doi: 10.3934/mfc.2022006
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, 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. ChenY.-R. NanH.-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. FanZ. 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. GuoJ. 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. JabliH. KhammariM. 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. JainS. 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. KammogneM. N. KountchouR. KengneA. T. AzarH. 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. KammogneV. 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. KralH. 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. MarkG. C. R. IrudayarajR. 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. MeraI. 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. MessadiA. MellitK. 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. MessadiA. MellitK. 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. MukherjeeR. 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. NguyenT.-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-PanaderoJ. Martinez-RomanA. 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. TsafackR. KengneA. CheukemJ. 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. WangJ. FanZ. WangB. 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. ChenY.-R. NanH.-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. FanZ. 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. GuoJ. 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. JabliH. KhammariM. 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. JainS. 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. KammogneM. N. KountchouR. KengneA. T. AzarH. 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. KammogneV. 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. KralH. 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. MarkG. C. R. IrudayarajR. 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. MeraI. 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. MessadiA. MellitK. 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. MessadiA. MellitK. 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. MukherjeeR. 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. NguyenT.-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-PanaderoJ. Martinez-RomanA. 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. TsafackR. KengneA. CheukemJ. 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. WangJ. FanZ. WangB. 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. 

Figure 1.  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 $
Figure 2.  Bifurcation diagram (a) and Lyapunov exponent (b)
Figure 3.  Time evolution of the trajectory $ x3(t) $ of the IFOC and the corresponding phase portrait in the plane $ (x1, x3) $
Figure 4.  Input membership functions of the fuzzy system
Figure 5.  External disturbance
Figure 6.  Histogram of external disturbance
Figure 7.  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
Figure 8.  Time evolution. (a) quadratic, (b) direct flux rotor, (c) speed of the rotor, (d) quadratic stator current under the ISMC
Figure 9.  The evolution of the $ u(t) $ under the ISMC
Figure 10.  Time evolution. (a) quadratic, (b) direct flux rotor, (c) speed of the rotor, (d) quadratic stator current under the FISMC
Figure 11.  Fuzzy integral sliding mode controller
Figure 12.  Performance index of the ISMC and FISMC
Figure 13.  IAE of ISMC and FISMC
Table 1.  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
[1]

Ramasamy Kavikumar, Boomipalagan Kaviarasan, Yong-Gwon Lee, Oh-Min Kwon, Rathinasamy Sakthivel, Seong-Gon Choi. Robust dynamic sliding mode control design for interval type-2 fuzzy systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1839-1858. doi: 10.3934/dcdss.2022014

[2]

Cecilia Cavaterra, Denis Enăchescu, Gabriela Marinoschi. Sliding mode control of the Hodgkin–Huxley mathematical model. Evolution Equations and Control Theory, 2019, 8 (4) : 883-902. doi: 10.3934/eect.2019043

[3]

Yuan Li, Ruxia Zhang, Yi Zhang, Bo Yang. Sliding mode control for uncertain T-S fuzzy systems with input and state delays. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 345-354. doi: 10.3934/naco.2020006

[4]

Dongyun Wang. Sliding mode observer based control for T-S fuzzy descriptor systems. Mathematical Foundations of Computing, 2022, 5 (1) : 17-32. doi: 10.3934/mfc.2021017

[5]

Xiang Dong, Chengcheng Ren, Shuping He, Long Cheng, Shuo Wang. Finite-time sliding mode control for UVMS via T-S fuzzy approach. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1699-1712. doi: 10.3934/dcdss.2021167

[6]

Yaobang Ye, Zongyu Zuo, Michael Basin. Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3). Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1823-1837. doi: 10.3934/dcdss.2022010

[7]

Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1447-1464. doi: 10.3934/dcdss.2020375

[8]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control and Related Fields, 2021, 11 (4) : 905-934. doi: 10.3934/mcrf.2020051

[9]

Li Chen, Yongyan Sun, Xiaowei Shao, Junli Chen, Dexin Zhang. Prescribed-time time-varying sliding mode based integrated translation and rotation control for spacecraft formation flying. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022131

[10]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183

[11]

Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275

[12]

Tiantian Mu, Jun-E Feng, Biao Wang. Pinning detectability of Boolean control networks with injection mode. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022089

[13]

Nasim Ullah, Ahmad Aziz Al-Ahmadi. A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty. Mathematical Foundations of Computing, 2020, 3 (2) : 81-99. doi: 10.3934/mfc.2020007

[14]

Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. Impulsive control of a symmetric ball rolling without sliding or spinning. Journal of Geometric Mechanics, 2010, 2 (4) : 321-342. doi: 10.3934/jgm.2010.2.321

[15]

Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058

[16]

Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261

[17]

Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889

[18]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[19]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control and Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026

[20]

Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3

[Back to Top]