The estimation of low and high-pass active filter parameters with
opposite charged system search algorithm
Hasan Temurtas
ß
Kütahya Dumlupınar University, Faculty of Engineering, Department of Computer Engineering, 43100 Kütahya, Turkey
a r t i c l e i n f o
Article history:Received 14 March 2019 Revised 29 March 2020 Accepted 20 April 2020 Available online 27 April 2020 Keywords:
Charged system search algorithm (CSS) Opposite charged system search algorithm (OCSS)
Sallen-Key topology Butterworth low and high-pass active filters
E24 standard series
a b s t r a c t
Algorithms are frequently used to solve problems that have a large search space and take a long time to be mathematically solved. They can later be improved with different improvement methods based on the structure and the type of the problem. In this study, the charged system search algorithm (CSS), which has been successfully implemented in the solutions of numerous engineering problems studied within the literature, was improved by introducing opposition-based learning method (OBL) to it in two differ-ent methods. With these improved algorithms, solutions were developed for 30-dimensional multimodal test functions in the first place and the results were discussed. In the second place, the parameters of active filters given below were determined from E24 standard series with developed approaches.
Filters are electronic circuits that enhance the wanted frequency components of electric signals applied on their inputs and remove harmonics and interferences from these signals. They are divided into two types; active and passive filters. Active filters are produced with transistors or op-amps. They are finan-cially more advantageous compared to passive filters. These filters are preferred espefinan-cially in low fre-quencies due to their low costs. Adjustable for a large frequency domain, active filters are very convenient in terms of size and weight and their designs are highly simple. They can easily be connected successively without affecting one another. In this study, the parameter values of Sallen-Key topology Butterworth low and high-pass active filters, which have an extensive area of use, were determined through improved algorithms.
My suggestions for the future studies are these: the effects of the opposite position learning approach on other heuristic algorithms can be analysed solving test functions and different engineering problems; different approaches other than the two approaches proposed in this study, can be developed for the opposite position learning concept; LPF and HPF designs solved in the study can be solved for different degrees and stages and finally new designs can be done for different filter types and different resistor and capacitor series that haven’t been handled in this study.
Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction
Multidimensional problems with large search spaces have been observed to take a long time to solve with numerical methods. Therefore, meta-heuristic algorithms are frequently used today to solve complicated problems that are very difficult or impossible to solve with numerical methods (Kaveh, 2016). Several algorithms have been developed in the last few years to solve complicated engineering problems. Among these algorithms that have been implemented within the literature, the main ones can be specified as genetic algorithm (GA) (Goldberg, 1989), differential evolution algorithm (DE) (Storn & Price, 1997), particle swarm optimization algorithm (PSO) (Kennedy & Eberhart, 1995), artificial bee colony
algorithm (ABC) (Karabog˘a & Basßtürk, 2007), ant colony optimiza-tion algorithm (ACO) (Dorigo & Di Caro, 1999), harmony search algorithm (HS) (Geem, Kim, & Loganathan, 2001), charged system search algorithm (CSS) (Kaveh, 2016; Kaveh & Talatahari, 2010a), and gravitational search algorithm (GSA) (Rashedi, Nezamabadi-pour, & Saryazdi, 2009; Özyön & Yasßar, 2018).
Meta-heuristic algorithms generally start searching for a solu-tion set with randomly created individual groups. Therefore, the layouts of the individuals in the initially-created population within the search space are highly crucial to obtain the optimal results. The solution cannot converge towards the optimal results in the case of the individuals in the initial population being placed near local minimums. This is a shortcoming that needs to be eliminated (Cura, 2008). Thus, the meta-heuristic algorithms found in the lit-erature need to be enhanced or improved through various methods
https://doi.org/10.1016/j.eswa.2020.113474
0957-4174/Ó 2020 Elsevier Ltd. All rights reserved. E-mail address:hasan.temurtas@dpu.edu.tr
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Expert Systems with Applications
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w ato overcome this shortcoming. Different methods are used in the literature to provide more improvement for the performances of meta-heuristic algorithms. One such method is the opposition-based learning method (Tizhoosh, 2005; Rahnamayan, Tizhoosh, & Salama, 2008).
Numerous meta-heuristic algorithms have been tried to be improved in the literature by introducing the opposition-based learning method (OBL) to them. Some of these are the studies of opposition-based differential evolution (Rahnamayan et al., 2008), opposition-based learning particle swarm optimization
(Omran, 2009), opposition-based harmony search (Singh,
Mykherje, & Ghoshal, 2015), opposition-based biogeography opti-mization algorithm (Ergezer, Simon, & Du, 2009), and opposition-based gravitational search algorithm (Özyön, Yasßar, Durmusß, & Temurtasß, 2015).
In this study, the meta-heuristic algorithm known as charged system search algorithm (CSS) in the literature was selected to be improved. The algorithm was developed in 2010 by Kaveh and Talahatari, who were inspired by Coulomb’s law and Gauss’s law in electromagnetics, and Newton’s laws of motion (Kaveh, 2016; Kaveh & Talatahari, 2010a). The literature contains studies in which CSS and its improved versions were successfully imple-mented in various engineering problems. Some of these can be specified as optimum design of skeletal structures (Kaveh & Talatahari, 2010b), optimum design of grillage systems (Kaveh & Talatahari, 2010c), geography and topology optimization of geode-sic domes (Kaveh & Talatahari, 2010a), optimization of lattice structures (Kaveh & Talatahari, 2010b, 2010d, 2012), and economic power distribution problems (Özyön, Temurtasß, Durmusß, & Kuvat, 2012; Özyön, Yasßar, Durmusß, Temurtasß, & Kuvat, 2012).
In this study, the concept of opposition-based learning was introduced to CSS in two different methods. The first of these methods entails that while half of the individuals in the initial pop-ulation are created randomly, the other half are placed on the opposite of those individuals. The solution set is searched within the initial population constituted by these individuals. In the sec-ond method, an individual is created for the solution set and another individual is placed on the opposite of that individual. These two individuals are compared and the optimal individual is included in the population of the solution set, while the other is discarded. This procedure continues as such until the pre-determined individual number is reached. In this way, faster learn-ing is sought after. The newly developed algorithms were applied to test functions having two different structures, namely unimodal and multimodal, and the obtained results were assessed.
Another important objective of the study is the practicability of the improved algorithms with regard to real-life engineering prob-lems. Therefore, the problem of determining the parameter values of active filters, playing a very crucial role in electronic engineering and frequently used in circuit designs, was selected to be solved. Filter circuits are widely used to pass a specific frequency band and weaken or hinder other frequencies in electronic systems. Fil-ters are designed in two main types; active and passive. In the pas-sive filter design, paspas-sive filter items such as resistors, capacitors, and coils are used, while in the active filter design, semiconductor circuit components such as transistors and integrated circuits are used additionally. These filters, which do not include any passive circuit components, are more advantageous than passive filters in terms of their costs. The output impedance of active filters is very low, while their input impedance is very high. Therefore, circuits and circuit components to be connected to the input and output of active filters cannot possibly be affected by one another. Accord-ingly, active filters can easily be connected in sequence
(Hiçdurmaz, Durmusß, Temurtasß & Özyön, 2016; Durmusß,
Hiçdurmaz, Temurtasß & Özyön, 2016).
The solution for the problem of determining active filter param-eters in the literature have been performed through numerous heuristic algorithms such as feedback algorithm (Hiçdurmaz et al., 2016; Durmusß et al., 2016), genetic algorithm (Horrocks & Spittle, 1993), parallel tabu search algorithm (Kalinli, 2006), evolu-tion algorithm (Vural, Yildirim, Kadioglu, & Basargan, 2012), parti-cle swarm optimization (De, Kar, Mandal, & Ghoshal, 2015; Vural, Bozkurt, & Yildirim, 2013), artificial bee colony optimization (Vural et al., 2013), vortex search algorithmDog˘an & Ölmez, 2015), aver-age differential evolution algorithm (Durmusß, 2018), and clonal selection algorithm (Jiang, Yang, & Gan, 2007).
In this study, the parameter values of Sallen-Key topology But-terworth low and high-pass active filters were determined with CSS and OCSS. In traditional approaches, circuit components are regarded to have ideal and unlimited values. The use of these val-ues in practice highly increases designing costs (Hiçdurmaz et al., 2016; Durmusß et al., 2016). In this study, E24 standard series val-ues were used to decrease designing costs. Optimization problems dealt by selecting values from a specific series are known as dis-crete problems in the literature. The optimization of these prob-lems is more complex and difficult compared to the solutions of continuous problems. The study is regarded to enrich the literature in this respect as well. In line with all these objectives, tenth-order low and high-pass active filters were designed with second-order series structures. The parameter values of these designed tenth-order filters were determined with CSS and OCSS. The obtained parameter values and the quality factors (Q) of the designed filters were calculated separately. These calculated values were com-pared to the values found in the literature and the consequent results were assessed.
The contribution of the study to expert and intelligent systems is the analysis of the effects of opposite position learning approach, whose effects on other optimization algorithms have been anal-ysed in literature previously, on charged system search algorithm (CSS). Besides, LPF and HPF design problems which have been solved in the study, are important problems for electronical engi-neering. The design of these problems by making selections from resistor and capacitor series takes a long time. The use of optimiza-tion algorithms in the design of the problems provides to save time significantly and to design easily
2. Charged system search algorithm (CSS)
Based on Coulomb’s law and Gauss’s law in electromagnetics and Newton’s laws of motion, CSS was developed in 2010 by Kaveh and Talahatari. This algorithm can be thought of as a multiple-agent approach in which each multiple-agent in the population is regarded as a charged particle (CP). A cube with an a diameter expressing the uniform charge density of each charged particle is assumed and expressed with the equations below (Kaveh, 2016; Kaveh & Talatahari, 2010a; Özyön, Temurtasß et al., 2012; Özyön, Yasßar et al., 2012).
qi¼
fitðiÞ fitworst
fitbest fitworst ; i¼ 1; 2; :::; N ð1Þ
In the equation, fitbest and fitworst represent the best and the worst fit values for all the particles, while fit(i) i. represents the fit-ness of the charged particle and N represents the total number of the charged particles (CPs). The initial places of the CPs in the search space are randomly determined and Eq.(2)was used to that end.
In the equation, xðoÞi;j determines the initial value of the i. variable for j. CP. xi,minand xi,maxare the minimum and the maximum per-missible values for the i. variable. randijis a number generated ran-domly at [0.1] range. The initial velocities of the charged particles are acknowledged as shown below(Kaveh & Talatahari, 2010a, 2011a, 2011b).
v
ðoÞi;j ¼ 0; i ¼ 1; 2; :::; N ð3Þ
Each CP applies a force to the other CPs according to Coulomb’s law. The magnitude of this force is proportional to the distance between the CPs within the cube and inversely proportional to the distance between the particles squared for the CPs outside the cube. These forces may be manifested as a pull or a push and calculated with the force parameter arij described as below
(Kaveh, 2016; Kaveh & Talatahari, 2010a, 2010b, 2010c, 2010d, 2011a, 2011b, 2012). arij¼ þ1 kt< randij 1 kt> randij ð4Þ
In the equation, the +1 value demonstrates that the force is manifested as a pull; while the1 value expresses that the force is manifested as a push. ktis the parameter checking the effect of the manifested force type. Generally, while the force manifested as a pull gathers the CPs in a specific position in the search space, the force manifested as a push tries to scatter the CPs. pij, which determines the probability for each CP to move toward the others, is given in Eq.(5)(Kaveh, 2016; Kaveh & Talatahari, 2010a; Özyön, Temurtasß et al., 2012; Özyön, Yasßar et al., 2012).
pij¼
1 fitðiÞfitbestfitðjÞfitðiÞ > rand _ fitðiÞ > fitðjÞ 0 else
(
ð5Þ
The consequently manifested force is calculated with Eq. (6)
(Kaveh, 2016; Kaveh & Talatahari, 2010a; Özyön, Temurtasß et al., 2012; Özyön, Yasßar et al., 2012).
Fj¼ X i;i–j qi a3riji1þ qi r2 ij i2 ! arijpij Xi Xj j¼ 1; 2; :::; N i1¼ 1; i2¼ 0 () rij< a i1¼ 0; i2¼ 1 () rij> a * ð6Þ
Here, Fjis the value of the force having an effect on j. CP. rijis the distance between two charged particles and described in Eq.(7)
(Kaveh, 2016; Kaveh & Talatahari, 2010a; Özyön, Temurtasß et al., 2012; Özyön, Yasßar et al., 2012).
rij¼ k Xi Xjk k Xi Xj =2 Xbestk þ
e
ð7ÞIn the equation, Xiand Xjare the places of i. CP and j. CP respec-tively. Xbestis the place of the best CP in question, while
e
is a small positive number taken to prevent ineffability. The consequently manifested forces and laws of motion determine the new places of the CPs. At this stage, each CP moves toward its new place under the effect of the consequently manifested forces and its previous velocity as shown below (Kaveh, 2016; Kaveh & Talatahari, 2010a; Özyön, Temurtasß et al., 2012; Özyön, Yasßar et al., 2012).Xj;new¼ randj1:ka:
Fj
mj:
D
t2þ rand
j2:kv:Vj;old:
D
tþ Xj;old ð8ÞVj;new¼Xj;new
D
Xt j;old ð9ÞHere, kais the acceleration coefficient and kvis the velocity coef-ficient checking the effect of the previous velocity. randj1and randj2 are two random numbers at [0.1] range distributed uniformly
throughout the series. If a CP moves toward the outside of the search place, its new place is corrected through the method of manual correction. In addition, a memory known as charged mem-ory is utilized to record the optimal results (Kaveh, 2016; Kaveh & Talatahari, 2010a; Özyön, Temurtasß et al., 2012; Özyön, Yasßar et al., 2012).
3. Opposition-based learning (OBL)
Heuristic algorithms start to calculate with an initial population to produce a solution. While the initial population is being deter-mined, the agents are generated in random places within the solu-tion space. However, starting the calculasolu-tion with an initial population embodying individuals whose fit values are better rather than randomly generated places may enable the population to produce a solution within a shorter timeframe. In line with this, the concept of opposition-based learning was put forward in 2005 by Tizhoosh. According to this approach whose definition was also stated by Tizhoosh, the opposite of any random number is gener-ally closer to the solution than the random number itself (Tizhoosh, 2005; Rahnamayan et al., 2008). Thus, it can be said that an initial population created along with the oppositional value of a number will need a smaller search space to converge toward the best solution. This procedure can accelerate the convergence. The concept of opposition-based learning is explained in the sections below.
3.1. Oppositional number
If the number x is a real number described at [a, b] range, the opposite of this number is defined according to Eq.(10)in accor-dance with the opposition theorem (Tizhoosh, 2005; Rahnamayan et al., 2008).
x ^
¼ a þ b x ð10Þ
3.2. Oppositional point
The statement in Eq.(10)can be generalized for multidimen-sional series. To that end, let a P¼ ðx1; x2; ::::; xdÞ point be defined in a D-dimensional space. Here, let x1; x2; ::::; xd2 R and xi2 ½ai; bi8i2 f1; 2; ::::; dg. The P
^
¼ ð x^1; x^2; ::::; x^dÞcomponents of the opposite of this point are defined in the equation below.
x ^
i¼ aiþ bi xi ð11Þ
3.3. Oppositional optimization
In the solution of an optimization problem, a defined point in a D-dimensional space like P¼ ðx1; x2; ::::; xdÞ may be likened to each candidate solution in a population. According to the definition of oppositional point, the opposite of this point is demonstrated as
P ^
¼ ð x^1; x^2; ::::; x^dÞ. In this regard, the newly defined opposi-tional point is one of the candidate solutions for the current prob-lem. In this case, when assessed according to their objective functions, the fit functions of both of the candidate solutions will be fðPÞ and f ð P^Þ. If f ð P^Þ > f ðPÞ for a better solution, the P^ individ-ual whose fit value is better replaces the P individindivid-ual (Tizhoosh, 2005; Rahnamayan et al., 2008).
4. Opposite charged system search algorithm (OCSS)
In this study, two different concepts of opposition-based learn-ing were produced which were estimated to improve the
perfor-mance of CSS. These two different concepts of opposition-based learning were separately integrated into CSS, and consequently, the opposite charged system search algorithm (OCSS) was obtained. These were named OCSS-1 and OCSS-2 and analyzed in the subsections of two different cases.
4.1. Case-1 (OCSS-1)
The first approach (OCSS-1) is based upon the prospect that while half of the agents are randomly assigned during the creation of the initial population of CSS, the other half are placed symmet-rically to those agents. In this approach, all the generated agents have joined in the population without being compared to one another.
4.2. Case-2 (OCSS-2)
In the second approach (OCSS-2), the opposites of all the agents are randomly generated in the initial population, which is then cre-ated with the agents whose fit values are higher. The objective of this procedure is to accelerate the convergence velocity of CSS by starting the search with the individuals closer to the solution of the problem, i.e., with higher fit values.
The following pseudo-code summarizes the OCSS algorithm:
Level 1: Initialization
Step 1: Initialize CSS algorithm parameters (N, kvinitial, kvson, kainitial, kason, r).
Step 2: Select algorithm type (CSS, OCSS-1, OCSS-2) Step 3: If the algorithm type is CSS then
Initialize CPs (Charged Particles) with random positions. Evaluate the values of the fitness function for the CPs.
Step 3: If the algorithm type is OCSS-1 then
Initialize half of CPs with random positions and the other half as their opposites.
Evaluate the values of the fitness function for the CPs. Step 4: If the algorithm type is OCSS-2 then
Initialize CPs with random positions and find their opposites. Evaluate the values of the fitness function for the CPs and
their opposites.
Replace each CP with its opposite if fitness function is better. Step 5: Set the initial velocities of CPs to zero.
Step 6: Compare the fitness function for the CPs with each other and sort increasingly.
Step 7: Store CMS (Charged Memory Size) number of the first CPs and their related values of the objective function in the CM (Charged Memory).
Level 2: Search
Step 8: Determine the probability of moving each CP toward others, and calculate the attracting force vector for each CP.
Step 9: Move each CP to the new position and find the velocities.
Step 10: If each CP exits from the allowable search space, correct its position.
Step 11: Evaluate and compare the values of the objective function for the new CPs, and sort them increasingly. Step 12: If some new CP vectors are better than the worst
ones in the CM, include the better vectors in the CM and exclude the worst ones from the CM.
Level 3: Terminating criterion controlling
Repeat ‘‘Search” level (Level 2) steps until a terminating criterion is satisfied.
4.3. Complexity analysis
This subsection presents the complexity of the proposed algo-rithms. While calculating the time complexity of the algorithms, the cost of each step given in the algorithm X was calculated asymptotically. Accordingly, the cost of each step and the complex-ity of the algorithm are as follows:
Level 1 is executed only once. Therefore:
o Step 1: Initializes the algorithm parameters. The complexity of Step 1 (T1) is O(1)
o Step 2: Selects the algorithm. So, The complexity of Step 2 (T2) is O(1)
o Step 3: Initializes N numbers of CP, each of which is D dimensional. The complexity of Step 3 (T3) is O(N*D) o Step 4: Initializes 2xN numbers of CP, each of which is D
dimensional. Therefore, the complexity of Step 4 (T4) is O (N*D) as in Step3.
o Step 5: Initializes N numbers of CP’s velocity for each dimen-sion. The complexity of Step 5 (T5) is O(N*D)
o Step 6: Sorts the CPs according to their fitness values by using Quick Sort algorithm. The complexity of Step 6 (T6) is O(NlogN)
o Step 7: Stores some good results in the CM (the size of CM is lower than N). So, the complexity of Step 7 (T7) is O(N) Level 2 is run up to the maximum number of iterations (itrmax).
Therefore, the cost of each step in Level 2 is multiplied by itrmax: o Step 8: Determines the probability of each CP (O(N)), and cal-culate the attracting force vector for each CP (O(N2)). Conse-quently, the complexity of Step 8 (T8) is O(itrmaxxN2) o Step 9: Updates the positions and velocities of the
popula-tion. So, the complexity of Step 9 (T9) is O(itrmaxxNxD) o Step 10: Pulls the CPs into search space boundaries. For the
worst case, the complexity of Step 10 (T10) is O(itrmaxxNxD) o Step 11: Evaluates (O(N)) and sorts O(NlogN) the population according to their fitness values. The complexity of Step 11 (T11) is O(itrmaxxNlogN)
o Step 12: Updates the CM. The complexity of Step 12 (T12) is O (itrmaxxN)
Level 2 is only checking termination condition and is run up to the maximum number of iterations (itrmax). Therefore, the cost of Level 3 (T13) is O(itrmax).
Consequently, the algorithm’s time complexity (T) is O (itrmax -xNxD), since the time complexity of the algorithm will be asymptotically summed up by the cost of all steps (T = T1+ T2+ T3+ T4+ T5+ T6+ T7+ T8+ T9+ T10+ T11+ T12+ T13).
5. Test functions
Five test functions were selected to assess the performances of the suggested algorithms (OCSS-1 and OCSS-2) for test functions. These functions are given inTable 1. In the table, the (D) value rep-resents the dimension of the function, while (S) reprep-resents the search space and fminrepresents the minimum value of the func-tion. These functions are multidimensional and multimodal having numerous local minimum points.
The test functions in the table were solved 30 times with the CSS, OCSS-1, and OCSS-2 algorithms. The parameter values used in these solutions are given inTable 2.
The program developed in MATLAB R2015b for the solution of the test functions was run for 1000 iterations (50000 function calls/FCall) for all the functions in a workstation with Intel Xeon E5-2637 v4 3.50 GHz and a 128 GB RAM hardware. The results from 30 stages for the multimodal functions are given inTable 3.
Regarding the best solutions obtained in 30 runs for the func-tions in the table, the graphs demonstrating the algorithm
conver-Table 1
Multidimensional and multimodal test functions.
Mathematical expression Function name D Search space (S) fmin
f1ðxÞ ¼ Pn i¼1x2i 10cosð2pxiÞ þ 10 Rastrigin 30 ½5:12; 5:12n 0 f2ðxÞ ¼40001 Pn i¼1x2i Qn i¼1cos pxiffii þ 1 Griewank 30 ½600; 600n 0 f3ðxÞ ¼ 20exp 0:2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n Pn i¼1x2i q exp1 n Pn i¼1cosð2pxiÞ þ 20 þ e Ackley 30 ½32; 32n 0 f4ðxÞ ¼pn 10sinðpy1Þ þ Pn1
i¼1ðyi 1Þ21þ 10sin2ðpyiþ1Þ
h i n o þPn1 i¼1uðxi; a; k; mÞ yi¼ 1 þxiþ14 ; a ¼ 10; k ¼ 100; m ¼ 4; uðxi; a; k; mÞ ¼ kðxi aÞm xi> a 0 a < xi< a kðxi aÞm xi< a 8 < : Penalized No.1 30 ½50; 50n 0 f5ðxÞ ¼ Pn
i¼1jxisinðxiÞ þ 0:1xij Alpine 30 ½10; 10n 0
Table 2 Parameter values.
Number of iterations (N) Number of agents (CP) Calling function (FCall) Run (R) Dimension (D)
1000 40 40,000 30 30
kvinitial kvend kainitial kaend e
0.9 0.1 0.9 0.1 1e-6
Table 3
Data regarding 30-D.
fmin CSS OCSS-1 OCSS-2
f1 0 Worst 2.235167e+02 1.878954e+02 1.904746e+02
Average 9.014736e+01 9.515336e+01 8.273491e+01
Best 1.711637e+01 1.698556e+01 1.851086e+01
Std 6.876254e+01 5.911733e+01 5.869286e+01
Time 5.65568 s 5.28999 s 5.27604 s
f2 0 Worst 4.551764e01 1.482566e02 2.949897e02
Average 1.755758e02 1.264975e03 3.783967e03
Best 1.243450e14 9.658940e15 1.632028e14
Std 8.145497e02 3.852342e03 6.744142e03
Time 7.66894 s 7.60492 s 6.40164 s
f3 0 Worst 2.000017e+01 2.000898e+01 2.000408e+01
Average 1.784372e+01 1.816707e+01 1.777334e+01
Best 4.563273e+00 3.133527e02 7.125512e01
Std 4.757680e+00 5.163457e+00 5.272744e+00
Time 5.58145 s 5.34602 s 5.40107 s
f4 0 Worst 1.400035e01 2.355793e01 3.427066e02
Average 1.176161e02 7.902365e03 2.191341e03
Best 1.142562e15 1.339815e15 1.288862e15
Std 3.558389e02 4.227894e02 7.815260e03
Time 5.93826 s 5.66874 s 6.85128 s
f5 0 Worst 2.378570e+00 3.501433e+00 8.276720e01
Average 1.766849e01 2.584762e01 5.892138e02
Best 1.735188e03 1.207354e03 3.772920e03
Std 5.449036e01 7.001590e01 1.845579e01
Time 7.40668 s 6.90424 s 6.15288 s
(a) (b)
gences based on the iteration number are given inFigs. 1a,2a,3a,
4a, and5a, while the box plots showing the distribution of the best values are given inFigs. 1b,2b,3b,4b, and5b.
Based on the graphs regarding the solution of the f1function, judging from the convergence curves, all three algorithms were observed to converge at the same time. On the basis of the box plots constituted by solving the functions 30 times, the OCSS-1 approach reached the best values in a more stable search structure. Based on the graphs regarding the solution of the f2function, judging from the convergence curves, the OCSS-1 approach was observed to converge earlier than the other two algorithms. On the basis of the box plots constituted by solving the function 30 times, the CSS approach expressed one value deviation for the f1
function, while the OCSS-1 approach reached the minimum value in the majority of 30 runs.
Based on the graphs regarding the solution of the f3function, judging from the convergence curves, the OCSS-1 approach is seen to be more superior to the CSS and OCSS-2 approaches. Both of the opposition-based approaches are observed to improve the original CSS algorithm in terms of convergence.
Based on the graphs regarding the solution of the f4function, the OCSS-1 algorithm reached the minimum value at about the 180th iteration, while the OCSS-2 and CSS algorithms reached the minimum level at about the 210th and 250th iteration respec-tively. On the basis of the box plots constituted by solving the func-tion 30 times, the CSS algorithm has three deviating values and the
(a) (b)
Fig. 2. The box plots and convergence curves from which the best results for f2 were obtained.
(a) (b)
Fig. 3. The box plots and convergence curves from which the best results for f3 were obtained.
(a) (b)
OCSS-1 approach has one deviating value, while the OCSS-2 algo-rithm has two deviating values.
Finally, based on the graphs regarding the solution of the f5 func-tion, judging from the convergence curves, the OCSS-2 approach was seen to converge more quickly. On the basis of the box plots consti-tuted by solving the function 30 times, while the CSS approach had five deviating values, the OCSS-2 approach had three. Upon analyz-ing the values and graphs obtained for five solved functions, the opposition-based learning structure was observed to provide a dis-tinct improvement for the original algorithm.
6. Sallen-Key topology Butterworth low pass active filter (LPF) A low-pass active filter is an electronic device that allows all the frequency components below a selected frequency value to pass and prevents or attenuates the pass of all the frequency components higher than the aforementioned frequency value. These filters are designed with varying structures in the literature. The second-order circuit plan of the Sallen-Key topology Butterworth low-pass
active filter used in this study is given inFig. 6(Hiçdurmaz et al., 2016; Pactitis, 2007; Sallen & Key, 1955; Karki, 2002).
The transfer function of the circuit is given in Eq.(12).
HLPFðsÞ ¼VVoiðsÞðsÞ¼ 1 1þsðR1þR2ÞC1þs2R1R2C1C2; s ¼ j2
p
f HLPFðf Þ ¼1ð2p 1 fÞ2 R1R2C1C2þj2pfðR1þR2ÞC1 ð12ÞThe standard form of the transfer function is described in Eq.
(13) (Hiçdurmaz et al., 2016; Pactitis, 2007; Sallen & Key, 1955; Karki, 2002). HLPFðf Þ ¼ 1 1 f FSFf c 2 þj f QFSFf c FSF¼ 1 2pfc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1R2C1C2 p ; Q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1R2C1C2 p ðR1þR2ÞC1 ð13Þ
In the equation, fcrepresents the cut-off frequency while FSF represents the frequency scaling factor and Q represents the qual-ity factor.
The amplitude response of the designed filter is described in Eq.
(14). HLPFðf Þ ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f FSFfc 2 2 þ f QFSFfc 2 s ð14Þ
A tenth-order LPF is constructed by cascading five second-order stages inFig. 7. To realize the tenth-order Butterworth LPF, FSF and Q values of each stage are given inTable 4(Hiçdurmaz et al., 2016). The designed filter circuit has a cut-off frequency of 10 kHz.
6.1. The simulation results of the 10th-order Butterworth LPF In the application of CSS and OCSS, each passive component of the filter belonging to each stage was encoded in the string form
(a) (b)
Fig. 5. The box plots and convergence curves from which the best results for f5 were obtained.
C
2C
1R
1R
2v
ov
iFig. 6. The circuit plan of the second-order Sallen-Key topology Butterworth LPF.
C21 C11 R11 R21 .... vi C25 C15 R15 R25 vo Stage 1 Stage 5
as shown inTable 5. In this case, the component values of the filter are successively adjusted by BSA until the error is minimized. The design error of the filter is the summation of the cost function errors of FSF and Q and given as (Hiçdurmaz et al., 2016).
Error1¼ P5 i¼1 FSFt;iFSFi j j FSFt;i ; Error2¼ P5 i¼1 Qt;iQi j j Qt;i
ErrorTotal¼ 0:5 Error1þ 0:5 Error2
ð15Þ
In the equation, FSFt,iis the targeted FSF, while Qt,iis the tar-geted Q. The objective here is to minimize the error.
The parameter values used in the solution of the problem were selected the same as the parameters used in the solution of the test functions. The numbers of iteration, agents in the population, and dimension (unknown) were taken as 1000, 40, and 20 respectively. The problem was solved 30 times with these parameters with the CSS and OCSS algorithms. E24 standard series used for the solution of the problem is given inTable 6, while the component values for the designed filter calculated by the algorithms are given inTable 7.
The FSF and Q values calculated for all the layers along with these component values calculated for the designed filter are given in
Table 8. The statistical values regarding the best solution values among 30 solutions are given inTable 9.
The gain curve graphs acquired in the solution of LPF with CSS and OCSS are given inFig. 8a and b. The graphs demonstrating the convergence curves belonging to the best solution value obtained with the algorithms are given inFig. 9a, and the box plots for 30 stages are presented inFig. 9b.
7. Sallen-Key topology Butterworth high pass active filter (HPF) In HPF, above its cut-off frequency fc inputs are transmitted to the output with no attenuation whereas below fc inputs are infi-nitely attenuated. A second-order unity-gain Sallen-Key HPF archi-tecture is given as [1].
The transfer function of the circuit is given in Eq.(16)(Durmusß et al., 2016).
Table 4
The FSF and Q values of the 10th-order Butterworth Filter.
Filter order Stage 1 Stage 2 Stage 3 Stage 4 Stage 5
FSF Q FSF Q FSF Q FSF Q FSF Q
10 1.000 0.5062 1.000 0.5612 1.000 0.7071 1.000 1.1013 1.000 3.1969
Table 5
Representing the component values in the string form.
Stage 1 . . . Stage 5
R11 C11 R21 C21 . . . R15 C15 R25 C25
Table 6
E24 standard series for the resistors and the capacitors. E24 Series, R, C
1.0 1.1 1.2 1.3 1.5 1.6 1.8 2.0
2.2 2.4 2.7 3.0 3.3 3.6 3.9 4.3
4.7 5.1 5.6 6.2 6.8 7.5 8.2 9.1
Table 7
Component values of best solutions for filter design (LPF).
Method Component Stage 1 Stage 2 Stage 3 Stage 4 Stage 5
CSS R1(kX) 1.50 1.10 5.10 3.00 1.10 R2(kX) 2.70 2.40 2.40 4.30 1.00 C1(nF) 7.50 8.20 3.00 2.00 2.40 C2(nF) 8.20 12.00 6.80 10.00 100.00 OCSS-1 R1(kX) 3.30 2.70 1.60 2.70 1.50 R2(kX) 1.80 6.80 2.00 2.70 2.40 C1(nF) 6.20 3.00 6.20 2.70 1.30 C2(nF) 6.80 4.70 13.00 13.00 56.00 OCSS-2 R1(kX) 3.00 3.30 7.50 2.70 6.80 R2(kX) 5.10 4.70 3.60 3.30 1.30 C1(nF) 3.90 3.60 2.00 2.40 0.62 C2(nF) 4.30 4.70 4.70 12.00 47.00 Table 8
Design parameters of filter design (LPF).
Method Stage1 Stage 2 Stage 3 Stage 4 Stage 5
FSF Q FSF Q FSF Q FSF Q FSF Q
Target 1.0000 0.5062 1.0000 0.5612 1.0000 0.7071 1.0000 1.1013 1.0000 3.1969
CSS 1.0085 0.5010 0.9875 0.5616 1.0072 0.7023 0.9909 1.1002 0.9795 3.2238
OCSS-1 1.0057 0.5005 0.9892 0.5645 0.9910 0.7195 0.9950 1.0971 0.9831 3.1931
HHPFðsÞ ¼VVoiðsÞðsÞ¼ 1 1þsR1C1C2C1þC2þ 1 s2 R1R2C1C2 ; s ¼ j2
p
f HHPFðf Þ ¼ 1 1 1 2pfpffiffiffiffiffiffiffiffiffiffiffiffiffiR1R2C1C2 2 j C1þC2 2pfR1C1C2 ð16ÞThe standard form of the transfer function is described in Eq.
(17) (Durmusß et al., 2016; Pactitis, 2007; Sallen & Key, 1955; Karki, 2002). HHPFðf Þ ¼ 1 1 f c FSFf 2 j f c QFSFf FSF¼ 2
p
fc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1R2C1C2 p ; Q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1R2C1C2 p ðR2ðC1þC2Þ ð17ÞThe amplitude response of the designed filter is described in Eq.
(18). HHPFðf Þ j j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 fc FSFf 2 2 þ fc QFSFf 2 s ð18Þ
A tenth-order HPF is constructed by cascading five second-order stages inFig. 10. To realize a tenth-order Butterworth HPF, FSF and Q values of each stage are given inTable 4. The designed filter cir-cuit has a cut-off frequency of 10 kHz.
7.1. Simulation results for the tenth-order Butterworth HPF
In the application of CSS and OCSS, each passive component of the filter belonging to each stage was encoded in the string form as shown inTable 5. In this case, the component values of the filter are successively adjusted by BSA until the error is minimized. The design error of the filter is the summation of the cost function errors of FSF and Q and given as Eq.(15).
Table 9
Values obtained from 30 stages (LPF).
Method Best error Mean error Worst error Std. Dev. Time (s)
CSS 4.248318e02 1.105586e01 6.622992e01 1.335668e01 6.84818
OCSS-1 4.365034e02 1.379336e01 6.192738e01 1.597862e01 6.92527
OCSS-2 3.452711e02 1.029345e01 8.003830e01 1.491788e01 7.25974
(a) (b)
Fig. 8. Total gain curves for computed solutions (LPF).
(a) (b)
Fig. 9. Convergence curves and box plots (LPF).
R
1R
2C
1C
2v
iv
oThe parameter values used in the solution of the problem were selected the same as the pre-determined parameters. The problem was solved 30 times with these parameters with CSS and OCSS. E24 standard series used to solve the problem is given inTable 6while the component values found by the algorithms for the designed HPF are presented inTable 10. The FSF and Q values calculated for all the layers along with these component values calculated for the designed filter are given inTable 11. The statistical values regarding the best solution values among 30 solutions are given inTable 12.
A tenth-order HPF is constructed by cascading five second-order stages inFig. 11. The gain curve graphs acquired in the solution of HPF with CSS and OCSS are given inFig. 12a and b. The graphs demonstrating the convergence curves belonging to the best solu-tion value obtained with the algorithms are given inFig. 13a, and the box plots for 30 stages are presented inFig. 13b.
The convergence speed in performance evaluation of the algo-rithms can be defined as the iteration number until finding the minimum value of the purpose function. As for convergence rate cr, is defined as the rate of iteration number nf, where the minimum
value is obtained, to maximum iteration number N, as in Eq.(19). (Pelusi, Mascella, & Tallini, 2018; Pelusi et al., 2018).
cr¼
nf
N ð19Þ
The convergence rates of the algorithms handled in the study are calculated approximately for LPF problem as CSS 0,449, OCSS-1 0,692 and OCSS-2 0,704, as for HPF problem as CSS 0,622, OCSS-OCSS-1 0,610 ve OCSS-2 0,645.
In order to compare such studies in the literature statistically, the paired sample t-test and the Wilcoxon tests are applied to the obtained results (García, Molina, Lozano, & Herrera, 2009). Since the parametric paired sample t-test cannot bear precise results in the case of a low data number, more sensitive and precise results can be acquired by utilizing the non-parametric Wilcoxon test. To determine the relationship between the algorithms, the best values from 30 stages were subjected to statistical assessment tests by using the sumrank and ranksum commands from the Mat-lab library. The obtained statistical test results are given in
Table 13.
Table 10
Component values of best solutions for filter design (HPF).
Method Component Stage 1 Stage 2 Stage 3 Stage 4 Stage 5
CSS R1(kX) 6.20 9.10 16.00 5.10 39.00 R2(kX) 5.60 6.80 8.20 1.00 0.91 C1(nF) 2.00 1.50 1.30 5.60 3.00 C2(nF) 3.60 2.70 1.50 9.10 2.40 OCSS-1 R1(kX) 4.30 7.50 3.60 9.10 15.00 R2(kX) 3.00 5.60 1.80 1.80 0.36 C1(nF) 2.40 2.00 6.20 3.00 6.20 C2(nF) 8.20 3.00 6.20 5.10 7.50 OCSS-2 R1(kX) 13.00 8.20 9.10 20.00 100.00 R2(kX) 12.00 6.20 2.70 1.60 2.40 C1(nF) 1.60 1.80 6.80 1.00 1.00 C2(nF) 1.00 2.70 1.50 8.20 1.10 Table 11
Design parameters of filter design (HPF).
Method Stage1 Stage 2 Stage 3 Stage 4 Stage 5
FSF Q FSF Q FSF Q FSF Q FSF Q Target 1.0000 0.5062 1.0000 0.5612 1.0000 0.7071 1.0000 1.1013 1.0000 3.1969 CSS 0.9934 0.5042 0.9947 0.5543 1.0050 0.6966 1.0129 1.0967 1.0044 3.2530 OCSS-1 1.0011 0.5010 0.9974 0.5669 0.9917 0.7071 0.9947 1.0858 0.9956 3.2129 OCSS-2 0.9927 0.5064 0.9876 0.5634 0.9947 0.7064 1.0178 1.1005 1.0209 3.2238 Table 12
Values obtained in 30 stages (HPF).
Method Best error Mean error Worst error Std. Dev. Time (s)
CSS 4.351205e02 1.036853e01 6.486634e01 1.129261e01 7.2027
OCSS-1 3.062424e02 1.437823e01 8.918727e01 1.747409e01 7.80453
OCSS-2 3.905782e02 1.502514e01 8.432651e01 1.806655e01 7.56954
8. Conclusion
In the first section of this study, the concept of opposition-based learning, one of the approaches utilized for enhancing the perfor-mance of heuristic algorithms in the literature, was applied to the charged system search algorithm (CSS) in two different meth-ods. In the first approach (OCSS-1), while half of the agents were assigned randomly during the constitution of the initial popula-tion, the other half were placed symmetrically to the first half. In the second approach (OCSS-2), opposites of the randomly consti-tuted agents were determined in the initial population, which was then created with the agents bearing higher fit values. The algorithm was improved in terms of performance and stability thanks to these approaches. This newly created algorithm was named the opposite charged system search algorithm (OCSS). For performance analysis, OCSS was applied to five multidimensional and multimodal test functions. Based on the results obtained, both of the developed approaches (OCSS-1 and OCSS-2) provided
improvement for CSS in terms of stability and performance. Upon comparing the approaches to each other, OCSS-1 was concluded to reach better values and embody a more stable structure than OCSS-2.
In the second section of the study, with regard to applying the newly developed algorithms to engineering problems, parameter values for Sallen-Key topology Butterworth low and high-pass active filters were determined with CSS and OCSS. E24 standard series values were utilized in this study to decrease designing costs. Tenth-order low and high-pass active filters were designed with series structures. The parameter values of these designed tenth-order filters were determined with CSS and OCSS. The obtained parameter values and the quality factors (Q) of the designed filters were calculated separately. These calculated values were compared to the values found in the literature and the results were accordingly assessed.
The charged system search algorithm (CSS) that has been han-dled in the study, has been applied to several test functions and complex problems since the day it was brought to literature and has obtained suitable solutions for these problems. But, in more complex fitness functions and high dimensional problems, as in several optimization algorithms, CSS also had problems such as sticking to local minimums and obtaining unsuccessful results. In order to eliminate these problems, in the study opposite learning based search strategies have been successfully proposed for CSS. Another important contribution is that with these developed algo-rithms, the acceptable solutions of the LP and HP filter design prob-lems, which are important designs for electrical-electronical engineering, have been brought to the literature.
(a) (b)
Fig. 13. Convergence curves and box plots (LPF).
(a) (b)
Fig. 12. Total gain curves for computed solutions (HPF).
Table 13
Statistical test results regarding the best values obtained for 30 stages in LPF and HPF solutions. Sumrank Ranksum LPF CSS vs OCSS-1 0.262299 0.311188 CSS vs OCSS-2 0.298944 0.115362 OCSS-1–OCSS-2 0.102011 0.016285 HPF CSS vs OCSS-1 0.490798 0.761828 CSS vs OCSS-2 0.171376 0.491783 OCSS-1–OCSS-2 0.991795 0.695215
My suggestions for the future studies are these: the effects of the opposite position learning approach on other heuristic algo-rithms can be analysed solving test functions and different engi-neering problems; different approaches other than the two approaches proposed in this study, can be developed for the oppo-site position learning concept; LPF and HPF designs solved in the study can be solved for different degrees and stages and finally new designs can be done for different filter types and different resistor and capacitor series that haven’t been handled in this study.
CRediT authorship contribution statement
Hasan Temurtasß: Conceptualization, Methodology, Data cura-tion, Writing - original draft, Validacura-tion, Writing - review & editing. Declaration of Competing Interest
The authors declare that they have no known competing finan-cial interests or personal relationships that could have appeared to influence the work reported in this paper.
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