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November  2022, 5(4): 295-314. doi: 10.3934/mfc.2022006

## Fuzzy-enhanced robust fault-tolerant control of IFOC motor with matched and mismatched disturbances

 1 Laboratory of Condensed Matter, Electronics and Signal Processing (LAMACETS), Department of Physic, Faculty of Sciences, University of Dschang, P. O. Box 67, Dschang, Cameroon 2 Laboratory of Mechanics, Materials and Structures, Faculty of Science, Department of Physics, University of Yaounde 1, P. O. Box 812 Yaounde, Cameroon 3 Laboratoire d'Automatique et Informatique Apliqueé (LAIA), IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

* Corresponding author: Alain Soup Tewa Kammogne

Received  November 2021 Revised  January 2022 Published  November 2022 Early access  February 2022

This paper focuses on the dynamical analysis of the permanent magnet asynchronous motor with the aim of subsequently designing effective robust control laws for the indirect field-oriented control (IFOC) devices. We first perform some tasks which demonstrate the existence of chaos phenomenon in the IFOC using relevant indicators such as phase portraits, bifurcations diagrams and Lyapunov exponents. Chaotic signature and some striking transitions are revealed such as period-doubling, torus, period-adding and chaos when an accessible parameter of the IFOC motor is changed. More interestingly, a certain range of the parameter space corresponds to the transient chaos. This behavior was not reported previously and can be considered as an enriching contribution. Secondly, due to the great interest to reduce the upper bound of uncertainties and interference, conventional sliding mode control (SMC) has been abundantly investigated for fault-tolerant control (FTC) systems. However, this approach presents several drawbacks in terms of overshoot, less robustness, transient state error, large chattering and speed of convergence that limit its use for industrial applications. For these reasons, the integral sliding mode control (ISMC) and the fuzzy sliding mode control (FISMC) are proposed to keep the IFOC motor in the regular operation zone. The optimal feedback gains and a sufficient condition are proposed for the stability of the overall IFOC system is drawn based on the linear quadratic regulator (LQR) method. To highlight the effectiveness and applicability of the proposed control scheme, numerical simulation results are presented. This analysis allows us a great knowledge of engineers for interpreting the operation of the IFOC motor. To highlight the effectiveness and the applicability of the proposed control scheme, numerical simulations results are presented and clearly demonstrated the feasibility of these techniques.

Citation: Jean Blaise Teguia, Alain Soup Tewa Kammogne, Stella Germaine Tsakoue Ganmene, Martin Siewe Siewe, Godpromesse Kenne. Fuzzy-enhanced robust fault-tolerant control of IFOC motor with matched and mismatched disturbances. Mathematical Foundations of Computing, 2022, 5 (4) : 295-314. doi: 10.3934/mfc.2022006
##### References:
 [1] A. T. Azar and Q. Zhu, Advances and Applications in Sliding Mode Control Systems, Springer, 2015. [2] M. Blanke, M. Kinnaert, J. Lunze, M. Staroswiecki and J. Schröder, Diagnosis and Fault-Tolerant Control, vol. 2, Springer, 2006. [3] K. Chau, J. Chen, C. Chan, J. K. Pong and D. Chan, Chaotic behavior in a simple DC drive, in Proceedings of Second International Conference on Power Electronics and Drive Systems, vol. 1, IEEE, 1997, 473–479. doi: 10.1109/PEDS. 1997.618750. [4] Q. Chen, Y.-R. Nan, H.-H. Zheng and X.-M. Ren, Full-order sliding mode control of uncertain chaos in a permanent magnet synchronous motor based on a fuzzy extended state observer, Chinese Physics B, 24 (2015), 110504.  doi: 10.1088/1674-1056/24/11/110504. [5] X. Fan, Z. Wang and Z. Shi, Event-triggered integral sliding mode control for uncertain fuzzy systems, Fuzzy Sets and Systems, 416 (2021), 47-63.  doi: 10.1016/j.fss.2020.09.002. [6] Y. Gao and K. Chau, Hopf bifurcation and chaos in synchronous reluctance motor drives, IEEE Transactions on Energy Conversion, 19 (2004), 296-302.  doi: 10.1109/TEC.2004.827012. [7] Z. Guo, J. Zhang and Q. Zhang, Research on efficiency optimization of current-fed asynchronous motor drive based on hybrid search method, Systems Science & Control Engineering, 7 (2019), 89-96.  doi: 10.1080/21642583.2019.1573440. [8] J. Hagel and C. Lhotka, A high order perturbation analysis of the Sitnikov problem, Celestial Mechanics and Dynamical Astronomy, 93 (2005), 201-228.  doi: 10.1007/s10569-005-0521-1. [9] N. Hemati, Strange attractors in brushless DC motors, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 41 (1994), 40-45.  doi: 10.1109/81.260218. [10] N. Jabli, H. Khammari, M. Mimouni and R. Dhifaoui, Bifurcation and chaos phenomena appearing in induction motor under variation of pi controller parameters, WSEAS Transactions on Systems, 9 (2010), 784-793. [11] J. K. Jain, S. Ghosh and S. Maity, Concurrent pi controller design for indirect vector controlled induction motor, Asian Journal of Control, 22 (2020), 130-142.  doi: 10.1002/asjc.1911. [12] C. N. Jones and J. Maciejowski, Reconfigurable flight control first year report, Department of Engineering, University of Cambridge. [13] A. S. T. Kammogne, M. N. Kountchou, R. Kengne, A. T. Azar, H. B. Fotsin and S. T. M. Ouagni, Polynomial robust observer implementation based passive synchronization of nonlinear fractional-order systems with structural disturbances, Frontiers of Information Technology & Electronic Engineering, 21 (2020), 1369-1386.  doi: 10.1631/FITEE.1900430. [14] A. S. T. Kammogne, V. F. Mawamba and J. Kengne, Robust prescribed-time stabilization for fuzzy sliding mode synchronization for uncertain chaotic systems, European Journal of Control, 59 (2021), 29-37.  doi: 10.1016/j.ejcon.2021.01.007. [15] C. Kral, H. Kapeller and F. Pirker, A stator and rotor fault detection technique for induction machines in traction applications for electric or hybrid electric vehicles, World Electric Vehicle Journal, 1 (2007), 184-189.  doi: 10.3390/wevj1010184. [16] Y. Kuroe and S. Hayashi, Analysis of bifurcation in power electronic induction motor drive systems, in 20th Annual IEEE Power Electronics Specialists Conference, IEEE, 1989, 923–930. doi: 10.1109/PESC. 1989.48578. [17] P. Mani, R. Rajan and Y. H. Joo, Integral sliding mode control for T–S fuzzy descriptor systems, Nonlinear Analysis: Hybrid Systems, 39 (2021), 100953, 14 pp. doi: 10.1016/j. nahs. 2020.100953. [18] A. P. Mark, G. C. R. Irudayaraj, R. Vairamani and K. Mylsamy, Dynamic performance analysis for different vector-controlled CSI-fed induction motor drives, Journal of Power Electronics, 14 (2014), 989-999.  doi: 10.6113/JPE.2014.14.5.989. [19] M. Mera, I. Salgado and I. Chairez, Robust observer-based controller design for state constrained uncertain systems: Attractive ellipsoid method, International Journal of Control, 93 (2020), 1397-1407.  doi: 10.1080/00207179.2018.1508853. [20] M. Messadi, A. Mellit, K. Kemih and M. Ghanes, CGPC control of chaos in a permanent magnet synchronous motor using the gradient conjugate and the genetic algorithm, Nonlinear Phenomena in Complex Systems, 17 (2014), 183-187. [21] M. Messadi, A. Mellit, K. Kemih and M. Ghanes, Predictive control of a chaotic permanent magnet synchronous generator in a wind turbine system, Chinese Physics B, 24 (2015), 010502.  doi: 10.1088/1674-1056/24/1/010502. [22] A. Mukherjee, R. Karmakar and A. K. Samantaray, Modelling of basic induction motors and source loading in rotor–motor systems with regenerative force field, Simulation Practice and Theory, 7 (1999), 563-576.  doi: 10.1016/S0928-4869(99)00019-1. [23] T. -B. -T. Nguyen, T. -L. Liao and J. -J. Yan, Adaptive sliding mode control of chaos in permanent magnet synchronous motor via fuzzy neural networks, Math. Probl. Eng., 2014, Art. ID 868415, 11 pp. doi: 10.1155/2014/868415. [24] T.-B.-T. Nguyen, T.-L. Liao and J.-J. Yan, Adaptive tracking control for an uncertain chaotic permanent magnet synchronous motor based on fuzzy neural networks, Journal of Vibration and Control, 21 (2015), 580-590.  doi: 10.1177/1077546313487761. [25] R. Puche-Panadero, J. Martinez-Roman, A. Sapena-Bano and J. Burriel-Valencia, Diagnosis of rotor asymmetries faults in induction machines using the rectified stator current, IEEE Transactions on Energy Conversion, 35 (2020), 213-221.  doi: 10.1109/TEC.2019.2951008. [26] A. S. K. Tsafack, R. Kengne, A. Cheukem, J. R. M. Pone and G. Kenne, Chaos control using self-feedback delay controller and electronic implementation in ifoc of 3-phase induction motor, Chaos Theory and Applications, 2 (2020), 40-48. [27] L. Wang, J. Fan, Z. Wang, B. Zhan and J. Li, Dynamic analysis and control of a permanent magnet synchronous motor with external perturbation, Journal of Dynamic Systems, Measurement, and Control, 138 (2016), 011003.  doi: 10.1115/1.4031726. [28] C. -l. Xia, Permanent Magnet Brushless DC Motor Drives and Controls, John Wiley & Sons, 2012. [29] Y. Zhang and J. Jiang, Bibliographical review on reconfigurable fault-tolerant control systems, Annual Reviews in Control, 32 (2008), 229-252.

show all references

##### References:
 [1] A. T. Azar and Q. Zhu, Advances and Applications in Sliding Mode Control Systems, Springer, 2015. [2] M. Blanke, M. Kinnaert, J. Lunze, M. Staroswiecki and J. Schröder, Diagnosis and Fault-Tolerant Control, vol. 2, Springer, 2006. [3] K. Chau, J. Chen, C. Chan, J. K. Pong and D. Chan, Chaotic behavior in a simple DC drive, in Proceedings of Second International Conference on Power Electronics and Drive Systems, vol. 1, IEEE, 1997, 473–479. doi: 10.1109/PEDS. 1997.618750. [4] Q. Chen, Y.-R. Nan, H.-H. Zheng and X.-M. Ren, Full-order sliding mode control of uncertain chaos in a permanent magnet synchronous motor based on a fuzzy extended state observer, Chinese Physics B, 24 (2015), 110504.  doi: 10.1088/1674-1056/24/11/110504. [5] X. Fan, Z. Wang and Z. Shi, Event-triggered integral sliding mode control for uncertain fuzzy systems, Fuzzy Sets and Systems, 416 (2021), 47-63.  doi: 10.1016/j.fss.2020.09.002. [6] Y. Gao and K. Chau, Hopf bifurcation and chaos in synchronous reluctance motor drives, IEEE Transactions on Energy Conversion, 19 (2004), 296-302.  doi: 10.1109/TEC.2004.827012. [7] Z. Guo, J. Zhang and Q. Zhang, Research on efficiency optimization of current-fed asynchronous motor drive based on hybrid search method, Systems Science & Control Engineering, 7 (2019), 89-96.  doi: 10.1080/21642583.2019.1573440. [8] J. Hagel and C. Lhotka, A high order perturbation analysis of the Sitnikov problem, Celestial Mechanics and Dynamical Astronomy, 93 (2005), 201-228.  doi: 10.1007/s10569-005-0521-1. [9] N. Hemati, Strange attractors in brushless DC motors, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 41 (1994), 40-45.  doi: 10.1109/81.260218. [10] N. Jabli, H. Khammari, M. Mimouni and R. Dhifaoui, Bifurcation and chaos phenomena appearing in induction motor under variation of pi controller parameters, WSEAS Transactions on Systems, 9 (2010), 784-793. [11] J. K. Jain, S. Ghosh and S. Maity, Concurrent pi controller design for indirect vector controlled induction motor, Asian Journal of Control, 22 (2020), 130-142.  doi: 10.1002/asjc.1911. [12] C. N. Jones and J. Maciejowski, Reconfigurable flight control first year report, Department of Engineering, University of Cambridge. [13] A. S. T. Kammogne, M. N. Kountchou, R. Kengne, A. T. Azar, H. B. Fotsin and S. T. M. Ouagni, Polynomial robust observer implementation based passive synchronization of nonlinear fractional-order systems with structural disturbances, Frontiers of Information Technology & Electronic Engineering, 21 (2020), 1369-1386.  doi: 10.1631/FITEE.1900430. [14] A. S. T. Kammogne, V. F. Mawamba and J. Kengne, Robust prescribed-time stabilization for fuzzy sliding mode synchronization for uncertain chaotic systems, European Journal of Control, 59 (2021), 29-37.  doi: 10.1016/j.ejcon.2021.01.007. [15] C. Kral, H. Kapeller and F. Pirker, A stator and rotor fault detection technique for induction machines in traction applications for electric or hybrid electric vehicles, World Electric Vehicle Journal, 1 (2007), 184-189.  doi: 10.3390/wevj1010184. [16] Y. Kuroe and S. Hayashi, Analysis of bifurcation in power electronic induction motor drive systems, in 20th Annual IEEE Power Electronics Specialists Conference, IEEE, 1989, 923–930. doi: 10.1109/PESC. 1989.48578. [17] P. Mani, R. Rajan and Y. H. Joo, Integral sliding mode control for T–S fuzzy descriptor systems, Nonlinear Analysis: Hybrid Systems, 39 (2021), 100953, 14 pp. doi: 10.1016/j. nahs. 2020.100953. [18] A. P. Mark, G. C. R. Irudayaraj, R. Vairamani and K. Mylsamy, Dynamic performance analysis for different vector-controlled CSI-fed induction motor drives, Journal of Power Electronics, 14 (2014), 989-999.  doi: 10.6113/JPE.2014.14.5.989. [19] M. Mera, I. Salgado and I. Chairez, Robust observer-based controller design for state constrained uncertain systems: Attractive ellipsoid method, International Journal of Control, 93 (2020), 1397-1407.  doi: 10.1080/00207179.2018.1508853. [20] M. Messadi, A. Mellit, K. Kemih and M. Ghanes, CGPC control of chaos in a permanent magnet synchronous motor using the gradient conjugate and the genetic algorithm, Nonlinear Phenomena in Complex Systems, 17 (2014), 183-187. [21] M. Messadi, A. Mellit, K. Kemih and M. Ghanes, Predictive control of a chaotic permanent magnet synchronous generator in a wind turbine system, Chinese Physics B, 24 (2015), 010502.  doi: 10.1088/1674-1056/24/1/010502. [22] A. Mukherjee, R. Karmakar and A. K. Samantaray, Modelling of basic induction motors and source loading in rotor–motor systems with regenerative force field, Simulation Practice and Theory, 7 (1999), 563-576.  doi: 10.1016/S0928-4869(99)00019-1. [23] T. -B. -T. Nguyen, T. -L. Liao and J. -J. Yan, Adaptive sliding mode control of chaos in permanent magnet synchronous motor via fuzzy neural networks, Math. Probl. Eng., 2014, Art. ID 868415, 11 pp. doi: 10.1155/2014/868415. [24] T.-B.-T. Nguyen, T.-L. Liao and J.-J. Yan, Adaptive tracking control for an uncertain chaotic permanent magnet synchronous motor based on fuzzy neural networks, Journal of Vibration and Control, 21 (2015), 580-590.  doi: 10.1177/1077546313487761. [25] R. Puche-Panadero, J. Martinez-Roman, A. Sapena-Bano and J. Burriel-Valencia, Diagnosis of rotor asymmetries faults in induction machines using the rectified stator current, IEEE Transactions on Energy Conversion, 35 (2020), 213-221.  doi: 10.1109/TEC.2019.2951008. [26] A. S. K. Tsafack, R. Kengne, A. Cheukem, J. R. M. Pone and G. Kenne, Chaos control using self-feedback delay controller and electronic implementation in ifoc of 3-phase induction motor, Chaos Theory and Applications, 2 (2020), 40-48. [27] L. Wang, J. Fan, Z. Wang, B. Zhan and J. Li, Dynamic analysis and control of a permanent magnet synchronous motor with external perturbation, Journal of Dynamic Systems, Measurement, and Control, 138 (2016), 011003.  doi: 10.1115/1.4031726. [28] C. -l. Xia, Permanent Magnet Brushless DC Motor Drives and Controls, John Wiley & Sons, 2012. [29] Y. Zhang and J. Jiang, Bibliographical review on reconfigurable fault-tolerant control systems, Annual Reviews in Control, 32 (2008), 229-252.
Phase portraits of system (2) in the plane $(x_{2},x_{1})$ obtained for some value of $k$ : (a) $k=1.2$, (b) $k=1.5$, (c) $k=3.2$, (d) $k=3.56$, (e) $k=3.6$ and (f) $k=3.67$
Bifurcation diagram (a) and Lyapunov exponent (b)
Time evolution of the trajectory $x3(t)$ of the IFOC and the corresponding phase portrait in the plane $(x1, x3)$
Input membership functions of the fuzzy system
External disturbance
Histogram of external disturbance
State trajectories of the IFOC when the controller is deactivated; (a) the direct axis of the rotor flux, (b) quadrature axis component of the rotor flux, (c) rotor speed error, (d) quadratic axis stator current
Time evolution. (a) quadratic, (b) direct flux rotor, (c) speed of the rotor, (d) quadratic stator current under the ISMC
The evolution of the $u(t)$ under the ISMC
Time evolution. (a) quadratic, (b) direct flux rotor, (c) speed of the rotor, (d) quadratic stator current under the FISMC
Fuzzy integral sliding mode controller
Performance index of the ISMC and FISMC
IAE of ISMC and FISMC
Fuzzy rules extracted for the TS fuzzy logic system
 $\dot s$ s NL NM NS Z PS PM PL NL -1 -1 -1 -1 -0.66 -0.33 0 NM -1 -1 -1 -0.66 -0.33 0 0.33 NS -1 -1 -0.66 -0.33 0 0.33 0.66 Z -1 -0.66 -0.33 0 0.33 0.33 1 PS -0.66 -0.33 0 0.33 0.66 1 1 PM -0.33 0 0.33 0.66 1 1 1 PL 0 0.33 0.66 1 1 1 1
 $\dot s$ s NL NM NS Z PS PM PL NL -1 -1 -1 -1 -0.66 -0.33 0 NM -1 -1 -1 -0.66 -0.33 0 0.33 NS -1 -1 -0.66 -0.33 0 0.33 0.66 Z -1 -0.66 -0.33 0 0.33 0.33 1 PS -0.66 -0.33 0 0.33 0.66 1 1 PM -0.33 0 0.33 0.66 1 1 1 PL 0 0.33 0.66 1 1 1 1
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