Membrane computing model for IIR filter design

Information Sciences 329 (2016) 164–176

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Membrane computing model for IIR filter design Jun Wang a,∗, Peng Shi b, Hong Peng c a b c

School of Electrical Engineering and Electronic Information, Xihua University, Chengdu, Sichuan 610039, China School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA 5005, Australia School of Computer and Software Engineering, Xihua University, Chengdu, Sichuan 610039, China

a r t i c l e

i n f o

Article history: Received 3 February 2015 Revised 4 August 2015 Accepted 14 September 2015 Available online 21 September 2015 Keywords: Adaptive IIR filter System identification Membrane computing Tissue-like membrane systems

a b s t r a c t Adaptive infinite impulse response (IIR) filter has been preferably used in modeling real-world systems because of its reduced number of coefficients and better performance over finite impulse response (FIR) filter. However, it is still a challenging problem how to design an optimal IIR filter due to its nonlinear and multimodal error surface. This paper introduces membrane computing to design an optimal IIR filter and proposes a novel design method that employs a tissue-like membrane system with ring-shaped topology structure. A modification of the different evolution mechanism was developed as evolution rules for objects according to the special membrane structure. Under the control of the object’s evolution-communication mechanism, the tissue-like membrane system can effectively find the global minima for an IIR filter design problem. The proposed method was evaluated over several benchmark IIR systems and was compared with several state-of-the-art evolutionary algorithms. The experimental results show that the proposed method has better identification performance compared with other algorithms. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Nowadays, adaptive digital filters have been applied in a wide range of areas such as signal processing, control, communication and image processing. Digital filters can be broadly classified into two types: finite impulse response (FIR) filters and infinite impulse response (IIR) filters [2,14]. FIR filter has a finite impulse response because its output is determined by current and previous inputs. However, current output of an IIR filter is calculated not only by current and previous inputs but also by previous outputs. As a result, IIR filter has an infinite impulse response with only finite number of parameters. It is well-known that the desired response can be approximated more effectively by the output of the filter that has both poles and zeros compared to one that has only zeros [25,26]. Therefore, an adaptive IIR filter with sufficient number of poles and zeros can exactly model an unknown pole-zero system. The optimal design of IIR filters is still a challenging optimization problem. Classical gradient-based algorithms such as steepest descent and Newton-like algorithms have been used to design the IIR filters [31], and they can efficiently determine the optimal solution of an unimodal objective function. However, error surface (typically mean square error between the desired response and the estimated filter output) of an IIR filter is often nonlinear and multimodal, so the gradient-based algorithms easily get stuck in local minimum and can not converge to its global minima [24]. Evolutionary algorithms, which use populationbased optimization techniques to search global minima, have exhibited a considerable advantage in solving the hard global ∗

Corresponding author. Tel.: +86 2887720369; fax +86 2887720369. E-mail address: [email protected], [email protected] (J. Wang).

http://dx.doi.org/10.1016/j.ins.2015.09.011 0020-0255/© 2015 Elsevier Inc. All rights reserved.

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optimization problems, such as cat swarm optimization (CSO) [1], ant colony optimization (ACO) [4], genetic algorithms (GA) [6], artificial bee colony (ABC) [9], particle swarm optimization (PSO) [11] and differential evolution (DE) [23]. In recent years evolutionary algorithms have been widely used for design problem of IIR filters. When considering global optimization techniques for IIR filter design, GA has attracted considerable attention since the filter modeled by it has the potential of obtaining global optimal solution [15,28,29,32]. Although many GAs have shown good performance for finding the promising regions of the search space, most of them often have two drawbacks: premature convergence and lack of good local search ability. To overcome these problems, other evolutionary algorithms have been successively developed for IIR filter design problem. Karaboga et al. [10] used ACO to deal with the filter design problem, however, the existing study has shown that ACO also can suffer premature convergence and search stagnation. PSO uses the velocity-position model to find the optimal solution for the optimization problem, which has simple implementation and a few parameters to control its convergence. Application of PSO in IIR filter design has been described in Krusienski et al. [12] and Luitel et al. [13]. And then, use of DE and ABC in IIR filter design has been presented in Karaboga [7], [8], respectively. Although PSO, DE and ABC algorithms have considerable performance advantage in solving global optimization problem compared with GA, the existing studies have indicated that they may face with the problems of premature convergence, stagnation and revisiting of the same solution over and again. In addition, Panda et al. [17] presented a CSO-based method for IIR filter design. Membrane computing was inspired from the structure and functioning of living cells as well as the cooperation of cells in tissues, organs and populations of cells [19]. Membrane computing is a class of distributed parallel computing models, known as membrane systems or P systems [16,18,22,30]. The advantages that membrane systems possess and the potentiality exhibited by membrane computing when its idea, method and model are used to solve difficult problems in a number of areas, have attracted much attention on membrane computing applications: for example, the membrane algorithm of solving global optimization problems [20,21,33]. The research results on a variety of optimization problems have indicated that compared to the existing evolutionary algorithms, membrane algorithm can offer a more competitive method due to its three advantages: better convergence, stronger robustness and better balance between exploration and exploitation. Based on the above consideration, main motivation behind this work is focusing on application of membrane computing in the optimal IIR filter design problem and proposing a novel membrane system-based method for IIR filter design. Thus, a tissue-like membrane system with ring-shaped membrane structure is developed to find the global optimal solution for the identification of IIR system. Different from the existing design methods, the tissue-like membrane system uses the object’s communication mechanism and a modified differential evolution mechanism to guarantee its superiority. To the best of our knowledge, this is the first attempt to use membrane computing model in digital IIR filter design. Based on the tissue-like membrane system, this paper presents a good, comprehensive set of results, and states arguments for the superiority of the method. Simulation results demonstrate the effectiveness and good performance of the proposed method. The rest of this paper is arranged as follows. Section 2 briefly introduce an IIR system, tissue-like membrane systems and the known differential evolution mechanism. In Section 3, the identification method based on tissue-like membrane systems is described in detail. In Section 4, experimental results carried out on some benchmark problems are presented. Finally, conclusion is drawn in Section 5. 2. Preliminaries 2.1. IIR system An IIR system can be expressed by a difference equation

d0 (k) = HS (z)x(k)

(1)

where x(k) and d0 (k) are input and output signals of the IIR system, respectively. HS (z) is the transfer function of the unknown system and is given by



HS (z) =

A(z) B(z)



(2)

A(z) and B(z) are Z-domain feed-forward and feed-back coefficient polynomials of the IIR system respectively, and they are expressed as follows

A(z) =

L 

ai z−i ,

B(z) = 1 −

i=0

M 

bi z−i

i=1

M and L are feed-back and feed-forward filter orders respectively, and M ≥ L. The overall output of the IIR system is given by

d(k) = d0 (k) + ν(k)

(3)

where ν (k) is an additive white Gaussian noise. Combining (1) and (3), we get



d(k) =



A(z) x(k) + ν(k) B(z)

(4)

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Fig. 1. Block diagram of system identification using an adaptive IIR filter.

Most applications of IIR filter in signal processing can be characterized as a system identification problem. Adaptive algorithms have been used to search the filter’s coefficients such that its input–output relationship matches closely to that of the unknown system. Fig. 1 shows the block diagram of system identification by an adaptive IIR filter. The adaptive filter can be described by a difference equation

y(k) = HF (z)x(k)

(5)

where HF (z) is the transfer function of the adaptive filter and is given by



HF (z) =

Aˆ (z) Bˆ(z)



(6)

Aˆ (z) and Bˆ(z) are feed-forward and feed-back coefficient polynomials respectively, and they are given by

Aˆ (z) =

L 

aˆi z−i ,

Bˆ(z) = 1 −

i=0

M 

bˆ i z−i

i=1

where aˆi and bˆ i denote the estimated feed-forward and feed-back coefficients, respectively. Transfer function HS (z) of IIR system can be identified by transfer function HF (z) of the adaptive filter. The identification task is formulated as an optimization problem where mean square error (MSE) is used as the cost function

min J(τ ) = min τ

τ

N 1 (d(k) − y(k))2 N

(7)

k=1

where τ = (aˆ0 , aˆ1 , . . . , aˆL , bˆ 1 , bˆ 2 , . . . , bˆ M ) denotes the filter’s coefficient vector, and N is the number of input samples. In this work, membrane systems will be used to identify the coefficient vector. 2.2. Tissue-like membrane systems In this section we briefly review the definition and inherent mechanism of tissue-like membrane systems. More detailed descriptions of tissue-like membrane systems can be found in literature [5,19]. Formally, a tissue-like membrane system (of degree q > 0) with symport/antiport rules is a construct

 = (O, w1 , . . . , wq , R1 , . . . , Rq , R , i0 )

(8)

where (1) (2) (3) (4) (5)

O is a finite alphabet, whose symbols are called objects; w1 , w2 , . . . , wq are initial multisets of objects in cells 1, 2, . . . , q, respectively; R1 , R2 , . . . , Rq are finite sets of evolution rules in the q cells, respectively; R is finite set of communication rules of the form (i, u/v, j), i = j, i, j = 1, 2, . . . , q, u, v ∈ O∗ ; i0 indicates the output region of the system.

Fig. 2 shows a tissue-like membrane system that consists of q cells. Each cell is surrounded by a cell membrane, and the outer region of the q cells is called the environment. Usually, each cell contains one or more objects, and w1 , w1 , . . . , wq describe the multisets of objects of the q cells. The tissue-like membrane system has two types of rules: evolution rule and communication rule. Evolution rule is of the form u → v, u, v ∈ O∗ , and application of that rule means that u will evolve to v. Communication rule of the form (i, u/v, j) is called the antiport rule, and application of that rule means that u in cell i and v in cell j are interchanged. Note that if either i = 0 or j = 0, the two objects are interchanged between a cell and the environment. If one of u or v in above rule is empty object λ, the rule is

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Fig. 2. Membrane structure of tissue-like membrane systems.

called the symport rule, for example, (i, u/λ, j), and application of such rule means that u will be communicated from cell i to cell j. The communication rules describe a virtual graph, where the q cells denote the nodes and the edges indicate if it is possible for pairs of cells to communicate directly. The tissue-like membrane system starts with initial multisets w1 , . . . , wq . And then, in each step, something happens: the objects in cells evolve and some of them are communicated. As usual in membrane systems, all cells that are parallel units work in parallel. The process is repeated until the halting condition is attached. When it halts, the system produces a final result in the output region. 2.3. Differential evolution mechanisms Differential evolution (DE) is one of the most powerful population-based stochastic optimization algorithms [27]. DE searches a global minima in a D-dimensional solution space by three evolution operations: mutation, crossover and selection [3,23]. Let Zi = (zi1 , zi2 , . . . , ziD ) be a chromosome/genome in the considered population. A donor vector Yi = (yi1 , yi2 , . . . , yiD ) can be created from original vectors in the population. There are five frequently used schemes to create the donor vector: (1) (2) (3) (4) (5)

“DE/rand/1”: Yi = Zr1 + F · (Zr2 − Zr3 ), “DE/best/1”: Yi = Zbest + F · (Zr1 − Zr2 ), “DE/current-to-best/1”: Yi = Zi + F · (Zbest − Zi ) + F · (Zr1 − Zr2 ), “DE/best/2”: Yi = Zbest + F · (Zr1 − Zr2 ) + F · (Zr3 − Zr4 ), “DE/rand/2”: Yi = Zr1 + F · (Zr2 − Zr3 ) + F · (Zr4 − Zr5 ).

The indices, r1 , r2 , r3 , r4 and r5 , are mutually exclusive integers randomly chosen and all are different from the base index, i. The scaling factor, F, is a positive control parameter for scaling the difference vectors. Zbest is the best individual vector with the best fitness in the population. To increase the diversity of the population, a crossover operation is applied. The DE-family has two crossover operations: exponential and binormal. We briefly describe the binormal crossover that we employ in the improved DE mechanism. To generate a trial vector Ri = (ri1 , ri2 , . . . , riD ), binormal crossover is performed on each of the D components whenever a randomly generated number between 0 and 1 is less than or equal to the Cr value:



ri j =

yi j ,

if randi j [0, 1] ≤ Cr or j = jrand ,

zi j ,

otherwise,

(9)

where randij [0, 1] is a uniformly distributed random number, and jrand ∈ {1, 2, . . . , D} is a randomly chosen index, which ensures that Ri gets at least one component from Yi . The selection operation is used to determine which of the target vector or trial vector will survive in the next generation. If the trial vector yields a better value of the objective function, it replace its target vector in the next generation; otherwise the parent is retained in the population:



Zi =

Ri ,

if f (Ri ) ≤ f (Zi ),

Zi ,

otherwise,

(10)

where f(·) is the objective function to be minimized. 3. Proposed identification method based on tissue-like membrane systems 3.1. A tissue-like membrane system with ring-sharped structure In this work, a tissue-like membrane system with ring-sharped structure is designed as the core component of the presented identification method for IIR systems. The tissue-like membrane system consists of the q cells that are labeled by 1, 2, . . . , q respectively, shown in Fig. 3(a). The region labeled by 0 is called the environment (outer region of the q cells), and it is also assigned as output region of the system. When the system halts, the object in the output region will be regarded as the optimal solution for the IIR design problem. The tissue-like membrane system is considered as a ring-sharped membrane structure, shown in 3(b). This consideration aims at developing a modification of differential evolution mechanism as evolution rules of objects. The q cells are arranged as an unidirectional ring, and each of them exchanges information with its neighboring cell.

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Fig. 3. Membrane structure of tissue-like membrane systems (a) and its ring-sharped topology (b).

Fig. 4. An example of object representation in cells.

3.1.1. Objects The tissue-like membrane system is used to optimize the filter’s coefficients for an IIR system: (a0 , a1 , . . . , aL ) and (b1 , b2 , . . . , bM ). Therefore, the object is considered to represent a coefficient vector of adaptive IIR filter. Thus, each object in cells can be formally described by

Z = (z1 , z2 , . . . , zD ) = (a0 , a1 , . . . , aL , b1 , . . . , bM )

(11)

where the dimension of the object is D = L + M + 1. As usual, each cell contains one or more objects. For the sake of simplicity, assume that the q cells have the same number of objects, n. Fig. 4 shows an example of object representation, where object Z corresponds to a coefficient vector of IIR filter. In addition, each cell memorizes a local best object (denoted by Zlbest ), which is the found object with the lowest MSE value in the cell. The environment (i.e., output region) stores a global best object (denoted by Zgbest ), which is the found object with the lowest MSE value in the q cells. Zgbest will be updated in each computing step, and it is also final computing result when the system halts. 3.1.2. Initialization Initially, the tissue-like membrane system generates n initial objects for each of the q cells, and these initial objects form multisets of objects in the q cells, w1 , w1 , . . . , wq . Suppose that A and B are the lower and upper bounds of coefficients of the considered IIR filter, respectively. To generate an initial object, D random real numbers in [A, B] are produced as its components, 0 ), which satisfy the following constraint: Z = (z10 , z20 , . . . , zD

A ≤ zi0 ≤ B,

i = 1, 2, . . . , D

3.1.3. Communication rules As usual, tissue-like membrane system uses communication rules to exchange the objects between two adjacent cells or between a cell and the environment. Each cell communicates its best object, Zlbest , into its subsequent cell in the ring-sharped topology. At the same time, Zlbest is also transmitted into the environment to update the global best object, Zgbest . The tissue-like membrane system has two types of communication rules: • Rule (i, Zlbest /λ, j), where j = i + 1 for ∀i ∈ {1, 2, . . . , q − 1} or j = 1 for i = n i , into its subsequent cell j. Thus, cell j will receive the object, called The rule communicates the local best object of cell i, Zlbest j

the external best object (denoted by Zebest ).

• Rule

i (i, Zlbest /λ, 0),

where i = 1, 2, . . . , q

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Fig. 5. Block diagram of the proposed identification method for IIR systems.

i The rule transmits the local best object of cell i, Zlbest , into the environment and update the global best object of the system, Zgbest . The updating strategy is described as follows:



Zgbest =





i Zlbest ,

i if f Zlbest < f (Zgbest )

Zgbest ,

otherwise

(12)

where f(·) denotes the fitness function of an object. Note that each cell has two best objects: the local best object, Zlbest , and the external best object, Zebest . The two best objects will participate in the evolution of objects in the system. This consideration can bring two benefits: (i) enhancing the diversity of objects in the system and (ii) balancing exploitation and exploration. 3.1.4. Evolution rules Each of the q cells uses evolution rules to evolve the objects of the system. In this work, differential evolution (DE) mechanism is introduced as evolution rules for objects. As described above, the differential evolution mechanism has three operations: mutation, crossover and selection. However, a modification of differential evolution mechanism is considered according to both ring-shaped membrane structure and communication mechanism of the tissue-like membrane system. The modification is achieved only on mutation operation, however, crossover and selection operations in original mechanism are still retained. The modified mutation operation is a variant of “DE/current-to-best/1”, and it is described by:



Yi = Zi + F · (Zlbest − Zi ) + F · (Zebest − Zi ) + F · Zr1 − Zri 2



(13)

where Zi is an original object in cell i and Yi is the created donor object; Zr1 and Zr2 are two objects chosen randomly from cell i; Zlbest and Zebest are two best objects in cell i described above. F is the scaling factor and is given by

F = 0.5 × (1 + rand(0, 1))

(14)

3.2. The identification method for IIR systems The proposed identification method is based on a tissue-like membrane system with ring-sharped structure, which is used to find the optimal coefficients for adaptive IIR filter. Fig. 5 gives the block diagram of the proposed identification method for IIR systems.

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For the IIR identification problem (7), the tissue-like membrane system employs an object to represent a feasible coefficient vector of adaptive IIR filter. The proposed method randomly generates n initial objects for each cell, and then executes the tissuelike membrane system. The objects in the system are evaluated by computing their MSE values, and local best object Zlbest of each cell is transmitted into its subsequent cell by communication rule. Next, the objects in cells evolve orderly by evolution rules (the modified mutation rule, crossover rule and selection rule). As usual in membrane computing, the q cells that are parallel units run in parallel. Under the control of evolution-communication mechanism, the tissue-like membrane system continuously evolves objects in the system and updates the global optimal object Zgbest in the environment until the halting condition is attached. For simply, the tissue-like membrane system uses a simple halting condition: maximum computing step, T. When the system halts, the global optimal object stored in the environment, Zgbest , is regarded as final computing result: the optimal coefficient vector of adaptive IIR filter. In the following, we briefly analyze the time complexity of the proposed method. The identification method consists of four main steps: initialization, object communication, object evolution (mutation, crossover and selection) and halting judgment. Note that the used tissue-like membrane system has q cells and each cell contains n objects. Let T be the maximum computing step number. From Fig. 5, it can be observed that initialization step contains double loop (q and n times), so its time complexity is O(qn). For object evolution step, there are triple loop (q, n and T times), therefore, its time complexity is O(qnT). The object communication step contains double loop (q and T times), so its time complexity is O(qT). For halting step, its time complexity is O(1). Therefore, the time complexity of the proposed method is O(qnT). 4. Experimental results and discussions In experiments, four benchmark IIR systems from second order to sixth order have been selected to demonstrate the applicability and effectiveness of the designed tissue-like membrane system (TMS, in short) for the identification of IIR systems. The input is a white signal with zero mean, unit variance and uniform distribution, and the additive noise is a Gaussian white signal with variance 10−3 . Input parameters of TMS were given: q=16, n=30, Cr =0.8 and T = 500. To compare the performance of TMS, experimental results of five evolutionary algorithms have been obtained, including GA, PSO, DE, ABC and CSO. Population size for the five algorithms was chosen: NP = 30. The parameters of GA were: pc =0.8 and pm =0.01. For PSO, standard velocity-position model was used, where the inertia weight was linearly decreased from 0.9 to 0.4 and the learning factors were chosen to c1 = c2 = 2.0. DE used the mutation operation of “DE/rand/1” and original crossover and selection operations: F = 0.5 and CR = 0.9. The parameter for ABC was: limit value=40. The parameters for CSO were: SMP = 5, SRD = 20%, CDC = 80%, MR = 0.9 and C = 2.0. In experiments, two performance measures, residual mean square errors (RMSE) and mean square deviation (MSD), have been used to compare the performance of TMS with GA, PSO, DE, ABC and CSO. RMSE is the steady state MSE value, while MSD is defined by

MSD =

N−1 1  ˆ i))2 (φ(i) − φ( N

(15)

i=0

where φ and φˆ are the desired and estimated parameter vectors respectively, and N is the total number of parameters to be estimated. 4.1. Case 1: identification using full order filter In experiments, four unknown IIR systems of four examples in appendix have been modeled by the corresponding full order filters, respectively. To illustrate the performance of the six methods for full order filter, convergence characteristic, the estimated and actual parameters, RMSE and MSD values are provide in Fig. 6, Tables 1 and 2, respectively. The presented results are the average values over 50 trials. Table 1 gives the average values of filter coefficients estimated by the six methods over 50 trials and actual values for the four examples. It can be observed from Table 1 that TMS provides the closest values to actual filter coefficients in comparison to other five methods for each example. The results illustrate that TMS can exploit the best filter coefficients for the four examples. Table 2 shows the results of six methods for four examples modeled by full order IIR filters: mean values and standard deviations of the six methods over 50 trials for each example. For example I, RMSE values of TMS, CS, ABC, DE, PSO and GA are 3.951e-5, 6.539e-5, 9.382e-5, 8.741e-5, 1.823e-4 and 2.056e-3, respectively, so TMS achieves the lowest RMSE value but GA has the highest one. For example II, TMS has the lowest RMSE value, 4.328e-5, second is CSO, 6.363e-5, followed by ABC, DE and PSO, the highest is GA, 3.275e-3. For example III, TMS has the lowest RMSE value, 3.643e-5, while GA has the highest RMSE value, 2.536e-2. In example IV, RMSE value of TMS is 4.982e-5, which is the lowest in the six methods. The comparison results show that TMS has the lowest RMSE value compared with other five methods for each of four examples, which indicates that TMS can achieve the best identification performance from the view of RMSE metric. From Table 2, it can be clear seen that MSD values of TMS, respectively, are 1.295e-6, 9.735e-6, 1.152e-5 and 3.721e-4, which are the lowest in the six methods for the four examples. The comparison results further indicate that TMS can achieve the better identification performance over other five methods. From Table 2, it can be observed that for the four examples, standard deviations of both RMSE and MSD of TMS are significantly lower than those of other five methods. The results illustrate that TMS is obviously superior to other five methods in terms of robustness.

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(a)

(b)

(c)

(d)

Fig. 6. Average convergence characteristics for four examples modeled using full order IIR filters: (a) example I; (b) example II; (c) example III; (d) example IV.

Fig. 6 shows average convergence characteristics of the six methods over 50 trials for each of four examples modeled by full order IIR filters. From Fig. 6, the following results can be observed: (1) TMS converges to the lowest MSE value in comparison to other five methods for each of four examples; (2) TMS has the fastest convergence speed, for example, it can be converged about 202th generation for examples I and II, 206th generation for example III and 300th generation for example IV; (3) For the four examples, ABC and DE have the similar convergence performance, but are better than PSO; (4) GA suffers from premature convergence for the four examples. Note that TMS uses a modified differential evolution mechanism as evolution rules for objects. Therefore, TMS can be viewed as a multi-population extension of DE. The results in Fig. 6, Tables 1 and 2 indicate that TMS is significantly superior to DE in terms of identification performance, robustness and convergence characteristic. The advantage is benefiting from the used membrane structure and inherent mechanism of TMS.

4.2. Case 2: identification using the reduced order filter For case 2, the four unknown IIR systems were modeled by four reduced order IIR filters, and TMS, CSO, ABC, DE, PSO and GA have been used to identify the coefficients of the filters, respectively. The results obtained by the six methods in terms of convergence characteristic and RMSE for the reduced order filters are provided in Fig. 7 and Table 3, respectively. The results are average values of the six methods over 50 trials for each IIR system. In example I, the second order system was modeled by a first order IIR filter. RMSE values of TMS, CSO, ABC, DE, PSO and GA are 1.534e-2, 1.815e-2, 2.517e-2, 2.829e-2, 3.881e-2 and 6.337e-2, respectively. The results clearly indicate that TMS has better identification performance in comparison to other five methods. It can be observed from Fig. 7(a) that TMS can converge within 220 generations and it converges to the lowest value in the six methods. Therefore, TMS is better than other five methods in terms of convergence characteristic. In example II, the third order system was modeled by a second order IIR filter. TMS has the lowest RMSE value, 1.284e-3, while GA obtains the largest RMSE value, 1.295e-2. So, identification ability of TMS is superior to that of other methods. From Fig. 7(b), it can be viewed that within 220 generations TMS can converge to the lowest RMSE value, 1.284e-3, whereas CSO, ABC, DE and PSO can converge after 300 generations.

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Table 1 Parameter estimation for four examples modeled using full order IIR filters. Example

Parameter

Actual

Estimated value (mean value over 50 trials)

value

TMS

GA

PSO

DE

ABC

CSO

I

a0 a1 b1 b2

0.05 −0.4 1.1314 −0.4

0.0495 −0.3984 1.1306 −0.2983

0.0877 −0.0411 1.1821 −0.3051

0.0536 −0.4184 1.0876 −0.2077

0.0472 −0.4035 1.1229 −0.4582

0.0481 −0.4042 1.1135 −0.2376

0.0493 −0.4021 1.1248 −0.2433

II

a0 a1 a2 b1 b2 b3

−0.2 −0.4 0.5 0.6 −0.25 0.2

−0.1986 −0.3954 0.4991 0.6079 −0.2351 0.2027

−0.2258 −0.2717 0.4643 0.7742 −0.4379 0.3206

−0.2106 −0.3649 0.4670 0.6123 −0.3134 0.2249

−0.2071 −0.3592 0.4782 0.6035 −0.1982 0.2047

−0.2093 −0.3927 0.4819 0.6048 −0.2021 0.2039

−0.2050 −0.3927 0.5038 0.6017 −0.2179 0.2031

III

a0 a1 a2 a3 b1 b2 b3 b4

1.0 −0.9 0.81 −0.729 −0.04 −0.2775 0.2101 −0.14

1.0017 −0.8905 0.8083 −0.7275 −0.0391 −0.2785 0.2133 −0.1428

1.0670 −0.7493 0.7214 −0.4350 −0.2308 −0.3064 0.1065 −0.0489

1.1587 −0.6562 0.3380 −0.9309 −0.6264 −0.6618 0.5165 −0.0067

0.9735 −0.8527 0.8312 −0.7285 −0.1147 −0.2961 0.1846 −0.1279

0.9673 −0.8631 0.8472 −0.7264 −0.1026 −0.2815 0.1933 −0.1325

0.9951 −0.8839 0.8206 −0.7253 −0.0506 −0.2930 0.1962 −0.1461

IV

a0 a1 a2 a3 a4 a5 b1 b2 b3 b4 b5

0.1084 0.5419 1.0837 1.0837 0.5419 0.1084 −0.9853 −0.9738 −0.3864 −0.1112 −0.0113

0.1076 0.5412 1.0825 1.0816 0.5422 0.1103 −0.9836 −0.9719 −0.3871 −0.1108 −0.0136

0.5083 0.7449 1.0303 1.0714 0.7067 0.3578 −0.6080 −0.9316 −0.3451 −0.3382 −0.1848

0.2484 0.3789 1.6960 1.4109 0.8467 0.2684 −1.0628 −0.7275 −0.4842 −0.3291 −0.2238

0.2077 0.4105 1.2194 1.1824 0.6391 0.1168 −1.0316 −0.8936 −0.4071 −0.2075 −0.0428

0.1833 0.4528 1.1792 1.3164 0.6985 0.1238 −1.0253 −0.8846 −0.4205 −0.1736 −0.0295

0.1038 0.5403 1.0813 1.0803 0.5447 0.1145 −0.9768 −0.9632 −0.3827 −0.1137 −0.0167

Table 2 Results of RMSE and MSD metrics for four examples modeled using full order IIR filters over 50 trials. Method

Measure

Example I

Example II

Example III

Example IV

TMS

RMSE

3.951e-5 (±5.183e-9) 1.295e-6 (±2.363e-9)

4.328e-5 (±2.573e-21) 9.735e-6 (±3.182e-15)

3.643e-5 (±8.302e-10) 1.152e-5 (±6.436e-9)

4.982e-5 (±8.562e-8) 3.721e-4 (±2.635e-4)

2.056e-3 (±3.228e-3) 8.743e-3 (±9.331e-3)

3.275e-3 (±3.186e-3) 1.473e-2 (±9.285e-3)

2.536e-2 (±1.631e-2) 8.483e-2 (±3.851e-2)

4.426e-2 (±5.338e-2) 7.568e-2 (±4.926e-2)

1.823e-4 (±5.316e-5) 1.935e-3 (±1.988e-3)

6.352e-5 (±4.563e-10) 1.226e-5 (±7.301e-8)

8.733e-5 (±2.627e-5) 4.762e-4 (±4.737e-4)

8.663e-5 (±1.613e-4) 4.613e-3 (±3.965e-3)

8.741e-5 (±2.358e-6) 6.219e-5 (±3.572e-4)

6.352e-5 (±2.198e-10) 1.225e-5 (±3.416e-8)

7.853e-5 (±3.625e-8) 3.289e-5 (±6.248e-6)

7.325e-5 (±5.364e-5) 1.514e-3 (±3.118e-3)

9.382e-5 (±1.215e-5) 8.265e-5 (±5.128e-4)

6.352e-5 (±3.382e-10) 1.224e-5 (±5.814e-10)

7.792e-5 (±5.118e-6) 3.352e-5 (±3.419e-5)

7.584e-5 (±9.382e-5) 8.793e-4 (±3.082e-3)

6.539e-5 (±3.447e-7) 1.349e-6 (±1.083e-5)

6.363e-5 (±1.743e-18) 1.224e-5 (±7.385e-12)

5.982e-5 (±8.368e-9) 1.568e-5 (±8.499e-8)

6.469e-5 (±3.225e-7) 4.812e-4 (±6.688e-4)

MSD GA

RMSE MSD

PSO

RMSE MSD

DE

RMSE MSD

ABC

RMSE MSD

CSO

RMSE MSD

J. Wang et al. / Information Sciences 329 (2016) 164–176

(a)

173

(b)

(c)

(d)

Fig. 7. Average convergence characteristics for four examples modeled using the reduced order IIR filters: (a) example I; (b) example II; (c) example III; (d) example IV. Table 3 Results of RMSE metrics for four examples modeled using the reduced order IIR filters over 50 trials. Method

Example I

Example II

Example III

Example IV

TMS

1.534e-2 (±2.281e-20)

1.284e-3 (±4.526e-21)

6.398e-3 (±8.251e-12)

7.853e-5 (±3.824e-8)

GA

6.337e-2 (±3.294e-2)

1.295e-2 (±6.731e-2)

6.132e-2 (±3.281e-2)

1.873e-1 (±3.528e-1)

PSO

3.881e-2 (±2.019e-2)

2.521e-3 (±2.378e-19)

8.581e-3 (±3.755e-3)

5.166e-4 (±1.246e-3)

DE

2.829e-2 (±1.983e-2)

1.394e-3 (±2.516e-19)

7.725e-3 (±4.619e-4)

8.935e-5 (±2.285e-4)

ABC

2.517e-2 (±2.142e-2)

1.413e-3 (±2.615e-19)

7.832e-3 (±5.328e-4)

9.137e-5 (±2.331e-4)

CSO

1.815e-2 (±5.16e-18)

1.387e-3 (±2.428e-19)

6.716e-3 (±3.025e-11)

7.898e-5 (±2.613e-5)

For example III, it can be obviously seen from Fig. 7(c) and Table 3 that TMS converges to the lowest RMSE value, 6.398e-3, within 280 generations. The results illustrate that TMS has stronger identification ability and better convergence characteristic compared with other five methods. For the high order IIR system in example IV, it can be seen from Fig. 7(d) and Table 3 that TMS, CSO, ABC and DE have approximate identification and convergence performances. However, TMS is slightly better than CSO, ABC and DE in terms of convergence characteristic and the RMSE metric. From Table 3, it can be obviously seen that TMS obtains the lowest standard deviations for the four IIR systems: 2.281e-20 for example I, 4.526e-21 for example II, 8.251e-12 for example III and 3.824e-8 for example IV. The results illustrate that TMS is more robust for the four IIR systems using the reduced order IIR filters.

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J. Wang et al. / Information Sciences 329 (2016) 164–176 Table 4 The p-values of Wilcoxon’s test for full order filter. TMS vs.

Example I

Example II

Example III

Example IV

GA PSO DE ABC CSO

2.01e-5 2.28e-4 2.35e-4 1.29e-3 1.45e-3

2.07e-5 2.33e-4 2.41e-4 1.51e-3 1.62e-3

2.25e-5 2.42e-4 2.58e-4 1.62e-3 1.84e-3

2.38e-5 2.56e-4 2.79e-4 1.87e-3 2.03e-3

Table 5 The p-values of Wilcoxon’s test for the reduced order filter. TMS vs.

Example I

Example II

Example III

Example IV

GA PSO DE ABC CSO

4.28e-3 0.0163 0.0185 0.0214 0.0236

4.37e-3 0.0197 0.0215 0.0236 0.0254

5.23e-3 0.0275 0.0293 0.0348 0.0369

5.43e-3 0.0342 0.0358 0.0398 0.0427

In addition, it can be obviously seen from Fig. 7 and Table 3 that compared with DE, TMS has better identification ability, stronger robustness and better convergence characteristic. The comparison results demonstrate that TMS outperforms its counterpart method, DE, in terms of identification performance, robustness and convergence characteristic for IIR system identification using the reduced order IIR filter. 4.3. Statistical significance test The Wilcoxon’s rank sum test is a nonparametric statistical significance test for independent samples. The statistical significance test has been conducted at the 5% significance level in the experiments. We have created six groups for each example, which were corresponding to the six methods (TMS, CSO, ABC, DE, PSO and GA), respectively. Each group consists of the RMSE values produced by the corresponding methods over 50 trials for the examples. The mean values of each group for full order filter and reduced order filter are provided in Tables 2 and 3, respectively. The comparison results show that mean values of TMS are better than those of other methods. To illustrate that the goodness is statistically significant, we have completed a statistical significance test on the four examples for five methods. Tables 4 and 5 give the p-values of two groups (one group corresponding to TMS and another group corresponding to some other method) for full order filter and reduced order filter, respectively. It is evident from Tables 4 and 5 that all p-values are less than 0.05 (5% significance level). This is a strong evidence against the null hypothesis, establishing significant superiority of the proposed IIR design method. 5. Conclusions This paper has discussed the use of tissue-like membrane systems to develop a novel method for IIR filter design. The new method has been applied to identify several benchmark IIR systems. The proposed method has been compared with five stateof-the-art evolutionary algorithm-based methods for the identification of IIR systems by the actual order and reduced order filters respectively. The comparison results demonstrate that the proposed method has a superior identification performance to GA, PSO, DE, ABC and CSO in terms of the convergence speed, RMSE and MSD. The study also indicates that compared to other evolutionary technique-based methods, membrane computing model is a competitive candidate for identification of IIR system. In the topic, the proposed method and the compared methods have the same goal, that is, finding the filter’s optimal coefficient for IIR filter design problem with a multimodal error surface. In comparison to the existing evolutionary algorithms, membrane systems use a different mechanism to exploit the optimal solution of IIR filter design problem: the object’s communication mechanism and the modified differential evolution mechanism. The two mechanisms can enhance the diversity of objects in the system and balance exploitation and exploration of solution during the optimal design problem is solved. In this work, application of membrane computing in the optimal IIR filter design problem has been discussed, however, the IIR filter considered is a kind of general IIR filters. In a lot of real-world engineering problems, some special filters have been widely used. For example, low pass (LP), high pass (HP), band pass (BP) and band stop (BS) filters have been widely used to solve the image processing/signal processing problems. So, how to design the optimal special filters based on membrane systems is our further work. In addition, the order of the IIR filter is assumed to be known/fixed in this work. However, it may be difficult to assign a most suitable order for an unknown IIR plant a priori. Therefore, how to determine the optimal order in the optimal IIR filter design is another further work. However, for the two special issues, the tissue-like membrane systems proposed in

J. Wang et al. / Information Sciences 329 (2016) 164–176

175

this work cannot directly used. Some specialized membrane systems should be required to deal with the two issues, such as membrane systems with cell division. Acknowledgment This work was partially supported by the National Natural Science Foundation of China (grant nos. 61170030 and 61472328), and Research Fund of Sichuan Science and Technology Project (no. 2015HH0057), China. Appendix A. Case study examples A.1. Example I The transfer function of the example is given by Shynk [26]



Hs (z) =

0.05 − 0.4z−1 1 − 1.1314z−1 + 0.25z−2



(A.1)

Case 1: This 2nd order system can be modeled using a 2nd order IIR filter. Hence the transfer function of the model is given by

 H f (z) =

a0 + a1 z−1 1 − b1 z−1 − b2 z−2

 (A.2)

Case 2: This 2nd order system can be modeled using a reduced order IIR filter. Hence the transfer function of the 1st order IIR filter is given by



H f (z) =

a0 1 − b1 z−1



(A.3)

A.2. Example II The transfer function of the example is given by Panda et al. [17]



Hs (z) =

−0.2 − 0.4z−1 + 0.5z−2 1 − 0.6z−1 + 0.25z−2 − 0.2z−3



(A.4)

Case 1: This 3rd order system can be modeled using a 3rd order IIR filter. Hence the transfer function of the model is given by

 H f (z) =

a0 + a1 z−1 + a2 z−2 1 − b1 z−1 − b2 z−2 − b3 z−3

 (A.5)

Case 2: This 3rd order system can be modeled using a reduced order IIR filter. Hence the transfer function of the 2nd order IIR filter is given by



H f (z) =



a0 + a1 z−1 . 1 − b1 z−1 − b2 z−2

(A.6)

A.3. Example III The transfer function of the example is given by Panda et al. [17]



Hs (z) =

1 − 0.9z−1 + 0.8z−2 − 0.729z−3 1 + 0.04z−1 + 0.2775z−2 − 0.2101z−3 + 0.14z−4

 (A.7)

Case 1: This 4th order system can be modeled using a 4th order IIR filter. Hence the transfer function of the model is given by

 H f (z) =

a0 + a1 z−1 + a2 z−2 + a3 z−3 1 − b1 z−1 − b2 z−2 − b3 z−3 − b4 z−4

 (A.8)

Case 2: This 4th order system can be modeled using a reduced order IIR filter. Hence the transfer function of the 3rd order IIR filter is given by



H f (z) =

a0 + a1 z−1 + a2 z−2 1 − b1 z−1 − b2 z−2 − b3 z−3



(A.9)

176

J. Wang et al. / Information Sciences 329 (2016) 164–176

A.4. Example IV The transfer function of the example is given by Krusienski et al. [12]



Hs (z) =

0.1084 + 0.5419z−1 + 1.0837z−2 + 1.0837z−3 + 0.5419z−4 + 0.1084z−5 1 + 0.9853z−1 + 0.9738z−2 − 0.3864z−3 + 0.1112z−4 + 0.0113z−5

 (A.10)

Case 1: This 5th order system can be modeled using a 5th order IIR filter. Hence the transfer function of the model is given by

 H f (z) =

a0 + a1 z−1 + a2 z−2 + a3 z−3 + a4 z−4 + a5 z−5 1 − b1 z−1 − b2 z−2 − b3 z−3 − b4 z−4 − b5 z−5

 (A.11)

Case 2: This 5th order system can be modeled using a reduced order IIR filter. Hence the transfer function of the 4th order IIR filter is given by



H f (z) =



a0 + a1 z−1 + a2 z−2 + a3 z−3 + a4 z−4 . 1 − b1 z−1 − b2 z−2 − b3 z−3 − b4 z−4

(A.12)

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