doi: 10.3934/mcrf.2022024
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Barrier Lyapunov functions-based adaptive neural tracking control for non-strict feedback stochastic nonlinear systems with full-state constraints: A command filter approach

Department of Electrical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad (FUM), Mashhad, Iran

*Corresponding author: Seyed Kamal Hosseini Sani

Received  January 2022 Revised  April 2022 Early access May 2022

In this paper, an adaptive neural network command filter controller is investigated for a class of non-strict feedback stochastic nonlinear systems with full-state constraints. By using the command filter approach and error compensation mechanism, the "explosion of complexity" problem caused by the backstepping method and the filtering errors are eliminated. In order to avoid excessive and burdensome computations and to ensure that the backstepping method works normally for non-strict feedback structures, neural networks are employed to approximate the unknown nonlinear functions that contain all the state variables of the system. Meanwhile, the barrier Lyapunov functions are constructed to ensure the constraints are not transgressed. Finally, based on the Lyapunov stability theorem, an adaptive neural tracking controller is presented to guarantee that all the signals of the closed-loop system are semi-global uniformly ultimately bounded (SGUUB) in probability, and the tracking error converges to a small neighborhood around the origin, besides the full-state constraints are not violated. The simulation results are given to confirm the effectiveness of the proposed control method.

Citation: Parisa Seifi, Seyed Kamal Hosseini Sani. Barrier Lyapunov functions-based adaptive neural tracking control for non-strict feedback stochastic nonlinear systems with full-state constraints: A command filter approach. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022024
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show all references

References:
[1]

K. J. Åström, Introduction to Stochastic Control Theory, Courier Corporation, 2012.

[2]

W.-J. ChangH.-Y. Qiao and C.-C. Ku, Sliding mode fuzzy control for nonlinear stochastic systems subject to pole assignment and variance constraint, Inform. Sci., 432 (2018), 133-145.  doi: 10.1016/j.ins.2017.12.016.

[3]

B. ChenX. P. LiuS. S. Ge and C. Lin, Adaptive fuzzy control of a class of nonlinear systems by fuzzy approximation approach, IEEE Transactions on Fuzzy Systems, 20 (2012), 1012-1021.  doi: 10.1109/TFUZZ.2012.2190048.

[4]

W. ChenL. JiaoJ. Li and R. Li, Adaptive nn backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40 (2010), 939-950.  doi: 10.1109/TSMCB.2009.2033808.

[5]

G. CuiW. Yang and J. Yu, Neural network-based finite-time adaptive tracking control of nonstrict-feedback nonlinear systems with actuator failures, Inform. Sci., 545 (2021), 298-311.  doi: 10.1016/j.ins.2020.08.024.

[6]

G. Cui, J. Yu and G. Song, Distributed consensus control for second-order stochastic nonlinear multiagent systems using command filter backstepping, In 2018 5th International Conference on Information, Cybernetics, and Computational Social Systems (ICCSS), IEEE, (2018), 105–110. doi: 10.1109/ICCSS.2018.8572405.

[7]

M.-Y. CuiZ.-J. WuX.-J. Xie and P. Shi, Modeling and adaptive tracking for a class of stochastic lagrangian control systems, Automatica, 49 (2013), 770-779.  doi: 10.1016/j.automatica.2012.11.013.

[8]

D. DeHaan and M. Guay, Extremum-seeking control of state-constrained nonlinear systems, Automatica, 41 (2005), 1567-1574.  doi: 10.1016/j.automatica.2005.03.030.

[9]

H. Deng and M. Krstić, Stochastic nonlinear stabilization-i: A backstepping design, Systems Control Lett., 32 (1997), 143-150.  doi: 10.1016/S0167-6911(97)00068-6.

[10]

K. Do, Backstepping control design for stochastic systems driven by lévy processes, Internat. J. Control, 95 (2022), 68-80.  doi: 10.1080/00207179.2020.1778793.

[11]

W. DongJ. A. FarrellM. M. PolycarpouV. Djapic and M. Sharma, Command filtered adaptive backstepping, IEEE Transactions on Control Systems Technology, 20 (2012), 566-580.  doi: 10.1109/TCST.2011.2121907.

[12]

L. FangL. MaS. Ding and J. H. Park, Finite-time stabilization of high-order stochastic nonlinear systems with asymmetric output constraints, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51 (2021), 7201-7213.  doi: 10.1109/TSMC.2020.2965589.

[13]

J. A. FarrellM. PolycarpouM. Sharma and W. Dong, Command filtered backstepping, IEEE Transactions on Automatic Control, 54 (2009), 1391-1395.  doi: 10.1109/TAC.2009.2015562.

[14]

E. Gilbert and I. Kolmanovsky, Nonlinear tracking control in the presence of state and control constraints: A generalized reference governor, Automatica, 38 (2002), 2063-2073.  doi: 10.1016/S0005-1098(02)00135-8.

[15]

B. HomayounM. M. Arefi and N. Vafamand, Robust adaptive backstepping tracking control of stochastic nonlinear systems with unknown input saturation: A command filter approach, Internat. J. Robust Nonlinear Control, 30 (2020), 3296-3313.  doi: 10.1002/rnc.4933.

[16]

B. HomayounM. M. ArefiN. Vafamand and S. Yin, Neuro-adaptive command filter control of stochastic time-delayed nonstrict-feedback systems with unknown input saturation, J. Franklin Inst., 357 (2020), 7456-7482.  doi: 10.1016/j.jfranklin.2020.04.042.

[17]

T. Hu and Z. Lin, Control Systems with Actuator Saturation: Analysis and Design, Springer Science & Business Media, 2001.

[18]

M. Hui-Fang and D. Na, Neural network-based adaptive state-feedback control for high-order stochastic nonlinear systems, Acta Automatica Sinica, 40 (2014), 2968-2972.  doi: 10.1016/S1874-1029(15)60002-7.

[19]

S. KhooJ. YinZ. Man and X. Yu, Finite-time stabilization of stochastic nonlinear systems in strict-feedback form, Automatica, 49 (2013), 1403-1410.  doi: 10.1016/j.automatica.2013.01.054.

[20]

M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., 1995.

[21]

H. LiL. BaiQ. ZhouR. Lu and L. Wang, Adaptive fuzzy control of stochastic nonstrict-feedback nonlinear systems with input saturation, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2185-2197.  doi: 10.1109/TSMC.2016.2635678.

[22]

Y. Li and S. Tong, Command-filtered-based fuzzy adaptive control design for mimo-switched nonstrict-feedback nonlinear systems, IEEE Transactions on Fuzzy Systems, 25 (2017), 668-681.  doi: 10.1109/TFUZZ.2016.2574913.

[23]

J. Liu, K. Feng, Y. Qu, A. Nawaz, H. Song and F. Wang, Stability analysis of t–s fuzzy coupled oscillator systems influenced by stochastic disturbance, Neural Computing and Applications, 33 (2021), 2549–2560, https://link.springer.com/article/10.1007/s00521-020-05116-x.

[24]

J. LiuB. XuD. ChenJ. LiX. Gao and G. Liu, Grid-connection analysis of hydro-turbine generator unit with stochastic disturbance, IET Renewable Power Generation, 13 (2018), 500-509.  doi: 10.1049/iet-rpg.2018.5678.

[25]

S.-J. LiuJ.-F. Zhang and Z.-P. Jiang, Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems, Automatica, 43 (2007), 238-251.  doi: 10.1016/j.automatica.2006.08.028.

[26]

Y.-J. LiuS. LuS. TongX. ChenC. P. Chen and D.-J. Li, Adaptive control-based barrier lyapunov functions for a class of stochastic nonlinear systems with full state constraints, Automatica, 87 (2018), 83-93.  doi: 10.1016/j.automatica.2017.07.028.

[27]

Q. Lu and X. Zhang, A mini-course on stochastic control, Control and Inverse Problems for Partial Differential Equations, 22 (2019), 171–254, arXiv: 1612.02523.

[28]

D. Q. MayneJ. B. RawlingsC. V. Rao and P. O. Scokaert, Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814.  doi: 10.1016/S0005-1098(99)00214-9.

[29]

Y. NiuY. Liu and T. Jia, Reliable control of stochastic systems via sliding mode technique, Optimal Control Appl. Methods, 34 (2013), 712-727.  doi: 10.1002/oca.2050.

[30]

Z. Pan and T. Basar, Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion, SIAM J. Control Optim., 37 (1999), 957-995.  doi: 10.1137/S0363012996307059.

[31]

W. QiG. Zong and H. R. Karimi, Sliding mode control for nonlinear stochastic semi-markov switching systems with application to srmm, IEEE Transactions on Industrial Electronics, 67 (2020), 3955-3966.  doi: 10.1109/TIE.2019.2920619.

[32]

J. QiuK. SunI. J. Rudas and H. Gao, Command filter-based adaptive nn control for mimo nonlinear systems with full-state constraints and actuator hysteresis, IEEE Transactions on Cybernetics, 50 (2020), 2905-2915.  doi: 10.1109/TCYB.2019.2944761.

[33]

K. Sachan and R. Padhi, Barrier lyapunov function based output-constrained control of nonlinear euler-lagrange systems, In 2018 15th International Conference on Control, Automation, Robotics and Vision (ICARCV), IEEE, (2018), 686–691. doi: 10.1109/ICARCV.2018.8581068.

[34]

W. Si and X. Dong, Barrier lyapunov function-based decentralized adaptive neural control for uncertain high-order stochastic nonlinear interconnected systems with output constraints, J. Franklin Inst., 355 (2018), 8484-8509.  doi: 10.1016/j.jfranklin.2018.09.034.

[35]

W. SiX. Dong and F. Yang, Adaptive neural tracking control for nonstrict-feedback stochastic nonlinear time-delay systems with full-state constraints, Internat. J. Systems Sci., 48 (2017), 3018-3031.  doi: 10.1080/00207721.2017.1367049.

[36]

W. SunS.-F. SuY. Wu and J. Xia, A novel adaptive fuzzy control for output constrained stochastic non-strict feedback nonlinear systems, IEEE Transactions on Fuzzy Systems, 29 (2021), 1188-1197.  doi: 10.1109/TFUZZ.2020.2969909.

[37]

D. SwaroopJ. K. HedrickP. P. Yip and J. C. Gerdes, Dynamic surface control for a class of nonlinear systems, IEEE Trans. Automat. Control, 45 (2000), 1893-1899.  doi: 10.1109/TAC.2000.880994.

[38]

K. P. TeeS. S. Ge and E. H. Tay, Barrier lyapunov functions for the control of output-constrained nonlinear systems, Automatica, 45 (2009), 918-927.  doi: 10.1016/j.automatica.2008.11.017.

[39]

S. TongY. Li and S. Sui, Adaptive fuzzy tracking control design for siso uncertain nonstrict feedback nonlinear systems, IEEE Transactions on Fuzzy Systems, 24 (2016), 1441-1454.  doi: 10.1109/TFUZZ.2016.2540058.

[40]

T. Van Nguyen, N. H. Thai, H. T. Pham, T. A. Phan, L. Nguyen, H. X. Le and H. D. Nguyen, Adaptive neural network-based backstepping sliding mode control approach for dual-arm robots, Journal of Control, Automation and Electrical Systems, 30 (2019), 512–521, https://link.springer.com/article/10.1007/s40313-019-00472-z.

[41]

F. WangB. ChenY. Sun and C. Lin, Finite time control of switched stochastic nonlinear systems, Fuzzy Sets and Systems, 365 (2019), 140-152.  doi: 10.1016/j.fss.2018.04.016.

[42]

F. WangL. ZhangS. Zhou and Y. Huang, Neural network-based finite-time control of quantized stochastic nonlinear systems, Neurocomputing, 362 (2019), 195-202.  doi: 10.1016/j.neucom.2019.06.060.

[43]

H.-q. WangB. Chen and C. Lin, Adaptive neural tracking control for a class of stochastic nonlinear systems, Internat. J. Robust Nonlinear Control, 24 (2014), 1262-1280.  doi: 10.1002/rnc.2943.

[44]

H. Wang and Q. Zhu, Adaptive output feedback control of stochastic nonholonomic systems with nonlinear parameterization, Automatica, 98 (2018), 247-255.  doi: 10.1016/j.automatica.2018.09.026.

[45]

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Figure 1.  System output $ y(t) $ and the reference signal $ y_d(t) $
Figure 2.  Tracking error $ z_1 $
Figure 3.  Phase portrait of $ z_1 $ and $ z_2 $
Figure 4.  System state $ x_2 $
Figure 5.  Control signal $ u $
Figure 6.  Adaptive laws $ \hat{\theta}_1 $ and $ \hat{\theta}_2 $
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