Skip to main content
SpringerLink
Log in

Neural learning control for discrete-time nonlinear systems in pure-feedback form

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

This study focuses on state-feedback and output-feedback neural learning control problems for discrete-time nonlinear systems in the pure-feedback form. First, an extended result for the exponential stability for a class of discrete-time linear time-varying (LTV) systems with n-step delays is proposed to verify the exponential convergence of estimated weights. Subsequently, both state-feedback and output-feedback adaptive neural network (NN) controllers are constructed by combining the classical n-step and n-step input-output predictors. After ensuring convergence of the system output to a recurrent reference signal, the radial basis function subvector of NN is verified to satisfy the persistent exciting condition using the system state equation and the implicit function theorem. By combining the extended stability corollary of an LTV system, the estimated weights are verified to exponentially converge to their optimal values. By constructing “learning rules” and using a “mod” function, the estimated weights with a convergent sequence are synthetically represented and stored as the experience knowledge, which is reused to construct neural learning controllers. The proposed neural learning controllers not only accomplish similar control tasks but also reduce the burden of online computation compared with the conventional adaptive NN controllers. Finally, simulation results are presented to demonstrate the effectiveness of the presented schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Narendra K S, Parthasarathy K. Identification and control of dynamical systems using neural networks. IEEE Trans Neural Netw, 1990, 1: 4–27

    Article  Google Scholar 

  2. Polycarpou M M. Stable adaptive neural control scheme for nonlinear systems. IEEE Trans Autom Control, 1996, 41: 447–451

    Article  MathSciNet  MATH  Google Scholar 

  3. Krstic M, Kanellakopoulos I, Kokotovic P V. Nonlinear design of adaptive controllers for linear systems. IEEE Trans Autom Control, 1994, 39: 738–752

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhang T P, Ge S S, Hang C C. Adaptive neural network control for strict-feedback nonlinear systems using backstepping design. Automatica, 2000, 36: 1835–1846

    Article  MathSciNet  MATH  Google Scholar 

  5. Wang M, Zhang S Y, Chen B, et al. Direct adaptive neural control for stabilization of nonlinear systems with time-varying delays. Sci China Inf Sci, 2010, 53: 800–812

    Article  MathSciNet  MATH  Google Scholar 

  6. Tong S C, Li Y M. Robust adaptive fuzzy backstepping output feedback tracking control for nonlinear system with dynamic uncertainties. Sci China Inf Sci, 2010, 53: 307–324

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhou Q, Zhao S Y, Li H Y, et al. Adaptive neural network tracking control for robotic manipulators with dead zone. IEEE Trans Neural Netw Learn Syst, 2019, 30: 3611–3620

    Article  MathSciNet  Google Scholar 

  8. Ge S S, Wang C. Adaptive NN control of uncertain nonlinear pure-feedback systems. Automatica, 2002, 38: 671–682

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang D, Huang J. Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form. Automatica, 2002, 38: 1365–1372

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang C, Hill D J, Ge S S, et al. An ISS-modular approach for adaptive neural control of pure-feedback systems. Automatica, 2006, 42: 723–731

    Article  MathSciNet  MATH  Google Scholar 

  11. He W, David A O, Yin Z, et al. Neural network control of a robotic manipulator with input deadzone and output constraint. IEEE Trans Syst Man Cybern Syst, 2016, 46: 759–770

    Article  Google Scholar 

  12. Chen M, Ren B B, Wu Q X, et al. Anti-disturbance control of hypersonic flight vehicles with input saturation using disturbance observer. Sci China Inf Sci, 2015, 58: 070202

    Article  MathSciNet  Google Scholar 

  13. Xu B, Yuan Y. Two performance enhanced control of flexible-link manipulator with system uncertainty and disturbances. Sci China Inf Sci, 2017, 60: 050202

    Article  Google Scholar 

  14. Zhang T P, Xia M Z, Yi Y, et al. Adaptive neural dynamic surface control of pure-feedback nonlinear systems with full state constraints and dynamic uncertainties. IEEE Trans Syst Man Cybern Syst, 2017, 47: 2378–2387

    Article  Google Scholar 

  15. Liu Y J, Tong S. Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems. Automatica, 2017, 76: 143–152

    Article  MathSciNet  MATH  Google Scholar 

  16. Dai S L, He S D, Wang M, et al. Adaptive neural control of underactuated surface vessels with prescribed performance guarantees. IEEE Trans Neural Netw Learn Syst, 2019, 30: 3686–3698

    Article  MathSciNet  Google Scholar 

  17. Chen M, Ge S S. Adaptive neural output feedback control of uncertain nonlinear systems with unknown hysteresis using disturbance observer. IEEE Trans Ind Electron, 2015, 62: 7706–7716

    Article  Google Scholar 

  18. Tong S C, Li Y M. Observer-based adaptive fuzzy backstepping control of uncertain nonlinear pure-feedback systems. Sci China Inf Sci, 2014, 57: 012204

    Article  MathSciNet  MATH  Google Scholar 

  19. Yeh P C, Kokotović P V. Adaptive control of a class of nonlinear discrete-time systems. Int J Control, 1995, 62: 303–324

    Article  MathSciNet  MATH  Google Scholar 

  20. Ge S S, Li G Y, Lee T H. Adaptive NN control for a class of strict-feedback discrete-time nonlinear systems. Automatica, 2003, 39: 807–819

    Article  MathSciNet  MATH  Google Scholar 

  21. Ge S S, Yang C G, Lee T H. Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time. IEEE Trans Neural Netw, 2008, 19: 1599–1614

    Article  Google Scholar 

  22. Wang M, Wang Z D, Dong H L, et al. A novel framework for backstepping-based control of discrete-time strict-feedback nonlinear systems with multiplicative noises. IEEE Trans Autom Control, 2021, 66: 1484–1496

    Article  MathSciNet  MATH  Google Scholar 

  23. Shao S Y, Chen M. Sliding-mode-disturbance-observer-based adaptive neural control of uncertain discrete-time systems. Sci China Inf Sci, 2020, 63: 149204

    Article  MathSciNet  Google Scholar 

  24. Liu Y J, Li S, Tong S C, et al. Adaptive reinforcement learning control based on neural approximation for nonlinear discrete-time systems with unknown nonaffine dead-zone input. IEEE Trans Neural Netw Learn Syst, 2019, 30: 295–305

    Article  Google Scholar 

  25. Wang M, Wang Z D, Chen Y, et al. Adaptive neural event-triggered control for discrete-time strict-feedback nonlinear systems. IEEE Trans Cybern, 2020, 50: 2946–2958

    Article  Google Scholar 

  26. Sahoo A, Xu H, Jagannathan S. Adaptive neural network-based event-triggered control of single-input single-output nonlinear discrete-time systems. IEEE Trans Neural Netw Learn Syst, 2016, 27: 151–164

    Article  MathSciNet  Google Scholar 

  27. Xu B, Yang C G, Shi Z K. Reinforcement learning output feedback NN control using deterministic learning technique. IEEE Trans Neural Netw Learn Syst, 2014, 25: 635–641

    Article  Google Scholar 

  28. Wang Z S, Liu L, Wu Y M, et al. Optimal fault-tolerant control for discrete-time nonlinear strict-feedback systems based on adaptive critic design. IEEE Trans Neural Netw Learn Syst, 2018, 29: 2179–2191

    Article  MathSciNet  Google Scholar 

  29. Wang M, Wang Z D, Chen Y, et al. Event-based adaptive neural tracking control for discrete-time stochastic nonlinear systems: a triggering-threshold compensation strategy. IEEE Trans Neural Netw Learn Syst, 2020, 31: 1968–1981

    Article  MathSciNet  Google Scholar 

  30. Kurdila A J, Narcowich F J, Ward J D. Persistency of excitation in identification using radial basis function approximants. SIAM J Control Optim, 1995, 33: 625–642

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang C, Hill D J. Learning from neural control. IEEE Trans Neural Netw, 2006, 17: 130–146

    Article  Google Scholar 

  32. Wang C, Wang M, Liu T F, et al. Learning from ISS-modular adaptive NN control of nonlinear strict-feedback systems. IEEE Trans Neural Netw Learn Syst, 2012, 23: 1539–1550

    Article  Google Scholar 

  33. Wang M, Wang C. Learning from adaptive neural dynamic surface control of strict-feedback systems. IEEE Trans Neural Netw Learn Syst, 2015, 26: 1247–1259

    Article  MathSciNet  Google Scholar 

  34. Dai S L, Wang C, Wang M. Dynamic learning from adaptive neural network control of a class of nonaffine nonlinear systems. IEEE Trans Neural Netw Learn Syst, 2014, 25: 111–123

    Article  Google Scholar 

  35. Wang M, Wang C, Shi P, et al. Dynamic learning from neural control for strict-feedback systems with guaranteed predefined performance. IEEE Trans Neural Netw Learn Syst, 2016, 27: 2564–2576

    Article  MathSciNet  Google Scholar 

  36. Abdelatti M, Yuan C Z, Zeng W, et al. Cooperative deterministic learning control for a group of homogeneous nonlinear uncertain robot manipulators. Sci China Inf Sci, 2018, 61: 112201

    Article  MathSciNet  Google Scholar 

  37. Wang M, Yang A L. Dynamic learning from adaptive neural control of robot manipulators with prescribed performance. IEEE Trans Syst Man Cybern Syst, 2017, 47: 2244–2255

    Article  Google Scholar 

  38. Dai S L, Wang M, Wang C. Neural learning control of marine surface vessels with guaranteed transient tracking performance. IEEE Trans Ind Electron, 2016, 63: 1717–1727

    Article  Google Scholar 

  39. He S D, Wang M, Dai S L, et al. Leader-follower formation control of USVs with prescribed performance and collision avoidance. IEEE Trans Ind Inf, 2019, 15: 572–581

    Article  Google Scholar 

  40. Wang C, Chen T R, Liu T F. Deterministic learning and data-based modeling and control. Acta Autom Sin, 2009, 35: 693–706

    Article  MathSciNet  MATH  Google Scholar 

  41. Chen W S, Hua S Y, Ge S S. Consensus-based distributed cooperative learning control for a group of discrete-time nonlinear multi-agent systems using neural networks. Automatica, 2014, 50: 2254–2268

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang J T, Yuan C Z, Wang C, et al. Composite adaptive NN learning and control for discrete-time nonlinear uncertain systems in normal form. Neurocomputing, 2020, 390: 168–184

    Article  Google Scholar 

  43. Fradkov A L, Evans R J. Control of chaos: methods and applications in engineering. Annu Rev Control, 2005, 29: 33–56

    Article  Google Scholar 

  44. Dai S L, He S, Ma Y, et al. Distributed cooperative learning control of uncertain multiagent systems with prescribed performance and preserved connectivity. IEEE Trans Neural Netw Learn Syst, 2021, 32: 3217–3229

    Article  MathSciNet  Google Scholar 

  45. Dai S L, He S D, Lin H, et al. Platoon formation control with prescribed performance guarantees for USVs. IEEE Trans Ind Electron, 2018, 65: 4237–4246

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by Guangdong Natural Science Foundation (Grant No. 2019B151502058), National Natural Science Foundation of China (Grant Nos. 61773169, 61890922, 61973129), National Science Fund for Distinguished Young Scholars (Grant No. 61825301), National Key Research and Development Program of China (Grant No. 2018AAA0101603), Guangzhou Science and Technology Project (Grant No. 201904010295), Science and Technology Planning Project of Guangdong Province (Grant No. 2020B1111010002), and Guangdong Marine Economic Development Project (Grant No. 2020018).

Author information

Authors and Affiliations

  1. School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510641, China

    Min Wang & Haotian Shi

  2. School of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Cong Wang

  3. State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, China

    Jun Fu

Authors
  1. Min Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haotian Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jun Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Min Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, M., Shi, H., Wang, C. et al. Neural learning control for discrete-time nonlinear systems in pure-feedback form. Sci. China Inf. Sci. 65, 122206 (2022). https://doi.org/10.1007/s11432-020-3138-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11432-020-3138-7

Keywords

Search

Navigation