Nonlinear Dynamics

, Volume 93, Issue 3, pp 1241–1259 | Cite as

Adaptive neural novel prescribed performance control for non-affine pure-feedback systems with input saturation

  • Yingyang WangEmail author
  • Jianbo Hu
  • Jianhao Wang
  • Xiaobo Xing
Original Paper


An error constraint control problem is considered for pure-feedback systems with non-affine functions being possibly in-differentiable. A new constraint variable is used to construct virtual control that guarantees the tracking error within the transient and steady-state performance envelopment. The new error transformation avoids non-differentiable problems and complex deductions caused by traditional error constraint approaches. A locally semi-bounded and continuous condition for non-affine functions is employed to ensure the controllability and transform the closed-loop system into a pseudo-affine form. An auxiliary system with bounded compensation term is proposed for nonlinear systems with input saturation. On the basis of backstepping technique, an adaptive neural controller is designed to handle unknown terms and circumvent repeated differentiations of virtual controls. The boundedness and convergence of the closed-loop system are proved by Lyapunov theory. Asymptotic tracking is achieved without violating control input constraint and error constraint. Two examples are performed to verify the theoretical findings.


Non-affine Input saturation Adaptive neural control Prescribed performance 



The authors would like to express their sincere thanks to anonymous reviewers for their helpful suggestions for improving the technical note.


This work was supported by National Natural Science Foundation of China [Grant number. 61603410 & 61573374], National Basic Research Program of China [Grant number. 2014CB744900] and Young Talent Fund of University Association for Science and Technology in Shaanxi, China [grant number. 20170107].

Compliance with ethical standards

Conflict of interest

The authors declare that there exists no conflict of interest.


  1. 1.
    Ngo, K.B., Mahony, R., Jiang, Z.P.: Integrator backstepping using barrier functions for systems with multiple state constraints. IEEE Conf. Decis. Control (2005). Google Scholar
  2. 2.
    Tee, K.P., Ge, S.S., Tay, E.H.: Barrier Lyapunov functions for the output-constrained nonlinear systems. Automatica 45(4), 918–927 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ren, B.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
  4. 4.
    Liu, Y.J., Tong, S.: Barrier Lyapunov Functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints. Automatica 64, 70–75 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kim, B.S., Yoo, S.J.: Approximation-based adaptive tracking control of nonlinear pure-feedback systems with time-varying output constraints. Int. J. Control Autom. Syst. 13(2), 257–265 (2015)CrossRefGoogle Scholar
  6. 6.
    Wang, C., Wu, Y., Yu, J.: Barrier Lyapunov functions-based adaptive control for nonlinear pure-feedback systems with time-varying full state constraints. Int. J. Control Autom. Syst. 15(16), 1–9 (2017)Google Scholar
  7. 7.
    Ilchmann, A., Trenn, S.: Input constrained funnel control with applications to chemical reactor models. Syst. Control Lett. 53(5), 361–375 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hackl, C.M., Ji, Y., Schroder, D.: Enhanced funnel-control with improved performance. IEEE Conf. Control Autom. 115(1), 1–6 (2007)Google Scholar
  9. 9.
    Hopfe, N., Ilchmann, A., Ryan, E.P.: Funnel control with saturation: linear MIMO systems. IEEE Trans. Autom. Control 55(2), 532–538 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hopfe, N., Ilchmann, A., Ryan, E.P.: Funnel control with saturation: nonlinear SISO systems. IEEE Trans. Autom. Control 55(9), 2177–2182 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ilchmann, A., Schuster, H.: PI-funnel control for two mass systems. IEEE Trans. Autom. Control 54(4), 918–923 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, Q., Tang, X., Nan, Y.: Finite-time neural funnel control for motor servo systems with unknown input constraint. J. Syst. Sci. Complex. 30(3), 579–594 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hackl, C.M.: Funnel Control. Springer, Berlin (2017)CrossRefGoogle Scholar
  14. 14.
    Han, S.I., Lee, J.M.: Fuzzy echo state neural networks and funnel dynamic surface control for prescribed performance of a nonlinear dynamic system. IEEE Trans. Ind. Electron. 61(2), 1099–1112 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, X., Wang, H., Gao, C., Chen, M.: Adaptive fuzzy funnel control for a class of strict feedback nonlinear systems. Neurocomputing 241, 71–80 (2017)CrossRefGoogle Scholar
  16. 16.
    Bechlioulis, C.P., Rovithakis, G.A.: Brief paper: Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica 45(2), 532–538 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bechlioulis, C.P., Rovithakis, G.A.: Robust adaptive fuzzy control of nonaffine systems guaranteeing transient and steady state error bounds. Int. J. Adapt. Control Signal Process. 26(7), 576–591 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Han, S.I., Lee, J.M.: Recurrent fuzzy neural network backstepping control for the prescribed output tracking performance of nonlinear dynamic systems. ISA Trans. 53(1), 33–43 (2014)CrossRefGoogle Scholar
  19. 19.
    Zhang, L., Tong, S., Li, Y.: Prescribed performance adaptive fuzzy output-feedback control of uncertain nonlinear systems with unmodeled dynamics. Nonlinear Dyn. 77(4), 1653–1665 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yoo, S.J.: Fault-tolerant control of strict-feedback non-linear time-delay systems with prescribed performance. IET Control Theory Appl. 7(11), 1553–1561 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Han, S.I., Lee, J.M.: Improved prescribed performance constraint control for a strict feedback non-linear dynamic system. IET Control Theory Appl. 7(14), 1818–1827 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhang, L., Sui, S., Li, Y., Tong, S.: Adaptive fuzzy output feedback tracking control with prescribed performance for chemical reactor of MIMO nonlinear systems. Nonlinear Dyn. 80(12), 1–13 (2015)zbMATHGoogle Scholar
  24. 24.
    Bechlioulis C.P., Kyriakopoulos K.J.: Robust model-free formation control with prescribed performance for nonlinear multi-agent systems. In: IEEE International Conference on Robotics and Automation, pp 1268–1273 (2015)Google Scholar
  25. 25.
    Theodorakopoulos, A., Rovithakis, G.A.: A simplified adaptive neural network prescribed performance controller for uncertain MIMO feedback linearizable systems. IEEE Trans. Neural Netw. Learn. Syst. 26(3), 589–605 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Bechlioulis, C.P., Doulgeri, Z., Rovithakis, G.A.: Guaranteeing prescribed performance and contact maintenance via an approximation free robot force/position controller. Automatica 48(2), 360–365 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Karayiannidis, Y., Doulgeri, Z.: Model-free robot joint position regulation and tracking with prescribed performance guarantees. Roboti. Auton. Syst. 60(2), 214–226 (2012)CrossRefGoogle Scholar
  28. 28.
    Kostarigka, A.K., Doulgeri, Z., Rovithakis, G.A.: Prescribed performance tracking for flexible joint robots with unknown dynamics and variable elasticity. Automatica 49(5), 1137–1147 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Na, J., Chen, Q., Ren, X., Guo, Y.: Adaptive prescribed performance motion control of servo mechanisms with friction compensation. IEEE Trans. Ind. Electron. 61(1), 486–494 (2013)CrossRefGoogle Scholar
  30. 30.
    Psomopoulou, E., Theodorakopoulos, A., Doulgeri, Z., Rovithakis, G.A.: Prescribed performance tracking of a variable stiffness actuated robot. IEEE Trans. Control Syst. 23(5), 1914–1926 (2015)CrossRefGoogle Scholar
  31. 31.
    Bechlioulis, C.P., Karras, G.C., Heshmati, A.S.: Trajectory tracking with prescribed performance for under actuated underwater vehicles under model uncertainties and external disturbances. IEEE Trans. Control Syst. 25(2), 429–440 (2017)CrossRefGoogle Scholar
  32. 32.
    Marantos, P., Bechlioulis, C.P., Kyriakopoulos, K.J.: Robust trajectory tracking control for small-scale unmanned helicopters with model uncertainties. IEEE Trans. Control Syst. 99, 1–12 (2017)Google Scholar
  33. 33.
    Liu, Y.J., Tong, S.: Adaptive fuzzy output-feedback control of pure-feedback uncertain nonlinear systems with unknown dead zone. IEEE Trans. Fuzzy Syst. 23(5), 1341–1347 (2015)Google Scholar
  34. 34.
    Wu, L.B., Yang, G.H.: Adaptive fault-tolerant control of a class of nonaffine nonlinear systems with mismatched parameter uncertainties and disturbances. Nonlinear Dyn. 82(3), 1–11 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wu, L.B., Yang, G.H., Wang, H., Wang, F.: Adaptive fuzzy asymptotic tracking control of uncertain nonaffine nonlinear systems with non-symmetric dead-zone nonlinearities. Int. J. Inf. Sci. 348, 1–14 (2016)MathSciNetGoogle Scholar
  36. 36.
    Liu, Y.: Adaptive tracking control for a class of uncertain pure-feedback systems. Int. J. Robust Nonlinear Control 26(5), 1143–1154 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hou, M., Zhang, Z., Deng, Z., Duan, G.: Global robust finite-time stabilization of unknown pure-feedback systems with input dead-zone non-linearity. IET Control Theory Appl. 10(2), 234–243 (2016)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Yu, Z.X., Yu, Z.Y.: Adaptive neural dynamic surface control for nonlinear pure-feedback systems with multiple time-varying delays: a Lyapunov-Razumikhin method. Asian J. Control 15(4), 1124–1138 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Jiang, B., Shen, Q., Shi, P.: Neural-networked adaptive tracking control for switched nonlinear pure-feedback systems under arbitrary switching. Pergamon Press 61, 119–125 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Li, X.Q., Liu, L., Fu, Z., Yuan, L.: Adaptive NN dynamic surface control for a class of uncertain pure-feedback systems. IEEE Int. Conf. Inf. Auto. (2017). Google Scholar
  41. 41.
    Yu, Z., Li, S., Yu, Z.: Adaptive neural control for a class of pure-feedback nonlinear time-delay systems with asymmetric saturation actuators. Neurocomputing 173, 1461–1470 (2016)CrossRefGoogle Scholar
  42. 42.
    Wang, M., Wang, C.: Neural learning control of pure-feedback nonlinear systems. Nonlinear Dyn. 79(4), 2589–2608 (2015)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Bartolini, G., Punta, E.: Sliding mode output-feedback stabilization of uncertain nonlinear nonaffine systems. Automatica 48(12), 3106–3113 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Zhou, J., Li, X.: Finite-time sliding mode control design for unknown non-affine pure-feedback systems. Math. Probl. Eng. (2015). Google Scholar
  45. 45.
    Zhang, Z., Duan, G., Hou, M.: Global finite time stabilization of pure-feedback systems with input dead zone nonlinearity. J. Frankl. I. (2017). doi: 10.1016/j.jfranklin.2017.12.040.Google Scholar
  46. 46.
    Wang, Y., Song, Y.: Fraction dynamic-surface-based neuro-adaptive finite-time containment control of multi-agent systems in non-affine pure-feedback form. IEEE Trans Neural Netw Learn Syst. 99, 678–689 (2017)CrossRefGoogle Scholar
  47. 47.
    Zou, A.M., Hou, Z.G., Tan, M.: Adaptive control of a class of nonlinear pure-feedback systems using fuzzy backstepping approach. IEEE Trans. Fuzzy Syst. 16(4), 886–897 (2008)CrossRefGoogle Scholar
  48. 48.
    Tong, S., Li, Y.: Adaptive fuzzy output feedback backstepping control of pure-feedback nonlinear systems via dynamic surface control technique. Int. J. Adapt. Control Signal Process. 27(7), 541–561 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Labiod, S., Guerra, T.M.: Indirect adaptive fuzzy control for a class of nonaffine nonlinear systems with unknown control directions. Int. J. Control Autom. Syst. 8(4), 903–907 (2010)CrossRefGoogle Scholar
  50. 50.
    Hou, M., Deng, Z., Duan, G.: adaptive control of uncertain pure-feedback nonlinear systems. Int. J. Syst. Sci. 48(10), 2137–2145 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Liu, Z., Dong, X., Xue, J., Li, H., Chen, Y.: Adaptive neural control for a class of pure-feedback nonlinear systems via dynamic surface technique. IEEE Trans. Neural Netw. Learn. Syst. 27(9), 1969–1975 (2016)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Sui, S., Tong, S., Li, Y.: Observer-based fuzzy adaptive prescribed performance tracking control for nonlinear stochastic systems with input saturation. Neurocomputing 158, 100–108 (2015)CrossRefGoogle Scholar
  53. 53.
    Esfandiari, K., Abdollahi, F., Talebi, H.A.: Adaptive control of uncertain nonaffine nonlinear systems with input saturation using neural networks. IEEE Trans. Neural Netw. Learn. Syst 26(10), 2311–2322 (2015)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Li, Y., Tong, S., Li, T.: Composite adaptive fuzzy output feedback control design for uncertain nonlinear strict-feedback systems with input saturation. IEEE Trans. Cybern. 45(10), 2299–2308 (2015)CrossRefGoogle Scholar
  55. 55.
    Li, Y., Tong, S., Li, T.: Hybrid fuzzy adaptive output feedback control design for uncertain MIMO nonlinear systems with time-varying delays and input saturation. IEEE Trans. Fuzzy Syst. 24(4), 841–853 (2016)CrossRefGoogle Scholar
  56. 56.
    Yang, Q., Chen, M.: Adaptive neural prescribed performance tracking control for near space vehicles with input nonlinearity. Neurocomputing 174, 780–789 (2016)CrossRefGoogle Scholar
  57. 57.
    Yang, Y., Ge, C., Wang, H., et al.: Adaptive neural network based prescribed performance control for teleportation system under input saturation. J. Franklin I. 352(5), 1850–1866 (2015)CrossRefGoogle Scholar
  58. 58.
    Zhou, Q., Wang, L., Wu, C., Li, H., Du, H.: Adaptive fuzzy control for nonstrict-feedback systems with input saturation and output constraint. IEEE Trans. Syst. Man Cybern. 47(1), 1–12 (2017)CrossRefGoogle Scholar
  59. 59.
    Hartman, E.J., Keeler, J.D., Kowalski, J.M.: Layered neural networks with Gaussian hidden units as universal approximations. Neurocomputing 2(2), 210–215 (1990)Google Scholar
  60. 60.
    Park, J., Sandberg, L.W.: Universal approximation using radial-basis-function networks. Neurocomputing 3(2), 246–257 (1990)Google Scholar
  61. 61.
    Wang, L.X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Equipment Management and Unmanned Aircraft Engineering CollegeAir Force Engineering UniversityXi’anChina

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