.
a Levenberg–Marquardt (LM) training algorithm.b Mean squared error (MSE).
1) Model selection: This is concerned with the number ofweights, and hence the number of hidden units and lay-ers. The more weights there are, relative to the number oftraining cases, the more over-fitting amplifies noise in thetargets [16]. Thus the model and training data set must becoherent with each other.
2) Regularization: This involves modifying the performancefunction, which is chosen normally to be MSE. It is possi-ble to improve generalization if we modify the performancefunction by adding a term that consists of the mean of thesum of squares of the network weights and biases [14].
3) Jittering: Jitter is artificial noise deliberately added to theinputs during training. Training with jitter is a form ofsmoothing related to kernel regression. It is also closely re-lated to regularization methods such as weight decay andridge regression [17].
4) Early stopping: It is one of effective methods for improv-ing generalization. In this technique the available data isdivided into three subsets. The first subset is the trainingset, which is used for computing the gradient and updatingthe network weights and biases. The second subset is thevalidation set. The error on the validation set is monitoredduring the training process. The validation error will nor-
178 A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182
Fig. 8. Discrete FRF peak values for the first mode in magenta stripes, andcorresponding three-dimensional surface obtained by neural network for thefirst accelerometer.
mally decrease during the initial phase of training, as doesthe training set error. However, when the network beginsto over-fit the data, the error on the validation set will typi-cally begin to rise. When the validation error increases for aspecified number of iterations, the training is stopped, andthe weights and biases at the minimum of the validation er-ror are returned [14].
5) Data normalization: Before training, it is often useful toscale the inputs and targets so that they always fall withina specified range (between −1 and 1 for training MLP NNwith hidden layer with ‘tan-sigmoid’ activation function).The data normalization can help in generalization ability ofthe NN very much [14].
3.3. Training neural networks for three-dimensional surfacefitting
To train NN for three-dimensional surface fit, Levenberg–Marquardt (LM) training algorithm that is a gradient-basedbatch training method is employed to obtain optimal weightsof the network (in one case Bayesian regularization is em-ployed) [14,18]. In addition, mean squared error (MSE) per-formance function is employed as an index for measuring con-sistency between input and output signals (for Bayesian regu-larization (BR), which is a modified version of LM algorithm,sum squared error (SSE) [14] is used as the index).
Having three dynamic mode shapes, piezoelectric actuatorposition optimization should be performed for two differentaccelerometers, which were introduced in Fig. 5. Thus, sixdifferent MLP NNs should be employed to fit the FRF peakdata. The input vector of the NNs consists of a data set con-cerning column (in X-direction, between 1 and 8) and row (inY-direction, between 1 and 6) in which each actuator is placed;while the output is the FRF peak value of the actuator. Notingthat the number of training data is limited (FRF peak data is pro-vided for only 48 possible actuator positions), to prevent dataover-fitting, a small MLP NN is employed to fit the FRF peak
Fig. 9. Discrete FRF peak values for the second mode in magenta stripes, andcorresponding three-dimensional surface obtained by neural network for thefirst accelerometer.
Fig. 10. Discrete FRF peak values for the third mode in magenta stripes, andcorresponding three-dimensional surface obtained by neural network for thefirst accelerometer.
data for each mode that uses three/four neurons in the hiddenlayer. Also, to have best data fitting and to increase generaliza-tion ability of the NNs, input and output data are normalizedbetween −1 and 1. Employing a gradient-based learning algo-rithm (LM) increases the chance for the learning algorithm tobe trapped in a local minimum (training data under-fitting). Inorder to avoid this difficulty, 50 NNs have been trained for eachmode shape and the best fitting is chosen.
For two accelerometers, Tables 3 and 4 summarize structure,number of parameters of NNs, learning algorithm, performancemeasure and its value, and maximum number of epochs al-lowed for NNs. Note that BR training algorithm is used onlyfor the second dynamic mode shape for accelerometer 1 to geta better generalization since BR, generally, provides better gen-eralization performance than early stopping [14]. The discreteFRF peak values (in stripes but no shading), and the corre-sponding surface (in green shading) for the three modes of twoaccelerometers are illustrated in Figs. 8 to 13 (for colors see theweb version of the article).
A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182 179
Fig. 11. Discrete FRF peak values for the first mode in magenta stripes, andcorresponding three-dimensional surface obtained by neural network for thethird accelerometer.
Fig. 12. Discrete FRF peak values for the second mode in magenta stripes, andcorresponding three-dimensional surface obtained by neural network for thethird accelerometer.
4. Piezoelectric actuator position optimization
The modal surfaces obtained in previous section, weresummed with weighting factors to yield a combined surface
Table 5Optimum actuator position under different weighting factors for the first ac-celerometer
Caseno.
Weighting Weighting factors Optimalposition[mm]
Mode 1 Mode 2 Mode 3 X Y
1 Evenly weighted(acceleration control)
1 1 1 142 119
2 Normalized to meanFRF peak value
0.82 1 0.34 148 122
3 Normalized to max.FRF peak value
1 0.35 0.23 145 117
Fig. 13. Discrete FRF peak values for the third mode in magenta stripes, andcorresponding three-dimensional surface obtained by neural network for thethird accelerometer.
Table 6Optimum actuator position under different weighting factors for the third ac-celerometer
Caseno.
Weighting Weighting factors Optimalposition[mm]
Mode 1 Mode 2 Mode 3 X Y
1 Evenly weighted(acceleration control)
1 1 1 119 115
2 Normalized to meanFRF peak value
0.82 1 0.34 143 116
3 Normalized to max.FRF peak value
1 0.35 0.23 128 112
(please see Tables 5 and 6 for the values of weighting factorsfor each mode). By employing these combined surface, a novelnumerical optimization algorithm that is inspired by invasivecolonizing behavior of weeds designated as invasive weed op-timization (IWO) algorithm [19], is employed to determine theoptimum position of the piezoelectric actuators on the fin. Itshould be noted that different optimization algorithms can beemployed to solve the problem of the optimal positioning ofthe pair of piezoelectric actuators on the smart fin. These algo-rithms can either use gradient information of the peak FRF fit-ted surfaces or not. Since NNs are employed to fit the FRF peakdata, gradient information of the surface can be obtained withno difficulty. However, it is not always possible to obtain gradi-ent information for optimal actuator positioning problems. Thisis due to the fact that in some cases the necessary data for de-termining the optimal position of actuators is obtained directlyfrom numerical analysis techniques or experiment. This is oneof the reasons for employing non-gradient-based search algo-rithms for solving optimal actuator/sensor placement problemsin recent years [20–22]. Thus, in this paper, IWO algorithm,which is a non-gradient-based search algorithm, is employed tofind the optimal solution; so that the method can be utilized ifthe gradient information is not available.
180 A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182
4.1. Invasive weed optimization (IWO) algorithm
In recent years there have been an extensive research con-ducted for studying bio-inspired numerical optimization algo-rithms like genetic algorithms (GAs) [23], ant colony opti-mization (ACO) [24], particle swarm optimization (PSO) [25],Memetic algorithms (MAs) [26], and so on [27]. These numer-ical optimization algorithms are categorized with non-gradient-based search algorithms; they are designated as direct searchalgorithms as well. The main advantage of these algorithms isthat they only use the objective function and constrain values tosteer towards the optimal solution. Since derivative informationis not used, the direct search methods are typically slow, requir-ing many function evaluations for convergence. For the samereason, they can also be applied to many problems without ap-plying major changes in the algorithm.
Invasive weed optimization algorithm, IWO, which is in-troduced in [19] for the first time, is a bio-inspired numericaloptimization algorithm that simply simulates natural behaviorof weeds in colonizing and finding suitable place for growthand reproduction. To model and simulate colonizing behaviorof weeds for introducing a novel optimization algorithm, somebasic properties of the process is considered:
1) A finite number of seeds are being dispread over the searcharea (initializing a population);
2) Every seeds grows to a flowering plant and produces seedsdepending on their fitness (reproduction);
3) The produced seeds are being randomly dispread over thesearch area and grow to new plants (spatial dispersal);
4) This process continues until maximum number of plants isreached; now only the plants with higher fitness can sur-vive and produce seeds, others are being eliminated (com-petitive exclusion). The course continues until maximumiterations is reached and hopefully the plant with best fit-ness it the closest to the optimal solution.
IWO has some distinctive properties in comparison with tra-ditional GAs (and other numerical search algorithms), like re-production, spatial dispersal, and competitive exclusion [19].In addition, no genetic operators are employed in the proposedalgorithm, which makes it more dissimilar to GAs. In Appen-dix A, pseudocode for IWO algorithm is introduced and somesimulations are reported to show the ability of the algorithmin locating the global minimum of two benchmark functions.Extensive simulations are reported to compare performance ofIWO algorithm with other algorithms like GAs, PSO, and MAsfor different low and high dimension functions in [19], where itis shown that IWO algorithm is a competitive for other numer-ical stochastic optimization algorithms.
4.2. Actuator position optimization using IWO
Several optimization cases with associated weighting fac-tors, including those corresponding to acceleration control, aregiven in Tables 5 and 6 for accelerometer 1 and 3 respectively.Optimal piezoelectric actuator positions for two accelerome-
ters are illustrated in Figs. 14 and 15 respectively. Case 2 usedweighting factors which equalized the mean peak FRF valuesof all modes, while case 3 used weighting factors which equal-ized the maximum peak FRF values from all modes. Note thatin the case of a single actuator, positions within 20 mm of theroot were excluded from the optimization due to the potential
Fig. 14. The optimum actuator pair positions for the first accelerometer.
Fig. 15. The optimum actuator pair positions for the third accelerometer.
Fig. 16. Combined FRF surface normalized to maximum FRF peak value forthe first accelerometer.
A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182 181
for interference with the fin attachment. For better illustration,the combined modal surface plot with weighting normalized tothe maximum FRF peak value (case 3) is shown in Fig. 16 forthe first accelerometer.
5. Conclusion
A novel actuator optimal positioning algorithm is developedbased on neural networks for a smart fin as a scaled modelof F/A-18 vertical tail. The optimization methodology allowedthe placement of piezoelectric actuator pairs for effective vibra-tion reduction over the entire structure. The frequency responsefunction (FRF) of the system is recorded and maximization ofthe FRF peaks is considered as the objective function of theoptimization algorithm to find the optimal placement of thepiezoelectric actuators on the smart fin. Totally, six multi-layerperceptron neural networks (MLP NN) are employed to per-form surface fitting to the discrete data generated by the finiteelement method (FEM). Then, IWO algorithm has been em-ployed to find the proper position of actuators. Results indicatean accurate surface fitting for the FRF peak data as well as anoptimal placement of the piezoelectric actuators for vibrationsuppression. The proposed algorithm is able to solve any ac-tuator/sensor optimal positioning problem on different flexiblesmart structures.
Acknowledgement
The authors wish to acknowledge the valuable remarks andsuggestions made by the anonymous reviewers of the paperwhich led to many improvements.
Appendix A. An introduction to invasive weedoptimization (IWO) algorithm
Invasive weed optimization, IWO, is a novel numerical sto-chastic optimization algorithm inspired from colonizing weeds.Weeds are plants whose vigorous, invasive habits of growthpose a serious threat to desirable, cultivated plants making thema threat for agriculture. Weeds have shown to be very robustand adaptive to change in environment. It is tried to mimic ro-bustness, adaptation and randomness of colonizing weeds ina simple but effective optimizing algorithm designated inva-sive weed optimization [19]. Pseudocode for IWO algorithmis given as follows:
A.1. Pseudocode for IWO algorithm
Begin;Generate random population of N solutions (weeds);For i = 1 to the maximum number of generations;
Compute maximum and minimum fitness in the colony;For each individual w ∈ N ;
Compute number of seeds of w, corresponding toits fitness;3
3 Any member of the population of plants is allowed to produce seeds de-pending on its own fitness and the colony’s lowest and highest fitness: number
Randomly distribute generated seeds over the searchspace with normal distribution around the parentplant (w);4
Add the generated seeds to the solution set, N ;End;If N > Nmax;5
Sort the population N in descending order of theirfitness;Truncate population of weeds with smaller fitnessuntil N = Nmax;
End If;Next i;
End;
A.2. Convergence of IWO algorithm
Convergence of IWO algorithm is demonstrated by employ-ing the algorithm for locating global minimum of two bench-mark examples: two dimension Sphere and Griewank func-tions [28]. Fig. 17 shows the process of colonizing weedsaround the point with best fitness for Sphere function. It canbe observed that the plants grow towards the optimal pointfrom the initialization area. In their progress towards the op-timal point, plants with lower (worse) fitness are being ex-cluded, and only weeds with higher (better) fitness are allowedto be reproduced, which leads in colonization about the opti-mal point. The final value of the fitness function for Spherefunction is found to be fitness (x0) = 2.4362e–8, for the point:x0 = [−0.1413e–3,−0.0662e–3]. It is known that the opti-mal value of the function is zero for the point [0,0] in x–y
plane [19].Finding the global minimum of the Griewank function is a
challenging problem, which is the main reason for being a fa-vorite benchmark for optimization algorithms. It is known thatthe function has only one global minimum at [0,0] in x–y planebut numerous local minima. Fig. 18 illustrates the process ofobtaining optimal solution of the problem. To demonstrate mer-its of proposed algorithm, same simulation is performed usingGA toolbox provided in MATLAB®, where the initial condi-tions and number of maximum agents where identical in bothsimulations. As depicted in Fig. 18, the proposed algorithm out-performed GA in finding the global minimum of the Griewankfunction [19].
of seeds each plant use increases linearly from minimum possible seed produc-tion to its maximum [19].
4 The generated seeds are being randomly distributed over the d dimensionalsearch space by normally distributed random numbers with mean equal to zero;but varying variance. This means that seeds will be randomly distributed suchthat they abode near to the parent plant. However, standard deviation (SD), σ ,of the random function will be reduced from a previously defined initial value,σinitial, to a final value, σfinal, in every step (generation) [19].
5 After reaching the maximum number of allowable plants, pmax, a mecha-nism for eliminating the plants with poor fitness in the generation activates. Inthis way, only plants with higher fitness survive and are allowed to replicate.The population control mechanism also is applied to their children to the endof a given run [19].
182 A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182
Fig. 17. Convergence of IWO to the optimal value of the Sphere function.
Fig. 18. Upper diagram: optimizing process of the Griewank function by IWOalgorithm vs. standard genetic algorithm. Lower diagram: Associated varianceof each generation.
References
[1] T.G. Ryall, R.W. Moses, M.A. Hopkins, D. Henderson, D.G. Zimicik,F. Nitzsche, Buffet load alleviation, in: Proc. of the Second AustralianCongress on Applied Mechanics, Australia, 1999, pp. 126–131, paper R-017.
[2] A. Yousefi-Koma, D.G. Zimicik, Applications of smart structures to air-craft for performance enhancement, Canadian Aeronautics and SpaceJournal 49 (4) (2003) 163–172.
[3] L.A. Meyn, K.D. James, Full-scale wind-tunnel studies of F/A-18 tail buf-fet, Journal of Aircraft 33 (3) (1996) 589–595.
[4] R.W. Moses, E. Pendleton, A comparison of pressure measurement be-tween a full-scale and 1/6-scale F/A-18 twin tail during buffet, in: 83rdMeeting of AGARD Structures and Materials Panel (SMP), on Loads andRequirements of Military Aircraft, Florence, Italy, 4–5 September 1996,AGARD-R-815, NATO Advisory Group for Aerospace Research and De-velopment, Neuilly-sur-Seine, France.
[5] E.F. Sheta, R.W. Moses, L.J. Huttsell, Active smart material control sys-tem for buffet alleviation, Journal of Sound and Vibration 292 (3–5) (2006)854–868.
[6] B. Lee, N. Valerio, Vortical flow structure near the F/A-18 LEX at highincidence, Journal of Aircraft (1994) 1221–1223.
[7] E.F. Sheta, V.J. Harrand, L.J. Huttsell, Active vortical flow control for alle-viation of twin-tail buffet of generic fighter aircraft, Journal of Fluids andStructures (2001) 769–789.
[8] R.W. Moses, Active vertical tail buffeting alleviation on an F/A-18 modelin a wind tunnel, in: Second Joint NASA/FAA/DOD Conference on AgingAircraft, Williamsburg, VA, 1998.
[9] A. Yousefi-Koma, Y. Chen, D.G. Zimcik, 2004. Development of an activecontrol system for a smart fin, in: Proceedings of the 7th Cansmart Inter-national Worskshop on Smart Materials and Structures, Montreal, QC.
[10] A. Yousefi-Koma, D.G. Zimcik, A. Mander, Experimental and theoreticalsystem identification of flexible structures with piezoelectric actuators, in:24th Congress of the International Council of the Aeronautical Sciences(ICAS2004), Yokohama, Japan, August 2004.
[11] M. Rahmoune, A. Benjeddou, R. Ohayon, D. Osmont, Finite elementmodeling of a smart structure plate system, in: Proceedings of the 7th In-ternational Conference on Adaptive Structures, Rome, Italy, 1996.
[12] A. Rader, Optimization of piezoelectric actuator configuration on a flex-ible fin for vibration control using genetic algorithms, MA Sc. Thesis,Department of Mechanical and Aerospace Engineering Carleton Univer-sity, Ottawa, Ontario, Canada, July 2005.
[13] A.A. Rader, F.F. Afagh, A. Yousefi-Koma, D.G. Zimcik, Optimization ofpiezoelectric actuator configuration on a flexible fin for vibration con-trol using genetic algorithms, Journal of Intelligent Material Systems andStructures (2006), submitted for publication.
[14] M. Hagan, Backpropagation, in: H. Demuth, M. Beale (Eds.), User’sGuide for the Neural Network Toolbox for MATLAB, ver. 3, The Math-works Inc., Natick, MA, 1998, Chapter 5.
[15] V. Prasad, B.W. Bequette, Nonlinear system identification and model re-duction using artificial neural networks, Computers and Chemical Engi-neering 27 (2003) 1741–1754.
[16] J.E. Moody, The effective number of parameters: An analysis of general-ization and regularization in nonlinear learning systems, in: J.E. Moody,S.J. Hanson, R.P. Lippmann (Eds.), Advances in Neural InformationProcessing Systems 4 (1992) 847–854.
[17] G. An, The effects of adding noise during backpropagation training on ageneralization performance, Neural Computation 8 (1996) 643–674.
[18] M.T. Hagan, M. Menhaj, Training feedforward networks with the Mar-quardt algorithm, IEEE Transactions on Neural Networks 5 (6) (1994)989–993.
[19] A. Reza Mehrabian, C. Lucas, A novel numerical optimization algorithminspired from weed colonization, Ecological Informatics 1 (4) (December2006) 355–366, doi:10.1016/j.ecoinf.2006.07.003.
[20] D. Halim, S.O. Reza Moheimani, An optimization approach to optimalplacement of collocated piezoelectric actuators and sensors on a thin plate,Mechatronics 13 (2003) 27–47.
[21] J.-H. Han, I. Lee, Optimal placement of piezoelectric sensors and actuatorsfor vibration control of a composite plate using genetic algorithms, SmartMaterials and Structure 8 (1999) 257–267.
[22] A.M. Sadri, J.R. Wright, R.J. Wynne, Modelling and optimal placement ofpiezoelectric actuators in isotropic plates using genetic algorithms, SmartMaterials and Structure 8 (1999) 490–498.
[23] J.H. Holland, Adoption in Natural and Artificial Systems, University ofMichigan, Ann Arbor, 1975.
[24] M. Dorigo, V. Maniezzo, A. Colorni, The ant system: optimization by acolony of cooperating agents, IEEE Transactions on Systems, Man, andCybernetics B 26 (1) (1996) 1–13.
[25] J. Kennedy, R. Eberhart, Particle swarm optimization, in: Proc. of theIEEE Int. Conf. on Neural Networks, Piscataway, NJ, 1995, pp. 1942–1948.
[26] P. Moscato, On evolution, search, optimization, genetic algorithms andmartial arts: towards memetic algorithms, Technical Report Caltech Con-current Computation Program, Report 826, California Institute of Tech-nology, Pasadena, CA, 1989.
[27] E. Elbeltagi, T. Hegazy, D. Grierson, Comparison among fiveevolutionary-based optimization algorithms, Advanced Engineering Infor-matics 19 (2005) 43–53.
[28] I.C. Trelea, The particle swarm optimization algorithm: convergenceanalysis and parameter selection, Information Processing Letters 85(2003) 317–325.
