# Mixed Modified Recurring Rogers-Szego Polynomials Neural Network Control with Mended Grey Wolf Optimization Applied in SIM Expelling System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. CVT Organized System and SIM Models

#### 2.2. SIM Expelling CVT Organized System

#### 2.3. MMRRSPNN Control with MGWO by Using Two Adjusted Factors

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Heydar, M.; Milad, S.; Mohammad, H.A.; Ravinder, K.; Shahaboddin, S. Modeling and efficiency optimization of steam boilers by employing neural networks and response-surface method (RSM). Mathematics
**2019**, 7, 629. [Google Scholar] - Shao, Y.E.; Lin, S.C. Using a time delay neural network approach to diagnose the out-of-control signals for a multivariate normal process with variance shifts. Mathematics
**2019**, 7, 959. [Google Scholar] [CrossRef] [Green Version] - Behzad, M.; Mahyar, G.; Mohammad, H.A.; Heydar, M.; Shahabddin, S. Moisture estimation in cabinet dryers with thin-layer relationships using a genetic algorithm and neural network. Mathematics
**2019**, 7, 1042. [Google Scholar] - Shih, P.C.; Chiu, C.Y.; Chou, C.H. Using dynamic adjusting NGHS-ANN for predicting the recidivism rate of commuted prisoners. Mathematics
**2019**, 7, 1187. [Google Scholar] [CrossRef] [Green Version] - Nagamani, G.; Ramasamy, S. Stochastic dissipativity and passivity analysis for discrete-time neural networks with probabilistic time-varying delays in the leakage term. Appl. Math. Comput.
**2016**, 289, 237–257. [Google Scholar] [CrossRef] - Nagamani, G.; Ramasamy, S. Dissipativity and passivity analysis for discrete-time T–S fuzzy stochastic neural networks with leakage time-varying delays based on Abel lemma approach. J. Frankl. Inst.
**2016**, 353, 3313–3342. [Google Scholar] [CrossRef] - Nagamani, G.; Ramasamy, S.; Mey-Base, A. Robust dissipativity and passivity based state estimation for discrete-time stochastic Markov jump neural networks with discrete and distributed time-varying delays. Neural Comput. Appl.
**2017**, 28, 717–735. [Google Scholar] [CrossRef] - Ramasamy, S.; Nagamani, G.; Radhika, T. Further results on dissipativity criterion for markovian jump discrete-time neural networks with two delay components via discrete wirtinger inequality approach. Neural Process. Lett.
**2017**, 45, 939–965. [Google Scholar] [CrossRef] - Lee, T.T.; Jeng, J.T. The Chebyshev polynomial-based unified model neural networks for functional approximation. IEEE Trans. Syst. Man Cybern. Part B
**1998**, 28, 925–935. [Google Scholar] - Lin, C.H. Composite Recurring Laguerre orthogonal polynomials neural network dynamic control for continuously variable transmission system using altered particle swarm optimization. Nonlinear Dyn.
**2015**, 81, 1219–1245. [Google Scholar] [CrossRef] - Lin, C.H. Comparative dynamic control for continuously variable transmission with nonlinear uncertainty using blend amend recurring Gegenbauer-functional-expansions neural network. Nonlinear Dyn.
**2017**, 87, 1467–1493. [Google Scholar] [CrossRef] - Ting, J.C.; Chen, D.F. Nonlinear backstepping control of SynRM drive systems using reformed recurring Hermite polynomial neural networks with adaptive law and error estimated law. J. Power Electron.
**2018**, 8, 1380–1397. [Google Scholar] - Szego, G. Beitrag zur theorie der thetafunktionen. Sitz Preuss. Akad. Wiss. Phys. Math. Ki.
**1926**, 19, 242–252. [Google Scholar] - Cirovic, V.; Aleksendric, D.; Mladenovic, D. Braking torque control using recurring neural networks. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2012**, 226, 754–766. [Google Scholar] [CrossRef] - Wong, W.C.; Chee, E.; Li, J.; Wang, X. Recurring neural network-based model predictive control for continuous pharmaceutical manufacturing. Mathematics
**2018**, 6, 242. [Google Scholar] [CrossRef] [Green Version] - Ting, J.C.; Chen, D.F. Novel mingled reformed recurring hermite polynomial neural network control system applied in continuously variable transmission system. J. Mech. Sci. Technol.
**2018**, 32, 4399–4412. [Google Scholar] [CrossRef] - Lin, C.H.; Chang, K.T. Admixed recurring Gegenbauer polynomials neural network with mended particle swarm optimization control system for synchronous reluctance motor driving continuously variable transmission system. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2020**, 234, 183–198. [Google Scholar] - Liu, C.; Huang, B.; Wang, Q. Control performance assessment subject to multi-objective user- specified performance characteristics. IEEE Trans. Control Syst. Technol.
**2011**, 19, 682–691. [Google Scholar] [CrossRef] - Cano-Izquierdo, J.M.; Ibarrola, J.; Kroeger, M. Control loop performance assessment with a dynamic neuro-fuzzy model (dFasArt). IEEE Trans. Autom. Sci. Eng.
**2012**, 9, 377–389. [Google Scholar] [CrossRef] - Kordestani, M.; Safavi, A.A.; Sharafi, N.; Saif, M. Novel multiagent model-predictive control performance indices for monitoring of a large-scale distributed water system. IEEE Syst. J.
**2018**, 12, 1286–1294. [Google Scholar] [CrossRef] - Kordestani, M.; Safavi, A.A.; Sharafi, N.; Saif, M. Fault tolerant control of Rhine-Meuse delta water system: A performance assessment based approach. In Proceedings of the International Conference on Power Generation Systems and Renewable Energy Technologies (PGSRET), Istanbul, Turkey, 26–27 August 2019; pp. 31–38. [Google Scholar]
- Emary, E.; Yamany, W.; Hassanien, A.E. Multi-objective gray-wolf optimization for attribute reduction. Procedia Comput. Sci.
**2015**, 1, 623–632. [Google Scholar] [CrossRef] [Green Version] - Mosavi, M.; Khishe, M.; Ghamgosar, A. Classification of sonar data set using neural network trained by gray wolf optimization. Neural Netw. World
**2016**, 26, 393–415. [Google Scholar] [CrossRef] [Green Version] - Khandelwal, A.; Bhargava, A.; Sharma, A.; Sharma, H. Modified grey wolf optimization algorithm for transmission network expansion planning problem. Arabian J. Sci. Eng.
**2018**, 43, 2899–2908. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Sultana, U.; Khairuddin, A.B.; Mokhtar, A.S. Grey wolf optimizer based placement and sizing of multiple distributed generation in the distribution system. Energy
**2016**, 111, 525–536. [Google Scholar] [CrossRef] - Parsian, A.; Ramezani, M.; Ghadimi, N. A hybrid neural network-gray wolf optimization algorithm for melanoma detection. Biomed. Res.
**2017**, 28, 3408–3411. [Google Scholar] - Duangjai, J.; Pongsak, P. Grey wolf algorithm with borda count for feature selection in classification. In Proceedings of the 3rd International Conference on Control and Robotics Engineering (ICCRE), Nagoya, Japan, 20–23 April 2018; pp. 238–242. [Google Scholar]
- Munoz, A.R.; Lipo, T.A. Dual stator winding induction machine drive. IEEE Trans. Ind. Appl.
**2000**, 36, 1369–1379. [Google Scholar] [CrossRef] - Ojo, O.; Davidson, I.E. PWM-VSI inverter assisted stand-alone dual stator winding induction generator. IEEE Trans. Ind. Appl.
**2000**, 36, 1604–1611. [Google Scholar] - Singh, G.K.; Nam, K.; Lim, S.K. A simple indirect field-oriented control scheme for multiphase induction machine. IEEE Trans. Ind. Electron.
**2005**, 52, 1177–1184. [Google Scholar] [CrossRef] - Lin, C.H.; Hwang, C.C. Multiobjective optimization design for a six-phase copper rotor induction motor mounted with a scroll compressor. IEEE Trans. Magn.
**2016**, 52, 9401604. [Google Scholar] [CrossRef] - Lin, C.H.; Hwang, C.C. Multi-objective optimization design using amended particle swarm optimization and Taguchi method for a six-phase copper rotor induction motor. Eng. Optim.
**2017**, 49, 693–708. [Google Scholar] [CrossRef] - Lin, C.H. Modelling and control of six-phase induction motor servo-driven continuously variable transmission system using blend modified recurring Gegenbauer orthogonal polynomial neural network control system and amended artificial bee colony optimization. Int. J. Numer. Model. Electron. Netw. Devices Fields
**2016**, 29, 915–942. [Google Scholar] [CrossRef] - Lin, C.H. A six-phase CRIM driving CVT using blend modified recurring Gegenbauer OPNN Control. J. Power Electron.
**2016**, 16, 1438–1454. [Google Scholar] [CrossRef] - Lin, C.H. Blend modified recurring Gegenbauer orthogonal polynomial neural network control for six-phase copper rotor induction motor servo-driven continuously variable transmission system using amended artificial bee colony optimization. Trans. Inst. Meas. Control
**2017**, 39, 921–950. [Google Scholar] [CrossRef] - Ting, J.C.; Chen, D.F. SynRM servo-drive CVT systems using MRRHPNN control with mend ACO. J. Power Electron.
**2018**, 18, 1409–1423. [Google Scholar] - Hong, C.W.; Chen, C.C. Dynamic performance simulation of a continuously variable transmission motorcycle for fuzzy autopilot design. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**1997**, 211, 477–490. [Google Scholar] [CrossRef] - Srivastava, N.; Haque, I. Transient dynamics of metal V-belt CVT: Effects of bandpack slip and friction characteristic. Mech. Mach. Theory
**2008**, 43, 457–479. [Google Scholar] [CrossRef] - Srivastava, N.; Haque, I. A review on belt and chain continuously variable transmissions (CVT): Dynamics and control. Mech. Mach. Theory
**2009**, 44, 19–41. [Google Scholar] [CrossRef] - Hu, J.; Guo, Z.; Peng, H. Research on regenerative braking control strategy of plug-in hybrid electric vehicle considering CVT ratio rate of change. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2017**, 232, 1931–1943. [Google Scholar] [CrossRef] - Xie, D.; Zhang, H.; Dong, C.; Liu, Z.; Yang, Z. Challenges of Power Engineering and Environment—A Theoretical Investigation on Experimental Model of Torsional Vibration for Turboset Shafting; Springer: Berlin/Heidelberg, Germany, 2007; pp. 568–573. [Google Scholar]
- Matyja, T.; Lazarz, B. Modeling the coupled flexural and torsional vibrations in rotating machines in transient states. J. Vibroeng.
**2015**, 16, 1911–1924. [Google Scholar] - Astrom, K.J.; Hagglund, T. PID Controller: Theory, Design, and Tuning; Instrument Society of America: Research Triangle Park, NC, USA, 1995. [Google Scholar]
- Hagglund, T.; Astrom, K.J. Revisiting the Ziegler-Nichols tuning rules for PI control. Asian J. Control
**2002**, 4, 364–380. [Google Scholar] [CrossRef] - Hagglund, T.; Astrom, K.J. Revisiting the Ziegler-Nichols tuning rules for PI control-part II: The frequency response method. Asian J. Control
**2004**, 6, 469–482. [Google Scholar] [CrossRef] - Gasper, G.; Rahman, M. Encyclopedia of Mathematics and Its Applications, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Astrom, K.J.; Wittenmark, B. Adaptive Control; Addison-Wesley: New York, NY, USA, 1995. [Google Scholar]
- Slotine, J.J.E.; Li, W. Applied Nonlinear Control; Prentice-Hall: Englewood Cliffs, NJ, USA, 1991. [Google Scholar]
- Lewis, F.L.; Campos, J.; Selmic, R. Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities; SIAM Frontiers in Applied Mathematics: Philadelphia, PA, USA, 2012. [Google Scholar]

**Figure 1.**Conformation of the six-phase induction motor (SIM) and continuously variable transmission (CVT) organized system: (

**a**) geometric components of the CVT system and (

**b**) geometric components of the CVT organized system.

**Figure 6.**Tested results for the SIM expelling CVT organized system at the tested Event E1 case by using the controller CT1: (

**a**) speed response, (

**b**) speed discrepancy response.

**Figure 7.**Tested results for the SIM expelling CVT organized system at the tested Event E2 case by using the controller CT1: (

**a**) speed response, (

**b**) speed discrepancy response.

**Figure 8.**Tested results for the SIM expelling CVT organized system by using the controller CT1: (

**a**) response of developed torque at the tested Event E1 case, (

**b**) response of developed torque at the tested Event E2 case.

**Figure 9.**Tested results for the SIM expelling CVT organized system at the tested Event E1 case by using the controller CT2: (

**a**) speed response, (

**b**) speed discrepancy response.

**Figure 10.**Tested results for the SIM expelling CVT organized system at the tested Event E2 case by using the controller CT2: (

**a**) speed response, and (

**b**) speed discrepancy response.

**Figure 11.**Tested results for the SIM expelling CVT organized system by using the controller CT2: (

**a**) response of developed torque ${\tau}_{a}$ at the tested Event E1 case, (

**b**) response of developed torque ${\tau}_{a}$ at the tested Event E2 case.

**Figure 12.**Tested results for the SIM expelling CVT organized system obtained at the tested Event E1 case by using the controller CT3: (

**a**) speed response, (

**b**) speed discrepancy response.

**Figure 13.**Tested results for the SIM expelling CVT organized system at the tested Event E2 case by using the controller CT3: (

**a**) speed response, (

**b**) speed discrepancy response.

**Figure 14.**Tested results for the SIM expelling CVT organized system by using the controller CT3: (

**a**) response of developed torque ${\tau}_{a}$ at the tested Event E1 case, (

**b**) response of developed torque ${\tau}_{a}$ at the tested Event E2 case.

**Figure 15.**Tested results at the tested Event E1 case by using the controller CT3: (

**a**) convergent response of learning rate ${\epsilon}_{1}$, (

**b**) convergent response of learning rate ${\epsilon}_{2}$.

**Figure 16.**Tested results at the tested Event E2 case by using the controller CT3: (

**a**) the convergent response of learning rate ${\epsilon}_{1}$, (

**b**) the convergent response of learning rate ${\epsilon}_{2}$.

**Figure 17.**Tested results at the tested Event E3 case by using the controller CT1: (

**a**) speed-adjusted response; (

**b**) current response in phase a.

**Figure 18.**Tested results at the tested Event E3 case by using the controller CT2: (

**a**) speed-adjusted response; (

**b**) current response in phase a.

**Figure 19.**Tested results at the tested Event E3 case by using the controller CT3: (

**a**) speed-adjusted response; (

**b**) current response in phase a.

Control System and Three Tested Cases | Controller CT1 | |||

Performance | Tested Event E1 Case | Tested Event E2 Case | Tested Event E3 Case | |

Maximum errors of ${e}_{a}$ | 88 r/min | 215 r/min | 398 r/min | |

Root mean square errors of ${e}_{a}$ | 45 r/min | 60 r/min | 51 r/min | |

Control System and Three Tested Cases | Controller CT2 | |||

Performance | Tested Event E1 Case | Tested Event E2 Case | Tested Event E3 Case | |

Maximum errors of ${e}_{a}$ | 69 r/min | 88 r/min | 198 r/min | |

Root mean square errors of ${e}_{a}$ | 30 r/min | 31 r/min | 27 r/min | |

Control System and Three Tested Cases | Controller CT3 | |||

Performance | Tested Event E1 Case | Tested Event E2 Case | Tested Event E3 Case | |

Maximum errors of ${e}_{a}$ | 40 r/min | 43 r/min | 69 r/min | |

Root mean square errors of ${e}_{a}$ | 20 r/min | 22 r/min | 17 r/min |

Control System | Control System CT1 | Control System CT2 | Control System CT3 | |
---|---|---|---|---|

Characteristic Performance | ||||

Vibration value in the control rule | Small (10% of nominal value at tested Event E2 case) | smaller (8% of nominal value at tested Event E2 case) | Smallest (6% of nominal value at tested Event E2 case) | |

Dynamic response | Slow (rising time as 2.0 sec at tested Event E2 case) | Fast (rising time as 1.8 sec at tested Event E2 case) | Faster (rising time 1.6 sec at tested Event E2 case) | |

Regulation capability for load torque disturbance | Poor (maximum error as 398 r/min at tested Event E3 case) | Good (maximum error as 198 r/min at tested Event E3 case) | Better (maximum error as 69 r/min at tested Event E3 case) | |

Convergent speed | Slow (settle time as 2.5 sec at tested Event E2 case) | Fast (settle time as 2.2 sec at tested Event E2 case) | Faster (settle time as 2.0 sec at tested Event E2 case) | |

Speed tracking error | Large (maximum error as 215 r/min at tested Event E2 case) | Middle (maximum error as 88 r/min at tested Event E2 case) | Small (maximum error as 43 r/min at tested Event E2 case) | |

Rejection ability for parameter disturbance | Poor (maximum error as 215 r/min at tested Event E2 case) | Good (maximum error as 88 r/min at tested Event E2 case) | Better (maximum error as 43 r/min at tested Event E2 case) | |

Two learning rates | None | Vary (two optimal learning rates) | Vary (two optimal learning rates) | |

Torque ripple (V-belt shaking action and torsional vibration variations) | Large (12% of nominal value at tested Event E2 case) | Smaller (10% of nominal value at tested Event E2 case) | Smallest (6% of nominal value at tested Event E2 case) |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, D.-F.; Shih, Y.-C.; Li, S.-C.; Chen, C.-T.; Ting, J.-C.
Mixed Modified Recurring Rogers-Szego Polynomials Neural Network Control with Mended Grey Wolf Optimization Applied in SIM Expelling System. *Mathematics* **2020**, *8*, 754.
https://doi.org/10.3390/math8050754

**AMA Style**

Chen D-F, Shih Y-C, Li S-C, Chen C-T, Ting J-C.
Mixed Modified Recurring Rogers-Szego Polynomials Neural Network Control with Mended Grey Wolf Optimization Applied in SIM Expelling System. *Mathematics*. 2020; 8(5):754.
https://doi.org/10.3390/math8050754

**Chicago/Turabian Style**

Chen, Der-Fa, Yi-Cheng Shih, Shih-Cheng Li, Chin-Tung Chen, and Jung-Chu Ting.
2020. "Mixed Modified Recurring Rogers-Szego Polynomials Neural Network Control with Mended Grey Wolf Optimization Applied in SIM Expelling System" *Mathematics* 8, no. 5: 754.
https://doi.org/10.3390/math8050754