Globally Robust Adaptive Critic Based Neuro-Integral Terminal Sliding Mode Technique with UDE for Nonlinear Systems

  • Deepika DeepikaEmail author
  • Shiv Narayan
  • Sandeep Kaur
Regular Paper


This paper presents an innovative neural network based optimal integral terminal sliding mode control framework for stabilization of an uncertain affine class of nonlinear systems. In literature, an uncertainty and disturbance estimator (UDE) has been successfully integrated with sliding mode control to diminish the influence of unknown system uncertainties through a low pass filter. However, this estimator causes an initial control signal to shoot up to a very large value which may undesirably affect the functioning of the connected actuators and sensing devices. To this end, this paper utilizes a single network continuous adaptive critic optimal technique which would significantly reduce the initial large control magnitude generated by UDE technique. To ensure the global robustness, an exponential bilateral decay function is employed in the proposed control law. Moreover, the excellent approximation characteristic of radial basis feed-forward neural network is also exploited to estimate the partial non-linear dynamics. The closed loop stability is also guaranteed with the proposed technique through Lyapunov’s principle. Finally, two practical examples are simulated to state the efficacy of the proposed method and comparison with the prior methods is also provided in the presented work.


Sliding mode control (SMC) Radial basis feed-forward neural network (RBFNN) Uncertainty and disturbance estimator (UDE) Lyapunov’s stability theorem Single network continuous adaptive critic (SNCAC) Optimal control 



  1. 1.
    Bertsekas, D. P. (2000). Dynamic programming and optimal control. Belmont. MA: Athena Scientific.Google Scholar
  2. 2.
    Tang, G. Y. (2005). Suboptimal control for nonlinear systems: a successive approximation approach. Systems & Control Letters,54(5), 429–434.MathSciNetCrossRefGoogle Scholar
  3. 3.
    He, B., Huang, S., & Wang, J. (2015). Product low-carbon design using dynamic programming algorithm. International Journal of Precision Engineering and Manufacturing-Green Technology,2(1), 37–42.CrossRefGoogle Scholar
  4. 4.
    Liu, D., Wang, D., Wang, F. Y., Li, H., & Yang, X. (2014). Neural-network-based online HJB solution for optimal robust guaranteed cost control of continuous-time uncertain nonlinear systems. IEEE Transactions on Cybernetics,44(12), 2834–2847.CrossRefGoogle Scholar
  5. 5.
    Werbos, P. (1992). Approximate dynamic programming for realtime control and neural modelling. Handbook of intelligent control: neural, fuzzy and adaptive approaches, 15, 493–525.Google Scholar
  6. 6.
    Padhi, R., Unnikrishnan, N., Wang, X., & Balakrishnan, S. N. (2006). A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems. Neural Networks,19(10), 1648–1660.CrossRefGoogle Scholar
  7. 7.
    Fang, J., Zhang, L., Long, Z., & Wang, M. Y. (2018). Fuzzy adaptive sliding mode control for the precision position of piezo-actuated nano positioning stage. International Journal of Precision Engineering and Manufacturing,19(10), 1447–1456.CrossRefGoogle Scholar
  8. 8.
    Mobayen, S., & Tchier, F. (2018). Composite nonlinear feedback integral sliding mode tracker design for uncertain switched systems with input saturation. Communications in Nonlinear Science and Numerical Simulation,65, 173–184.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu, J., Peng, Q., Huang, Z., Liu, W., & Li, H. (2018). Enhanced sliding mode control and online estimation of optimal slip ratio for railway vehicle braking systems. International Journal of Precision Engineering and Manufacturing,19(5), 655–664.CrossRefGoogle Scholar
  10. 10.
    Xiu, C., Hou, J., Xu, G., & Zang, Y. (2017). Improved fast global sliding mode control based on the exponential reaching law. Advances in Mechanical Engineering,9(2), 1687814016687967.CrossRefGoogle Scholar
  11. 11.
    Park, S. C., Lee, J. M., & Han, S. I. (2018). Tracking error constrained terminal sliding mode control for ball-screw driven motion systems with state observer. International Journal of Precision Engineering and Manufacturing,19(3), 359–366.CrossRefGoogle Scholar
  12. 12.
    Zhong, Q. C., & Rees, D. (2004). Control of uncertain LTI systems based on an uncertainty and disturbance estimator. Journal of Dynamic Systems, Measurement, and Control,126(4), 905–910.CrossRefGoogle Scholar
  13. 13.
    Balakrishnan, S. N., & Biega, V. (1996). Adaptive-critic-based neural networks for aircraft optimal control. Journal of Guidance, Control and Dynamics,19(4), 893–898.CrossRefGoogle Scholar
  14. 14.
    Han, D., & Balakrishnan, S. N. (2002). Adaptive critic-based neural networks for agile missile control. Journal of Guidance, Control and Dynamics,25(2), 404–407.CrossRefGoogle Scholar
  15. 15.
    Wang, D., & Mu, C. (2017). Adaptive-critic-based robust trajectory tracking of uncertain dynamics and its application to a spring–mass–damper system. IEEE Transactions on Industrial Electronics,65(1), 654–663.CrossRefGoogle Scholar
  16. 16.
    Jiang, Y., & Jiang, Z. P. (2014). Robust adaptive dynamic programming and feedback stabilization of nonlinear systems. IEEE Transactions on Neural Networks and Learning Systems,25(5), 882–893.CrossRefGoogle Scholar
  17. 17.
    Gao, W., Jiang, Y., Jiang, Z. P., & Chai, T. (2016). Output-feedback adaptive optimal control of interconnected systems based on robust adaptive dynamic programming. Automatica,72, 37–45.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yang, X., Liu, D., Wei, Q., & Wang, D. (2016). Guaranteed cost neural tracking control for a class of uncertain nonlinear systems using adaptive dynamic programming. Neurocomputing,198, 80–90.CrossRefGoogle Scholar
  19. 19.
    Kumar, S., Padhi, R., & Behera, L. (2008). Continuous-time single network adaptive critic for regulator design of nonlinear control affine systems. IFAC Proceedings Volumes,41(2), 8797–8802.CrossRefGoogle Scholar
  20. 20.
    Mitra, A., & Behera, L. (2015). Continuous-time single network adaptive critic based optimal sliding mode control for nonlinear control affine systems. In 2015 34th Chinese Control Conference (CCC) (pp. 3300–3306). IEEE.Google Scholar
  21. 21.
    Das, M., & Mahanta, C. (2014). Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions,53(4), 1191–1198.CrossRefGoogle Scholar
  22. 22.
    Das, M., & Mahanta, C. (2016, January). Disturbance observer based optimal second order sliding mode controller for nonlinear systems with mismatched uncertainty. In 2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI) (pp. 361–365). IEEE.Google Scholar
  23. 23.
    Talole, S. E., & Phadke, S. B. (2008). Model following sliding mode control based on uncertainty and disturbance estimator. Journal of Dynamic Systems, Measurement, and Control,130(3), 034501.CrossRefGoogle Scholar
  24. 24.
    Suryawanshi, P. V., Shendge, P. D., & Phadke, S. B. (2014). Robust sliding mode control for a class of nonlinear systems using inertial delay control. Nonlinear Dynamics,78(3), 1921–1932.CrossRefGoogle Scholar
  25. 25.
    Kuperman, A., & Zhong, Q. C. (2011). Robust control of uncertain nonlinear systems with state delays based on an uncertainty and disturbance estimator. International Journal of Robust and Nonlinear Control,21(1), 79–92.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Deepika, Kaur, S., & Narayan, S. (2018). Fractional order uncertainty estimator based hierarchical sliding mode design for a class of fractional order non-holonomic chained system. ISA Transactions,77, 58–70.CrossRefGoogle Scholar
  27. 27.
    Mirzaeinejad, H. (2018). Robust predictive control of wheel slip in antilock braking systems based on radial basis function neural network. Applied Soft Computing,70, 318–329.CrossRefGoogle Scholar
  28. 28.
    Deepika, D., Narayan, S., & Kaur, S. (2019). Robust finite time integral sliding mode tracker for nth-order non-affine non-linear system with uncertainty and disturbance estimator. Mathematics and Computers in Simulation,156, 364–376.MathSciNetCrossRefGoogle Scholar

Copyright information

© Korean Society for Precision Engineering 2019

Authors and Affiliations

  1. 1.Department of Electrical EngineeringPunjab Engineering College (Deemed to be University)ChandigarhIndia

Personalised recommendations