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Nonlinear Dynamics

, Volume 95, Issue 3, pp 2443–2459 | Cite as

Prescribed performance-barrier Lyapunov function for the adaptive control of unknown pure-feedback systems with full-state constraints

  • Longsheng ChenEmail author
  • Qi Wang
Original Paper
  • 224 Downloads

Abstract

In this paper, an adaptive state-feedback control technique is proposed for a class of unknown pure-feedback systems. A remarkable feature is that not only the problem of full-state constraints and prescribed performance tracking is solved together, but also the design is an approximation-free control scheme for pure-feedback systems with completely unknown nonlinearities. These properties will lead to a difficult task for designing a stable controller. To this end, a novel prescribed performance-barrier Lyapunov function is developed to guarantee that all the state constraints are not violated and the tracking error is preserved within a specified prescribed performance bound at all times, simultaneously. Then, by utilizing the mean value theorem, Nussbaum gain technique, a low-pass filter and a novel bounded estimation approach at each step of back-stepping procedure, a novel adaptive dynamics surface control scheme is developed to remove the difficulties of pure-feedback characteristic, unknown nonlinearities, unknown control direction and “explosion of complexity”, which can guarantee that the proposed design is universal and low-complexity. Moreover, it is proved that all the signals in the closed-loop system are global uniformly bounded. Two simulation studies are worked out to illustrate the performance of the proposed approach.

Keywords

Unknown pure-feedback systems Adaptive dynamic surface control Prescribed performance Barrier Lyapunov function Full-state constraints 

Notes

Acknowledgements

This research was supported by the Aeronautical Science Foundation of China (2015ZC56007) and the Educational Commission of Jiangxi Province of China (GJJ150707, GJJ170610)

References

  1. 1.
    Song, H.T., Zhang, T., Zhang, G.L., et al.: Robust dynamic surface control of nonlinear systems with prescribed performance. Nonlinear Dyn. 76(1), 599–608 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ge, S.S., Li, G.Y., Lee, T.H.: Adaptive NN control for a class of strict-feedback discrete-time nonlinear systems. Automatica 39(5), 807–819 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Li, Y.H., Sheng, Q., Zhang, X.Y.: Robust and adaptive backstepping control for nonlinear systems using RBF neural network. IEEE Trans. Neural Netw. 15(3), 693–701 (2004)Google Scholar
  4. 4.
    Chen, M., Zhou, Y.L., Guo, W.W.: Robust tracking control for uncertain MIMO nonlinear systems with input saturation using RWNNDO. Neurocomputing 144, 436–447 (2014)Google Scholar
  5. 5.
    Li, T.S., Tong, S.C., Feng, G.: A novel robust adaptive fuzzy tracking control for a class of nonlinear MIMO systems. IEEE Trans. Fuzzy Syst. 18, 150–160 (2010)Google Scholar
  6. 6.
    Tong, S.C., Li, Y.M.: Adaptive fuzzy output feedback tracking backstepping control of strict- feedback nonlinear systems with unknown dead zones. IEEE Trans. Fuzzy Syst. 20(1), 168–180 (2012)Google Scholar
  7. 7.
    Zhou, Q., Wu, C.W., Jing, X.J., et al.: Adaptive fuzzy backstepping dynamic surface control for nonlinear inputdelay systems. Neurocomputing 199, 58–65 (2016)Google Scholar
  8. 8.
    Ge, S.S., Wang, C.: Adaptive NN control of uncertain nonlinear pure-feedback systems. Automatica 38(4), 671–682 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ren, B.B., Ge, S.S., Su, C.Y., et al.: Adaptive neural control for a class of uncertain nonlinear systems in pure feedback form with hysteresis input. IEEE Trans. Syst. Man Cybern. Part B Cybern. 30(2), 431–442 (2009)Google Scholar
  10. 10.
    Wang, H.Q., Chen, B., Lin, C.: Adaptive neural tracking control for a class of perturbed pure-feedback nonlinear systems. Nonlinear Dyn. 72(1–2), 207–220 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Shen, Q.K., Shi, P., Zhang, T.P., et al.: Novel neural control for a class of uncertain pure-feedback systems. IEEE Trans. Neural Netw. Learn. Syst. 25(4), 718–727 (2014)Google Scholar
  12. 12.
    Wang, H.Q., Liu, X.P., Liu, K.F., et al.: Adaptive neural control for a general class of pure-feedback stochastic nonlinear systems. Neurocomputing 135, 348–356 (2014)Google Scholar
  13. 13.
    Wang, M., Wang, C.: Neural learning control of pure feedback nonlinear systems. Nonlinear Dyn. 79(4), 2589–2608 (2015)MathSciNetGoogle Scholar
  14. 14.
    Liu, Y.J., Ton, S.C., Wang, W.: Adaptive fuzzy output tracking control for a class of uncertain nonlinear systems. Fuzzy Sets Syst. 160(19), 2727–2754 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang, F., Liu, Z., Zhang, Y., et al.: Adaptive fuzzy control for a class of stochastic pure-feedback nonlinear systems with unknown hysteresis. IEEE Trans. Fuzzy Syst. 24(1), 140–152 (2016)Google Scholar
  16. 16.
    Boskvoic, J.D., Chen, L.J., Mehra, R.K.: Adaptive control design for nonaffine models arising in flight control. J. Guid. Control Dyn. 27(2), 209–217 (2004)Google Scholar
  17. 17.
    Wang, D., Huang, J.: Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form. Automatica 38(8), 1365–1372 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lee, C.H., Chung, B.R.: Adaptive backstepping controller design for nonlinear uncertain systems using fuzzy neural systems. Int. J. Syst. Sci. 43(10), 1855–1869 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Tong, S.C., Li, Y.M., Shi, P.: Observer-based adaptive fuzzy backstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems. IEEE Trans. Fuzzy Syst. 20(4), 771–783 (2012)Google Scholar
  20. 20.
    Swaroop, D., Hedrick, J.K., Yip, P.P., et al.: Dynamic surface control for a class of nonlinear systems. IEEE Trans. Autom. Control 45(10), 1893–1899 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Niu, B., Li, H., Qin, T., Karimi, H.R.: Adaptive NN dynamic surface controller design for nonlinear pure-feedback switched systems with time-delays and quantized input. IEEE Trans. Syst. Man Cybern. Syst. 99, 1–13 (2017)Google Scholar
  22. 22.
    Wang, D.: Neural network-based adaptive dynamic surface control of uncertain nonlinear pure-feedback systems. Int. J. Robust Nonlinear Control 21, 527–541 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zhao, Q.C., Lin, Y.: Adaptive dynamic surface control for pure-feedback systems. Int. J. Robust Nonlinear Control 22, 1647–1660 (2012)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Wang, M., Liu, X.P., Shi, P.: Adaptive neural control of pure-feedback nonlinear time-delay systems via dynamic surface technique. IEEE Trans. Syst. Man Cybern. Part B Cybern. 41(6), 1682–1692 (2011)Google Scholar
  25. 25.
    Liu, S.G., Sun, X.X., Xie, W.J., et al.: Adaptive dynamic surface control for a class of pure-feedback nonlinear systems. Inf. Control 41(3), 301–306 (2012)Google Scholar
  26. 26.
    Sun, G., Wang, D., Li, X.Q., et al.: A DSC approach to adaptive neural network tracking control for pure-feedback nonlinear systems. Appl. Math. Comput. 219, 6224–6235 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Li, Y.M., Li, T.S., Tong, S.C.: Adaptive neural networks output feedback dynamic surface control design for MIMO pure-feedback nonlinear systems with hysteresis. Neurocomputing 198, 58–68 (2016)Google Scholar
  28. 28.
    Tong, S.C., Li, Y.M.: Adaptive fuzzy output feedback backstepping control of pure-feedback nonlinear systems via dynamic surface control technique. Int. J. Adapt. Control Signal Process. 27, 541–561 (2013)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Bechlioulis, C.P., Rovithakis, G.A.: Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans. Autom. Control 53(9), 2090–2099 (2008)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Bechlioulis, C.P., Rovithaki, G.A.: Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities. IEEE Trans. Autom. Control 56(9), 2224–2230 (2011)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Na, J.: Adaptive prescribed performance control of nonlinear systems with unknown dead zone. Int. J. Adapt. Control Signal Process. 27(5), 426–446 (2013)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Zhang, L.L., Sui, S., Li, Y.M., et al.: Adaptive fuzzy output feedback tracking control with prescribed performance for chemical reactor of MIMO nonlinear systems. Nonlinear Dyn. 80(1–2), 945–957 (2015)zbMATHGoogle Scholar
  33. 33.
    Bechlioulis, C.P., Rovithakis, G.A.: A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems. Automatica 50(4), 1217–1226 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Choi, Y.H., Yoo, S.J.: Decentralized approximation-free control for uncertain large-scale pure-feedback systems with unknown time-delayed nonlinearities and control directions. Nonlinear Dyn. 85(2), 1053–1066 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Chen, L.S., Wang, Q.: Adaptive dynamic surface control for unknown pure feedback non-affine systems with multiple constraints. Nonlinear Dyn. 21(2), 109–116 (2017)MathSciNetGoogle Scholar
  36. 36.
    Tee, K.P., Ge, S.S., Tay, E.H.: Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4), 918–927 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Ren, B.B., Ge, S.S., Tee, K.P., et al.: Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function. IEEE Trans. Neural Netw. 21(8), 1339–1345 (2010)Google Scholar
  38. 38.
    Tee, K.P., Ren, B.B., Ge, S.S.: Control of nonlinear systems with time-varying output constraints. Automatica 47(11), 2511–2516 (2011)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Qiu, Y.N., Liang, X.G., Dai, Z.Y., et al.: Backstepping dynamic surface control for a class of nonlinear systems with time-varying output onstraints. IET Control Theory Appl. 9(15), 2312–2319 (2015)MathSciNetGoogle Scholar
  40. 40.
    Zhou, Q., Wang, L.J., Wu, C.W., et al.: Adaptive fuzzy control for nonstrict-feedback systems with input saturation and output constraint. IEEE Trans. Syst. Man Cybern. 47(1), 1–12 (2017)Google Scholar
  41. 41.
    Liu, Y.J., Tong, S.C.: Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems. Automatica 76, 143–152 (2017)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Liu, Y.J., Tong, S.C., Philip Chen, C.L.: Neural network control-based adaptive learning design for nonlinear systems with full-state constraints. IEEE Trans. Neural Netw. Learn. Syst. 27(7), 1562–1571 (2016)MathSciNetGoogle Scholar
  43. 43.
    Tee, K.P., Ge, S.S.: Control of nonlinear systems with partial state constraints using a barrier Lyapunov function. Int. J. Control 84(12), 2008–2023 (2011)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Kim, B.S., Yoo, S.J.: Adaptive control of nonlinear pure-feedback systems with output constraints: integral barrier Lyapunov functional approach. Int. J. Control Autom. Syst. 13(1), 249–256 (2015)Google Scholar
  45. 45.
    Liu, Y.J., Tong, S.C.: Barrier Lyapunov functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints. Automatica 64, 70–75 (2016)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Ngo, K.B., Mahony, R., Jiang, Z.P.: Integrator backstepping using barrier functions for systems with multiple state constraints. In: Proceedings of the 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05, pp. 8306–8312. IEEE 2005 (2005)Google Scholar
  47. 47.
    Nussbaum, R.D.: Some remarks on a conjecture in parameter adaptive control. Syst. Control Lett. 3(5), 243–246 (1983)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Ge, S.S., Fan, H., Tong, H.L.: Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients. IEEE Trans. Syst. Man Cybern. Part B Cybern. 34(1), 499–516 (2004)Google Scholar
  49. 49.
    Ryan, E.P.: A universal adaptive stabilizer for a class of nonlinear systems. Syst. Control Lett. 16(3), 209–218 (1991)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Aircraft EngineeringNanchang Hangkong UniversityNanchangChina

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