Abstract In this paper, a novel design of the flower pollination algorithm is presented for model identification problems in nonlinear active noise control systems. The recently introduced flower pollination based heuristics is implemented to minimize the mean squared error based merit/cost function representing the scenarios of active noise control system with linear/nonlinear and primary/secondary paths based on the sinusoidal signal, random and complex random signals as noise interferences. The flower pollination heuristics based active noise controllers are formulated through exploitation of nonlinear filtering with Volterra series. The comparative study on statistical observations in terms of accuracy, convergence and complexity measures demonstrates that the proposed meta-heuristic of flower pollination algorithm is reliable, accurate, stable as well as robust for active noise control system. The accuracy of the proposed nature inspired computing of flower pollination is in good agreement with the state of the art counterpart solvers based on variants of genetic algorithms, particle swarm optimization, backtracking search optimization algorithm, fireworks optimization algorithm along with their memetic combination with local search methodologies. Moreover, the central tendency and variation based statistical indices further validate the consistency and reliability of the proposed scheme mimic the mathematical model for the process of flower pollination systems.

The trend of exploiting the potential of bio/nature-inspired soft computing techniques is growing in the research community due to their extensive use in optimization problems arising in engineering, science and technology [

The ANC is a fundamental problem in control engineering and has been studied extensively with both traditional and different local/global optimization techniques [

The design of FPA based intelligent computing paradigm is presented for an effective solution of nonlinear ANC systems.

Mean squared error based merit function with nonlinear Volterra series filtering is formulated.

The accurate and robust performance of the FPA based ANC for various noise interferences in the case of different primary and secondary path scenarios prove the efficacy of the approach.

Central tendency and variation based statistical indices validate the consistency and reliability of the proposed scheme.

The rest of the manuscript is prepared as: ANC model is given in Section 2. The design approach is described in Section 3. Section 4 presents the results and the comparative studies with state of the art counterparts, and conclusions are given in Section 5.

The conventional block diagram of ANC based controller is given in

The methodology for ANC modeling with FPA consists of two phases; (1) formulating fitness function (2) presenting optimization mechanism based on FPA. The detailed flowchart in terms of process block structure is shown in

Block diagram of proposed ANC controller is given in

here

The input noise interference or source signal s(k) and output of nonlinear adaptive Volterra filtering b(k) with length

Accordingly, Volterra filtering of type 2 (VF-T2) with

Similarity, for Volterra filtering of type 3 (VF-T3) in case of the length of the Volterra filter

In case of

The fitness or merit function for ANC model is given as:

for

Here

In the case of perfect model, one has fitness function

The FPA is a mathematical model inspired by the process of pollination dynamics in flowers during the reproduction mechanism [

Global pollination carried out via biotic/cross pollination procedures with the help of insects, birds and bees to transport the pollens.

Abiotic or self-pollination process is adapted for efficient local search.

Flower fidelity process based reproduction probability.

Switching probability between 0 and 1 is exploited for feasible local and global pollination process [

The impressive swarm based optimization characteristics of FPA is exploited by the scholars from different fields [

where,

here

here,

The results of detailed ANC experimentations are presented here for multiple independent executions of the FPA. Three ANC problems are implemented based on different lengths (

While, in case of LSP, the transfer function is defined as

The NPP transfer function is given as:

Let q

The simulations are conducted in Matlab R2017b running under Windows 10 environment on DESKTOP-73HVB7M, with Intel(R) Core(TM) i7-4790 CPU@3.60 GHz, 16-GB RAM.

In this problem, FPA based ANC system is exploited for Case 1: ANC for LPP and NSP (ANC-LPP-NSP), Case 2: ANC for NPP and LSP (ANC-NPP-LSP) and, Case 3: ANC for NPP and NSP (ANC-NPP-NSP). The ANC primary/secondary paths are defined in

Reliable inferences on the outcome of ANC are presented for hundred independent trials of the FPA and result in the form of graphical representation of the statistics are given in

The performance of the FPA is further examined through histogram plots and statistical measures of minimum (MIN), mean, and standard deviation (STD). The histogram plots are provided in

ANC system with VF-TI | Index | Statistical indices | ||
---|---|---|---|---|

Min | Mean | STD | ||

Sinusoidal noise | Case 1 | 8.64E-05 | 9.10E-05 | 4.10E-06 |

Case 2 | 8.53E-05 | 8.65E-05 | 1.10E-06 | |

Case 3 | 3.16E-04 | 3.18E-04 | 1.83E-06 | |

Random noise | Case 1 | 6.97E-06 | 2.32E-05 | 1.23E-05 |

Case 2 | 6.42E-06 | 1.83E-05 | 8.55E-06 | |

Case 3 | 6.55E-06 | 2.33E-05 | 1.34E-05 | |

Complex Random noise | Case 1 | 1.32E-09 | 6.43E-07 | 2.71E-06 |

Case 2 | 2.95E-05 | 2.98E-05 | 7.43E-07 | |

Case 3 | 4.02E-05 | 4.08E-05 | 8.62E-07 |

The computational complexity of the FPA based ANC controllers is evaluated via mean time of execution required for the optimization and results for mean along with STD are tabulated in

Index | Sinusoidal | Random | Complex random | |||
---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | |

Case 1 | 93.479 | 0.333 | 85.145 | 0.386 | 76.974 | 2.959 |

Case 2 | 52.696 | 0.349 | 85.864 | 0.498 | 56.501 | 0.371 |

Case 3 | 50.839 | 0.297 | 80.377 | 0.217 | 56.708 | 0.355 |

In problem 2, FPA based ANC system is implemented for Case 1: ANC for LPP and NSP (ANC-LPP-NSP), Case 2: ANC for NPP and LSP (ANC-NPP-LSP) and, Case 3: ANC for NPP and NSP (ANC-NPP-NSP).

Graphical representation of the statistical outcomes for hundred independent trials of the FPA based ANC for each case of different noise interferences are given in

ANC system with VF-T2 | Index | Statistical indices | ||
---|---|---|---|---|

Min | Mean | STD | ||

Sinusoidal noise | Case 1 | 1.07E-05 | 2.73E-05 | 3.91E-05 |

Case 2 | 8.19E-06 | 1.42E-05 | 4.32E-06 | |

Case 3 | 2.00E-05 | 4.79E-05 | 1.39E-04 | |

Random noise | Case 1 | 4.19E-05 | 1.17E-04 | 7.34E-05 |

Case 2 | 1.79E-05 | 1.07E-04 | 7.15E-05 | |

Case 3 | 2.44E-05 | 1.45E-04 | 1.04E-04 | |

Complex random noise | Case 1 | 3.96E-04 | 1.49E-01 | 1.42E-01 |

Case 2 | 1.08E-05 | 1.37E-04 | 1.71E-04 | |

Case 3 | 4.78E-04 | 2.03E-01 | 1.31E-01 |

The performance of the FPA based ANC systems is further investigated through histogram plots and STATISTICAL operators and it is observed that the results of random VF-T2 are better than that of complex random but inferior to sinusoidal VF-T2. One may decipher that relatively better accuracy is attained for ANC system based sinusoidal and random noise signals. While the results of ANC with sinusoidal noise are consistently found better than random noise scenarios.

The computational complexity analyses for the optimization of FPA based ANC is evaluated based on mean time and STD. The results of complexity are given in

Index | Sinusoidal noise | Random noise | Complex random noise | |||
---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | |

Case 1 | 71.902 | 1.456 | 113.904 | 1.405 | 72.157 | 1.205 |

Case 2 | 71.008 | 1.381 | 114.081 | 1.615 | 79.680 | 1.264 |

Case 3 | 68.994 | 1.167 | 113.046 | 0.856 | 79.737 | 1.439 |

In this problem, FPA based ANC system is exploited for different primary/secondary path scenarios. The proposed FPA based ANC are conducted for hundred independent trials and graphical representation of the statistics in sort and unsorted plots are given in

ANC system with VF-TI | Index | Statistical indices | ||
---|---|---|---|---|

MIN | Mean | STD | ||

Sinusoidal noise | Case 1 | 5.81E-01 | 7.19E-01 | 6.40E-02 |

Case 2 | 9.28E-06 | 3.40E-05 | 2.41E-05 | |

Case 3 | 1.26E-05 | 5.99E-03 | 1.11E-02 | |

Random noise | Case 1 | 2.57E-04 | 1.23E-03 | 7.33E-04 |

Case 2 | 1.55E-04 | 9.20E-04 | 4.70E-04 | |

Case 3 | 3.40E-04 | 1.43E-03 | 9.16E-04 | |

Complex random noise | Case 1 | 1.51E-01 | 6.42E-01 | 7.25E-02 |

Case 2 | 1.64E-03 | 1.38E-02 | 6.75E-03 | |

Case 3 | 5.16E-01 | 6.82E-01 | 4.25E-02 |

The computational complexity analyses for the optimization of FPA based ANC is also evaluated based on mean execution time and STD, and results are provided in

Index | Sinusoidal | Random | Complex random | |||
---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | |

Case 1 | 10.699 | 0.004 | 14.275 | 0.691 | 45.523 | 3.773 |

Case 2 | 13.638 | 0.445 | 21.570 | 0.392 | 58.928 | 0.711 |

Case 3 | 13.131 | 0.447 | 18.866 | 0.377 | 59.225 | 0.785 |

The computational complexity of FPA based ANC is examined with counterpart optimization solvers. The computational complexity on mean execution time index of BSA and BSA-SQP results for sinusoidal noise signal are lie around

Comparative studies of FPA results for ANC systems are made with reported studies based on adaptive genetic algorithm AGA [

A novel design of nature-inspired heuristic of FPA is presented for the identification problem in nonlinear ANC with interferences. Different ANC scenarios by considering linear/nonlinear and primary/secondary paths are evaluated by determining coefficients of three different Volterra filters, i.e., VF-T1, VF-T2 and VF-T3. The performance of the FPA based ANC is verified through consistently achieving reasonable gauges of statistical operators in terms of accuracy, convergence and complexity measures. The performance is further validated via histogram analysis to prove that the FPA based ANC systems are reliable, accurate, stable and robust but the performance of the VF-T3 is comparatively better. The accuracy of FPA based ANC is in good agreement with state of the art counterpart solvers based on GA, PSO, BSA and FWA along with their hybrid with local search. In the future, one may explore to enhance the performance of ANC system by implementation of recently introduced fractional derivative definition [