Optimal design of FIR fractional order differentiator using cuckoo search algorithm

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Optimal design of FIR fractional order differentiator using cuckoo search algorithm Manjeet Kumar ⇑, Tarun Kumar Rawat Electronics and Communication Engineering Division, Netaji Subhas Institute of Technology, Sector-3, Dwarka, New Delhi 110078, India

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a r t i c l e

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Article history: Available online xxxx

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Keywords: Cuckoo search algorithm Fractional order differentiator Meta-heuristics Genetic algorithm Lévy flight

a b s t r a c t In this paper, a new meta-heuristic optimization algorithm, called cuckoo search algorithm (CSA) is applied to determine the optimal coefficients of the finite impulse response-fractional order differentiator (FIR-FOD) problem. CSA is based on lifestyle and unique parasitic behavior in egg laying and breeding of some cuckoo species along with Lévy flight behavior of some birds and fruit flies. The CSA is capable of solving linear and nonlinear optimization problems. The proposed CSA method prevents the local minima problem encountered in conventional FIR-FOD design method. A novel weighted least square (WLS) fitness function is adopted to improve the response of the FOD to a great extent. The proposed CSA based method has alleviated from inherent drawbacks of premature convergence and stagnation unlike genetic algorithm (GA). To verify the effectiveness of the proposed FIR-FOD based on the cuckoo search algorithm, different set of initial population is tested by simulation. Simulation results affirm that the proposed fractional order differentiator design approach using CSA outperforms the genetic algorithm in terms design accuracy (magnitude and phase error), fast convergence rate and optimal solution. The simulation results confirmed that the proposed FOD using CSA outperforms the FOD designed using evolutionary algorithm like GA and conventional FOD design methods such as radial basis function (RBF) interpolation method and DCT interpolation method. Ó 2014 Elsevier Ltd. All rights reserved.

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1. Introduction

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In recent years, design of fractional order digital differentiator has become an important research topic in the field of digital signal processing. Fractional order digital differentiator calculates the time-derivative of any applied signal. From the last few decades, the concept of fractional derivative has received great attention in many applications of engineering, science and technology including image processing (Das, 2008), automatic control (Miller & Ross, 1993; Mbodje & Montseny, 1995), fluid dynamics (Miller & Ross, 1993), electromagnetic theory (Engheta, 1997), electrical networks and probability (Fenander, 1998). Fractional calculus (Das, 2008; Miller & Ross, 1993) is concerned with the generalization of integral and derivative to a non-integer orders. To obtain the flexible designing of digital differentiators, we generalize the integer order derivative to non-

44 45 46 47 48 49 50 51 52 53 54 55 56

n

integer order. The integer order derivative Dn f ðxÞ ¼ d dxf ðxÞ (nth order n derivative of the function f ðxÞ) has been generalized to fractional

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⇑ Corresponding author. Tel.: +91 8860010669.

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E-mail addresses: [email protected] (M. Kumar), [email protected] (T.K. Rawat).

order derivative Da f ðxÞ ¼ d dxf ðxÞ a , where n is an integer and a is a real number (Miller & Ross, 1993; Oldham & Spanier, 1974). The ideal frequency response of digital fractional order differentiator is given by

Hid ðwÞ ¼ ðjwÞ

a

ð1Þ

where a is considered as a fractional number in the range ½0:5; 0:5 and x 2 ½0; 1 is the normalized frequency. The fractional order differentiator design process can be described as an optimization problem, where the design requirement is described in terms of error function which should be minimized. The design process of fractional order differentiator involves four steps. First, define a desired ideal frequency response of fractional order differentiator. Second, choose a class of system (FIR or IIR). Third, choose an optimality criteria that control the behavior of the design problem. Finally, applying an optimization method to find the optimal system coefficients. The design methods for digital fractional order differentiators (DFOD) have been intensively researched in the signal processing literature for over almost two decades; see (Krishna, 2011; Vinagre, Podlubny, Hernandez, & Feliu, 2001) and references therein. The significant research in the field of continuous-time

http://dx.doi.org/10.1016/j.eswa.2014.12.020 0957-4174/Ó 2014 Elsevier Ltd. All rights reserved.

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FOD has been cited by several authors. Carlson et al. presented a third order Newton process to approximate ð1=sÞ1=n for fractional capacitor (Carlson & Halijak, 1964). Roy presented a method for realization of immittance with constant argument that approximates the fractional operator. The method is used for approximating the immittance as s0:5 based on continued fraction expansion (Dutta Roy, 1967). Chareff et al. reported a graphical method to approximate a single fractional power pole transfer function. The method is used for multiple-fractal system which consists of a number of fractional power poles (Chareff, Sun, Tsao, & Onaral, 1992). Matsuda and Fujii applied a H1 control theory to design a broadband compensator using the wave absorbing control method (Matsuda & Fujii, 1993). Oustaloup et al. investigated complex noninteger differentiator by a recursive distribution of complex zeros and poles (Oustaloup, Levron, Mathieu, & Nanot, 2000). Barbosa et al. proposed the use of the least square method to approximate the fractional order differentiator and integrator (Barbosa, Machado, & Silva, 2006). After the effective use of continuous-time FOD, the research trend has been towards the use of discrete-time fractional order differentiator. Tseng proposed a fractional order differentiator based on Taylor series expansion in which fractional derivative of a digital signal is computed using filtering technique (Tseng, 2001). Vinagre et al. reported continued fraction expansion and recursive Tustin transformation methods for discretizing the continuous-time fractional order differentiator (Vinagre, Chen, & Petras, 2003). Samadi et al. presented FOD for polynomial signal using Newton series expansion. This method gives exact fractional derivatives of polynomial signal (Samadi, Ahmad, & Swamy, 2004). Tseng applied logarithm and Taylor series expansion to approximate the variable fractional order integrator and differentiator (Tseng, 2008). Tseng also presented FOD design using fractional sample delay and the design accuracy was improved at higher frequency region (Tseng, 2006). Tseng and Lee investigated the design of FOD using interpolation techniques such as radial basis function (Tseng & Lee, 2010a), DFT interpolation (Tseng & Lee, 2010b), DCT interpolation (Tseng & Lee, 2013a), and DST interpolation (Tseng & Lee, 2013b) to approximate the fractional derivative of a signal. Khare et al. applied DCT-III interpolation to design FOD (Khare, Kumar, & Rawat, 2014). Chen et al. reported a fractional order Savitzky–Golay differentiator for estimating the fractional order derivative of a contaminated signal (Chen, Chen, & Xue, 2011). Tseng and Lee investigated the design of quadrantally even symmetric and odd symmetric linear phase filters based on fractional derivative constraints (Tseng & Lee, 2013c). Tseng and Lee also investigated fractional derivative constrained in the complex domain for 1-D and 2-D FIR filters design (Tseng & Lee, 2014). These conventional techniques use the unimodal fitness function to approximate the ideal FOD. In these techniques, minimization of a fitness function is accomplished by gradient based algorithms. The drawbacks of these conventional optimization techniques are: (i) Locally optimal filter coefficients are obtained, (ii) Do not provide a suitable solution to high-dimensional optimization problem, and (iii) Requires linear fitness function. It can be concluded that conventional techniques are suitable for unimodal fitness function with limited search space. Thus, there is a demand of robust and global optimization technique. The conventional optimization algorithms do not provide a suitable solution to high-dimensional optimization problems because the search size increases exponentially with the increase in problem size. Solving these problems with conventional techniques is not practical. However, when the complex and multimodal fitness function is used, gradient based algorithms cannot converge to the minimum. In this condition, heuristic optimization algorithms are employed to achieve a global optimal solution in solving difficult optimization problems.

Recently, a number of nature inspired meta-heuristic algorithms (Yang, 2011) have been developed for the purpose of robust and global optimization. These algorithms have the capability to solve large problems in less time and in a robust manner. It is shown by many researchers that these algorithms are suitable to solve complex computational problems. Some of the commonly used nature inspired optimization techniques for filter design are as follows: genetic algorithm is inspired by the Darwin’s ‘‘Survival of the fittest’’ strategy (Goldberg, 1989; Boudjelaba, Ros, & Chikouche, 2014; Upadhyay, Kumar, & Rawat, 2014), Ant Colony optimization simulates the ant food search behavior (Karaboga, Kalinli, & Karaboga, 2004; Hai-bin, Dao-bo, & Xiu-fen, 2006), Particle Swarm Optimization (PSO) simulates the behavior of bird flocking (Sun, Zhao, & Zhao, 2009; Lin, Chang, & Hsieh, 2008), Differential Evolution depends on appropriately choosing trial vector generation strategies (Storn & Price, 1997), bee swarm intelligence mimics the honey collection behavior of honey bee (Karaboga & Akay, 2009), Simulated Annealing is used for thermodynamic effect (Davidson & Harel, 1996), Cat Swarm optimization is based on the behavior of cat for tracking and seeking of an object (Panda, Pradhan, & Majhi, 2011), Artificial Immune System mimics the biological immune system (Kalinli & Karaboga, 2005), Firefly Algorithm is inspired by flashing characteristics of fireflies (Yang, 2009; Kavousi-Fard, Samet, & Marzbani, 2014), Bat Algorithm is based on the echolocation behavior of bats (Yang & Gandomi, 2012) etc. The efficiency of meta-heuristic algorithms can be

Fig. 1. Flowchart of the cuckoo search algorithm.

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Fig. 2. Comparison of 4th order FIR-FOD using GA and CSA (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

Table 1 Control parameters of GA and CSA for optimizing FIR-FOD. Parameters

GA

CSA

Initial population Maximum iterations Tolerance

20 50–200

25 100–1000

105 0.001

105 –

1 1 1 Feasible population 0.90 Scattered Rank Roulette Constraint dependent –

1 1 – – – – – – – 0.25

Maximum iteration/Best solution

Maximum iteration

Initial value of all coefficients Lower bound Upper bound Elite count Creation function Crossover fraction Crossover function Scaling function Selection function Mutation function Discovering rate of alien eggs Stopping criteria

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attributed to the fact that they imitate the best features in nature, especially the selection of the fittest in biological systems which have evolved by natural selection over millions of years. These soft-computing techniques are even easier to implement as compared to other conventional optimization techniques. Also, the

results obtained are better which make meta-heuristic algorithms very attractive. To date, none of the published literature has applied natureinspired meta-heuristic optimization algorithm for the optimization of fractional order differentiator problem. In this paper, the capability of global search and optimal robust solution finding feature of GA and CSA is investigated for the design of FOD. The genetic algorithm exhibit the problem of premature convergence, stagnation and revisiting the same solution over and again. To overcome these problems, the cuckoo search algorithm (CSA) is applied to determine the optimal solution of fractional order differentiator design problem. Various statistical results and simulations provide a comparison with GA, radial basis function interpolation method (Tseng & Lee, 2010a) and DCT interpolation method (Tseng & Lee, 2013a) to validate the superiority of the proposed CSA based method. In an overall comparison of the optimal solution found for FODs, CSA is observed to outperform GA by a noticeable value. The execution time reported for the proposed CSA based method has been given to ensure that in the required time frame CSA based method can efficiently find optimal solutions for the practical applications. Some of the unique features of cuckoo search algorithm (Yang & Deb, 2009) which makes it popular in very less time for optimization problems are: (i) The number of parameters required in the cuckoo search algorithm are less than other nature inspired optimization techniques, thus it could be implemented to a wider range of optimization problems, (ii) Due to the elitism property

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of cuckoo search the best solution always remains in the search space, (iii) Furthermore, the convergence rate is insensitive to the parameters. This means that we do not have to fine tune these parameters for a specific problem. Thus, cuckoo search is more generalized and robust for many optimization problems, comparing with other meta-heuristic algorithms, and (iv) Combination with Lévy flight makes the randomization more efficient as the step length is heavy tailed, and any large step is possible. In very short duration, it is implemented on various kinds of optimization problems and has outperformed other nature inspired optimization schemes. Some of the applications of the cuckoo search algorithm are: engineering design problems such as welded beam design, spring design optimization (Yang & Deb, 2010), milling optimization problem (Yildiz, 2013), multivariable controller design (Rajabioun, 2011), cantilever beam design, corrugated bulkhead design, tubular column design, parameter identification of structures, structural optimization (Gandomi, Yang, & Alavi, 2013), optimization of PCB track length, robotics manipulator (Sharma, Rana, & Kumar, 2014), and feedback system identification (Patwardhan, Patidar, & George, 2014). The rest of the paper is organized as follows: Section 2 deals with the brief description of fractional derivative and some standard definitions of integral–differential operator. In Section 3, problem formulation for the design of FIR-FOD and WLS based cost function is presented. Section 4, focuses on the basics of GA and CSA and how FIR-FOD is optimized using these algorithms. Section 5 presents the implementation of GA and CSA for the design of

FIR-FOD. In Section 6, the simulation results are demonstrated and interpreted in detail. Finally, the paper is concluded in Section 7.

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2. Fractional Derivative and definition

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For the last two decades, the concept of fractional calculus is found useful in many applications of signal processing. The unique feature of fractional calculus is its ability to generalize the integral and differential operators to noninteger order. The generalized continuous integral–differential Davis operator is (Miller & Ross, 1993) 8

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a

d a>0 > < dta ; a 1; a¼0 a Dt ¼ > :Rt a ðd s Þ ; a<0 a

ð2Þ 245

a

where a Dt denotes integral–differential operator to calculate the

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ath order fractional differentiation and integration of the input sig-

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nal with respect to time, t and a is the initial condition of the operation. Some of the popular definitions for this integral–differential operation are Riemann–Liouville (R–L), Grünwald–Letnikov (G–L) and the Caputo definitions (Miller & Ross, 1993; Oldham & Spanier, 1974). In this paper, the Grünwald–Letnikov definition for the fractional order computation is used, which is as follows

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a

a Dt

sðtÞ ¼ lim D!0

1 X ð1Þk C a k¼0

Da

k

sðt  kDÞ

ð3Þ

Fig. 3. Comparison of 8th order FIR-FOD using GA and CSA (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

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Fig. 4. Comparison of 10th order FIR-FOD using GA and CSA (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

Table 2 Optimized coefficients for FIR-FOD with different order filter. Filter order (N  1)

Algorithm

Optimized coefficients ðhk Þ 0 6 k 6 N

4

GA CSA

0:1427 0:7498

8

GA

0:0066 0:0119 0:0073 0:0322 0:0699 0:7892 0:9549 0:1804 0:0532 0:2137 0:9443 0:7854 0:1043 0:0891 0:0053 0:0244 0:0038 0:0096

CSA 10

GA

0:9999 0:7365 0:0522 0:0758 0:9946 0:1122 0:0828 0:0062

0:0637 0:2948 0:8388 0:8125 0:3571 0:0529 0:0803 0:0424 0:0453 0:0183 0:0124 0:2823 0:2235 0:6294 0:9904 0:2184 0:0695 0:1140 0:0068 0:0171 0:0206 0:0027

CSA

Table 3 Comparison summary of magnitude and phase error for different order FIR-FOD using GA and CSA. Filter order (N  1)

Magnitude error (m ) GA

Magnitude error (m ) CSA

Phase error (p ) GA

Phase error (p ) CSA

5

4

5:0650  10

0:0490

0:0269

8

2:6005  105

1:9257  105

0:0082

0:0029

10

2:1357  105

1:0894  105

0:0051

0:0012

5

2:2225  10

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where C ak is the binomial coefficient. The value of C ak is given by using the relation between Euler’s Gamma function and factorial, defined as

C ak ¼

 

a

( ¼ 262

k

¼

Cða þ 1Þ Cðk þ 1ÞCða  k þ 1Þ

1

k¼1

aða1Þða2Þðakþ1Þ 123k

kP1

ð4Þ

J ¼ Jm þ Jp

ð5Þ

where J m and Jp is the fitness function of optimal magnitude response and optimal phase response, respectively. These are defined as

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where CðÞ is the gamma function. The result of fractional derivative depends on the bound of the operator a. A common value for this bound a ¼ 0.

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3. Problem formulation

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In this paper, for the purpose of optimal design of fractional order differentiator, a novel weighted least square fitness function, J, is adopted:

1

2

W 1 ðxÞjabsðHid ðxÞÞ  absðHðxÞÞj dx

ð9Þ

0

and

Jp ¼ 267

268

270 271 272

273

275

N1 X hðnÞzn

ð6Þ

n¼0

where hðnÞ are the filter coefficients to be optimized and the corresponding frequency response is N1 X HðxÞ ¼ hðnÞejxn

279 281 282 283 284

287

289 1

2

W 2 ðxÞjphaseðHid ðxÞÞ  phaseðHðxÞÞj dx

ð10Þ

0

where W 1 ðxÞ is a nonnegative weighting function for optimal magnitude response and W 2 ðxÞ is a nonnegative weighting function for optimal phase response. Substituting Eqs. (9) and (10) into Eq. (8), we obtain the weighted least square fitness function J:

J¼

278

288

Z

Let the transfer function of FIR-FOD of length N be expressed as,

HðzÞ ¼

277

285

Z

Jm ¼

ð8Þ

276

Z

1

h

W 1 ðxÞjabsðHid ðxÞÞ  absðHðxÞÞj

291 292 293 294 295

296

2

0

ð7Þ

n¼0

i þW 2 ðxÞjphaseðHid ðxÞÞ  phaseðHðxÞÞj2 dx

ð11Þ

Fig. 5. Comparison of 4th order FIR-FOD using CSA with worst, average and best condition (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

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In this paper, the fitness function in Eq. (11) is minimized using genetic and cuckoo search algorithm and the corresponding optimized filter coefficients are used for the design of fractional order differentiator.

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4. Nature inspired optimization algorithms

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Meta-heuristic algorithms form an important part of global optimization algorithms and soft computing. These are classified as, population based and trajectory based algorithm. In this paper, we are implementing two population based meta-heuristic algorithms namely, genetic algorithm and cuckoo search algorithm for the purpose of designing and performance comparison of FIRFOD.

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4.1. Genetic algorithm

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Genetic algorithm is based on Charles Darwin’s principle of ‘‘Survival of the fittest’’. Initially, a set of coefficient chromosomes is randomly selected. These chromosomes are encoded as binary strings called genotypes. The genetic operations such as selection, crossover and mutation are applied on each individual genotype chromosome to produce a new generation of offspring chromosomes. Corresponding to each genotype, there is a decimal equivalent, phenotype, which is used to evaluate the fitness

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function. According to the problem under consideration, each individual in the population is assigned, by means of a fitness function, a measure of its goodness. Best fitted chromosomes, called the elite chromosomes are transmitted as it is to the next generation. With each generation, better solutions are obtained. Some important terms used in GA are as follows.

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Selection: The basic idea of selection is that it should be related to fitness. It uses a probability distribution for selection in which the selection probability of a given string is proportional to its fitness. Crossover: Crossover refers to replacing some of the genes of one parent with that of the other parent. Crossover fraction decides the amount of genes replaced. Mutation: It is another important genetic operation, which includes choosing a subset of genes randomly and then changing the allele value of the chosen gene. It simply means complementing the chosen bits, in case of binary strings. Elite count: This is the count which denotes the number of genes which are best fitted and thus transmitted as it is, to the next generation.

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The steps involved in GA are as follows: Step 1: Initialize the chromosome string of the population (N p ) and maximum iteration cycle (N i ).

Fig. 6. Comparison of 8th order FIR-FOD using CSA with worst, average and best condition (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

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Step 2: Decoding the strings and evaluate the error fitness values. Step 3: Selection of the elite strings in order of increasing the error fitness value from the minimum value. Step 4: Copying the elite strings over the non-selected chromosome strings. Step 5: Crossover and mutation generate the offsprings. Step 6: Genetic cycle updating. Step 7: The iteration stops when the maximum number of iterations reached or the convergence of minimum error fitness value is achieved. The chromosome string corresponding to the minimum error fitness value gives the optimal solution to the problem.

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4.2. Cuckoo search algorithm

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In order to understand the cuckoo search algorithm, cuckoo’s unique breeding behavior and the concept of the Lévy flights is needed to be understand. Next subsections illustrate these concepts in detail.

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and if alien eggs remain unidentified then they grow up as young ones. It is also proved that a cuckoo chick can mimic the call of host chick to have more access to food and take care of the host bird. But if the host identifies the alien member then either kill it or abandons the nest.

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4.2.2. Lévy Flights Lévy flights are the forward steps taken by living beings, such as, birds, insects, animals etc. in search of their food. These steps are random and depend on current location and the transition probability to the next location. The direction of random forward walk follows a probability which can be modelled mathematically. This random walk is derived from Lévy distribution with an infinite variance and mean. Such behavior has been applied to optimization and optimal search, and the results obtained show its promising capability. For generating/choosing a random new nest following Lévy flight based formula is used,

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vy ðkÞ xl ¼ xi þ a0  Le 362 363 364 365 366 367

4.2.1. Cuckoo’s lifestyle of reproduction Cuckoos are the birds with a unique style of reproduction. They follow brood parasitism, that is, they depend on other birds for hatching their eggs. Parasitic cuckoo tries to find a nest where the host bird has recently laid eggs. Cuckoo use this nest to hide its own eggs. Now, the host bird takes care of all the eggs together

ð12Þ

where, a0 ða0 > 0Þ is the step size related to the problem specified. This equation represents a random walk which is a Markov chain which means its next step depends on the current location and the transition probability. Lévy ðkÞ follows the Lévy distribution with an infinite variance and infinite mean (Yang & Deb, 2009).

Fig. 7. Comparison of 10th order FIR-FOD using CSA with worst, average and best condition (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

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To apply Cuckoo Search algorithm, three basic rules are followed (Yang & Deb, 2009): (i) Each cuckoo bird lays one egg at a time and choose a host nest randomly to hide it. (ii) The nests with the best host environment are carried over to the next generations. (iii) Available host nests are fixed in number, and the probability of identifying alien egg of the host bird is P a 2 ½0; 1. Based on these three basic rules, CSA is applied for the design of fractional order differentiator. The steps involved in CSA are as follows: Step 1: Randomly generate an initial population of n host nests xi (i ¼ 1; 2; . . . ; n) and initialize the maximum number of iterations (N i ). Step 2: Generate a new nest using the Lévy flights given by vy ðkÞ. Compute the fitness value, f l , of xl ¼ xi þ a0  Le the new nests. Step 3: Compute fitness value, f n , of a randomly generated host nest, xi , and Compare these values with the fitness value (f l ) of the new generated nests by Lévy flight. Step 4: For a minimization problem, if f n > f l , initial host nests xi is replaced by new nests, xl , generated by Lévy flights. Step 5: A fraction pa of the worst nests obtained after Step 4 are abandoned and new nests (xn ) using a random flight are built. Step 6: Compute the fitness of all the new nests. Step 7: Updating the best nest xp of the generation.

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Step 8: The best nest obtained until the current iteration, xb , is replaced by the best nest xp of the generation, if the fitness value, f p , of nest xp is less than the fitness value, f b , of xb for the minimization problem. Step 9: Repeat Steps 2–8 until the maximum number of iterations reached and best nest xb gives the optimal solution to the problem.

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5. Implementation of nature inspired algorithms for FIR-FOD design

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This section discusses the implementation steps of optimization algorithms used in this paper for the problem of FIR-FOD design.

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5.1. Implementation of GA for the FIR-FOD design problem

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To design an (N  1) th order FIR-FOD, find out the best fitted Ncoefficient chromosomes, hð0Þ; hð1Þ; . . . ; hðN  1Þ, using genetic algorithm. The steps of GA as implemented for the optimization of the coefficients are as follows:

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Step 1: Initialize the maximum number of iteration (N i ¼ 200). The maximum and minimum value of the coefficients chromosomes are +1 and 1, respectively. Step 2: Set an initial population size (N þ x), i.e., number of solution coefficients initially, which are later converged to N number of solution coefficients. Here, x P 0. Small popula-

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Fig. 8. Comparison of different order FIR-FOD using CSA (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

Q2 Please cite this article in press as: Kumar, M., & Rawat, T. K. Optimal design of FIR fractional order differentiator using cuckoo search algorithm. Expert Q1 Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.12.020

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tion size leads to poor results whereas large size leads in more simulation time consumption. So, there must be a trade-off in selecting this parameter. Here, we have taken the chromosome string of population, N p ¼ 25. Select an initial random solution set, i.e., initial set of N coefficient chromosomes. Here, for N ¼ 7, the initial set of chromosome coefficients is taken as ½0:5; 1; 0:5; 0; 0:5; 1; 0:5; for N ¼ 9, it is set to ½0:5; 1; 0:5; 1; 0; 0:5; 0:5; 1; 0:5; and for N ¼ 11, it is taken as ½0:5; 0:5; 1; 0:5; 1; 0; 0:5; 1; 0:5; 1; 0:5. Decoding the strings and evaluate the error fitness values J as given in Eq. (11) and the weighted function W 1 ðxÞ is set to 0:8 and W 2 ðxÞ to 0.2 for the selected chromosome string. Selection of the elite strings in order of increasing the error fitness value from the minimum value. Here, elite count is set to 2. Crossover is applied between two chromosomes to produce offspring chromosomes. Here, the crossover fraction taken is 0.8. Mutation is applied to prevent redundancy in offsprings. Here, the constraint dependent mutation function is chosen. Genetic cycle gets updated. The fitness function is evaluated for each coefficient and least fitted coefficients are left at each iteration.

Step 8: The iteration stops when the maximum number of iterations reached or the convergence of minimum error fitness value is achieved. The chromosome string corresponding to minimum error fitness value vector gives the optimal coefficients. These optimum coefficients are employed in Q3 the design of FIR-FOD (see Fig. 1).

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5.2. Implementation of CSA for the FIR-FOD design problem

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To design an (N  1) th order FIR-FOD, find out N best nests, hð0Þ; hð1Þ; . . . ; hðN  1Þ, using the cuckoo search algorithm. The flowchart of the cuckoo search algorithm is shown in Fig. 2. The procedure, used in this paper is outlined as:

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Step 1: Randomly generate an initial population of n ¼ 25 host nests xi (i ¼ 1; 2; . . . ; n) and define the maximum number of iterations (N i ¼ 1000). Total N number of coefficients are required ðN  1Þth order filter. The maximum and minimum value of the coefficients are þ1 and 1, respectively. Set the probability of discovering alien eggs, Pa ¼ 0:25. The weighted function W 1 ðxÞ is set to 0:8 and W 2 ðxÞ to 0.2. Step 2: Generate a new nest using the Lévy flights given by vy ðkÞ. Compute the fitness value, f l , of xl ¼ xi þ a0  Le the new nests given by Eq. (11).

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Fig. 9. Comparison of 4th order FIR-FOD using GA with worst, average and best condition (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

Q2 Please cite this article in press as: Kumar, M., & Rawat, T. K. Optimal design of FIR fractional order differentiator using cuckoo search algorithm. Expert Q1 Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.12.020

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Step 3: Compute fitness value, f n , of a randomly generated host nest, xi , and Compare these values with the fitness value (f l ) of the new generated nests by Lévy flight. Step 4: For a minimization problem, if f n > f l , initial host nests xi is replaced by new nests, xl , generated by Lévy flight. Step 5: A fraction pa of the worst nests obtained after Step 4 are abandoned and new nests (xn ) using a random flight are built. Step 6: Compute the fitness of all the new nests. Step 7: Updating the best nest xp of the generation. Step 8: The best nest obtained until the current iteration, xb , is replaced by the best nest xp of the generation, if the fitness value, f p , of nest xp is less than the fitness value, f b , of xb for the minimization problem. Step 9: Repeat Steps 2–8 until the maximum number of iterations reached and the best nest xb gives the optimal coefficients. These optimum coefficients are employed for the design of FIR-FOD.

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6. Simulation results

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This section explains the design of FIR fractional order differentiator using genetic algorithm and cuckoo search algorithm. In order to demonstrate the effectiveness of the proposed FOD design method, several examples of FIR-FOD are constructed using GA and CSA algorithms. Simulation has been performed extensively to design the FIR-FOD of the order of 4, 8 and 10, respectively.

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Absolute magnitude error, absolute phase error and convergence rate are the parameters taken into consideration to evaluate the performance of the designed fractional order differentiator. Statistical analysis has also been done for comparing the performance of CSA with GA by varying the number of iterations and running the simulation 100 times in order to obtain the optimal solution for CSA and GA. The WLS error in the magnitude response is,

m ¼

Z

1

2

W 1 ðxÞjabsðHid ðxÞÞ  absðHðxÞÞj dx

ð13Þ

0

and the value of the WLS error of the phase response is,

p ¼

Z

1

2

W 2 ðxÞjphaseðHid ðxÞÞ  phaseðHðxÞÞj dx

515 516 517 518 519 520

521 523 524

525

ð14Þ

0

527

Fitness factor denotes the number of iterations required to obtain the best fitted coefficients so as to converge the error value to a minimum.

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Example 1. In this example, we implement genetic algorithm and cuckoo search algorithm to design the FIR-FOD. The design parameters are, filter length N ¼ 5; 9; 11, normalized frequency x 2 ½0; 1 and fractional order a ¼ 0:5. In order to adapt to these specifications, weighting function W 1 ðxÞ is set to 0.8 and W 2 ðxÞ is set to 0.2. To design the FIR-FOD using genetic algorithm, following parameters are taken, population size ¼ 20, elite counts ¼ 2, crossover fraction ¼ 0:8, maximum iterations ¼ 200, lower bound is 1

531

Fig. 10. Comparison of 8th order FIR-FOD using GA with worst, average and best condition (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

Q2 Please cite this article in press as: Kumar, M., & Rawat, T. K. Optimal design of FIR fractional order differentiator using cuckoo search algorithm. Expert Q1 Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.12.020

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Fig. 11. Comparison of 10th order FIR-FOD using GA with worst, average and best condition (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

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and upper bound is 1. Scaling function is chosen as type rank, selection function is taken as Roulette type, the constraint dependent mutation function is selected and crossover function is of scattered type. Now, the fitness function as described in Eq. (11) is evaluated iteratively and checked at each step whether optimization is met or not. At each iteration, the genetic operations such as selection, crossover and mutation are employed. To implement CSA, the parameters chosen here are, initial populations of host nests ¼ 25, maximum iterations ¼ 1000, the filter coefficient range is from 1 to 1 and the probability of abandoning worst nests Pa ¼ 0:25. Now, the fitness function as described in Eq. (11) is evaluated iteratively and checked at each step whether optimization is met or not. At each iteration, fitness value is checked. Extensive simulations were run at least 100 times so as to obtain the best solutions. The control parameters of the genetic algorithm and the cuckoo search algorithms are listed in Table 1. Fig. 2 shows the comparison of the 4th order FIR fractional order differentiator obtained by both the algorithms. Comparison is done on the basis of the following parameters: magnitude response, phase response, absolute magnitude error and absolute phase error. Zoomed in curves is also given in Fig. 2(a) and (b) to make the overlapping curves clearly visible. Fig. 3 shows the comparison of 8th order FIR-FOD obtained by both the algorithms. Fig. 4 demonstrates the comparison of 10th order FIR-FOD obtained by both the algorithms. The optimized coefficients for the

designed FIR-FOD with the order of 5, 8 and 10 have been calculated by GA and CSA and are reported in Table 2. Three performance parameters, absolute magnitude error, absolute phase error and convergence rate are investigated for the validation of the proposed work. It is observed that the magnitude error m ; 5:0650  105 and 2:2225  105 are obtained form GA and CSA, respectively for 4th order filter design. The phase error p , obtained for the same filter are 0.0490 and 0.0269, respectively. Table 3 summarizes the magnitude and phase error obtained by GA and CSA of filter orders 5, 8 and 10. Based on the observations of Table 3, we conclude that the magnitude and phase error obtained are less in case of cuckoo search algorithm as compared to the genetic algorithm. Thus, it is evident from Table 3, that magnitude and phase error decrease to an accountable value with an increase in filter order. Figs. 5–7 show the comparison of FIR-FOD obtained by using cuckoo search algorithm of order 4, 8 and 10, respectively with worst, average and the best condition. Fig. 8 depicts the comparison of different order FIR-FOD obtained by using cuckoo search algorithm with the best condition. It is observed that the magnitude error m , 2:6005  105 and 1:9257  105 are obtained for GA and CSA, respectively for 8th order FIR-FOD design. The phase error p , obtained for the same filter are 0.0082 and 0.0029, respectively. Figs. 9–11 show the comparison of FIR fractional order differentiator obtained by using genetic algorithm of order 4, 8 and 10, respectively with worst, average and the best condition.

Q2 Please cite this article in press as: Kumar, M., & Rawat, T. K. Optimal design of FIR fractional order differentiator using cuckoo search algorithm. Expert Q1 Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.12.020

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Fig. 12. Comparison of different order FIR-FOD using GA (a) magnitude response (b) phase response (c) absolute magnitude error (d) absolute phase error.

Table 4 Optimized coefficients for different order FIR-FOD with worst, average and best condition. Filter order (N  1)

Condition

Algorithm 0 6 k 6 N

Coefficients (hk ),

4

Worst

GA CSA GA CSA GA CSA

0:1199 0:0841 0:0932 0:2224 0:7498 0:1427

GA

0:2818 0:9235 0:7122 0:3291 0:1672 0:0117 0:0255 0:0460 0:0024 0:2245 0:3956 0:0405 0:5150 0:9977 0:0321 0:1872 0:3211 0:0051 0:0370 0:1109 0:4300 1:0000 0:4142 0:3829 0:2278 0:0354 0:0073 0:3701 0:0753 0:1495 0:3228 1:0000 0:3475 0:0839 0:0892 0:2795 0:2137 0:9443 0:7854 0:1043 0:0891 0:0053 0:0244 0:0038 0:0096 0:0066 0:0119 0:0073 0:0322 0:0699 0:7892 0:9549 0:1804 0:0532

Average Best 8

Worst

CSA Average

GA CSA

Best

GA CSA

10

Worst

GA CSA

Average

GA CSA

Best

GA CSA

0:2367 0:8400 0:8887 0:1379 0:1113 0:9937 0:7512 0:4967 0:0910 1:0000 0:7479 0:0220 0:9369 0:7443 0:3014 0:1467 0:9946 0:1122 0:0828 0:0062 0:9999 0:7365 0:0522 0:0758

0:0668 0:0983 0:1874 0:0142 0:2768 0:0218 0:5629 0:4096 0:6750 0:4913 0:5585 0:5272 0:9961 0:2472 0:1628 0:2215 0:1472 0:0915 0:2684 0:0303 0:1192 0:0479 0:0116 0:0248 0:0016 0:0239 0:0016 0:2482 0:4469 0:2550 0:9541 0:5151 0:2593 0:0425 0:2688 0:3973 0:1079 0:4310 0:8823 0:0759 0:3945 0:1168 0:1822 0:3210 0:2823 0:2235 0:6294 0:9904 0:2184 0:0695 0:1140 0:0068 0:0171 0:0206 0:0027 0:0637 0:2948 0:8388 0:8125 0:3571 0:0529 0:0803 0:0424 0:0453 0:0183 0:0124

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Table 5 Comparison summary of minimum and maximum values of magnitude and phase error for different order FIR-FOD using GA and CSA with worst, average and best condition. Filter order (N  1)

Condition

Value

4

Worst

Max Min

Average Best

8

Worst Average Best

10

Worst Average Best

Max Min Max Min Max Min Max Min Max Min Max Min Max Min Max Min

Magnitude error (m ) GA

Magnitude error (m ) CSA

Phase error (p ) GA

Phase error (p ) CSA

44:9302 0:0410

44:5597 0:0121

44:9037 0:0251

44:5268 0:0160

44:9257 0:0049

44:5836 0:0026

44:9179 0:0319

44:8827 0:0163

44:8595 0:0213

44:7413 0:0082

44:7413 0:0082

44:1902 0:0029

44:8542 0:0318

44:8246 0:0242

44:8205 0:0193

44:8094 0:0176

44:7589 0:0051

44:0135 0:0012

0:5511

0:4433

1:6616  104 0:3781

4:4330  105 0:3145

9:2681  105 0:3461

3:8247  105 0:2828

5:0650  105

2:2225  105

0:4232

0:4047

2:2162  104 0:3244

3:7274  105 0:2632

3:8474  105 0:2613

2:7889  105 0:2354

2:6005  105

1:9257  105

0:7672

0:2251

5:7535  104 0:3162

3:2016  105 0:1680

1:7767  104 0:2840

2:3332  105 0:1525

2:1357  105

1:0894  105

Fig. 13. Convergence curve for different order FIR-FOD using genetic algorithm (a) 4th order (b) 8th order (c) 10th order.

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Fig. 12 shows the comparison of different order FIR-FOD obtained by using genetic algorithm with the best condition. It is observed that the magnitude error 2:1357  105 and

1:0894  105 are obtained for GA and CSA, respectively for 10th order FIR-FOD design. The phase error p , obtained for the same filter are 0.0051 and 0.0012, respectively. The filter coefficients hk

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Fig. 14. Convergence curve for different order FIR-FOD using cuckoo search algorithm (a) 4th order (b) 8th order (c) 10th order.

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Fig. 15. Comparison of different order FIR-FOD using CSA, radial basis function interpolation method (Tseng & Lee, 2010) and DCT interpolation method (Tseng & Lee, 2013).

Fig. 16. Comparison of different order FIR-FOD using genetic algorithm, radial basis function interpolation method (Tseng & Lee, 2010) and DCT interpolation method (Tseng & Lee, 2013).

for the designed FIR-FOD with the order of 5, 8 and 10 are reported in Table 4 with different conditions. Worst condition is obtained with 100 iteration, average condition is obtained with 500 iteration

and best condition is obtained with 1000 iteration. Table 5 summarizes the magnitude and phase error obtained by the GA and the CSA with the order of 5, 8 and 10 and worst, average and the best

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condition. From Table 5, we conclude that the magnitude and phase error are less in case of the cuckoo search algorithm as compared to the genetic algorithm in all condition. In order to compare the algorithms in terms of error fitness value, Figs. 13 and 14 show the convergence rate of error fitness obtained when GA and CSA are employed, respectively. The convergence profiles are shown for the designed FIR-FOD of order 4, 8 and 10 with the best condition. Similar plots have also been obtained for the FIR-FOD with worst and average condition, which are not shown here. All optimization techniques have been simulated in MATLAB 7.11 version on Intel core (TM) i3 processor, 3.20 GHz with 4 GB RAM. It is also noticed that the proposed algorithm, CSA has a faster convergence rate in terms of error fitness value compare to that of the GA for obtaining the optimal results. As shown in Figs. 13 and 14, GA converges to the minimum error fitness value in 137 s; CSA converges to the minimum error fitness in 105 s. All the proposed fractional order digital differentiator exactly match the magnitude responses in close vicinity of ideal values. Here, the result obtained by 10th order FOD obviously lead in performance to their 8th and 4th order counterparts. Example 2. In this example, we evaluate the performance of the designed FOD using GA and CSA with the existing FOD based on radial basis function interpolation method and DCT interpolation method. Note that we compare both GA and CSA with the FOD that reported was by Tseng and Lee (2010a) and Tseng and Lee (2013a). Fig. 15 shows the comparison of magnitude response of proposed fractional order differentiator designed using CSA with existing FOD designed using the radial basis function interpolation method and the DCT interpolation method. Comparison of magnitude response of proposed FOD based on GA with existing FOD based on interpolation method are shown in Fig. 16. Fig. 17 shows the comparative magnitude plots for the designed 10th order FOD using CSA and GA and existing interpolation methods. It is evident from these three figures that the results of the cuckoo search algorithm perform better in terms of magnitude response compared to that of the genetic algorithm and other existing FOD. These simulation results affirm the effectiveness of the cuckoo search algorithm, in terms of accuracy and convergence rate. Lesser number of control parameters make this algorithm simpler and more generic as compared to the genetic algorithm.

Fig. 17. Comparison of FIR-FOD using genetic algorithm and cuckoo search algorithm, radial basis function interpolation method (Tseng & Lee, 2010) and DCT interpolation method (Tseng & Lee, 2013).

The simulations as well as statistical results shows that the CSA has significant improvement in performance compare to that of the radial basis function interpolation method and the DCT interpolation method. From the figures and tables, one can finally infer that the CSA based FIR-FOD design approach is the best among the literature available for this purpose.

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7. Conclusions

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In this paper, the FIR fractional order differentiator problem has been formulated as a CSA based optimization problem. The cuckoo search algorithm is used to design fractional order differentiators. Extensive simulations have been carried out to evaluate how well CSA has performed in finding the optimal coefficients. In comparison with GA and conventional interpolation based methods such as radial basis function and DCT, CSA outperformed in terms of magnitude and phase error. Furthermore, CSA converges very fast to the optimal coefficients and reaches the minimum error fitness value in low execution time. Tuning of a large number of parameters became a difficult task in GA. The implementation of CSA has increased the flexibility in the designing of FIR-FOD as it requires less number of parameters to be tuned. Furthermore, the computational complexity of the proposed CSA based FIR-FOD is reduced significantly due to the reduction in filter order compared to the conventional methods. Since, the performance of the CSA depends on the probability of identifying alien egg, P a , finding this probability for different problems is a critical requirement for high quality solutions. There is not an explicit formula to determine P a . Hence it requires extensive simulations to find the optimal solutions for a given problem. The CSA algorithm is free from the shortcoming of premature convergence exhibited by other optimization algorithms. The balance between the randomization and least number of control parameters make CSA superior over its counterparts. One of the future plan is to improve the overall performance of the proposed FIR-FOD without affecting the user friendly feature and within a predetermined number of iterations to obtain an optimal solution in multidimensional search space with multi objective strategy. Further, the proposed method needs to be explored as a future scope, for the designing of Hilbert transformer and 2D fractional order differentiator. The suitability of the proposed fractional order differentiator for the different image processing applications can also be investigated.

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Q2 Please cite this article in press as: Kumar, M., & Rawat, T. K. Optimal design of FIR fractional order differentiator using cuckoo search algorithm. Expert Q1 Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.12.020

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