Neural Computing and Applications

, Volume 24, Issue 3–4, pp 637–648 | Cite as

Deterministic learning and neural control of a class of nonlinear systems toward improved performance

Original Article


A deterministic learning theory was recently presented which states that an appropriately designed adaptive neural controller can learn the system internal dynamics while attempting to control a class of nonlinear systems in normal form. In this paper, we further investigate deterministic learning of the class of nonlinear systems with relaxed conditions, and neural control of the class of system toward improved performance. Firstly, without the assumption on the upper bound of the derivative of the unknown affine term, an adaptive neural controller is proposed to achieve stability and tracking of the plant states to that of the reference model. When output tracking is achieved, a partial PE condition is satisfied, and deterministic learning from adaptive neural control of the class of nonlinear systems is implemented without the priori knowledge on the upper bound of the derivative of the affine term. Secondly, by utilizing the obtained knowledge of system dynamics, a neural controller with constant RBF networks embedded is presented, in which the learned knowledge can be effectively exploited to achieve stability and improved control performance. Simulation studies are included to demonstrate the effectiveness of the results.


Deterministic learning Input-to-state stability Small-gain theorem Adaptive neural control Learning control 


  1. 1.
    Kwan C, Lewis FL (2000) Robust backstepping control of nonlinear systems using neural network. IEEE Trans Syst Man Cybernet Part A 30(6):753–766CrossRefGoogle Scholar
  2. 2.
    Wang D, Huang J (2002) Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form. Automatica 38(8):1365–1372CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Lewis FL, Yesildirek A, Liu K (1996) Multilayer neural-net robot controller with guaranteed tracking performance. IEEE Trans Neural Netw 7(2):388–398CrossRefGoogle Scholar
  4. 4.
    Polycarpou MM, Mears MJ (1998) Stable adaptive tracking of uncertain systems using nonlinearly parameterized on-line approximators. Int J Control Autom Syst 70(3):363–384CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Narendra KS, Parthasarathy K (1990) Identification and control of dynamic systems using neural networks. IEEE Trans Neural Netw 1(1):4–27CrossRefGoogle Scholar
  6. 6.
    Rovithakis George A (2001) Stable adaptive neuro-control design via Lyapunov function derivative estimation. Automatica 37(8):1213–1221CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chen FC, Khalil HK (1995) Adaptive control of a class of nonlinear discrete time systems using neural networks. IEEE Trans Autom Control 40(5):791–801CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Spooner JT, Passino KM (1996) Stable adaptive control using fuzzy systems and neural networks. IEEE Trans Fuzzy Syst 4(3):339–359CrossRefGoogle Scholar
  9. 9.
    Sanner RM, Slotine JE (1992) Gaussian networks for direct adaptive control. IEEE Trans Neural Netw 3(6):837–863CrossRefGoogle Scholar
  10. 10.
    Wang M, Chen B, Zhang Siying (2009) Adaptive neural tracking control of nonlinear time-delay systems with disturbances. Int J Adapt Control Signal Process 23(11):1031–1049CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Mears Mark J, Polycarpou Marios M (2003) Stable neural control of uncertain multivariable systems. International Journal of Adaptive Control and Signal Process 17(6):447–466CrossRefMATHGoogle Scholar
  12. 12.
    Christofides PD, Teel AR (1996) Singular perturbations and input-to-state stability. IEEE Trans Autom Control 41(11):1645–1650CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Shilnikov LP et al (2001) Methods of qualitative theory in nonlinear dynamics, part II. World Scientific, SingaporeGoogle Scholar
  14. 14.
    Chen L, Narendra Kumpati S (2003) Intelligent control using multiple neural networks. Int J Adapt Control Signal Process 17(6):417–430CrossRefMATHGoogle Scholar
  15. 15.
    Fu Y, Chai T, Yue H (2008) Intelligent control using multiple models and neural networks. Int J Adapt Control Signal Process 22(5):495–509CrossRefMathSciNetGoogle Scholar
  16. 16.
    Zhang T, GE SS, Hang CC (1999) Design and performance analysis of a direct adaptive controller for nonlinear systems. Automatica 35(11):1809–1817CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Zhang T, GE SS, Hang CC (2000) Adaptive neural networks control for strict-feedback nonlinear systems using backstepping design. Automatica 36(12):1835–1846MATHMathSciNetGoogle Scholar
  18. 18.
    Zhang T, Ge SS, Hang CC (2000) Stable adaptive control for a class of nonlinear systems using a modified Lyapunov function. IEEE Trans Autom Control 45(1):129–132CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Ge SS, Wang C (2002) Direct adaptive NN control of a class of nonlinear systems. IEEE Trans Neural Netw 13(1):214–221CrossRefMathSciNetGoogle Scholar
  20. 20.
    Ge SS, Wang C (2002) Adaptive NN control of uncertain nonlinear pure-feedback systems. Automatica 38(4):671–682CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Bechlioulis CP, Rovithakis GA (2009) Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica 45(2):532–538CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Wang C, Hill DJ, Ge SS, Chen GR (2006) An iss-modular approach for adaptive neural control of pure-feedback systems. Automatica 42(5):723–731CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    wang C, Hill DJ (2006) Learning from neural control. IEEE Trans Neural Netw 17(1):130–146CrossRefGoogle Scholar
  24. 24.
    Liu T, Wang C, Hill David J (2009) Learning from neural control of nonlinear systems in normal form. Syst Control Lett 58(9):633–638CrossRefMATHGoogle Scholar
  25. 25.
    Wang C, Hill DJ (2009) Deterministic learning theory for identification, recognition and control. CRC Press, Boca RatonGoogle Scholar
  26. 26.
    Jiang ZP, Teel AR, Praly L (1994) Small-gain theorem for iss systems and applications. Math Control Signals Syst 7(2):95–120CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Arcak M, Teel A (2002) Input-to-state stability for a class of Lurie systems. Automatica 38(11):497–500CrossRefMathSciNetGoogle Scholar
  28. 28.
    Wang M, Wang C, Zhang SY (2009) Direct adaptive neural control of completely non-affine pure-feedback monlinear systems with small-gain approach. In: Proceedings of 21st IEEE Chinese control and decision conference, Guilin, China, 395–400Google Scholar
  29. 29.
    Krstic M, Kanellakopoulos I, Kokotovic PV (1995) Nonlinear and adaptive control design. Wiley, New YorkGoogle Scholar
  30. 30.
    Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  31. 31.
    Wang C, Chen G, Ge SS (2003) Smart neural control of uncertain nonlinear systems. Int J Adapt Control Signal Process Spec Issue Parameter Adapt Learn Comput Intell Syst 17:467–488CrossRefMATHGoogle Scholar
  32. 32.
    Sontag E (1989) Smooth stabilization implies comprime factorizztion. IEEE Trans Autom Control 34(4):435–443CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Kurdila AJ, Narcowich FJ, Ward JD (1995) Persistency of excitation in identification using radial basis function approximants. SIAM J Control Optim 33(2):625–642CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kar Indrani, Behera L (2009) Direct adaptive neural control for affine nonlinear systems. Appl Soft Comput 9(2):756–764CrossRefGoogle Scholar
  35. 35.
    Choi JY, Farrell JA (2000) Nonlinear adaptive control using networks of piecewise linear approximators. IEEE Trans Neural Netw 11(2):390–401CrossRefGoogle Scholar
  36. 36.
    Huaguang Zhang, Yongbin Quan (2001) Modeling, identification, and control of a class of nonlinear systems. IEEE Trans Fuzzy Syst 9(2):349–354CrossRefGoogle Scholar
  37. 37.
    Huaguang Zhang, Zhenwei Liu, Guang-Bin Huang, Zhanshan Wang (2010) Novel weighting-delay-based stability criteria for recurrent neural networks with time-varying delay. IEEE Trans Neural Netw 21(1):91–106CrossRefGoogle Scholar
  38. 38.
    Wang Dan, Huang Jie (2005) Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form. IEEE Trans Neural Netw 16(1):195–202CrossRefGoogle Scholar
  39. 39.
    Zomaya Albert Y (1994) Reinforcement learning for the adaptive control of nonlinear systems. IEEE Trans Syst Man Cybernet 24(2):357–363CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  1. 1.School of Automation and the Center for Control and OptimizationSouth China University of TechnologyGuangzhouPeople’s Republic of China
  2. 2.Department of Electrical and Computer Engineering, Six Metrotech Center, Polytechnic InstituteNew York UniversityBrooklynUSA

Personalised recommendations