Adaptive neural control of unknown non-affine nonlinear systems with input deadzone and unknown disturbance

  • Shuang Zhang
  • Linghuan Kong
  • Suwen Qi
  • Peng Jing
  • Wei HeEmail author
  • Bin Xu
Original Paper


In this paper, an adaptive neural scheme is developed for unknown non-affine nonlinear systems with input deadzone and internal/external unknown disturbance. With the help of mean value theorem and implicit function theorem, the control problem that the system input cannot be expressed in a linear form can be solved. The unknown input deadzone is approximated by neural networks. The immeasurable states are estimated by a high-gain observer such that output feedback control is obtained. The approximation error of both neural networks and the unknown internal/external disturbance is considered as an overall disturbance which is compensated by a novel disturbance observer. Via Lyapunov’s stability theory, it can be proved that all the state signals are uniformly bounded ultimately. The transient response performance can be improved by tuning the control parameters, and the steady-state error converges to any small neighborhood of zero. Simulation examples are carried out to verify the effectiveness of the proposed method.


Non-affine nonlinear systems Input deadzone Disturbance observer Neural networks Adaptive control 



This work was supported by the National Natural Science Foundation of China under Grants 61873297 and 61873206 and 61622308 and Science and Technology on Space Intelligent Control Laboratory (ZDSYS-2017-05) and the Fundamental Research Funds for the China Central Universities of USTB under Grant FRF-BD-17-002A and China Postdoctoral Science Foundation funded Project No. 2018M630074 and Fundamental Research Funds of Shenzhen Science and Technology Project (JCYJ20160229172341417).


  1. 1.
    Li, Z., Su, C.Y., Wang, L., Chen, Z., Chai, T.: Nonlinear disturbance observer-based control design for a robotic exoskeleton incorporating fuzzy approximation. IEEE Trans. Ind. Electron. 62(9), 5763–5775 (2015)CrossRefGoogle Scholar
  2. 2.
    Li, Z., Su, C.Y., Li, G., Su, H.: Fuzzy approximation-based adaptive backstepping control of an exoskeleton for human upper limbs. IEEE Trans. Fuzzy Syst. 23(3), 555–566 (2014)CrossRefGoogle Scholar
  3. 3.
    Li, Z., Huang, Z., He, W., Su, C.Y.: Adaptive impedance control for an upper limb robotic exoskeleton using biological signals. IEEE Trans. Ind. Electron. 64, 1664–1674 (2017)CrossRefGoogle Scholar
  4. 4.
    Hamdy, M., Ramadan, A., Abozalam, B.: A novel inverted fuzzy decoupling scheme for mimo systems with disturbance: a case study of binary distillation column. J. Intell. Manuf. 29(8), 1859–1871 (2018)CrossRefGoogle Scholar
  5. 5.
    Xu, B., Sun, F.: Composite intelligent learning control of strict-feedback systems with disturbance. IEEE Trans. Cybern. 48, 730–741 (2018)CrossRefGoogle Scholar
  6. 6.
    Ning, X., Yang, Y., Li, Z., Gui, M., Fang, J.: Ephemeris corrections in celestial/pulsar navigation using time differential and ephemeris estimation. J. Guid. Control Dyn. 2, 1–8 (2017)Google Scholar
  7. 7.
    Peng, K., Zhang, K., You, B., Dong, J., Wang, Z.: A quality-based nonlinear fault diagnosis framework focusing on industrial multimode batch processes. IEEE Trans. Ind. Electron. 63, 2615–2624 (2016)Google Scholar
  8. 8.
    Wang, H., Wang, C., Chen, W., Liang, X., Liu, Y.: Three-dimensional dynamics for cable-driven soft manipulator. IEEE/ASME Trans. Mechatron. 22(1), 18–28 (2017)CrossRefGoogle Scholar
  9. 9.
    Peng, K., Zhang, K., Dong, J., You, B.: Quality-relevant fault detection and diagnosis for hot strip mill process with multi-specification and multi-batch measurements. J. Frankl. Inst. 352(3), 987–1006 (2015)CrossRefGoogle Scholar
  10. 10.
    Wang, H., Zhang, R., Chen, W., Liang, X., Pfeifer, R.: Shape detection algorithm for soft manipulator based on fiber bragg gratings. IEEE/ASME Trans. Mechatron. 21(6), 2977–2982 (2016)CrossRefGoogle Scholar
  11. 11.
    Cui, R., Chen, L., Yang, C., Chen, M.: Extended state observer-based integral sliding mode control for an underwater robot with unknown disturbances and uncertain nonlinearities. IEEE Trans. Ind. Electron. 64, 6785–6795 (2017)CrossRefGoogle Scholar
  12. 12.
    Yang, C., Jiang, Y., Li, Z., He, W., Su, C.Y.: Neural control of bimanual robots with guaranteed global stability and motion precision. IEEE Trans. Ind. Electron. 13, 1162–1171 (2017)Google Scholar
  13. 13.
    Yang, C., Wang, X., Long, C., Ma, H.: Neural-learning-based telerobot control with guaranteed performance. IEEE Trans. Cybern. 47(10), 3148–3159 (2017)CrossRefGoogle Scholar
  14. 14.
    He, W., Ge, W., Li, Y., Liu, Y.J., Yang, C., Sun, C.: Model identification and control design for a humanoid robot. IEEE Trans. Syst. Man Cybern. Syst. 47(1), 45–57 (2017)CrossRefGoogle Scholar
  15. 15.
    Park, J.H., Huh, S.H., Kim, S.H., Seo, S.J., Park, G.T.: Direct adaptive controller for nonaffine nonlinear systems using self-structuring neural networks. IEEE Trans. Neural Netw. 16(2), 414–22 (2005)CrossRefGoogle Scholar
  16. 16.
    Boulkroune, A., M’Saad, M., Farza, M.: Adaptive fuzzy tracking control for a class of mimo nonaffine uncertain systems. Neurocomputing 93(2), 48–55 (2012)CrossRefGoogle Scholar
  17. 17.
    Park, J.H., Park, G.T., Kim, S.H., Moon, C.J.: Direct adaptive self-structuring fuzzy controller for nonaffine nonlinear system. Fuzzy Sets Syst. 153(3), 429–445 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, X., Zhang, H., Sun, Q., Luo, Y.: Adaptive dynamic programming-based optimal control of unknown nonaffine nonlinear discrete-time systems with proof of convergence. Neurocomputing 91(2), 48–55 (2012)CrossRefGoogle Scholar
  19. 19.
    He, W., Li, Z., Chen, C.L.P.: A survey of human-centered intelligent robots: issues and challenges. IEEE/CAA J. Autom. Sin. 4(4), 602–609 (2017)CrossRefGoogle Scholar
  20. 20.
    Wang, D., Liu, D., Li, H., Luo, B., Ma, H.: An approximate optimal control approach for robust stabilization of a class of discrete-time nonlinear systems with uncertainties. IEEE Trans. Syst. Man Cybern. Syst. 46(5), 713–717 (2016)CrossRefGoogle Scholar
  21. 21.
    Wang, D., He, H., Liu, D.: Adaptive critic nonlinear robust control: a survey. IEEE Trans. Cybern. 47(10), 3429–3451 (2017)CrossRefGoogle Scholar
  22. 22.
    Luo, B., Wu, H.N., Li, H.X.: Adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro-dynamic programming. IEEE Trans. Neural Netw. Learn. Syst. 26(4), 684 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Luo, B., Huang, T., Wu, H.N., Yang, X.: Data-driven h\(^{\infty }\) control for nonlinear distributed parameter systems. IEEE Trans. Neural Netw. Learn. Syst. 26(11), 2949–2961 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Niu, B., Liu, Y., Zong, G., Han, Z., Fu, J.: Command filter-based adaptive neural tracking controller design for uncertain switched nonlinear output-constrained systems. IEEE Trans. Cybern. 47, 3160–3171 (2017)CrossRefGoogle Scholar
  25. 25.
    Wang, N., Qian, C., Sun, J.C., Liu, Y.C.: Adaptive robust finite-time trajectory tracking control of fully actuated marine surface vehicles. IEEE Trans. Control Syst. Technol. 24, 1454–1462 (2016)CrossRefGoogle Scholar
  26. 26.
    He, W., Ge, S.S., Li, Y., Chew, E., Ng, Y.S.: Neural network control of a rehabilitation robot by state and output feedback. J. Intell. Robot. Syst. 80(1), 15–31 (2015)CrossRefGoogle Scholar
  27. 27.
    Liu, Y.J., Li, J., Tong, S., Chen, C.L.P.: Neural network control-based adaptive learning design for nonlinear systems with full-state constraints. IEEE Trans. Neural Netw. Learn. Syst. 27, 1562–1571 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Peng, Z., Wang, D., Zhang, H., Sun, G.: Distributed neural network control for adaptive synchronization of uncertain dynamical multiagent systems. IEEE Trans. Neural Netw. Learn. Syst. 25, 1508–1519 (2014)CrossRefGoogle Scholar
  29. 29.
    Pan, Y., Yu, H.: Biomimetic hybrid feedback feedforward neural-network learning control. IEEE Trans. Neural Netw. Learn. Syst. 28, 1481–1487 (2017)CrossRefGoogle Scholar
  30. 30.
    Zhao, S., Liang, H., Du, P., Qi, S.: Adaptive nn finite-time tracking control of output constrained nonlinear system with input saturation. Nonlinear Dyn. 92(2), 1–12 (2018)zbMATHGoogle Scholar
  31. 31.
    Zhang, S., Dong, Y., Ouyang, Y., Yin, Z., Peng, K.: Adaptive neural control for robotic manipulators with output constraints and uncertainties. IEEE Trans. Neural Netw. Learn. Syst. 99, 1–11 (2018)Google Scholar
  32. 32.
    Chen, C.L.P., Liu, Z.: Broad learning system: an effective and efficient incremental learning system without the need for deep architecture. IEEE Trans. Neural Netw. Learn. Syst. 29, 10–24 (2018)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Cui, R., Yang, C., Li, Y., Sharma, S.: Adaptive neural network control of auvs with control input nonlinearities using reinforcement learning. IEEE Trans. Syst. Man Cybern. Syst. 47(6), 1019–1029 (2017)CrossRefGoogle Scholar
  34. 34.
    Yang, H., Liu, J.: An adaptive rbf neural network control method for a class of nonlinear systems. IEEE/CAA J. Autom. Sin. 5(2), 457–462 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Liu, D., Xu, Y., Wei, Q., Liu, X.: Residential energy scheduling for variable weather solar energy based on adaptive dynamic programming. IEEE/CAA J. Autom. Sin. 5(1), 36–46 (2018)CrossRefGoogle Scholar
  36. 36.
    Xu, B., Wang, D., Zhang, Y., Shi, Z.: Dob-based neural control of flexible hypersonic flight vehicle considering wind effects. IEEE Trans. Ind. Electron. 64, 8676–8685 (2017)CrossRefGoogle Scholar
  37. 37.
    Xu, B., Shou, Y.: Composite learning control of mimo systems with applications. IEEE Trans. Ind. Electron. 65, 6414–6424 (2018)CrossRefGoogle Scholar
  38. 38.
    Sun, C., Xia, Y.: An analysis of a neural dynamical approach to solving optimization problems. IEEE Trans. Autom. Control 54(8), 1972–1977 (2009)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Cui, R., Ren, B., Ge, S.S.: Synchronised tracking control of multi-agent system with high order dynamics. IET Control Theory Appl. 6(5), 603–614 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Chen, M., Ge, S.S., Ren, B.: Adaptive tracking control of uncertain mimo nonlinear systems with input constraints. Automatica 47(3), 452–465 (2011)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Ren, B., Ge, S.S., Tee, K.P., Lee, T.H.: Adaptive neural control for output feedback nonlinear systems using a barrier lyapunov function. IEEE Trans. Neural Netw. 21(8), 1339–1345 (2010)CrossRefGoogle Scholar
  42. 42.
    Dai, S.L., Wang, C., Luo, F.: Identification and learning control of ocean surface ship using neural networks. IEEE Trans. Ind. Inf. 8, 801–810 (2012)CrossRefGoogle Scholar
  43. 43.
    Dai, S.L., Wang, M., Wang, C.: Neural learning control of marine surface vessels with guaranteed transient tracking performance. IEEE Trans. Ind. Electron. 63(3), 1717–1727 (2016)CrossRefGoogle Scholar
  44. 44.
    Wang, F.Y., Zheng, N.N., Cao, D., Martinez, C.M., Li, L., Liu, T.: Parallel driving in cpss:a unified approach for transport automation and vehicle intelligence. IEEE/CAA J. Autom. Sin. 4(4), 577–587 (2017)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Wang, H., Zhu, Q.: Finite-time stabilization of high-order stochastic nonlinear systems in strict-feedback form. Automatica 54, 284–291 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Wang, H., Zhu, Q.: Global stabilization of stochastic nonlinear systems via \(c^1\) and \(c^{\infty }\) controllers. IEEE Trans. Autom. Control 62, 5880–5887 (2017)CrossRefGoogle Scholar
  47. 47.
    Xiong, S., Zhu, Q.: Decentralized risk-sensitive design for large-scale stochastic interconnected systems with time-varying delays. J. Frankl. Inst. 353(7), 1527–1552 (2016)MathSciNetCrossRefGoogle Scholar
  48. 48.
    He, W., Huang, H., Chen, Y., Xie, W., Feng, F., Kang, Y., Sun, C.: Development of an autonomous flapping-wing aerial vehicle. Sci. China (Inf. Sci.) 60(6), 063201 (2017)CrossRefGoogle Scholar
  49. 49.
    Guo, Q., Zhang, Y., Celler, B.G., Su, S.W.: Backstepping control of electro-hydraulic system based on extended-state-observer with plant dynamics largely unknown. IEEE Trans. Ind. Electron. 63(11), 6909–6920 (2016)CrossRefGoogle Scholar
  50. 50.
    Zhang, Z., Xu, S., Zhang, B.: Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity. IEEE Trans. Autom. Control 59(5), 1336–1341 (2014)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Hamdy, M., Abd-Elhaleem, S., Fkirin, M.A.: Time-varying delay compensation for a class of nonlinear control systems over network via \(h_{\infty }\) adaptive fuzzy controller. IEEE Trans. Syst. Man Cybern. Syst. 47, 2114–2124 (2017)CrossRefGoogle Scholar
  52. 52.
    Deng, H., Li, H.X., Wu, Y.H.: Feedback-linearization-based neural adaptive control for unknown nonaffine nonlinear discrete-time systems. IEEE Trans. Neural Netw. 19, 1615–1625 (2008)CrossRefGoogle Scholar
  53. 53.
    Wang, H., Sun, W., Liu, P.X.: Adaptive intelligent control of nonaffine nonlinear time-delay systems with dynamic uncertainties. IEEE Trans. Syst. Man Cybern. Syst. 47, 1474–1485 (2017)CrossRefGoogle Scholar
  54. 54.
    Meng, W., Yang, Q., Si, J., Sun, Y.: Adaptive neural control of a class of output-constrained nonaffine systems. IEEE Trans. Cybern. 46, 85–95 (2016)CrossRefGoogle Scholar
  55. 55.
    Meng, T., He, W.: Iterative learning control of a robotic arm experiment platform with input constraint. IEEE Trans. Ind. Electron. 65(1), 664–672 (2017)CrossRefGoogle Scholar
  56. 56.
    Tao, G., Kokotovic, P.V.: Adaptive control of plants with unknown hysteresis. IEEE Trans. Autom. Control 40(2), 212–220 (1995)zbMATHGoogle Scholar
  57. 57.
    Wang, X.S., Su, C.Y., Hong, H.: Robust adaptive control of a class of nonlinear systems with unknown dead-zone. Automatica 40(3), 407–413 (2004)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Selmic, R., Lewis, F.: Deadzone compensation in motion control systems using neural networks. IEEE Trans. Autom. Control 45(4), 602–613 (2000)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Zhang, Z., Xu, S., Zhang, B.: Exact tracking control of nonlinear systems with time delays and dead-zone input. Automatica 52(52), 272–276 (2015)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Chen, C.L.P., Wen, G.X., Liu, Y.J., Liu, Z.: Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems. IEEE Trans. Cybern. 46(7), 1591–1601 (2016)CrossRefGoogle Scholar
  61. 61.
    Zhou, Q., Li, H., Wang, L., Lu, R.: Prescribed performance observer-based adaptive fuzzy control for nonstrict-feedback stochastic nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst. 48(10), 1747–1758 (2018)CrossRefGoogle Scholar
  62. 62.
    Zhang, S., He, W., Huang, D.: Active vibration control for a flexible string system with input backlash. IET Control Theory Appl. 10(7), 800–805 (2016)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Ma, Z., Tong, S., Li, Y.: Adaptive output feedback fault-tolerant control for mimo non-affine non-linear systems based on disturbance observer. IET Control Theory Appl. 10(18), 2422–2436 (2016)MathSciNetCrossRefGoogle Scholar
  64. 64.
    He, W., Huang, H., Ge, S.S.: Adaptive neural network control of a robotic manipulator with time-varying output constraints. IEEE Trans. Cybern. 47, 3136–3147 (2017)CrossRefGoogle Scholar
  65. 65.
    Chen, Z., Li, Z., Chen, C.L.P.: Adaptive neural control of uncertain mimo nonlinear systems with state and input constraints. IEEE Trans. Neural Netw. Learn. Syst. 28, 1318–1330 (2017)CrossRefGoogle Scholar
  66. 66.
    Yang, B.J., Calise, A.J.: Adaptive control of a class of nonaffine systems using neural networks. IEEE Trans. Neural Netw. 18, 1149–1159 (2007)CrossRefGoogle Scholar
  67. 67.
    Wang, Y., Hu, J., Wang, J., Xing, X.: Adaptive neural novel prescribed performance control for non-affine pure-feedback systems with input saturation. Nonlinear Dyn. 93(3), 1241–1259 (2018)CrossRefGoogle Scholar
  68. 68.
    Ge, S.S., Hang, C.C., Zhang, T.: Adaptive neural network control of nonlinear systems by state and output feedback. IEEE Trans. Syst. Man Cybern. Part B Cybern. 29(6), 818–828 (1999)CrossRefGoogle Scholar
  69. 69.
    Ge, S.S., Hang, C.C., Tong, H.L., Zhang, T.: Stable Adaptive Neural Network Control, vol. 13. Springer, Berlin (2001)Google Scholar
  70. 70.
    Li, S., Yang, J., Chen, W.H., Chen, X.: Disturbance Observer-Based Control: Methods and Applications. CRC Press Inc, Boca Raton (2014)Google Scholar
  71. 71.
    Liu, Y.J., Wang, W.: Adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems. Inf. Sci. 177(18), 3901–3917 (2007)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Ge, S.S., Zhang, J.: Neural network control of nonaffine nonlinear system with zero dynamics by state and output feedback. IEEE Trans. Neural Netw. 14(4), 900–918 (2003)CrossRefGoogle Scholar
  73. 73.
    Wu, X., Gao, D.: Fault tolerance control of sofc systems based on nonlinear model predictive control. Int. J. Hydrog. Energy 42(4), 2288–2308 (2017)CrossRefGoogle Scholar
  74. 74.
    Yang, C., Deconinck, G., Gui, W.: An optimal power-dispatching control system for the electrochemical process of zinc based on backpropagation and hopfield neural networks. IEEE Trans. Ind. Electron. 50(5), 953–961 (2003)CrossRefGoogle Scholar
  75. 75.
    Patre, P.M., Mackunis, W., Kaiser, K., Dixon, W.E.: Asymptotic tracking for uncertain dynamic systems via a multilayer nn feedforward and rise feedback control structure. IEEE Trans. Autom. Control 53(9), 2180–2185 (2008)CrossRefGoogle Scholar
  76. 76.
    Lian, K.Y., Liu, P., Chiang, T.S., Chiu, C.S.: Adaptive synchronization design for chaotic systems via a scalar driving signal. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(1), 17–27 (2002)CrossRefGoogle Scholar
  77. 77.
    Pan, Y., Yu, H.: Composite learning from adaptive dynamic surface control. IEEE Trans. Autom. Control 61(9), 2603–2609 (2016)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Pan, Y., Yu, H.: Composite learning robot control with guaranteed parameter convergence. Automatica 89, 398–406 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Shuang Zhang
    • 1
    • 2
  • Linghuan Kong
    • 1
    • 2
  • Suwen Qi
    • 3
  • Peng Jing
    • 1
    • 2
  • Wei He
    • 1
    • 2
    Email author
  • Bin Xu
    • 4
    • 5
  1. 1.School of Automation and Electrical EngineeringUniversity of Science and Technology BeijingBeijingChina
  2. 2.Institute of Artificial IntelligenceUniversity of Science and Technology BeijingBeijingChina
  3. 3.Department of Biomedical Engineering, School of MedicineShenzhen UniversityShenzhenChina
  4. 4.Research Institute of Northwestern Polytechnical University in ShenzhenShenzhenChina
  5. 5.School of AutomationNorthwestern Polytechnical UniversityXi’anChina

Personalised recommendations