Generic placeholder image

Recent Advances in Electrical & Electronic Engineering

Editor-in-Chief

ISSN (Print): 2352-0965
ISSN (Online): 2352-0973

Research Article

Adaptive Neural Dynamic Surface Control for the Chaotic PMSM System with External Disturbances and Constrained Output

Author(s): Zhang Junxing , Wang Shilong, Li Shaobo* and Zhou Peng

Volume 13, Issue 6, 2020

Page: [894 - 905] Pages: 12

DOI: 10.2174/2352096513666200108115327

Price: $65

Abstract

Background: This article studies the issue of adaptive neural dynamic surface control for the chaotic permanent magnet synchronous motor system with constrained output, external disturbances and parameter perturbations.

Methods: Firstly, a virtual controller and two practical controllers are created based on the backstepping framework. In the process of creating controllers, adaptive technique and radial basis function neural networks are used to handle unknown parameters and nonlinearities, respectively. The nonlinear damping items are applied to overcome external disturbances. The barrier Lyapunov function is used to prevent the violation of system output constraint. Meanwhile, the first-order filter to eliminate the “explosion of complexity” of traditional back stepping has been introduced. Then, it is proved that all the closed-loop signals are uniform ultimate asymptotic stability and the tracking error converges to a small set of origin.

Results: The effectiveness and robustness of the developed approach are illustrated by numerical simulations.

Conclusion: The raised control scheme is a useful tool for enhancing the performance of the chaotic PMSM system with external disturbances, constrained output and parameter perturbations.

Keywords: Chaos PMSM system, RBF neural network, dynamic surface control, nonlinear damping, constrained output, Sliding Mode Control (SMC).

Graphical Abstract

[1]
D.K. Hong, W. Hwang, J.Y. Lee, and B.C. Woo, "Design, Analysis, and Experimental Validation of a Permanent Magnet Synchronous Motor for Articulated Robot Applications", IEEE Trans. Magn., vol. 54, pp. 1-4, 2018.
[http://dx.doi.org/10.1109/TMAG.2017.2752080]]
[2]
J. Cuenot, S. Zaim, B. Nahid-Mobarakeh, S. Pierfederici, E. Monmasson, R. Meuret, and F. Meibody-Tabar, "Overall Size Optimization of a High-Speed Starter Using a Quasi-Z-Source Inverter", IEEE Transactions on Transportation Electrification, vol. 3, pp. 891-900, 2017.
[http://dx.doi.org/10.1109/TTE.2017.2738022]
[3]
X. Wu, R. Tian, and X. Rui, "Adaptive robust speed control for continuously variable transmission in wind turbine under grid faults", J. Renew. Sustain. Energy, vol. 10, 2018.063302, .
[http://dx.doi.org/10.1063/1.5046393]
[4]
S.K. Kommuri, M. Defoort, H.R. Karimi, and K.C. Veluvolu, "A Robust Observer-Based Sensor Fault-Tolerant Control for PMSM in Electric Vehicles", IEEE Trans. Ind. Electron., vol. 63, pp. 7671-7681, 2016.
[http://dx.doi.org/10.1109/TIE.2016.2590993]
[5]
W. Mao, and G. Liu, "Development of an Adaptive Fuzzy Sliding Mode Trajectory Control Strategy for Two-axis PMSM-Driven Stage Application", International Journal of Fuzzy Systems,. Jan 2019.16
[http://dx.doi.org/10.1007/s40815-018-0596-y]
[6]
X. Guo, S. Du, Z. Li, F. Chen, K. Chen, and R. Chen, "Analysis of Current Predictive Control Algorithm for Permanent Magnet Synchronous Motor Based on Three-Level Inverters", IEEE Access, vol. 7, pp. 87750-87759, 2019.
[http://dx.doi.org/10.1109/ACCESS.2019.2925686]
[7]
T. Deng, Z. Su, J. Li, P. Tang, X. Chen, and P. Liu, "Advanced Angle Field Weakening Control Strategy of Permanent Magnet Synchronous Motor", IEEE Trans. Vehicular Technol., vol. 68, pp. 3424-3435, 2019.
[http://dx.doi.org/10.1109/TVT.2019.2901275]
[8]
H. Qiu, B. Tang, R. Wang, G. Zhang, C. Yang, and G. Cui, "Optimization and Calculation of Equivalent Thermal Network Method in the Temperature Field Research of Permanent Magnet Servo Motor", Recent Adv. Electr. Electron. Eng., vol. 9, pp. 241-248, 2016.
[9]
G. Qi, and J. Hu, "Force Analysis and Energy Operation of Chaotic System of Permanent-Magnet Synchronous Motor", Int. J. Bifurcat. Chaos,, vol. 27, 2018.1750216, .
[http://dx.doi.org/10.1142/S0218127417502169]
[10]
Z. Sun, L. Si, Z. Shang, and J. Lei, "Finite-time synchronization of chaotic PMSM systems for secure communication and parameters identification", Optik (Stuttg.), vol. 157, pp. 43-55, 2018.
[http://dx.doi.org/10.1016/j.ijleo.2017.09.057]
[11]
X. Chen, J. Hu, Z. Peng, and C. Yuan, "Bifurcation and chaos analysis of torsional vibration in a PMSM-based driven system considering electromechanically coupled effect", Nonlinear Dyn., vol. 88, pp. 277-292, 2017.
[http://dx.doi.org/10.1007/s11071-017-3419-z]
[12]
S. Lu, and X. Wang, "Observer-Based Command Filtered Adaptive Neural Network Tracking Control for Fractional-Order Chaotic PMSM", IEEE Access, vol. 7, pp. 88777-88788, 2019.
[http://dx.doi.org/10.1109/ACCESS.2019.2926526]
[13]
Y. Wu, and G. Li, "Adaptive disturbance compensation finite control set optimal control for PMSM systems based on sliding mode extended state observer", Mech. Syst. Signal Process., vol. 98, pp. 402-414, 2018.
[http://dx.doi.org/10.1016/j.ymssp.2017.05.007]
[14]
A.S.D. Paula, and M.A. Savi, "A multiparameter chaos control method based on OGY approach", Chaos Solitons Fractals, vol. 40, pp. 1376-1390, 2009.
[http://dx.doi.org/10.1016/j.chaos.2007.09.056]
[15]
M.F. Danca, "Random parameter-switching synthesis of a class of hyperbolic attractors", Chaos, vol. 18, no. 3, . 2008.033111
[http://dx.doi.org/10.1063/1.2965524] [PMID: 19045449]
[16]
H. Jian, Q. Yang, and L. Hui, "Adaptive robust nonlinear feedback control of chaos in PMSM system with modeling uncertainty", Applied Mathematical Modelling, vol. 40, 2016. p. S0307904X16302207,.
[17]
J. Hu, L. Liu, D. Ma, and N. Ullah, "Adaptive nonlinear feedback control of chaos in permanent-magnet synchronous motor system with parametric uncertainty", Proc. Inst. Mech. Eng., C J. Mech. Eng. Sci., vol. 229, no. 12, pp. 2314-2323, 2015.
[http://dx.doi.org/10.1177/0954406214557344]
[18]
C.S. Tang, and Y.H. Dai, Finite-time stability control of permanent magnet synchronous motor chaotic system with parameters uncertain..
[19]
N. Vafamand, and S. Khorshidi, "Robust Polynomial Observer-Based Chaotic Synchronization for Non-ideal Channel Secure Communication: An SOS Approach", Iran. J. Sci. Technol. Trans. Electr. Eng., vol. 42, pp. 83-94, 2018.
[http://dx.doi.org/10.1007/s40998-018-0047-7]
[20]
M.M. Mardani, N. Vafamand, M. Shokrian Zeini, M. Shasadeghi, and A. Khayatian, "Sum-of-Squares-Based Finite-Time Adaptive Sliding Mode Control of Uncertain Polynomial Systems With Input Nonlinearities", Asian J. Control, vol. 20, pp. 1658-1662, 2018.
[http://dx.doi.org/10.1002/asjc.1625]
[21]
M.S. Sadeghi, N. Vafamand, and M.H. Khooban, "LMI-based Stability Analysis and Robust Controller Design for a Class of Nonlinear Chaotic Power Systems", J. Franklin Inst., vol. 353, no. 13, pp. 2835-2858, 2016.
[http://dx.doi.org/10.1016/j.jfranklin.2016.04.021]
[22]
S. Mobayen, "Finite-Time Stabilization of a Class of Chaotic Systems with Matched and Unmatched Uncertainties: An LMI Approach", Complexity, vol. 21, pp. 14-19, 2016.
[http://dx.doi.org/10.1002/cplx.21624]
[23]
Pai and Ming-Chang, "Chaotic sliding mode controllers for uncertain time-delay chaotic systems with input nonlinearity", Appl. Math. Comput., vol. 271, pp. 757-767, 2015.
[http://dx.doi.org/10.1016/j.amc.2015.09.058]
[24]
Q. Li, and S. Liu, "Adaptive Modified Function Projective Lag Synchronization of Memristor-Based Five-Order Chaotic Circuit Systems", Adv. Math. Phys., vol. 2017, pp. 1-8, 2017.
[http://dx.doi.org/10.1155/2017/1843179]
[25]
W. Liu, J. Lu, S. Xu, Y. Li, and Z. Zhang, "Sampled-data controller design and stability analysis for nonlinear systems with input saturation and disturbances", Appl. Math. Comput., vol. 360, pp. 14-27, 2019.
[http://dx.doi.org/10.1016/j.amc.2019.04.024]
[26]
R. Behinfaraz, S. Ghaemi, and S. Khanmohammadi, "Adaptive synchronization of new fractional-order chaotic systems with fractional adaption laws based on risk analysis", Math. Methods Appl. Sci., vol. 42, no. 6, pp. 1772-1785, 2019.
[http://dx.doi.org/10.1002/mma.5471]
[27]
Y. Wu, J. Braselton, Y. Jin, and A.E. Shahat, "Adaptive control of bi-directionally coupled Lorenz systems with uncertainties", J. Franklin Inst., vol. 356, no. 3, pp. 1287-1301, 2018.
[http://dx.doi.org/10.1016/j.jfranklin.2018.10.029]
[28]
H. Tirandaz, and A. Karamimollaee, Combination Synchronization of Multiple Chaotic Systems with Uncertain Parameters Using Adaptive Hybrid Modified Projective Control Method, vol. 49, p. 59, 2018.
[29]
J. Wang, X. Chen, and J. Fu, "Adaptive finite-time control of chaos in permanent magnet synchronous motor with uncertain parameters", Nonlinear Dyn., vol. 78, pp. 1321-1328, 2014.
[http://dx.doi.org/10.1007/s11071-014-1518-7]
[30]
D.A. Prousalis, C.K. Volos, I.N. Stouboulos, and I.M. Kyprianidis, "Hyperchaotic memristive system with hidden attractors and its adaptive control scheme", Nonlinear Dyn., vol. 90, pp. 1-14, 2017.
[http://dx.doi.org/10.1007/s11071-017-3758-9]
[31]
M.R. Mufti, H. Afzal, F. Ur-Rehman, W. Aslam, and M.I. Qureshi, "Transmission Projective Synchronization of Multiple Non-Identical Coupled Chaotic Systems Using Sliding Mode Control", IEEE Access, vol. 7, pp. 17847-17861, 2019.
[http://dx.doi.org/10.1109/ACCESS.2019.2895067]
[32]
J. Yang, A. Mu, and N. Li, "Dynamical Analysis and Stabilization of Wind Turbine Drivetrain via Adaptive Fixed-Time Terminal Sliding Mode Controller", Mathematical Problems in Engineering,, vol. 2019, . Jul 2019, pp. 1-14.22
[http://dx.doi.org/10.1155/2019/8982028]
[33]
G. Maeng, and H.C. Han, "Adaptive sliding mode control of a chaotic nonsmooth-air-gap permanent magnet synchronous motor with uncertainties", Nonlinear Dyn., vol. 74, pp. 571-580, 2013.
[http://dx.doi.org/10.1007/s11071-013-0989-2]
[34]
O. Mofid, S. Mobayen, and M.H. Khooban, "Sliding mode disturbance observer control based on adaptive synchronization in a class of fractional‐order chaotic systems", Int. J. Adapt. Control Signal Process., vol. 33, no. 3, pp. 462-474, 2019.
[35]
M.K. Shukla, and B.B. Sharma, "Control and Synchronization Of A Class Of Uncertain Fractional Order Chaotic Systems Via Adaptive Backstepping Control", Asian J. Control, vol. 20, pp. 707-720, 2018.
[http://dx.doi.org/10.1002/asjc.1593]
[36]
M.K. Shukla, and B.B. Sharma, "Backstepping based stabilization and synchronization of a class of fractional order chaotic systems", Chaos Solitons Fractals, vol. 102, pp. 274-284, 2017.
[http://dx.doi.org/10.1016/j.chaos.2017.05.015]
[37]
J. Yu, J. Lei, and L. Wang, "Backstepping synchronization of chaotic system based on equivalent transfer function method", Optik (Stuttg.), vol. 130, pp. 900-913, 2017.
[http://dx.doi.org/10.1016/j.ijleo.2016.11.007]
[38]
R. Luo, and Y. Zeng, "The control of chaotic systems with unknown parameters and external disturbance via backstepping-like scheme", Complexity, vol. 21, pp. 573-583, 2016.
[http://dx.doi.org/10.1002/cplx.21771]
[39]
K. Kemih, M. Halimi, M. Ghanes, H. Fanit, and H. Salit, "Control and synchronization of Chaotic Attitude Control of Satellite with Backstepping controller", Eur. Phys. J. Spec. Top., vol. 223, pp. 1579-1589, 2014.
[http://dx.doi.org/10.1140/epjst/e2014-02180-4]
[40]
J. Tu, H. He, and P. Xiong, "Adaptive backstepping synchronization between chaotic systems with unknown Lipschitz constant", Appl. Math. Comput., vol. 236, pp. 10-18, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.03.012]
[41]
Y. Li, "Finite time command filtered adaptive fault tolerant control for a class of uncertain nonlinear systems", Automatica, vol. 106, pp. 117-123, 2019.
[http://dx.doi.org/10.1016/j.automatica.2019.04.022]
[42]
F. Zouari, A. Ibeas, A. Boulkroune, J. Cao, and M.M. Arefi, "Neuro-adaptive tracking control of non-integer order systems with input nonlinearities and time-varying output constraints", Inf. Sci., vol. 485, pp. 170-199, 2019.
[http://dx.doi.org/10.1016/j.ins.2019.01.078]
[43]
S. HeD, "Lin and J. Wang, “Robust terminal angle constraint guidance law with autopilot lag for intercepting maneuvering targets", Nonlinear Dyn., vol. 81, no. 1-2, pp. 881-892, .
[http://dx.doi.org/10.1007/s11071-015-2037-x]
[44]
D. Zhang, and G. Duan, "Distributed fixed-time consensus tracking for high-order uncertain non-linear multi-agent systems with switching topologies", IET Control Theory Appl., vol. 13, no. 11, pp. 1761-1772, 2019.
[http://dx.doi.org/10.1049/iet-cta.2018.5892]
[45]
C. Guo, Y. Han, H. Yu, and J. Qin, "Spatial Path-Following Control of Underactuated AUV With Multiple Uncertainties and Input Saturation", IEEE Access, vol. 7, pp. 98014-98022, 2019.
[http://dx.doi.org/10.1109/ACCESS.2019.2928897]
[46]
Y. Liu, and H. Li, "Adaptive asymptotic tracking using barrier functions", Automatica, vol. 98, pp. 239-246, 2018.
[http://dx.doi.org/10.1016/j.automatica.2018.09.017]
[47]
Q. Guo, Y. Zhang, B.G. Celler, and S.W. Su, "State-Constrained Control of Single-Rod Electrohydraulic Actuator With Parametric Uncertainty and Load Disturbance", IEEE T. Contr. Syst. T., vol. 26, no. 6, pp. 2242-2249, 2018.
[http://dx.doi.org/10.1109/TCST.2017.2753167]
[48]
H. BenslimaneA. Boulkroune and H. Chekireb, "Iterative Learning Control for Strict-Feedback Nonlinear Systems with Both Structured and Unstructured Uncertainties", Arab. J. Sci. Eng., vol. 41, no. 9, pp. 3683-3694, 2016.
[http://dx.doi.org/10.1007/s13369-015-1901-9]
[49]
Y.YuC. Sun, and Z. Jiao, "Robust-decentralized tracking control for a class of uncertain MIMO nonlinear systems with time-varying delays", Int. J. Robust Nonlin., vol. 24, no. 18, pp. 3474-3490, 2014.
[http://dx.doi.org/10.1002/rnc.3068]
[50]
S. Gao, Y. Wang, H. Dong, B. Ning, and H. Wang, "Controlling uncertain Genesio–Tesi chaotic system using adaptive dynamic surface and nonlinear feedback", Chaos Solitons Fractals, vol. 105, pp. 180-188, 2017.
[http://dx.doi.org/10.1016/j.chaos.2017.10.030]
[51]
Z. Yang, and H. Sugiura, "Robust nonlinear control of a three-tank system using finite-time disturbance observers", Control Eng. Pract., vol. 84, pp. 63-71, 2019.
[http://dx.doi.org/10.1016/j.conengprac.2018.11.013]
[52]
H. Khebbache, S. Labiod, and M. Tadjine, "Adaptive sensor fault-tolerant control for a class of multi-input multi-output nonlinear systems: Adaptive first-order filter-based dynamic surface control approach", ISA Trans., vol. 80, pp. 89-98, 2018.
[http://dx.doi.org/10.1016/j.isatra.2018.07.037] [PMID: 30097181]
[53]
Z. Yang, "Robust control of nonlinear semi-strict feedback systems using finite-time disturbance observers", Int. J. Robust Nonlinear Control, vol. 27, no. 17, pp. 3582-3603, 2017.
[http://dx.doi.org/10.1002/rnc.3756]
[54]
G. FuL, Ou and W. Zhang, , "Robust adaptive tracking control of MIMO nonlinear systems in the presence of actuator hysteresis", Int. J. Syst. Sci., vol. 47, no. 10, pp. 2359-2369, 2016.
[http://dx.doi.org/10.1080/00207721.2014.994051]
[55]
M. Sellali, A. Betka, S. Drid, A. Djerdir, L. Allaoui, and M. Tiar, "Novel control implementation for electric vehicles based on fuzzy-back stepping approach", Energy, vol. 178, pp. 644-655, 2019.
[http://dx.doi.org/10.1016/j.energy.2019.04.146]
[56]
L. ShengG. Xiaojie and Z. Lanyong, "Robust Adaptive Backstepping Sliding Mode Control for Six-Phase Permanent Magnet Synchronous Motor Using Recurrent Wavelet Fuzzy Neural Network", IEEE Access, vol. 5, pp. 14502-14515, 2017.
[http://dx.doi.org/10.1109/ACCESS.2017.2721459]
[57]
W. Chang, and S. Tong, "Adaptive fuzzy tracking control design for permanent magnet synchronous motors with output constraint", Nonlinear Dyn., vol. 87, no. 1, pp. 291-302, 2017.
[http://dx.doi.org/10.1007/s11071-016-3043-3]
[58]
J. Yu, Y. Ma, H. Yu, and C. Lin, "Reduced-order observer-based adaptive fuzzy tracking control for chaotic permanent magnet synchronous motors", Neurocomputing, vol. 214, pp. 201-209, 2016.
[59]
S. LuX, Wang, and Y. Li, , "Adaptive neural network control for fractional-order PMSM with time delay based on command filtered backstepping", AIP Adv., vol. 9, no. 5, p. 55105, . 2019.
[60]
K.A. Abuhasel, F.F.M. El-Sousy, M.F. El-Naggar, and A. Abu-Siada, "Adaptive RCMAC Neural Network Dynamic Surface Control for Permanent-Magnet Synchronous Motors Driven Two-Axis X-Y Table", IEEE Access, vol. 7, pp. 38068-38084, 2019.
[http://dx.doi.org/10.1109/ACCESS.2019.2906113]
[61]
J. Yu, P. Shi, W. Dong, B. Chen, and C. Lin, "Neural network-based adaptive dynamic surface control for permanent magnet synchronous motors", IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 3, pp. 640-645, 2015.
[http://dx.doi.org/10.1109/TNNLS.2014.2316289] [PMID: 25720014]
[62]
J. Yu, H. Yu, B. Chen, J. Gao, and Y. Qin, "Direct adaptive neural control of chaos in the permanent magnet synchronous motor", Nonlinear Dyn., vol. 70, no. 3, pp. 1879-1887, 2012.
[http://dx.doi.org/10.1007/s11071-012-0580-2]
[63]
S. Luo, J. Wang, Z. Shi, and Q. Qiu, "Output Feedback Adaptive Dynamic Surface Control of Permanent Magnet Synchronous Motor with Uncertain Time Delays via RBFNN", Discrete Dyn. Nat. Soc., vol. 2014, pp. 1-12, 2014.
[http://dx.doi.org/10.1155/2014/315634]
[64]
N. Vafamand, M.H. Khooban, A. Khayatian, and F. Blabbjerg, "Design of robust double–fuzzy-summation nonparallel distributed compensation controller for chaotic power systems", J. Dyn. Syst. Meas. Control, vol. 140, no. 3, p. 03114, 2018.
[http://dx.doi.org/10.1115/1.4037527]
[65]
"[1] N. Vafamand, S. Khorshidi and A. Khayatian, “Secure communication for non-ideal channel via robust TS fuzzy observer-based hyperchaotic synchronization", Chaos Solitons Fractals, vol. 112, pp. 116-124, 2018.
[http://dx.doi.org/10.1016/j.chaos.2018.04.035]
[66]
S. Gao, H. Dong, B. Ning, T. Tang, and Y. Li, "Nonlinear mapping-based feedback technique of dynamic surface control for the chaotic PMSM using neural approximation and parameter identification", IET Control Theory Appl., vol. 12, pp. 819-827, 2018.
[http://dx.doi.org/10.1049/iet-cta.2017.0550]
[67]
S. Luo, "Nonlinear Dynamic Surface Control of Chaos in Permanent Magnet Synchronous Motor Based on the Minimum Weights of RBF Neural Network", Abstr. Appl. Anal., vol. 2014, pp. 1-9, 2014.
[http://dx.doi.org/10.1155/2014/609340]
[68]
S. Luo, and R. Gao, "Chaos control of the permanent magnet synchronous motor with time-varying delay by using adaptive sliding mode control based on DSC", J. Franklin Inst., vol. 355, no. 10, pp. 4147-4163, 2018.
[http://dx.doi.org/10.1016/j.jfranklin.2018.04.031]
[69]
Y. Liu, J. Yu, H. Yu, C. Lin, and L. Zhao, "Barrier Lyapunov Functions-Based Adaptive Neural Control for Permanent Magnet Synchronous Motors With Full-State Constraints", IEEE Access, vol. 5, pp. 10382-10389, 2017.
[http://dx.doi.org/10.1109/ACCESS.2017.2713419]
[70]
L. Zhong, B.P. Jin, Y.H. Joo, Z. Bo, and G. Chen, "Bifurcations and chaos in a permanent-magnet synchronous motor", Circuits & Systems I Fundamental Theory & Applications IEEE Transactions on,, vol. 49. 2002,, no. 3, pp. 383-387.
[http://dx.doi.org/10.1109/81.989176]
[71]
S. Luo, "Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain", Chaos, vol. 24, no. 3, . 2014.033135
[http://dx.doi.org/10.1063/1.4895810] [PMID: 25273215]
[72]
Y. Guo, and J. Liu, "Neural network based adaptive dynamic surface control for flight path angle",
[73]
S. Wang, H. Yu, X. Gao, and N. Wang, "Adaptive Barrier Control for Nonlinear Servomechanisms with Friction Compensation", Complexity, vol. 2018, pp. 1-10, 2018.
[http://dx.doi.org/10.1155/2018/8925838]
[74]
S. Luo, J. Wang, S. Wu, and K. Xiao, "Chaos RBF dynamics surface control of brushless DC motor with time delay based on tangent barrier Lyapunov function", Nonlinear Dyn., vol. 78, pp. 1193-1204, 2014.
[http://dx.doi.org/10.1007/s11071-014-1507-x]
[75]
H. Deng, and M. Krstic, "Output-feedback Stochastic Nonlinear Stabilization", IEEE Trans. Automat. Contr., vol. 44, pp. 328-333, 1999.
[http://dx.doi.org/10.1109/9.746260]
[76]
K. Wang, Y. Liu, X. Liu, Y. Jing, and G.M. Dimirovski, "Study on TCP/AQM network congestion with adaptive neural network and barrier Lyapunov function", Neurocomputing, vol. 363, pp. 27-34, 2019.
[http://dx.doi.org/10.1016/j.neucom.2019.08.024]
[77]
C. Xi, and J. Dong, "Adaptive reliable guaranteed performance control of uncertain nonlinear systems by using exponent-dependent barrier Lyapunov function", Int. J. Robust Nonlin., vol. 29, no. 4, pp. 1051-1062, 2019.
[http://dx.doi.org/10.1002/rnc.4422]
[78]
L. TangA. Chen and D. Li, "Time-Varying Tan-Type Barrier Lyapunov Function-Based Adaptive Fuzzy Control for Switched Systems With Unknown Dead Zone", IEEE Access, vol. 7, pp. 110928-110935, 2019.
[http://dx.doi.org/10.1109/ACCESS.2019.2934117]
[79]
G. Xia, J. Xue, C. Sun, and B. Zhao, "Backstepping Control Using Barrier Lyapunov Function for Dynamic Positioning Control System with Passive Observer", Math. Probl. Eng., vol. 2019, pp. 1-9, 2019.
[http://dx.doi.org/10.1155/2019/8709369]

Rights & Permissions Print Cite
© 2024 Bentham Science Publishers | Privacy Policy