Adaptive State Observers for Incrementally Quadratic Nonlinear Systems with Application to Chaos Synchronization

  • Hongzhi Zhang
  • Wei ZhangEmail author
  • Younan Zhao
  • Mingming Ji
  • Lixin Huang


This paper is concerned with the problem of adaptive state observer design for a class of incrementally quadratic nonlinear systems with parameter uncertainty. Its nonlinearity satisfies the condition of incremental quadratic constraints which contains many commonly encountered nonlinearities in existing literature as some special cases. A circle criterion-based adaptive observer is first designed to expand the freedom of observer design. Based on the incremental quadratic constraint condition, sufficient conditions ensuring asymptotic stability of estimation error dynamics are established in terms of linear matrix inequalities. Finally, the adaptive observer is applied to the synchronization design of chaotic systems with parameter uncertainty, which fully demonstrates the effectiveness of the proposed observer.


Adaptive state observers Nonlinear systems Incremental quadratic constraints Circle criterion-based observers Chaos synchronization 



This work is supported by the National Natural Science Foundation of China under Grant No. 61603241 and the Project of Science and Technology Commission of Shanghai Municipality under Grant No. 17030501400.


  1. 1.
    M. Abbaszadeh, H.J. Marquez, Nonlinear observer design for one-sided Lipschitz systems, in Proceedings of the 2010 American Control Conference, pp. 5284–5289 (2010)Google Scholar
  2. 2.
    B. Açıkmeşe, M. Corless, Observers for systems with nonlinearities satisfying incremental quadratic constraints. Automatica 47(7), 1339–1348 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    I. Ahmad, M. Shafiq, M. Shahzad, Global finite-time multi-switching synchronization of externally perturbed chaotic oscillators. Circuits Syst. Signal Process. 37(12), 5253–5278 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Arcak, P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities. IEEE Trans. Autom. Control 46(7), 1146–1150 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. Ayati, H. Khaloozadeh, A stable adaptive synchronization scheme for uncertain chaotic systems via observer. Chaos Soliton. Fract. 42(4), 2473–2483 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Barbata, M. Zasadzinski, H.S. Ali, H. Messaoud, Exponential observer for a class of one-sided Lipschitz stochastic nonlinear systems. IEEE Trans. Autom. Control 60(1), 259–264 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    M. Benallouch, M. Boutayeb, M. Zasadzinski, Observer design for one-sided Lipschitz discrete-time systems. Syst. Control Lett. 61(9), 879–886 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S. Bowong, J.J. Tewa, Unknown inputs adaptive observer for a class of chaotic systems with uncertainties. Math. Comput. Model. 48(11), 1826–1839 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    L. Cao, H. Li, N. Wang, Q. Zhou, Observer-based event-triggered adaptive decentralized fuzzy control for nonlinear large-scale systems. IEEE Trans. Fuzzy Syst. 27(6), 1201–1214 (2018)CrossRefGoogle Scholar
  11. 11.
    L. Cao, Q. Zhou, G. Dong, Observer-based adaptive event-triggered control for nonstrict-feedback nonlinear systems with output constraint and actuator failures. IEEE Trans. Syst. Man Cybern. Syst. (2019).
  12. 12.
    A. Chakrabarty, M.J. Corless, G.T. Buzzard, S.H. Żak, A.E. Rundell, State and unknown input observers for nonlinear systems with bounded exogenous inputs. IEEE Trans. Autom. Control 62(11), 5497–5510 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Y. Cho, R. Rajamani, A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Trans. Autom. Control 42(4), 534–537 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    L. D’Alto, M. Corless, Incremental quadratic stability. Numer. Algebra Control Optim. 3(1), 175–201 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    H. Dimassi, A. Loria, Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication. IEEE Trans. Circuits Syst. I, Reg. Papers 58(4), 800–812 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    H. Dimassi, A. Loría, S. Belghith, A new secured transmission scheme based on chaotic synchronization via smooth adaptive unknown-input observers. Commun. Nonlinear Sci. Numer. Simulat. 17(9), 3727–3739 (2012)CrossRefzbMATHGoogle Scholar
  17. 17.
    M. Ekramian, F. Sheikholeslam, S. Hosseinnia, M.J. Yazdanpanah, Adaptive state observer for Lipschitz nonlinear systems. Syst. Control Lett. 62(4), 319–323 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    H. Li, Y. Gao, P. Shi, H.K. Lam, Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity. IEEE Trans. Autom. Control 61(9), 2745–2751 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    X. Li, D. Ho, J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems. Automatica 99, 361–368 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    W. Li, Z. Liu, J. Miao, Adaptive synchronization for a unified chaotic system with uncertainty. Commun. Nonlinear Sci. Numer. Simulat. 15(10), 3015–3021 (2010)CrossRefGoogle Scholar
  21. 21.
    X. Li, J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64, 63–69 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    X. Li, J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay. IEEE Trans. Autom. Control 63(1), 306–311 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    H. Liang, Z. Zhang, C. K. Ahn, Event-triggered fault detection and isolation of discrete-time systems based on geometric technique. IEEE Trans. Circuits Syst. II, Exp. Briefs (2019).
  24. 24.
    H. Liang, Y. Zhang, T. Huang, H. Ma, Prescribed performance cooperative control for multiagent systems with input quantization. IEEE Trans. Cybern. (2019).
  25. 25.
    H. Ma, H. Li, H. Liang, G. Dong, Adaptive fuzzy event-triggered control for stochastic nonlinear systems with full state constraints and actuator faults. IEEE Trans. Fuzzy Syst. (2019).
  26. 26.
    L. Pecora, T. Carroll, Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    M. Pourgholi, V.J. Majd, A nonlinear adaptive resilient observer design for a class of Lipschitz systems using LMI. Circuits Syst. Signal Process. 30(6), 1401–1415 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    R. Rajamani, Observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 43(3), 397–401 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    H. Su, H. Wu, X. Chen, Observer-based discrete-time nonnegative edge synchronization of networked systems. IEEE Trans. Neural Netw. Learn. Syst. 28(10), 2446–2455 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    H. Su, H. Wu, X. Chen, M.Z.Q. Chen, Positive edge consensus of complex networks. IEEE Trans. Syst. Man Cybern. Syst. 48(12), 2242–2250 (2018)CrossRefGoogle Scholar
  31. 31.
    H. Su, H. Wu, J. Lam, Positive edge-consensus for nodal networks via output feedback. IEEE Trans. Autom. Control 64(3), 1244–1249 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    G. Tao, A simple alternative to the Barbalat lemma. IEEE Trans. Autom. Control 42(5), 698 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    S.J. Theesar, P. Balasubramanian, Secure communication via synchronization of Lure systems using sampled-data controller. Circuits Syst. Signal Process. 33(1), 37–52 (2014)CrossRefGoogle Scholar
  34. 34.
    E. Wu, X. Yang, Generalized lag synchronization of neural networks with discontinuous activations and bounded perturbations. Circuits Syst. Signal Process. 34(7), 2381–2394 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    R. Wu, W. Zhang, F. Song, Z. Wu, W. Guo, Observer-based stabilization of one-sided Lipschitz systems with application to flexile link manipulator. Adv. Mech. Eng. 7(12), 1–8 (2015)CrossRefGoogle Scholar
  36. 36.
    A. Zemouche, M. Boutayeb, A unified \(H_\infty \) adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities. Syst. Control Lett. 58(4), 282–28 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    L. Zhang, H. Liang, Y. Sun, C. K. Ahh, Adaptive event-triggered fault detection for semi-Markovian jump systems with output quantization. IEEE Trans. Syst. Man Cybern. Syst. (2019).
  38. 38.
    Z. Zhang, H. Liang, C. Wu, C.K. Ahn, Adaptive event-triggered output feedback fuzzy control for nonlinear networked systems with packet dropouts and random actuator failure. IEEE Trans. Fuzzy Syst. (2019).
  39. 39.
    W. Zhang, H. Su, H. Wang, Z. Han, Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations. Commun. Nonlinear Sci. Numer. Simulat. 17(12), 4968–4977 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    W. Zhang, H. Su, F. Zhu, G.M. Azar, Unknown input observer design for one-sided Lipschitz nonlinear systems. Nonlinear Dyn. 79(2), 1469–1479 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    W. Zhang, H. Su, F. Zhu, M. Wang, Observer-based \(H_{\infty }\) synchronization and unknown input recovery for a class of digital nonlinear systems. Circuits Syst. Signal Process. 32, 2867–2881 (2013)MathSciNetCrossRefGoogle Scholar
  42. 42.
    W. Zhang, H. Su, F. Zhu, D. Yue, A note on observers for discrete-time Lipschitz nonlinear systems. IEEE Trans. Circuits Syst. II, Exp. Briefs 59(2), 123–127 (2012)CrossRefGoogle Scholar
  43. 43.
    W. Zhang, Y. Zhao, M. Abbaszadeh, M. Ji, Full-order and reduced-order exponential observers for discrete-time nonlinear systems with incremental quadratic constraints. J. Dyn. Syst. Meas. Control 141, 041005-1-9 (2019)CrossRefGoogle Scholar
  44. 44.
    Y. Zhao, W. Zhang, H. Su, J. Yang, Observer-based synchronization of chaotic systems satisfying incremental quadratic constraints and its application in secure communication. IEEE Trans. Syst. Man Cybern. Syst. (2019).
  45. 45.
    Y. Zhao, W. Zhang, W. Zhang, F. Song, Exponential reduced-order observers for nonlinear systems satisfying incremental quadratic constraints. Circuits Syst. Signal Process. 37(9), 3725–3738 (2018)MathSciNetCrossRefGoogle Scholar
  46. 46.
    F. Zhu, Z. Han, A note on observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 47(10), 1751–1754 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory of Intelligent Control and RoboticsShanghai University of Engineering ScienceShanghaiChina
  2. 2.School of Information Technology and Mechanical and Electrical IntegrationShanghai Zhongqiao CollegeShanghaiChina

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