Optimal positioning of piezoelectric actuators on a ... - Science Direct

Aerospace Science and Technology 11 (2007) 174–182 www.elsevier.com/locate/aescte

Optimal positioning of piezoelectric actuators on a smart fin using 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 Available online 16 January 2007

Abstract In this paper a novel approach is developed for optimization of piezoelectric actuators in vibration suppression. A scaled model of a vertical tail of F/A-18 is developed in which piezoelectric actuators are bounded to the surface. The frequency response function (FRF) of the system is then recorded and maximization of the FRF peaks is considered as the objective function of the optimization algorithm to enhance the actuator authority on the mode, which assigns the optimal placement of the pair of piezoelectric actuators on the smart fin. Six multi-layer perceptron neural networks are employed to perform surface fitting to the discrete data generated by the finite element method (FEM). Invasive weed optimization (IWO), a novel numerical stochastic optimization algorithm, is then employed to maximize the FRF peak which in due reduces the vibration of the smart fin. Results indicated an accurate surface fitting for the FRF peak data as well as the optimal placement of the piezoelectric actuators for vibration suppression. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Smart structures; Optimal actuator placement; Vibration control; Buffet suppression; Neural networks; Invasive weed optimization

1. Introduction All air vehicles create a type of vortex, which is an energeticswirling mass of air, called a trailing vortex off each wingtip when the plane is in motion. These trailing vortices can be thought of as small tornados that grow larger as they extend behind the plane. A vortex generator is really nothing more than a miniature wing-like device designed specifically to create a vortex. Even though a vortex creates drag, it can also provide advantages that outweigh its negative impact. One such advantage is the ability of a vortex to speed up the flow of air over a wing and allow a plane to reach a higher angle of attack than it would be able to otherwise. For high-performance twin-tail aircraft (HPTTA) such as the F/A-18 and F-15, buffet induction * Corresponding author. Tel.: +98 (021) 4431 1321; fax: +98 (021) 4433 6073. E-mail addresses: [email protected] (A.R. Mehrabian), [email protected] (A. Yousefi-Koma). 1 A. Reza Mehrabian is a graduate student. 2 A. Yousefi-Koma is an assistant Professor and the director.

1270-9638/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2007.01.001

tail vibrations occur when unsteady pressures associated with separated flow, or vortices, excite the vibration modes of the vertical-fin-structural assemblies (see Fig. 1) [1,2]. At high angles of attack, flow separates at the leading edge of the wings, and vortices are generated at different locations such as the wing fuselage interface or the leading edge extensions. This phenomenon, along with the aeroelastic coupling of the tail structural assembly, results in vibrations that can shorten the fatigue life of the empennage assembly and limit the flight envelope due to the large amplitude of the fin vibrations. This is a significant problem particularly for the F/A-18 aircraft, which requires frequent inspection to prevent catastrophic failure. There have been numerous studies on monitoring and controlling the buffet loads on both scaled model and actual aircraft by the use of active vibration control [3–5]. There are essentially two major techniques to control the tail buffet problem: flow control or structural control. The flow control methods aim into modifying the vertical flow-field around the vertical tails to reduce the buffet loads. Passive flow control [6] and active flow control [7] methods have been proposed, but only the passive methods received the most attention [5]. Active

A.R. Mehrabian, A. Yousefi-Koma / Aerospace Science and Technology 11 (2007) 174–182

Fig. 1. Vortex bursting observed during smoke flow visualization tests on NASA’s F-18 HARV.

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 simple scaled model of the vertical tail fin of an F/A-18 fighter jet, which approximately replicated the first two natural frequencies of the full-scale vertical fin [10]. This model included a flexible aluminum fin, with a thickness of 1 mm, fixed at the base. A total of 24 piezo-ceramic actuators are bonded onto both sides (12 on each side) of the aluminum plate. Two accelerometers are used to monitor the dynamic response of the fin tip. Due to the integration of actuators and sensors with the host structure, it is usually very cumbersome, if not impossible, to develop a mathematical model for a complex smart structure. Thus, FEM is used to predict the structural response. For instance an electromechanical coupling effect of piezoelectric materials is employed to establish a FEM model of a flexible plate with piezoelectric sensors and actuators [10]. This paper presents a new technique for optimal positioning of actuators on a smart structure. Neural networks are employed to find an optimal 3-dimensional surface for the Frequency Response Function (FRF) peak data obtained from a finite element model of a flexible aircraft fin. This is performed on a complete set, where the FRF was measured at 48 points on the smart fin. Neural networks are shown to be a suitable algorithm for obtaining a proper surface fit. Finally, the position of a single pair of piezoelectric actuator on the fin is optimized using weighting factors on each of the modal surfaces. These weighting factors were dependent on the desired control authority over each mode to determine the optimum actuator location. In this study a novel numerical optimization algorithms called invasive weed optimization (IWO) algorithm, which is inspired from colonizing behavior of weeds, is employed to find the optimal position of piezoelectric actuators. 2. Fin geometry and finite element modeling Finite element methods are often used to model and predict the dynamic response of a structure with integrated sensors and actuators. An excellent early example of this is the study by Rahmoune et al. [11], where the electromechanical coupling effect of piezoelectric materials was used to establish a finite element model of a flexible plate with bonded piezoelectric

175

Fig. 2. Experimental setup of the flexible fin with piezoelectric actuators (National Research Council Canada (NRC)). Table 1 Material properties of the structure components of the smart fin Property

Aluminum 2024-T3

PZT BM500

Density [Kg/m3 ] Elastic module [MPa] Thermal expansion [µm.◦ C] Thickness [mm] Charge constant, d31 [pC/N]

2796 73.0 23.2 1.02 –

7650 64.5 – 0.50 175

Fig. 3. FEM model of the fin in experimental configuration [12].

sensors and actuators. In the present paper, PATRAN is used to develop the FE model of the flexible fin, while NASTRAN is employed as the FEM solver with thermal load analogy for piezoelectric actuators. The conventional assumptions associated with FE modeling of smart structures such as the assumption of a perfect bond and neglect of the glue stiffness and mass apply to the model in this study [12,13]. The finite element model was based on the experimental apparatus, which consisted of a flexible aluminum fin with a thickness of 1 mm fixed at the fin root shown in Fig. 2. The material properties of the structure components are given in Table 1. Shell and solid elements were employed in PATRAN for the aluminum plate and PZT, respectively. The FE model equivalent to the experimental setup is presented in Fig. 3. Natural frequencies and damping ratios of the integrated smart structure are presented in Table 2. Structural damping obtained from experiment was also incorporated into the FEM model. The effects of the gluing and wiring of actuators resulted in higher natural frequencies recorded for

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Table 2 Modal frequencies and damping ratios of the smart fin Mode 1 2 3 4 5

Experiment

Frequency [Hz] FEM

Error [%]

Damping ratio from experiment

17.9 53.8 83.5 126.0 172.0

17.0 49.1 80.4 121.1 166.4

−5.3 −8.7 −3.7 −3.9 −3.3

0.016 0.012 0.025 0.008 0.016 Fig. 6. Sample FRF of the fin tip acceleration.

49.1 Hz, and 73 Hz vs. 80.4 Hz). A typical example of the FRF of the fin is shown in Fig. 6 [12,13]. Phase angles were irrelevant for the optimization of actuator configuration, but were considered in the development of an active control system for the fin by Yousefi-Koma et al. [9]. 3. Surface fitting by neural networks 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 in Fig. 4. For both the experimental and finite element models, five accelerometers were used to measure the dynamic response of the integrated smart structure. These were positioned so that they could measure relatively high responses from the first three dynamic modes of the smart fin. Fig. 5 shows position of two accelerometers (numbers 1 and 3) employed in this study. For the actuator position optimization, the same finite element model with the 48 possible actuator locations (6 rows of 8 actuators) as shown in Fig. 5 was used. For each vibration mode, the resulting FRF at the accelerometer was measured once for each of the 48 possible actuator positions (on each side of the fin). Each 25×25 mm actuator was excited within a range of 2 Hz around each of the first three fundamental frequencies of the integrated 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 optimization had 47 passive actuators for each FRF measurement, the compromise between stiffness and mass contributed by the passive actuators, in fact resulted in less than 10% difference in fundamental frequency for all three modes as compared with a single actuator on the fin (17.5 Hz vs. 17.0 Hz, 47 Hz vs.

In order to optimize the position of a single pair of piezoelectric actuators continuously within the actuator test area, a continuous fitness function describing the actuation authority of a pair of actuators anywhere within the test area is required. This fitness function may be developed using a genetic algorithm designed to generate the best three-dimensional polynomial surface fit of the piezoelectric actuator FRF peak values within the test area, and to a limited degree by extrapolation, outside of the test area as well [12,13]. Next, a second genetic algorithm can be used to perform the optimization of actuator position. The main difficulty in the proposed method is tuning a large number of coefficients for the three-dimensional polynomial surface to fit the FRF peak data, which makes the process of data fitting complicated and time consuming. For instance, a 9th order polynomial needs 55 coefficients to be optimized to fit the FRF peak data, while there is no guarantee that a polynomial 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 approximator that can approximate any function with a finite number of discontinuities, arbitrary well, given sufficient neurons in the hidden layer [14]. Therefore, a MLP NN is employed to generate 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 input layer (also called ‘input vector’), ending with an output layer and having a number of ‘hidden’ layers in between. Each layer consists of a series of linear or nonlinear nodes. The output of each node in a nonlinear layer is a nonlinear transformation of the weighted sum of the outputs of the nodes of the previous layer. The weights that connect outputs of one layer to the next are the parameters that characterize the system model. Fig. 7 illustrates and MLP NN which consists of a single layer of S perceptron neurons connected to R inputs through a set of weights wi,j that is the strength of the connection from the j th input to the ith neuron. The MLP shown in Fig. 7 consists of


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Table 3 Neural network properties for three-dimensional FRF peak data fitting for accelerometer 1

Fig. 7. Architecture of a two-layer perceptron neural-network with a single hidden layer with tan-sigmoid activation function and a linear output layer.

two layers and one input vector. The input transfer function of the first layer (also called ‘hidden layer’) is ‘tan-sigmoid’ and transfer function for the second layer is ‘pure-linear’. The hidden layer performs nonlinear summation of the input signals while the second layer only applies a linear summation on the output signal of the first layer. System modeling using NN involves ‘training’ the neural networks using a set of historical data. Training is an off-line optimization procedure that modifies the weights of the neural network in order to minimize the error (in some norm) between the network output and the desired output value over the entire data set. A single data set is usually processed by the optimization procedure for many times before an acceptable fit is obtained between NN output prediction and the actual output measurement. Each cycle over the data set is termed an epoch. The trained NN is then used as a model for prediction/estimation [15]. 3.2. Improving generalization One of the problems that arise in connection learning by NN is over or under-fitting of the provided training examples. NN, like other flexible nonlinear estimation methods such as kernel regression and smoothing splines, can suffer from either underfitting or over-fitting. A network that is not sufficiently complex can fail to detect fully the signal in a complicated data set, leading to under-fitting. Under-fitting can also occur when gradientbased learning methods is used. In this case, for a NN provided with sufficient number of layers and weights, the optimization (learning) algorithm may fail to find the global minimum of NN weights and stuck in a local minimum far from the global minima resulting in large errors on the training data set. A network that is too complex may fit the noise, not just the signal, leading to over-fitting. Over-fitting is especially dangerous because it can easily lead to predictions that are far beyond the range of the training data with many of the common types of NN. Over-fitting can also produce wild predictions in MLP even with noise-free data. There are different approaches to overcome generalization difficulties. The best way to avoid over-fitting is to use lots of training data. For noise-free data, five times as many training cases as weights may be sufficient to prevent over-fitting. But the number of weights for fear of under-fitting cannot be reduced arbitrarily. Given a fixed amount of training data, there are different approaches to avoiding under-fitting and over-fitting, and hence getting good generalization:

Network No. of parameters Learning algorithm Performance measure/value Stopping criteria (maximum number of epochs)

Dynamic mode shape 1



2-3-1 13 LMa MSEb /4.523e-4

2-4-1 17 BRc SSEd /0.0079

2-4-1 17 LMa MSEb /0.0013

500

500

500

Note the all neural networks used are two-layer perceptron with tan-sigmoid activation 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 4 Neural network properties for three-dimensional FRF peak data fitting for accelerometer 3

Network No. of parameters Learning algorithm Performance measure/value Maximum number of epochs allowed




2-3-1 13 LMa MSEb /5.276e-4

2-4-1 17 LMa MSEb /0.0200

2-4-1 17 LMa MSEb /0.0013

500

500

500

Note the all neural networks used are two-layer perceptron with tan-sigmoid activation 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 of weights, and hence the number of hidden units and layers. The more weights there are, relative to the number of training cases, the more over-fitting amplifies noise in the targets [16]. Thus the model and training data set must be coherent with each other. 2) Regularization: This involves modifying the performance function, which is chosen normally to be MSE. It is possible to improve generalization if we modify the performance function by adding a term that consists of the mean of the sum of squares of the network weights and biases [14]. 3) Jittering: Jitter is artificial noise deliberately added to the inputs during training. Training with jitter is a form of smoothing related to kernel regression. It is also closely related to regularization methods such as weight decay and ridge regression [17]. 4) Early stopping: It is one of effective methods for improving generalization. In this technique the available data is divided into three subsets. The first subset is the training set, which is used for computing the gradient and updating the network weights and biases. The second subset is the validation set. The error on the validation set is monitored during the training process. The validation error will nor-

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Fig. 8. Discrete FRF peak values for the first mode in magenta stripes, and corresponding three-dimensional surface obtained by neural network for the first accelerometer.

Fig. 9. Discrete FRF peak values for the second mode in magenta stripes, and corresponding three-dimensional surface obtained by neural network for the first accelerometer.

mally decrease during the initial phase of training, as does the training set error. However, when the network begins to over-fit the data, the error on the validation set will typically begin to rise. When the validation error increases for a specified number of iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned [14]. 5) Data normalization: Before training, it is often useful to scale the inputs and targets so that they always fall within a specified range (between −1 and 1 for training MLP NN with hidden layer with ‘tan-sigmoid’ activation function). The data normalization can help in generalization ability of the NN very much [14]. 3.3. Training neural networks for three-dimensional surface fitting To train NN for three-dimensional surface fit, Levenberg– Marquardt (LM) training algorithm that is a gradient-based batch training method is employed to obtain optimal weights of the network (in one case Bayesian regularization is employed) [14,18]. In addition, mean squared error (MSE) performance function is employed as an index for measuring consistency between input and output signals (for Bayesian regularization (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 actuator position optimization should be performed for two different accelerometers, which were introduced in Fig. 5. Thus, six different MLP NNs should be employed to fit the FRF peak data. The input vector of the NNs consists of a data set concerning column (in X-direction, between 1 and 8) and row (in Y-direction, between 1 and 6) in which each actuator is placed; while the output is the FRF peak value of the actuator. Noting that the number of training data is limited (FRF peak data is provided for only 48 possible actuator positions), to prevent data over-fitting, a small MLP NN is employed to fit the FRF peak

Fig. 10. Discrete FRF peak values for the third mode in magenta stripes, and corresponding three-dimensional surface obtained by neural network for the first accelerometer.

data for each mode that uses three/four neurons in the hidden layer. Also, to have best data fitting and to increase generalization ability of the NNs, input and output data are normalized between −1 and 1. Employing a gradient-based learning algorithm (LM) increases the chance for the learning algorithm to be trapped in a local minimum (training data under-fitting). In order to avoid this difficulty, 50 NNs have been trained for each mode shape and the best fitting is chosen. For two accelerometers, Tables 3 and 4 summarize structure, number of parameters of NNs, learning algorithm, performance measure and its value, and maximum number of epochs allowed for NNs. Note that BR training algorithm is used only for the second dynamic mode shape for accelerometer 1 to get a better generalization since BR, generally, provides better generalization performance than early stopping [14]. The discrete FRF peak values (in stripes but no shading), and the corresponding surface (in green shading) for the three modes of two accelerometers are illustrated in Figs. 8 to 13 (for colors see the web version of the article).


Fig. 11. Discrete FRF peak values for the first mode in magenta stripes, and corresponding three-dimensional surface obtained by neural network for the third accelerometer.

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Fig. 13. Discrete FRF peak values for the third mode in magenta stripes, and corresponding three-dimensional surface obtained by neural network for the third accelerometer. Table 6 Optimum actuator position under different weighting factors for the third accelerometer Case no.

Weighting

Weighting factors

Mode 1

Mode 2

Mode 3

X

Y

1

Evenly weighted (acceleration control) Normalized to mean FRF peak value Normalized to max. FRF peak value

1

1

1

119

115

0.82

1

0.34

143

116

1

0.35

0.23

128

112

2 3

Fig. 12. Discrete FRF peak values for the second mode in magenta stripes, and corresponding three-dimensional surface obtained by neural network for the third accelerometer.

4. Piezoelectric actuator position optimization The modal surfaces obtained in previous section, were summed with weighting factors to yield a combined surface Table 5 Optimum actuator position under different weighting factors for the first accelerometer Case no.

Weighting

Weighting factors

Mode 1

Mode 2

Mode 3

X

Y

1

Evenly weighted (acceleration control) Normalized to mean FRF peak value Normalized to max. FRF peak value

1

1

1

142

119

0.82

1

0.34

148

122

1

0.35

0.23

145

117

2 3

Optimal position [mm]

Optimal position [mm]

(please see Tables 5 and 6 for the values of weighting factors for each mode). By employing these combined surface, a novel numerical optimization algorithm that is inspired by invasive colonizing behavior of weeds designated as invasive weed optimization (IWO) algorithm [19], is employed to determine the optimum position of the piezoelectric actuators on the fin. It should be noted that different optimization algorithms can be employed to solve the problem of the optimal positioning of the pair of piezoelectric actuators on the smart fin. These algorithms can either use gradient information of the peak FRF fitted surfaces or not. Since NNs are employed to fit the FRF peak data, gradient information of the surface can be obtained with no difficulty. However, it is not always possible to obtain gradient information for optimal actuator positioning problems. This is due to the fact that in some cases the necessary data for determining the optimal position of actuators is obtained directly from numerical analysis techniques or experiment. This is one of the reasons for employing non-gradient-based search algorithms for solving optimal actuator/sensor placement problems in recent years [20–22]. Thus, in this paper, IWO algorithm, which is a non-gradient-based search algorithm, is employed to find the optimal solution; so that the method can be utilized if the gradient information is not available.

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4.1. Invasive weed optimization (IWO) algorithm In recent years there have been an extensive research conducted for studying bio-inspired numerical optimization algorithms like genetic algorithms (GAs) [23], ant colony optimization (ACO) [24], particle swarm optimization (PSO) [25], Memetic algorithms (MAs) [26], and so on [27]. These numerical optimization algorithms are categorized with non-gradientbased search algorithms; they are designated as direct search algorithms as well. The main advantage of these algorithms is that they only use the objective function and constrain values to steer towards the optimal solution. Since derivative information is not used, the direct search methods are typically slow, requiring many function evaluations for convergence. For the same reason, they can also be applied to many problems without applying major changes in the algorithm. Invasive weed optimization algorithm, IWO, which is introduced in [19] for the first time, is a bio-inspired numerical optimization algorithm that simply simulates natural behavior of weeds in colonizing and finding suitable place for growth and reproduction. To model and simulate colonizing behavior of weeds for introducing a novel optimization algorithm, some basic properties of the process is considered:

ters are illustrated in Figs. 14 and 15 respectively. Case 2 used weighting factors which equalized the mean peak FRF values of all modes, while case 3 used weighting factors which equalized the maximum peak FRF values from all modes. Note that in the case of a single actuator, positions within 20 mm of the root were excluded from the optimization due to the potential

Fig. 14. The optimum actuator pair positions for the first accelerometer.

1) A finite number of seeds are being dispread over the search area (initializing a population); 2) Every seeds grows to a flowering plant and produces seeds depending on their fitness (reproduction); 3) The produced seeds are being randomly dispread over the search area and grow to new plants (spatial dispersal); 4) This process continues until maximum number of plants is reached; now only the plants with higher fitness can survive and produce seeds, others are being eliminated (competitive exclusion). The course continues until maximum iterations is reached and hopefully the plant with best fitness it the closest to the optimal solution. IWO has some distinctive properties in comparison with traditional GAs (and other numerical search algorithms), like reproduction, spatial dispersal, and competitive exclusion [19]. In addition, no genetic operators are employed in the proposed algorithm, which makes it more dissimilar to GAs. In Appendix A, pseudocode for IWO algorithm is introduced and some simulations are reported to show the ability of the algorithm in locating the global minimum of two benchmark functions. Extensive simulations are reported to compare performance of IWO algorithm with other algorithms like GAs, PSO, and MAs for different low and high dimension functions in [19], where it is shown that IWO algorithm is a competitive for other numerical stochastic optimization algorithms.

Fig. 15. The optimum actuator pair positions for the third accelerometer.

4.2. Actuator position optimization using IWO Several optimization cases with associated weighting factors, including those corresponding to acceleration control, are given in Tables 5 and 6 for accelerometer 1 and 3 respectively. Optimal piezoelectric actuator positions for two accelerome-

Fig. 16. Combined FRF surface normalized to maximum FRF peak value for the first accelerometer.


for interference with the fin attachment. For better illustration, the combined modal surface plot with weighting normalized to the maximum FRF peak value (case 3) is shown in Fig. 16 for the first accelerometer. 5. Conclusion A novel actuator optimal positioning algorithm is developed based on neural networks for a smart fin as a scaled model of F/A-18 vertical tail. The optimization methodology allowed the placement of piezoelectric actuator pairs for effective vibration reduction over the entire structure. The frequency response function (FRF) of the system is recorded and maximization of the FRF peaks is considered as the objective function of the optimization algorithm to find the optimal placement of the piezoelectric actuators on the smart fin. Totally, six multi-layer perceptron neural networks (MLP NN) are employed to perform surface fitting to the discrete data generated by the finite element method (FEM). Then, IWO algorithm has been employed to find the proper position of actuators. Results indicate an accurate surface fitting for the FRF peak data as well as an optimal placement of the piezoelectric actuators for vibration suppression. The proposed algorithm is able to solve any actuator/sensor optimal positioning problem on different flexible smart structures. Acknowledgement The authors wish to acknowledge the valuable remarks and suggestions made by the anonymous reviewers of the paper which led to many improvements. Appendix A. An introduction to invasive weed optimization (IWO) algorithm Invasive weed optimization, IWO, is a novel numerical stochastic optimization algorithm inspired from colonizing weeds. Weeds are plants whose vigorous, invasive habits of growth pose a serious threat to desirable, cultivated plants making them a threat for agriculture. Weeds have shown to be very robust and adaptive to change in environment. It is tried to mimic robustness, adaptation and randomness of colonizing weeds in a simple but effective optimizing algorithm designated invasive weed optimization [19]. Pseudocode for IWO algorithm is given as follows:

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Randomly distribute generated seeds over the search space with normal distribution around the parent plant (w);4 Add the generated seeds to the solution set, N ; End; If N > Nmax ;5 Sort the population N in descending order of their fitness; Truncate population of weeds with smaller fitness until N = Nmax ; End If; Next i; End; A.2. Convergence of IWO algorithm Convergence of IWO algorithm is demonstrated by employing the algorithm for locating global minimum of two benchmark examples: two dimension Sphere and Griewank functions [28]. Fig. 17 shows the process of colonizing weeds around the point with best fitness for Sphere function. It can be observed that the plants grow towards the optimal point from the initialization area. In their progress towards the optimal point, plants with lower (worse) fitness are being excluded, and only weeds with higher (better) fitness are allowed to be reproduced, which leads in colonization about the optimal point. The final value of the fitness function for Sphere function is found to be fitness (x0 ) = 2.4362e–8, for the point: x0 = [−0.1413e–3, −0.0662e–3]. It is known that the optimal 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 favorite benchmark for optimization algorithms. It is known that the function has only one global minimum at [0, 0] in x–y plane but numerous local minima. Fig. 18 illustrates the process of obtaining optimal solution of the problem. To demonstrate merits of proposed algorithm, same simulation is performed using GA toolbox provided in MATLAB® , where the initial conditions and number of maximum agents where identical in both simulations. As depicted in Fig. 18, the proposed algorithm outperformed GA in finding the global minimum of the Griewank function [19].

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 to its fitness;3 3 Any member of the population of plants is allowed to produce seeds depending on its own fitness and the colony’s lowest and highest fitness: number

of seeds each plant use increases linearly from minimum possible seed production to its maximum [19]. 4 The generated seeds are being randomly distributed over the d dimensional search space by normally distributed random numbers with mean equal to zero; but varying variance. This means that seeds will be randomly distributed such that 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, p max , a mechanism for eliminating the plants with poor fitness in the generation activates. In this way, only plants with higher fitness survive and are allowed to replicate. The population control mechanism also is applied to their children to the end of a given run [19].

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Fig. 17. Convergence of IWO to the optimal value of the Sphere function.

Fig. 18. Upper diagram: optimizing process of the Griewank function by IWO algorithm vs. standard genetic algorithm. Lower diagram: Associated variance of each generation.

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