Kybernetika 55 no. 5, 831-851, 2019

Tracking control design for nonlinear polynomial systems via augmented error system approach and block pulse functions technique

Bassem Iben Warrad, Mohamed Karim Bouafoura and Naceur Benhadj BraiekDOI: 10.14736/kyb-2019-5-0831


In this paper, tracking control design for a class of nonlinear polynomial systems is investigated by augmented error system approach and block pulse functions technique. The proposed method is based on the projection of the close loop augmented system and the associated linear reference model that it should follow over a basis of block pulse functions. The main advantage of using this tool is that it allows to transform the analytical differential calculus into an algebraic one relatively easy to solve. The developments presented have led to the formulation of a linear system of algebraic equations depending only on parameters of the feedback control. Once the control gains are determined by solving the latter optimization problem in least square sense, the practical stability of the closed loop augmented system is checked through given conditions. A double inverted pendulums benchmark is used to validate the proposed tracking control method.


tracking control, nonlinear polynomial systems, augmented error system approach, block pulse functions, practical stability


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  1. O. Balatif, M. Rachik, E. Labriji and Z. Rachik: Optimal control problem for a class of bilinear systems via block pulse functions. J. Math. Control Inform. 34 (2017), 973-986.   DOI:10.1093/imamci/dnw005
  2. M. M. Belhaouane and N. Benhadj Braiek: Design of stabilizing control for synchronous machines via polynomial modelling and linear matrix inequalities approach. Int. J. Control, Automat. Systems 9 (2011), 425-436.   DOI:10.1007/s12555-011-0301-5
  3. N. Benhadj Braiek: Feedback stabilization and stability domain estimation of nonlinear systems. J. Franklin Inst. 332 (1995), 183-193.   DOI:10.1016/0016-0032(95)00038-x
  4. J. W. Brewer: Kronecker products and matrix calculus in systems theory. IEEE Trans. Circuits Systems 25 (1978), 772-781.   DOI:10.1109/tcs.1978.1084534
  5. W. Chen, Sh. S. Ge, J. Wu and M. Gong: Globally stable adaptive backstepping neural network control for uncertain strict-feedback systems with tracking accuracy known a priori. IEEE Tran. Neural Networks Learning Systems 26 (2015), 1842-1854.   DOI:10.1109/tnnls.2014.2357451
  6. M. Chen, Q. X. Wu and R. X. Cui: Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISA Trans. 52 (2013), 198-206.   DOI:10.1016/j.isatra.2012.09.009
  7. Y. Cui, H. Zhang, Y. Wang and W. Gao: Adaptive control for a class of uncertain strict-feedback nonlinear systems based on a generalized fuzzy hyperbolic model. Fuzzy Sets Systems 302 (2016), 52-64.   DOI:10.1016/j.fss.2015.11.015
  8. Y. Cui, H. Zhang, Q. Qu and C. Luo: Synthetic adaptive fuzzy tracking control for MIMO uncertain nonlinear systems with disturbance observer. Neurocomputing 249 (2017), 191-201.   DOI:10.1016/j.neucom.2017.03.064
  9. A. Deb, S. Roychoudhury and G. Sarkar: Analysis and Identification of Time Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions. Theory and Algorithms with MATLAB. Studies in Systems, Decision and Control, 2016.   DOI:10.1007/978-3-319-26684-8
  10. A. Deb, G. Sarkar and A. Sengupta: Triangular Orthogonal Functions for the Analysis of Continuous Time Systems. Anthem Press, 2012.   DOI:10.7135/upo9781843318118
  11. H. Ghorbel, A. El Hajjaji, M. Souissi. and M. Chaabane: Robust tracking control for TS fuzzy systems with unmeasurable premise variables: Application to tank system. J. Dynamic Systems, Measurement Control 136 (2014), 4.   DOI:10.1115/1.4026467
  12. D. Ginoya, P. D. Shendge and S. B. Phadke: Disturbance observer based sliding mode control of nonlinear mismatched uncertain systems. Comm. Nonlin. Sci. Numer. Simul. 26 (2015), 98-107.   DOI:10.1016/j.cnsns.2015.02.008
  13. L. T. Gruyitch: Nonlinear Systems Tracking. CRC Press. Taylor and Francis Group, 2016.   DOI:10.1201/b19258
  14. C. Hwang and Y. P. Shih: Optimal control of delay systems via block pulse functions. J. Optim. Theory Appl. 45 (1985), 101-112.   DOI:10.1007/bf00940816
  15. E. Jarzebowska: Model-Based Tracking Control of Nonlinear Systems. CRC Press. Taylor and Francis Group, 2012.   DOI:10.1201/b12262
  16. W. J. Jemai, H. Jerbi and M. N. Abdelkrim: Nonlinear state feedback design for continuous polynomial systems. Int. J. Control Automat. Systems 9 (2011), 566-573.   DOI:10.1007/s12555-011-0317-x
  17. V. Lakshmikantham, S. Leela and A. A. Martynyuk: Practical Stability of Nonlinear Systems. World Scientific Publishing, 1990.   DOI:10.1142/1192
  18. H. R. Marzban: Parameter identification of linear multi-delay systems via a hybrid of block-pulse functions and Taylor's polynomials. Int. J. Control 90 (2016), 504-518.   DOI:10.1080/00207179.2016.1186288
  19. Y. Qun Han: Adaptive tracking control of nonlinear systems with dynamic uncertainties using neural network. Int. J. Systems Sci. 49 (2018), 1391-1402.   DOI:10.1080/00207721.2018.1453955
  20. M. Rehan, K. Shik Hong and Sh. Sam Ge: Stabilization and tracking control for a class of nonlinear systems. Nonlinear Analysis: Real World Appl. 12 (2011), 1786-1796.   DOI:10.1016/j.nonrwa.2010.11.011
  21. F. Rotella and G. Dauphin-Tanguy: Nonlinear systems: Identification and optimal control. Int. J. Control 48 (1988), 525-544.   DOI:10.1080/00207178808906195
  22. X. Shao and H. Wang: A Novel Method of Robust Trajectory Linearization Control Based on Disturbance Rejection. Math. Problems Engrg. 2014 (2014), 1-10.   DOI:10.1155/2014/129247
  23. Y. Tang, N. Li, M. Liu, Y. Lu and W. Wang: Identification of fractional-order systems with time delays using block pulse functions. Mech. Systems Signal Process. 91 (2017), 382-394.   DOI:10.1016/j.ymssp.2017.01.008
  24. S. Tong, T. Wang and H. X. Li: Fuzzy robust tracking control for uncertain nonlinear systems. Int. J. Approx. Reason. 30 (2002), 73-90.   DOI:10.1016/s0888-613x(02)00061-0
  25. H. Wang, P. Shi, H. Li and Q. Zhou: Adaptive neural tracking control for a class of nonlinear systems with dynamic uncertainties. IEEE Trans. Cybernet. 47 (2017), 3075-3087.   DOI:10.1109/tcyb.2016.2607166
  26. B. Iben Warrad, M. K. Bouafoura and N. Benhadj Braiek: Tracking control synthesis of nonlinear polynomial systems. In: 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Colmar 2015.   CrossRef
  27. B. Iben Warrad, M. K. Bouafoura and N. Benhadj Braiek: Static output tracking control for non-linear polynomial time-delay systems via block-pulse functions. J. Chinese Inst. Engineers 41 (2018), 194-205.   DOI:10.1080/02533839.2018.1459849
  28. S. Xingling and W. Honglun: Trajectory Linearization Control Based Output Tracking Method for Nonlinear Uncertain System Using Linear Extended State Observer. Asian J. Control 18 (2016), 316-327.   DOI:10.1002/asjc.1053
  29. Y. Xu, J. Zhang and F. Liao: Augmented error system approach to control design for a class of neutral systems. Advances Diff. Equations (2015), 1-17.   DOI:10.1186/s13662-015-0596-2
  30. Y. Yu and Y. Sh. Zhong: Semiglobal Robust Backstepping Output Tracking for Strict-feedback Form Systems with Nonlinear Uncertainty. Int. J. Control Automat. Systems 9 (2011), 366-375.   DOI:10.1007/s12555-011-0219-y
  31. Y. Yu and Y. Zhong: Semi-global robust output tracking for non-linear uncertain systems in strict-feedback form. IET Control Theory Appl. 6 (2012), 751-759.   DOI:10.1186/s13662-015-0596-2
  32. H. Zhang, C. Li and X. Liao: Stability analysis and h infinity controller design of fuzzy large-scale systems based on piecewise Lyapunov functions. IEEE Trans. Syst. Man Cybernet. 36 (2006), 685-698.   DOI:10.1109/tsmcb.2005.860133