Design of hybrid nature-inspired heuristics with application to active noise control systems

Abstract

In this study, nature-inspired computational intelligence is exploited for active noise control (ANC) systems using variants of particle swarm optimization (PSO) algorithm and its memetic combination with efficient local search technique based on active-set (AS), interior-point (IP), Nelder–Mead (NM) and sequential quadratic programming (SQP) algorithms. In ANC, filtered extended least mean square algorithm is normally used for finding the optimal parameters of the linear finite-impulse response filter, which is more likely to trap in local minima (LM). The issue of LM problem is effectively handled with competence of nature-inspired heuristics by developing four variants of memetic computing approaches based on PSO-NM, PSO-AS, PSO-IP, and PSO-SQP in order to adapt the design variables of ANC with linear and nonlinear primary and secondary paths by taking input noise interferences of pure sinusoidal, random and complex random types. The comparative studies of proposed schemes through statistical performance indices have established the worth of the schemes in terms of accuracy, convergence and complexity parameters.

1 Introduction

Acoustic noise problems become more and more evident as number of industrial equipments is increasing with the time. The passive approaches are normally used for acoustic noise control in machines with the help of enclosures, barriers and silencers to attenuate the undesired noise [1, 2]. These passive techniques have their own advantages and limitations but usually considered to be very inefficient to reduce noise at low frequencies, while active noise control (ANC) is a popular field in adaptive signal processing (ASP) for removal of noise at low frequencies. The primary purpose of ANC system is to introduce anti-noise signal with the help of secondary sources for noise cancelation. Standard operation of active noise control is based on the principle of electroacoustic and electromechanical systems to cancel the noise from the primary source through superposition technique [3, 4]. In this manner, ASP procedures are used to analyze the waveform of the background noise and develop an inverted signal to cancel the required noise [5,6,7]. In ANC, the filtered-X least mean square (FXLMS) technique is used to generate anti-snoring noise by estimating the secondary path through adaptation of optimal weights of the finite-impulse response (FIR) filter [8,9,10,11]. By interfering with noise, the anti-snoring noise signals help to create a silent zone around the snored [12]. Following the same concept, S. V. Narasimhan et al. proposed variable step size algorithms for feedforward/feedback ANC systems [13,14,15].

The standard schematic of feedforward ANC system [16] through adaptive algorithm is shown in Fig. 1 in which one microphone is used to pick up reference signal x(n), while the other microphone is to pick up the residual noise e(n). Moreover, a speaker is used to propagate canceling signal y(n) generated by adaptive noise control filter W(n). Here x(n) is filtered through an estimated secondary path Ŝ(n) before being supplied to least mean square (LMS) algorithm for adapting the weights. However, such adaptive schemes are normally based on FXLMS which has many issues such as more prone to get stuck in local minima during its iterative mechanism and also have an additional requirement of identification of secondary path before adaptation process starts. These problems are addressed effectively by formulation of alternative adaptive strategies in the domain of intelligent computing based on evolutionary and swarm intelligence techniques [17,18,19]. It is already proven that the classical FXLMS algorithm in case of both FIR and Volterra filtering-based ANC controller performs degradedly as compared with biologically/nature-inspired heuristics [17,18,19]. However, besides the well-established worth of nature-inspired heuristics [20,21,22,23,24,25,26,27], these methodologies in the framework of memetic computing approaches based on variants of particle swarm optimization (PSO) algorithm integrated with different efficient local search techniques have yet not exploited to overcome the limitation of FXLMS and accordingly to improve the performance of ANC systems. The aim of this study is to develop an alternate, accurate and reliable hybrid computing frameworks (HCFs) for adaptive feedforward ANC system using variants of PSO algorithms supported with rapid local search methods.

Fig. 1

Generic form of feedforward ANC system

The strength of HCFs based on bioinspired and nature-inspired methodologies hybrid with efficient local search algorithms is applied effectively for finding the near-optimal solutions of many problems arising in different applications of engineering science and technology. For example, integration of global and local search solvers is used to solve nonlinear benchmark models [28], nonlinear Flierl–Petviashivili system [29], Navier–Stokes equations [30], programming problems [31, 32], Bratu’s-type equations [33, 34], gas turbines [35], differential equations [36], Troesch’s problem [37, 38], rotating electrical machines [39], antenna positioning problem [40], flower pollination algorithm [41, 42], nanofluidics [43], induction motor models [44], thermodynamic studies [45], nonlinear system identification [46], Volterra–Fredholm integro-differential equations [47] and astrophysics models [48]. Recently, new variants of swarm intelligence techniques are developed for the segmentation of nonstationary signals [49], MRI brain images classification [50], speaker recognition system [51], and fractal image compression [52]. These are the motivation factors for the authors to investigate HCFs-based nature-inspired heuristics to solve the optimization problem of ANC systems reliably and effectively.

In this paper, computational intelligence approaches based on variants of PSO hybrid with Nelder–Meads (NM) simplex method, interior-point (IP), active-set (AS) and sequential quadratic programming (SQP) algorithms are designed for ANC systems without identification of secondary path. Variants of PSO are formulated by taking different number of particles in the swarm and number of flights of the swarm to extract the potential of the approach in terms of accuracy, robustness, convergence and computational complexity. Optimization capabilities for the variants of PSO are further strengthened by hybridization with rapid local search algorithms. The design schemes of PSO, PSO with NM (PSO-NM), PSO with AS (PSO-AS), PSO with IP (PSO-IP) and PSO with SQP (PSO-SQP) are evaluated to prove the noise attenuation performance for ANC system by taking sinusoidal, random and complex random input noise interferences for number of cases of linear and nonlinear primary and secondary paths. Reliability and effectiveness of the schemes are validated through the results of statistical analyses for seven case studies of ANC systems, and each study the parameters of system are optimized with seven variants of PSO hybrid with four different local search algorithms for sufficient large number of independent runs.

Rest of the paper is organized as follows: A brief description of system model taken for ANC study is given in Sect. 2. Section 3 is about proposed research methodology developed for the attenuation of undesired noise disturbances with the help of hybrid PSO-based ANC controllers. The mathematical framework, formulation of fitness function and procedure used for training of design parameters are also given in this section. In Sect. 4, results of simulation are presented for the proposed schemes in case of seven ANC experimental scenarios with noise variations and different linear and nonlinear, primary and secondary paths. Additionally, results of seven case studies are also presented in this section for each experiment by taking different number of flights and elements in the swarm. In Sect. 5, conclusions for the numerical experimentations are listed along with potential future research directions.

2 System model for ANC

System model for the cancelation of unwanted noise interference generated by machine in a duct is shown in Fig. 2. The hybrid intelligence approaches based on variants of PSO are applied for ANC system given in Fig. 2. The aim of ANC system is to transmit artificial noise for the cancelation of unwanted noise by the speaker in the channel to create a silence zone. It is a common industrial application with purpose to enable human hearing in a noisy environment. In the ANC model of Fig. 2, u(n) represents the unwanted noise interference, d(n) is the measured noise by microphone, v(n) denotes the output signal, and v′(n) denotes the output of the speaker to cancel the measured noise. Additionally, P(z) represents the primary path based on transfer function from location of undesired noise interferences to the microphone and S(z) denotes the secondary path from the speaker to the microphone. An FIR filter is represented with W(z). In brief for the given system model, the adaptation procedure based on variants of PSO algorithm hybridized with rapid local search techniques instead of the conventional or traditional FXLMS methods for learning of the design parameters, i.e., weights of ANC system. Four hybrid algorithms based on PSO-NM, PSO-AS, PSO-IP and PSO-SQP are developed for training of unknown weights of ANC system.

Fig. 2

Block diagram of ANC system with variants of PSO algorithm

3 Methodology

Proposed ANC controllers are presented here based on hybrid PSO algorithm for finding the design variables of ANC systems. Four variants of hybrid PSO based on PSO-NM, PSO-IP, PSO-AS and PSO-SQP are designed and evaluated. The proposed methodology consists of two parts: firstly, the construction of fitness function for ANC system, while in second part optimization of fitness function carried out with PSO and its memetic variants.

The mathematical modeling is presented for the formulation of fitness function for adaptive ANC system in Fig. 2 for FIR filter-type heuristic algorithms.

3.1 Construction of fitness function

Construction of fitness function for ANC system for the adaptive meta-heuristic artificial intelligence techniques including PSO, genetic algorithm and simulated annealing based on FIR filter type with length of coefficient L is given as:

$$\varvec{W}_{i} \left( n \right) = \left( {w_{i} \left( n \right),w_{i} \left( {n - 1} \right), \ldots ,w_{i} \left( {n - L + 1} \right)} \right), \quad i = 1,2, \ldots ,p,$$

where w_i(n) be the coefficient vector of the ith element of swarm of FIR filter W at time n, and p represents total number of particles in a swarm. The governing equation for filtering of noise u(n) for the swarm of length p in terms of the real coefficients of FIR filter is written as: [19]

$$\left( {\begin{array}{*{20}c} {v_{1 } \left( n \right)} \\ {v_{2 } \left( n \right)} \\ \vdots \\ {v_{p } \left( n \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {w_{1} (n)} & {w_{1} (n - 1)} & \cdots & {w_{1} (n - L + 1)} \\ {w_{2} (n)} & {w_{2} (n - 1)} & \cdots & {w_{2} (n - L + 1)} \\ \vdots & \vdots & \cdots & \vdots \\ {w_{p} (n)} & {w_{p} (n - 1)} & \cdots & {w_{p} (n - L + 1)} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {u(n)} \\ {u(n - 1)} \\ \vdots \\ {u(n - L + 1)} \\ \end{array} } \right),$$

(1)

and

$$\left( {\begin{array}{*{20}c} {v_{1 }^{\prime } \left( n \right)} \\ {v_{2}^{\prime } \left( n \right)} \\ \vdots \\ {v_{p}^{\prime } \left( n \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {v_{1} (n)} & {v_{1} (n - 1)} & \cdots & {v_{1} (n - L + 1)} \\ {v_{2} (n)} & {v_{2} (n - 1)} & \cdots & {v_{2} (n - L + 1)} \\ \vdots & \vdots & \cdots & \vdots \\ {v_{p} (n)} & {v_{p} (n - 1)} & \cdots & {v_{p} (n - L + 1)} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {S_{1} } \\ {S_{2} } \\ \vdots \\ {S_{L} } \\ \end{array} } \right),$$

(2)

where $\left( {s_{1} ,s_{2} , \ldots ,s_{L} } \right)^{\text{T}}$ represents the impulse response of the secondary path S(z).

The fitness or objective function for ANC system is constructed by defining the following error function as:

$$\varepsilon_{i} (n) = d(n) - v_{i}^{\prime } (n),\quad i = 1,2, \ldots ,p.$$

(3)

Then

$$\left( {\begin{array}{*{20}c} {O_{1} \left( n \right)} \\ {O_{2} \left( n \right)} \\ \vdots \\ {O_{p} \left( n \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\hbox{max} \left( {\left| {\varepsilon_{1} (n)} \right|} \right.} & {,\left| {\varepsilon_{1} (n - 1)} \right|} & \cdots & {,\left| {\varepsilon_{1} (n - L + 1)} \right|} \\ {\hbox{max} \left( {\left| {\varepsilon_{2} (n)} \right|} \right.} & {,\left| {\varepsilon_{2} (n - 1)} \right|} & \cdots & {,\left| {\varepsilon_{2} (n - L + 1)} \right|} \\ \vdots & \vdots & \cdots & \vdots \\ {\hbox{max} \left( {\left| {\varepsilon_{p} (n)} \right|} \right.} & {,\left| {\varepsilon_{p} (n - 1)} \right|} & \cdots & {,\left| {\varepsilon_{p} (n - L + 1)} \right|} \\ \end{array} } \right).$$

(4)

The residual error function of ANC system is defined by the fitness or merit function.

The residual noise of the ANC system is minimized by defining the fitness functions of p particles in the swarm as:

$$f_{i} = \frac{1}{{O_{i} }},\quad i = 1,2, \ldots ,p$$

(5)

It is quite clear that smaller the value of O_i, better will be the fitness values and consequently the minimum value for residual error of the ANC system.

3.2 Learning of design variables

The procedures adapted for training of unknown variables in order to minimize the residual error of ANC system are presented here with the help of variants of PSO techniques hybrid with NM, AS, IP and SQP algorithms.

PSO is an optimization technique that belongs to the class of population-based search algorithm. The population is described as swarm, and an individual candidate solution in the swarm is termed as a particle. Any ith particle in the swarm has the following attributes associated with it. Its ith position and velocity are denoted by x_i, and y_i, respectively, and the ith best position is denoted by p_i.

Particle velocity and the position are updated as follows:

$$y_{im} (n) = y_{im} (n - 1) + r_{1} \cdot \varphi_{1} \cdot \left( {p_{lm} - x_{im} (n - 1)} \right) + r_{2} \cdot \varphi_{2} \cdot \left( {p_{gm} - x_{im} (n - 1)} \right),$$

(6)

$$x_{im} (n) = x_{im} (n - 1) + y_{im} (n),$$

(7)

where $\varphi_{1}$ and $\varphi_{2}$ are positive constants, representing the values of collective intelligence and individual intelligence factor, r₁ and r₂ are uniformly distributed random numbers between 0 and 1, index m represents the mth element of the ith particle, while, p_lm and p_gm represent the mth element of local and global best particle of the swarm, respectively.

The better performance of global search algorithm based on PSO is attained at the finishing stage of predefined cycles by hybridizing with local search algorithm for rapid refinement. In this study, global best particles of PSO technique are taken as initial weights for local search algorithms including NM, AS, IP and SQP algorithms and these hybrid PSO algorithms are used as ANC controllers to reduce the residual error of the system.

3.3 Schematic of proposed methodology

The proposed scheme is applied to ANC problem with three input noise variations, seven experimental scenarios based on linear and nonlinear primary and secondary paths, seven cases based on different number of flights and particles in the swarm for each experiment. The mathematical model to develop fitness function, optimization procedure with PSO along with its hybrid variants and comparative studies based on minimum value of residual error are shown in graphical abstract of proposed scheme as shown in Fig. 3.

Fig. 3

Work flow diagram of proposed design methodology

3.4 Implementation of design methodology

Software routines for standard PSO algorithm available at MATLAB central [53] are used for the optimization of the fitness function defined in Eq. (5). While for NM, IP, AS and SQP algorithms, the built-in functions in MATLAB optimization toolbox are used, five optimization algorithms are used for ANC system to reduce the unwanted noise disturbances, i.e., PSO, PSO-NM, PSO-AS, PSO-IP and PSO-SQP. The values or setting of necessary parameter for the variables for each algorithm is listed in Table 1. These settings are done with care, a lot of experimentations, experience and knowledge of the problem.

Table 1 Parameters setting for algorithms

The details of the steps in the hybrid PSO approaches with local search algorithms are provided below:

• Step 1: Initialization The initial swarm is created with the help of particles based on randomly generated bounded real values with as many number of elements as number of design parameters in ANC model. Setting of bounds, declarations and initializing the options of PSO tools ‘psooptimset’ are listed in Table 1, e.g., number of flights, and values of local and global social acceleration constants.

• Step 2: Fitness evaluation Determine the fitness value for each particle of the swarm using the relation given in (5).

• Step 3: Termination criteria Terminate the execution of PSO program:

  • If predefined fitness value is achieved.

  • If predefined number of flights are executed.

  • If predefined values of tolerances are achieved, i.e., function tolerances (TolFun) and nonlinear constraints tolerance (TolCon).

  If any of termination conditions satisfy, then go to step 5; otherwise, continue.

• Step 4: Updating swarm position and velocity New swarm at each cycle is created by using the relations given in Eqs. (6) and (7) for updating velocity and position of the particle,

  Go to step 2 and continue to proceed with updated swarm.

• Step 5: Storages of PSO parameters Store the values of best particle of PSO algorithm, along with their fitness value, execution time (ET) and number of flights used in the algorithms.

• Step 6: Hybridization Nelder–Mead (NM), interior-point (IP), active-set (AS) and sequential quadratic programming (SQP) algorithms are used as a hybrid local search mechanism with PSO technique for fine-tuning of results as per following procedure:

  • 1. Initialization MATLAB built-in routines ‘fmincon’ and ‘fminsearch’ available in optimization tool box are invoked with settings, bounds, declarations and initialization of parameter as listed in Table 1 for all four algorithms. The initial parameters of local search algorithms are the best particle of PSO algorithm.

  • 2. Evaluation of fitness Calculate the fitness using Eq. (5).

  • 3. Termination criteria Terminate the iterative process for updating the unknown weights if

    • Predefined number of flights is executed.

    • Predefined value of the tolerances achieved, i.e., value of ‘TolFun,’ ‘TolCon’ or X-tolerance (TolX).

    • Predefined maximum numbers of functions are evaluated.

      If termination criteria satisfy, then go to step 7.

  • 4. Updating variables Step increment in any of local search algorithm, NM, AS, IP and SQP, then go to step 6(b) for updating the weights.

• Step 7: Storage of hybrid algorithm parameters Store the optimal values of weight vectors by NM, AS, IP and SQP methods, their fitness, number of cycles consumed, execution time taken for this run of each algorithm.

• Step 8: Reliable comparative analyses Repeat the whole procedure from steps 1–7 for sufficient large number of independent runs for PSO and its memetic variants to generate large data set. The reliable comparative analysis between the algorithms is made with the help of statistical analyses on given large data set.

4 Results and discussion

In this section, detail simulation studies are presented for the proposed schemes for seven experimental scenarios of active noise control system by taking various linear and nonlinear primary and secondary paths.

The performance of the proposed PSO-based ANC is demonstrated by considering the following experimental setup. The coefficient length L = 20 of FIR-type filter is taken in the developed PSO-based ANC. The effectiveness of ANC is verified using different kinds of primary path P(z) and secondary path S(z). The linear functions associated with the primary and secondary paths are given by the following relations:

$$P_{\text{linear}} = Z^{ - 5} - 0.3Z^{ - 6} + 0.2Z^{ - 7} ,$$

(8)

$$S_{\text{linear}} = Z^{ - 2} + 1.5Z^{ - 3} - Z^{ - 4} .$$

(9)

The nonlinear transfer functions associated with primary and secondary paths are taken with assumption that the primary noise d(n) at the canceling point is based on third-order polynomial function as:

$$t\left( n \right) = x\left( {n - 3} \right) - 0.3\left( {n - 4} \right) + 0.2\left( {n - 4} \right),$$

(10)

$$d\left( n \right) = t\left( {n - 2} \right) + 0.8t\left[ {\left( {n - 4} \right)} \right]^{2} +\,0.04t\left[ {\left( {n - 4} \right)} \right]^{3} .$$

(11)

While the anti-noise signal $v^{\prime } (n)$ at the v(n) canceling point for nonlinear secondary path is taken as:

$$r(n) = 0.06\tanh \left( {1.5v(n\left. ) \right)} \right.,$$

(12)

$$v^{\prime } \left( n \right) = r\left( {n - 2} \right) + 1.5r\left( {n - 3} \right) - r\left( {n - 4} \right).$$

(13)

Experimental studies are carried out on Dell Latitute D830, laptop computer with dual 2.0-GHz central processing units and 2-GB dynamic random access memory in MATLAB.

Simulation studies have been carried out with four variants of PSO with local search algorithms, i.e., PSO-IPMs, PSO-NM, PSO-IP, PSO-SQP, based controllers to enhance the performance of ANC. The parameter settings used for PSO, IP, NM, SQP algorithms are listed in Table 1.

Seven scenarios for the experimentation are designed to examine the performance of the design schemes. Each experimental study consists of seven cases by taking different number of particles in a swarm and number of flights in the algorithm as given in Table 2, to provide an intelligent guess for PSO-based ANC controller for better trade-off between accuracy and complexity.

Table 2 Cases for each experimental study

4.1 ANC experimental scenario 1

In this experiment, ANC system with linear primary path and nonlinear secondary path is taken as given in Eqs. (8) and (13), respectively. The unwanted noise measurement is assumed to be a 200-Hz pure sine wave.

Five different optimization algorithms including four memetic variants of PSO, i.e., PSO-NM, PSO-AS, PSO-IP and PSO-SQP, are used to attenuate the undesired noise of ANC system of experiment 1 for all seven cases as per procedure given in the last section. In order to analyze in-depth performance of the proposed schemes, the procedure of reduction of unwanted noise is carried out for 100 independent runs of each algorithm for each case. The results in terms of fitness values in dB are listed in Table 3 based on value of statistical operators, the best, i.e., the maximum fitness, mean, i.e., average fitness and standard deviation (STD), i.e., spread around mean. It is seen that the sufficient large values of dB are achieved by each algorithm which validate the effectiveness of proposed design scheme for the reduction of unwanted noise. The results of PSO algorithm are relatively inferior from any of memetic variants of PSO with local search techniques. While no noticeable difference is seen in the results for all four memetic versions in case of best values, from the mean values it is observed that performance of the PSO-SQP is better than the rest of the algorithms around 5–10 dB. Smaller values of STD for PSO-SQP further validated the consistently accurate performance of algorithm in each case study. The values of fitness are plotted in Fig. 4 for cases 1–4 and in Fig. 5 for cases 5–7 against 100 independent runs of each algorithm. To broaden the small variation in the results, the fitness values are also plotted in sorted manner in Figs. 4 and 5 for all seven cases. It is seen that in case 1 the values of fitness for PSO, PSO-NM, PSO-AS, PSO-IP and PSO-SQP are consistently above 15, 20, 22, 25 and 40 dB, respectively. Therefore, the performance of PSO-SQP is superior from the rest of algorithms and same inferences are drawn from other six cases of this experiment. Additionally, by increasing the number of flights, i.e., cases 1–4, and number of particles in a swarm, i.e., cases 5–7, results of PSO improve consistently which can be seen through column values of Table 3. The improvement of local search algorithm is evident through row values and no guarantee of superior result through column values for the hybrid optimization mechanism due to flight independence of local search.

Table 3 Performance of PSO and its memetic variants for ANC experimental scenario 1

Fig. 4

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c, e and g are on left, while the sorted figures b, d, f and h are on right for cases 1, 2, 3 and 4 of experiment 1, respectively

Fig. 5

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c and e are on left, while the sorted figures b, d and f are on right for cases 5, 6 and 7 of experiment 1, respectively

The computational complexity of the design PSO variants is determined in terms of values of mean execution time (ET) and mean number of flights/cycle (Flts) taken by the optimization procedure. The mean values of ET and Flts are given in Table 4 for each case study. It is seen that with the increase of number of flights, i.e., from 50 to 400, the ET values of each algorithm increase; however, no noticeable difference is observed in the values of fitness in dB that can be seen in Table 4. It is seen that all memetic algorithms take almost double ET time than that of PSO, but this aspect is overshadowed due to their brilliance performance. While comparing the complexity of memetic algorithms, the most efficient algorithm is found to be PSO-SQP.

Table 4 Computational complexity (CM) in terms of execution time in seconds (ET(s)) and flights (Flts) for proposed algorithms in experiment 1

4.2 ANC experimental scenario 2

In the experiment study, nonlinear primary path (11) and linear secondary path (9) are taken, while the unwanted narrow band noise is assumed to be the 200-Hz pure sine wave.

All five variants of PSO algorithms are used to attenuate the undesired noise of ANC system of experiment 2 for all seven cases, and the results are given in Table 5. The results of PSO algorithm are found relatively inferior from memetic variants for this experimental study also. There is no noticeable difference in the results for all four memetic versions in case of best values; however, from the mean values, it is observed that performance of the PSO-SQP is better than the rest of the algorithms around 5–10 dB. The values of fitness are plotted in sorted and unsorted form in Fig. 6 for cases 1–4 and in Fig. 7 for cases 5–7 against 100 independent runs of each algorithm. It is seen that in case 1 the values of fitness for PSO, POS-NM, PSO-AS, PSO-IP and PSO-SQP are around 15–25, 20–35, 20–50, 20–50 and 50 dB, respectively. In general, the performance of PSO-SQP algorithm is better from the rest of algorithms in this experiment in terms of accuracy and convergence.

Table 5 Performance of PSO and its memetic variants for ANC system taken in experiment 2

Fig. 6

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c, e and g are on left, while the sorted figures b, d, f and h are on right for cases 1, 2, 3 and 4 of experiment 2, respectively

Fig. 7

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c and e are on left, while the sorted figures b, d and f are on right for cases 5, 6 and 7 of experiment 2, respectively

Results of complexity operators in terms of mean values of ET and Flts are tabulated in Table 6 for all seven cases of ANC experimental scenario 2. It is observed that with the increase of number of flights, i.e., from 50 to 400, the ET value of each algorithm increases for this case study as well. It is seen that all memetic algorithms take around double ET than that of PSO, but this aspect can be overshadowed due to their brilliance in accuracy. In general, hybrid scheme PSO-SQP is found efficient for each case.

Table 6 Computational complexity (CM) in terms of execution time (ET) and flights (Flts) for proposed algorithms in experiment 2

4.3 ANC experimental scenario 3

In this case, both primary and secondary paths are nonlinear as given in Eq. (11) and (13), while the unwanted narrow band noise is assumed to be the 200-Hz pure sine wave.

The PSO algorithm and four of its memetic variants are used to attenuate the undesired noise of ANC system of experiment 3 for all seven cases, and the results are given in Table 7. The results of PSO algorithm are found relatively inferior from memetic variants for this experimental study, while there is no noticeable difference in the results of hybrid computing algorithms for the best values, but, in case of mean values, the performance of PSO-SQP is better up to the level of 5–7 dB than the rest. The values of fitness are plotted for 100 independent runs, and results for cases 1–4 and 5–7 are shown in FigS. 8 and 9, respectively. It is observed that in case 1 the values of fitness for PSO, POS-NM, PSO-AS, PSO-IP and PSO-SQP are around 12–28, 22–35, 20–42, 18–42 dB and above 40 dB, respectively. The performance of PSO-SQP algorithms is found better from the rest in this case study. The values of complexity indices through mean ET and Flts are tabulated in Table 8 for each case. It is seen that with an increase in number of flights complexity of the all the algorithms grows. Additionally, memetic algorithms take bit longer ET than that of PSO, but this aspect is compensated due to their superior performance. In complexity parameter, PSO-SQP algorithms are found relatively better from all other algorithms.

Table 7 Performance of PSO and its memetic variants for ANC system taken in experiment 3

Fig. 8

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c, e and g are on left, while the sorted figures b, d, f and h are on right for cases 1, 2, 3 and 4 of experiment 3, respectively

Fig. 9

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c and e are on left, while the sorted figures b, d and f are on right for cases 5, 6 and 7 of experiment 3, respectively

Table 8 Computational complexity (CM) in terms of execution time (ET) and flights (Flts) for proposed algorithms in experiment 3

4.4 ANC experimental scenario 4

In fourth experimental scenario, we compare the results obtained from linear primary (8) and nonlinear secondary (13) paths of reducing random noise within 200–300 Hz.

Five optimization algorithms based on PSO, PSO-NM, PSO-AS, PSO-IP and PSO-SQP are used to attenuate the undesired noise of ANC system of this study for all seven variations, and results are given in Table 9. The results of PSO algorithm are found relatively inferior from memetic variants for this experimental study, and in all five algorithm best values of fitness are obtained for PSO-NM with mean values better from result around 2–5 dB. The values of fitness are plotted in sorted and unsorted form in Fig. 10 for cases 1–4 and in Fig. 11 for cases 5–7 for 100 independent runs of each algorithm. It is observed that in case 1 the values of fitness for PSO, POS-NM, PSO-AS, PSO-IP and PSO-SQP are consistently above 45, 47, 30, 40 and 40 dB, respectively. The performance of PSO-NM algorithms is better from the rest of algorithms in this study. Complexity metrics based on mean values of ET and Flts are tabulated in Table 10 for all seven cases of ANC experimental scenario 4. It is seen that all memetic algorithms take more time to execute than that of PSO, but this aspect overshadows due to their brilliance accuracy and convergence. The most efficient algorithms in this study are PSO-SPQ but will slight degrade accuracy from PSO-NM for each case.

Table 9 Performance of PSO and its memetic variants for ANC system taken in experiment 4

Fig. 10

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c, e and g are on left, while the sorted figures b, d, f and h are on right for cases 1, 2, 3 and 4 of experiment 4, respectively

Fig. 11

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c and e are on left, while the sorted figures b, d and f are on right for cases 5, 6 and 7 of experiment 4, respectively

Table 10 Computational complexity (CM) in terms of execution time (ET) and flights (Flts) for proposed algorithms in experiment 4

4.5 ANC experimental scenario 5

In fifth experiment, nonlinear primary path (11) and linear secondary path (9) with the 200–300-Hz unwanted random noise are used.

All five optimization algorithms are used to attenuate the undesired noise of ANC system of experiment 5 for each case, and the results are given in Table 11. The results of PSO algorithm are found slightly inferior from memetic variants for each case, while no noticeable difference is seen in the values of dB for all four memetic versions in case of best values, but the mean values performance of the PSO-NM is better than the rest of the algorithms with gain of 4–5 dB. The values of fitness are plotted for number of independent runs in Figs. 12 and 13. It is observed that in case 1 the values of fitness for PSO, POS-NM, PSO-AS, PSO-IP and PSO-SQP are consistently above 45, 47, 35, 35 and 40 dB, respectively. In general, the performance of PSO-NM algorithms is better from the rest of algorithms in this study. Complexity analysis using mean values of ET and Flts is listed in Table 12 for all seven cases of ANC experimental scenario 5. It is seen that all memetic algorithms take more ET than that of PSO, but this aspect overshadows due to their brilliance performance. In general, the most efficient algorithm is found to be PSO-IP in all four memetic approaches of this scenario.

Table 11 Performance of PSO and its memetic variants for ANC system taken in experiment 5

Fig. 12

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c, e and g are on left, while sorted figures b, d, f and h are on right for cases 1, 2, 3 and 4 of experiment 5, respectively

Fig. 13

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c and e are on left, while the sorted figures b, d and f are on right for cases 5, 6 and 7 of experiment 5, respectively

Table 12 Computational complexity (CM) in terms of execution time (ET) and flights (Flts) for proposed algorithms in experiment 5

4.6 ANC experimental scenario 6

In sixth experiment, nonlinear primary path (11) and secondary path (13) with the 200–300 Hz unwanted random noise are taken for the study.

The PSO algorithm along with four memetic variants is used to attenuate the undesired noise of ANC system of experiment 6 for all seven cases, and the results are given in Table 13. Results show that there is no noticeable difference in performance between PSO algorithm and its memetic variants for this experimental study; however, the mean values of the PSO-NM are better than the rest of the algorithms around 3–8 dB. The values of fitness are plotted in sorted and unsorted form in Figs. 14 and 15 for 100 independent runs of each algorithm. It is observed that in case 1 the values of fitness for PSO, POS-NM, PSO-AS, PSO-IP and PSO-SQP are consistently above 47, 50, 34, 43 and 42 dB, respectively. In general, the performance of PSO-NM algorithms is better from the rest of algorithms. Complexity analysis in terms of mean values of ET and Flts is conducted, and results are tabulated in Table 14 for all seven cases of ANC experimental scenario 6. It is seen that with the increase of number of flights, i.e., from 50 to 400, the ET value of each algorithm increases for this case study as well. The memetic algorithms take more ET than that of PSO. Generally, in memetic approaches the relatively efficient algorithm is found to be PSO-IP with relatively accurate algorithm PSO-NM for each case of this experimental study.

Table 13 Performance of PSO and its memetic variants for ANC system taken in experiment 6

Fig. 14

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c, e and g are on left, while the sorted figures b, d, f and h are on right for cases 1, 2, 3 and 4 of experiment 6, respectively

Fig. 15

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c and e are on left, while the sorted figures b, d and f are on right for cases 5, 6 and 7 of experiment 6, respectively

Table 14 Computational complexity (CM) in terms of execution time (ET) and flights (Flts) for proposed algorithms in experiment 6

4.7 ANC experimental scenario 7

In seventh experiment, the complex random noise with nonlinear primary (11) and secondary paths (13) is taken for the study. The complex random noise is based on white noise sequences generated through Gaussian-distributed complex random process.

All five algorithms are applied to attenuate the undesired noise of ANC system of experiment 7 for all seven cases, and the results are given in Table 15. The results of PSO-NM algorithm are found relatively better from rest for this experimental study in terms of mean values with a gain of 2–6 dB. The values of fitness for 100 independent runs are plotted in Figs. 16 and 17 for each case. It is observed that in case 1 the values of fitness for PSO, POS-NM, PSO-AS, PSO-IP and PSO-SQP are consistently above 40, 43, 30, 36 and 37 dB, respectively. In general, the performance of PSO-NM algorithms is better from the rest of algorithms in this experimental study as well. Computational complexity through mean values of ET and Flts is performed, and results are given in Table 16 for all seven cases of ANC experimental scenario 7. It is seen that ET of PSO-NM is very high as compared to all other memetic algorithms, but aspect overshadows due to brilliance performance of PSO-NM in terms of accuracy and convergence. In general, the most efficient algorithm among memetic approaches is found to be PSO-SQP for each case.

Table 15 Performance of PSO and its memetic variants for ANC system taken in experiment 7

Fig. 16

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c, e and g are on left, while the sorted figures b, d, f and h are on right for cases 1, 2, 3 and 4 of experiment 7, respectively

Fig. 17

Plot of fitness for 100 independent runs of the algorithms, unsorted figures a, c and e are on left, while the sorted figures b, d and f are on right for cases 5, 6 and 7 of experiment 7, respectively

Table 16 Computational complexity (CM) in terms of execution time (ET) and flights (Flts) for proposed algorithms in experiment 7

It is necessary to analyze the performance of proposed methodologies by variation of secondary path during the process of attenuation of unwanted noise interference. Besides the nonlinear secondary path defined in (12) and (13), another nonlinear secondary path in the system is taken as:

$$r(n) = 0.96\sinh \left( {1.9v(n\left. ) \right)} \right.,$$

(14)

$$v^{\prime } \left( n \right) = r\left( {n - 2} \right) + 1.5r\left( {n - 3} \right) - r\left( {n - 4} \right).$$

(15)

An experiment of ANC for attenuation of random noise interference with linear primary path but different nonlinear secondary paths is performed by the proposed methodology, such that, in case of first half of total iterations, the secondary path is taken as defined in (12) and (13), while, for remaining iterations, the change in the secondary path as given in (14) and (15) is made. The results of SQP, IP and AS algorithms for the experiment are shown in the form of learning curves in Fig. 18, and it is seen that all three algorithms are capable to deal with this variation in ANC system effectively. The similar behavior is observed for all variants of proposed methodology proposed in the present studies, which established that proposed memetic algorithm is accurate and stable by variation in the secondary paths.

Fig. 18

Performance of proposed algorithms for ANC with different nonlinear secondary paths

The comparison of the proposed results with conventional controllers based on FxLMS and VFxLMS algorithms is also made to analyze exhaustive performance. The conventional controller for ANC systems is designed through FxLMS and VFxLMS algorithms for which the secondary path modeling is essential for better performance and has a problem of premature convergence, i.e., stuck in local minima [17,18,19]. The ANC performance of the controller based on FxLMS for pure sinusoidal and random noise interferences is reported around 34 and 18 dB, respectively, while the respective results of VFxLMS are around 35 and 23 dB [19]. Both the algorithms fail for ANC systems with complex random noise interferences, whereas proposed controller based on PSO, PSO-NM, PSO-AS, PSO-IP, PSO-SQP, is applicable for all noise interferences of pure sinusoidal, random and complex random scenarios, with almost gain of 5–20 dB. The more gain in the performance is observed by PSO-SQP and PSO-NM algorithms and then rest of the methodologies, while the PSO-SQP has additional advantages of reduced complexity.

5 Conclusions

Following conclusions are drawn based on numerical experimentations conducted in this study.

Design of reliable and effective ANC controllers with the help of PSO and its memetic variants is based on PSO-NM, PSO-AS, PSO-IP and PSO-SQP. These algorithms are used viably for the optimization of residual error-based fitness function for the seven cases in each experimental scenario based on different number of particles in the swarm and number of flights executed by the swarm. The consistent performance of all five proposed algorithms is observed for each case of all seven ANC experimental scenarios based on linear and nonlinear primary and secondary paths by taking the sinusoidal, random and complex random input noises. It is observed that with the increase in number of flights or number of particles in the swarm, the performance improves a little but at the cost of more computational requirements. Comparative analyses on the results of hybrid PSO algorithms show that there is no noticeable difference of precision in them based on best values; however, the results of PSO-SQP in terms of mean values are superior for ANC with sinusoidal noise, while in case on ANC with random or complex random noises the performance of PSO-NM is relatively better in case of mean fitness values from rest in each case of all experimental studies. The complexity analyses in terms of mean values of ET and number of iterations show that hybrid approaches take bit longer than that of PSO algorithm, but this aspect is overshadowed due to their brilliant performance in each case of all ANC experimentation studies. In the hybrid PSO algorithms, the least computational extensive algorithm is found to be PSO-SQP for all taken scenarios. On the basis of statistical performance indices in terms of mean and standard deviation values for the fitness, i.e., accuracy, and execution time or iterations, i.e., complexity, generally the superior algorithm is found to be PSO-SQP, while the results of PSO-NM algorithm are slightly better in few cases, but in these cases complexity of PSO-NM is much higher than that of PSO-SQP results.

Use of meta-heuristic artificial intelligence techniques in the field of signal processing looks to be promising area of research, and future recommendations based on this study are given below:

• Change of optimization algorithm may result in improvement in the performance of ANC controller; in this regard, few alternatives can be exploited such as variants of ant/bee colony, differential evolution, fractional-order Darwinian PSO, firework algorithms, backtrack search technique and genetic programming integrated with efficient local search techniques.

• The proposed schemes can be exploited in the development of real-time ANC equipment with an improved performance used in both consumer electronics and defense industries.

• Exploration and exploitation should be carried out in application of strong mathematical foundation of fractional calculus in designing a new fractional variant of Volterra or kernel filters for ANC systems, and consequently, better performance may achieve.

• The proposed methods are developed effectively for feedforward ANC systems, and therefore, these schemes should be explored for feedback ANC systems as well as multichannel ANC models.