Improved Robust Adaptive Control Law for a Class of Uncertain Nonlinear Systems and Its Application to Chaotic Systems

  • Valiollah GhaffariEmail author
  • Abolhassan Razminia
  • Mojtaba Mirzaei
Research Paper


An adaptive robust control scheme is proposed for a class of unknown nonlinear systems. For achieving this purpose, a class of uncertain nonlinear systems is firstly considered. Then, the regulation and tracking problems are solved with the well-known Lyapunov stability theory. It is shown that although the system dynamic may not be fully or partially known, some adaptive control laws can be mathematically derived by using the proposed control method. Hence, by the proposed control method, the closed-loop system would be globally stabilized in the sense of the Lyapunov stability theory. Finally, the proposed robust adaptive control law is numerically applied into the Lorenz and Van der Pol chaotic systems. The efficiency of the suggested approach is shown by numerical simulation via comparison with an existing method.


Lyapunov stability theory Robust adaptive control Uncertain systems Chaotic systems 


  1. Arefi MM (2016) Adaptive robust stabilization of Rossler system with time-varying mismatched parameters via scalar input. J Comput Nonlinear Dyn 11:041024CrossRefGoogle Scholar
  2. Asemani MH, Vatankhah R (2017) NON-PDC observer-based T–S fuzzy tracking controller design and its application in CHAOS control. Asian J Control 19:969–982MathSciNetCrossRefGoogle Scholar
  3. Åström KJ, Wittenmark B (2013) Adaptive control. Courier Corporation, MineolazbMATHGoogle Scholar
  4. Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design. Springer, BerlinCrossRefGoogle Scholar
  5. Berezowski M (2001) Effect of delay time on the generation of chaos in continuous systems. One-dimensional model. Two-dimensional model–tubular chemical reactor with recycle. Chaos, Solitons Fractals 12:83–89CrossRefGoogle Scholar
  6. Che Y, Liu B, Li H, Lu M, Wang J, Wei X (2017) Robust stabilization control of bifurcations in Hodgkin-Huxley model with aid of unscented Kalman filter. Chaos, Solitons Fractals 101:92–99MathSciNetCrossRefGoogle Scholar
  7. Faramin M, Ataei M (2016) Chaotic attitude analysis of a satellite via Lyapunov exponents and its robust nonlinear control subject to disturbances and uncertainties. Nonlinear Dyn 83:361–374MathSciNetCrossRefGoogle Scholar
  8. Geiyer D, Kauffman JL (2015) Chaotic control of a piezomagnetoelastic beam for improved energy harvesting. Act Passiv Smart Struct Integr Syst 2015:943100Google Scholar
  9. Ghaffari V, Karimaghaee P (2012) Design of adaptive discrete time controller for a class of nonlinear systems. Nonlinear Stud 19:149–159MathSciNetzbMATHGoogle Scholar
  10. Hua C, Guan X (2004) Adaptive control for chaotic systems. Chaos Solitons Fractals 22:55–60MathSciNetCrossRefGoogle Scholar
  11. Ioannou PA, Sun J (1996) Robust adaptive control. Prentice-Hall, Upper Saddle RiverzbMATHGoogle Scholar
  12. Khalil HK (2003) Nonlinear systems. Prentice Hall, Englewood CliffsGoogle Scholar
  13. Khan W, Lin Y, Khan SU, Ullah N (2018) Quantized adaptive decentralized control for interconnected nonlinear systems with actuator faults. Appl Math Comput 320:175–189MathSciNetzbMATHGoogle Scholar
  14. Krstic M, Kanellakopoulos I, Kokotovic PV (1995) Nonlinear and adaptive control design. Wiley, HobokenzbMATHGoogle Scholar
  15. Li J, Yue H (2015) Adaptive fuzzy tracking control for stochastic nonlinear systems with unknown time-varying delays. Appl Math Comput 256:514–528MathSciNetzbMATHGoogle Scholar
  16. Liu C, Sun Z, Ye D, Shi K (2018) Robust adaptive variable structure tracking control for spacecraft chaotic attitude motion. IEEE Access 6:3851–3857CrossRefGoogle Scholar
  17. Louodop P, Fotsin H, Bowong S (2012) A strategy for adaptive synchronization of an electrical chaotic circuit based on nonlinear control. Phys Scr 85:025002CrossRefGoogle Scholar
  18. Luo R (2015) The robust adaptive control of chaotic systems with unknown parameters and external disturbance via a scalar input. Int J Adapt Control Signal Process 29:1296–1307MathSciNetCrossRefGoogle Scholar
  19. Mascolo S (1997) Backstepping design for controlling Lorenz chaos. In: Proceedings of the 36th IEEE conference on decision and control, pp 1500–1501Google Scholar
  20. Mierczyński J (2015) Lower estimates of top Lyapunov exponent for cooperative random systems of linear ODEs. Proc Am Math Soc 143:1127–1135MathSciNetCrossRefGoogle Scholar
  21. Mobayen S, Ma J (2018) Robust finite-time composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and time-delay. Chaos, Solitons Fractals 114:46–54MathSciNetCrossRefGoogle Scholar
  22. Mousavi SH, Khayatian A (2013) Adaptive control for a class of hysteretic systems. J Comput Nonlinear Dyn 8:011003CrossRefGoogle Scholar
  23. Nepomuceno EG, Martins SA, Lacerda MJ, Mendes EM (2018) On the use of interval extensions to estimate the largest Lyapunov exponent from chaotic data. Math Probl Eng 2018:6909151MathSciNetCrossRefGoogle Scholar
  24. Ni J, Liu L, Liu C, Hu X, Li S (2017) Fast fixed-time nonsingular terminal sliding mode control and its application to chaos suppression in power system. IEEE Trans Circuits Syst II Express Briefs 64:151–155CrossRefGoogle Scholar
  25. Noroozi N, Roopaei M, Karimaghaee P (2009) Adaptive control and synchronization in a class of partially unknown chaotic systems. Chaos Interdiscipl J Nonlinear Sci 19:023121MathSciNetCrossRefGoogle Scholar
  26. Noroozi N, Roopaei M, Karimaghaee P, Safavi AA (2010) Simple adaptive variable structure control for unknown chaotic systems. Commun Nonlinear Sci Numer Simul 15:707–727MathSciNetCrossRefGoogle Scholar
  27. Pan L, Zhou W, Fang JA, Li D (2010) Analysis of linear and adaptive feedback synchronization in a new unified chaotic system. Int J Adapt Control Signal Process 24:708–716MathSciNetzbMATHGoogle Scholar
  28. Pang Z, Jin D (2016) Experimental verification of chaotic control of an underactuated tethered satellite system. Acta Astronaut 120:287–294CrossRefGoogle Scholar
  29. Roopaei M, Zolghadri Jahromi M (2008) Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control. Chaos Interdiscip J Nonlinear Sci 18:033133MathSciNetCrossRefGoogle Scholar
  30. Shen Z, Li J (2017) Chaos control for a unified chaotic system using output feedback controllers. Math Comput Simul 132:208–219MathSciNetCrossRefGoogle Scholar
  31. Tong S, Zhang L, Li Y (2016) Observed-based adaptive fuzzy decentralized tracking control for switched uncertain nonlinear large-scale systems with dead zones. IEEE Trans Syst Man Cybern Syst 46:37–47CrossRefGoogle Scholar
  32. Tran X-T, Kang H-J (2015) Robust adaptive chatter-free finite-time control method for chaos control and (anti-) synchronization of uncertain (hyper) chaotic systems. Nonlinear Dyn 80:637–651MathSciNetCrossRefGoogle Scholar
  33. Wang Z, Tian Q, Hu H, Flores P (2016) Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance. Nonlinear Dyn 86:1571–1597CrossRefGoogle Scholar
  34. Wang P, Jin W, Su H (2018) Synchronization of coupled stochastic complex-valued dynamical networks with time-varying delays via aperiodically intermittent adaptive control. Chaos Interdiscip J Nonlinear Sci 28:043114MathSciNetCrossRefGoogle Scholar
  35. Xi X, Mobayen S, Ren H, Jafari S (2018) Robust finite-time synchronization of a class of chaotic systems via adaptive global sliding mode control. J Vib Control 24:3842–3854MathSciNetCrossRefGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Electrical Engineering, School of EngineeringPersian Gulf UniversityBushehrIran
  2. 2.Hydro-Aeronautical Research CenterShiraz UniversityShirazIran

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