Skip to main content

Advertisement

SpringerLink
Log in

A concept of coupled chaotic synchronous observers for nonlinear and adaptive observers-based chaos synchronization

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, new control approaches for synchronization the master and the slave chaotic systems is established by means of novel coupled chaotic synchronous observers and coupled chaotic adaptive synchronous observers. The simultaneous estimation of the master and the slave systems’ states is accomplished, by means of the proposed observers for each of the master and the slave systems, to produce error signals between these estimated states. This estimated synchronization error signal and the state-estimation errors converge to the origin by means of a specific observers-based feedback control signal to ensure synchronization as well as state estimation. Using Lyapunov stability theory, nonadaptive and adaptive control laws and properties of nonlinearities, a convergence condition for the state-estimation errors and the estimated synchronization error is developed in the form of nonlinear matrix inequalities. Solution of the resulted inequality constraints using a two-step approach is presented, which provides the necessary and sufficient condition to obtain values of the observer gain and controller gain matrices. Further, a method requiring less computational efforts for solving the matrix inequalities for obtaining the observer and the controller gain matrices using decoupling technique is also proposed. Numerical simulation of the proposed synchronization technique for FitzHugh–Nagumo neuronal systems is illustrated to elaborate efficaciousness of the proposed observers-based control methodologies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990). doi:10.1103/PhysRevLett.64.1196

    Article  MathSciNet  MATH  Google Scholar 

  2. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Westview Press, USA (1994)

    Google Scholar 

  3. Chadli, M., Zelinka, I.: Chaos synchronization of unknown inputs Takagi–Sugeno fuzzy: application to secure communications. Comput. Math. Appl. 68, 2142–2147 (2014). doi:10.1016/j.camwa.2013.01.013

    Article  MathSciNet  Google Scholar 

  4. Gonzalez-Miranda, J.M.: Synchronization and Control of Chaos. An Introduction for Scientists and Engineers. Imperial College Press, UK (2004). ISBN 9781860944888

  5. Carroll, T., Pecora, L.: Synchronizing chaotic circuits. IEEE Trans. Circuits Syst. 38, 453–456 (1991). doi:10.1109/31.75404

    Article  MATH  Google Scholar 

  6. Yassen, M.T.: Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos Soliton Fract. 26, 913–920 (2005). doi:10.1016/j.chaos.2005.01.047

    Article  MathSciNet  MATH  Google Scholar 

  7. Ali, A.: Synchronization and secure communication of uncertain chaotic systems based on full-order and reduced-order output-affine observers. Appl. Math. Comput. 219, 10000–10011 (2013). doi:10.1016/j.amc.2013.03.133

    MathSciNet  MATH  Google Scholar 

  8. Beyhan, S.: Runge–Kutta model-based nonlinear observer for synchronization and control of chaotic systems. ISA Trans. 52, 501–509 (2013). doi:10.1016/j.isatra.2013.04.005

    Article  Google Scholar 

  9. Ramirez, J.P., Fey, R.H.B., Nijmeijer, H.: Synchronization of weakly nonlinear oscillators with Huygens’ coupling. Chaos: an interdisciplinary. J. Nonlinear Sci. 23, 033118 (2013). doi:10.1063/1.4816360

    MATH  Google Scholar 

  10. Rehan, M., Hong, K.-S.: Robust synchronization of delayed chaotic FitzHugh–Nagumo neurons under external electrical stimulation. Comput. Math. Methods Med. 2012, Article ID 230980 (2012). doi:10.1155/2012/230980

  11. Zaheer, M.H., Rehan, M., Mustafa, G., Ashraf, M.: Delay-range-dependent chaos synchronization approach under varying time-lags and delayed nonlinear coupling. ISA Trans. 53, 1716–1730 (2014). doi:10.1016/j.isatra.2014.09.007

    Article  Google Scholar 

  12. Jeong, S.C., Ji, D.H., Park, J.H., Won, S.C.: Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer. Appl. Math. Comput. 219, 5984–5995 (2013). doi:10.1016/j.amc.2012.12.017

    MathSciNet  MATH  Google Scholar 

  13. Njah, A.N.: Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques. Nonlinear Dyn. 61, 1–9 (2010). doi:10.1007/s11071-009-9626-5

    Article  MathSciNet  MATH  Google Scholar 

  14. Yang, C.C.: Adaptive control and synchronization of identical new chaotic flows with unknown parameters via single input. Appl. Math. Comput. 216, 1316–1324 (2010). doi:10.1016/j.amc.2010.02.026

    MathSciNet  MATH  Google Scholar 

  15. Yang, C.C.: Adaptive synchronization of Lü hyperchaotic system with uncertain parameters based on single-input controller. Nonlinear Dyn. 63, 447–454 (2011). doi:10.1007/s11071-010-9814-3

    Article  MathSciNet  Google Scholar 

  16. Yang, J., Zhu, F.: Synchronization for chaotic systems and chaos-based secure communications via both reduced-order and step-by-step sliding mode observers. Commun. Nonlinear. Sci. Numer. Simul. 18, 926–937 (2013). doi:10.1016/j.cnsns.2012.09.009

    Article  MathSciNet  MATH  Google Scholar 

  17. Mbe, E.S.K., Fotsin, H.B., Kengne, J., Woafo, P.: Parameters estimation based adaptive generalized projective synchronization (GPS) of chaotic Chua’s circuit with application to chaos communication by parametric modulation. Chaos Solitons Fract. 61, 27–37 (2014). doi:10.1016/j.chaos.2014.02.004

    Article  Google Scholar 

  18. Liu, B., Wang, L., Jin, Y.-H., Huang, D.-X., Tang, F.: Control and synchronization of chaotic systems by differential evolution algorithm. Chaos Solitons Fract. 34, 412–419 (2007). doi:10.1016/j.chaos.2006.03.033

    Article  MATH  Google Scholar 

  19. Filal, R.L., Benrejeb, M., Borne, P.: On observer-based secure communication design using discrete-time hyperchaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19(15), 1424–1432 (2014). doi:10.1016/j.cnsns.2013.09.005

    Article  MathSciNet  Google Scholar 

  20. Boutayeb, M., Darouach, M., Rafaralahy, H.: Generalized state-space observers for chaotic synchronization and secure communication. IEEE Trans. Circuits Syst. 49, 345–349 (2002). doi:10.1109/81.989169

    Article  MathSciNet  Google Scholar 

  21. Yang, J., Zhu, F.: Synchronization for chaotic systems and chaos-based secure communications via both reduced-order and step-by-step sliding mode observers. Commun. Nonlinear Sci. Numer. Simul. 18, 926–937 (2013). doi:10.1016/j.cnsns.2012.09.009

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, Y.N., Chen, L., Cai, Z.S., Zhao, X.Z.: Experimental study of chaos synchronization in the Belousov–Zhabotinsky chemical system. Chaos Solitons Fract. 22, 767–771 (2004). doi:10.1016/j.chaos.2004.03.023

    Article  MATH  Google Scholar 

  23. Steinmetz, P.N., Roy, A., Fitzgerald, P.J.: Attention modulates synchronized neuronal firing in primate somatosensory cortex. Nature 404, 457–490 (2000). doi:10.1038/35004588

    Article  Google Scholar 

  24. Meffo, L.P., Woafo, P., Domnganga, S.: Cluster states in a ring of four coupled semiconductor lasers. Commun. Nonlinear Sci. Numer. Simul. 12, 942–952 (2007). doi:10.1016/j.cnsns.2005.10.002

    Article  MATH  Google Scholar 

  25. Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50, 1645–1662 (1990). doi:10.1137/0150098

    Article  MathSciNet  MATH  Google Scholar 

  26. Angeles, R., Nijmeijer, H.: Mutual synchronization of robots via estimated state feedback: a cooperative approach. IEEE T. Contr. Syst. Technol. 12, 542–554 (2004). doi:10.1109/TCST.2004.825065

    Article  Google Scholar 

  27. Kuhnert, L., Agladze, K.I., Krinsky, V.I.: Image processing using light sensitive chemical waves. Nature 337, 244–247 (1989). doi:10.1038/337244a0

    Article  Google Scholar 

  28. Ömer, M., Ercan, S.: Observer based synchronization of chaotic systems. Phys. Rev. E 54, 4803–4811 (1996). doi:10.1103/PhysRevE.54.4803

    Article  Google Scholar 

  29. Heagy, J.F., Carroll, T., Pecora, L.: Synchronous chaos in coupled oscillator systems. Phys. Rev. E 50, 1874–1885 (1994). doi:10.1103/PhysRevE.50.1874

    Article  Google Scholar 

  30. Senejohnnya, D.M., Delavari, H.: Active sliding observer scheme based fractional chaos synchronization. Commun. Nonlinear Sci. Numer. Simul. 17(11), 4373–4383 (2012). doi:10.1016/j.cnsns.2012.03.004

    Article  MathSciNet  MATH  Google Scholar 

  31. Bagheri, P., Shahrokhi, M., Salarieh, H.: Adaptive observer-based synchronization of two non-identical chaotic systems with unknown parameters. J. Vib. Control. (2015). doi:10.1177/1077546315580052

  32. Jeong, S.C., Ji, D.H., Park, J.H., Won, S.C.: Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer. Appl. Math. Comput. 219, 5984–5995 (2013). doi:10.1016/j.amc.2012.12.017

  33. Grassi, G.: Observer-based hyperchaos synchronization in cascaded discrete-time systems. Chaos Solitons Fract. 40, 1029–1039 (2009). doi:10.1016/j.chaos.2007.08.060

    Article  MathSciNet  MATH  Google Scholar 

  34. Wen, S., Zeng, Z., Huang, T.: Observer-based synchronization of memristive systems with multiple networked input and output delays. Nonlinear Dyn. 78, 541–554 (2014). doi:10.1007/s11071-014-1459-1

    Article  MathSciNet  MATH  Google Scholar 

  35. Raghavan, S., Hedrick, J.K.: Observer design for a class of nonlinear systems. Int. J. Control 59, 515–528 (1994). doi:10.1080/00207179408923090

    Article  MathSciNet  MATH  Google Scholar 

  36. Rajamani, R.: Observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 43, 397–401 (1998). doi:10.1109/9.661604

    Article  MathSciNet  MATH  Google Scholar 

  37. Zeitz, M.: The extended Luenberger observer for nonlinear systems. Syst. Control Lett. 9, 149–156 (1987). doi:10.1016/0167-6911(87)90021-1

    Article  MathSciNet  MATH  Google Scholar 

  38. Huang, H., Huang, T., Chen, X., Qian, C.: Exponential stabilization of delayed recurrent neural networks: a state estimation based approach. Neural Netw. 48, 153–157 (2013). doi:10.1016/j.neunet.2013.08.006

    Article  MATH  Google Scholar 

  39. Che, Y.-Q., Wang, J., Chan, W.-L., Tsang, K.-M.: Chaos synchronization of coupled neurons under electrical stimulation via robust adaptive fuzzy control. Nonlinear Dyn. 61, 847–857 (2010). doi:10.1007/s11071-010-9691-9

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang, C.-C., Lin, C.-L.: Robust adaptive sliding mode control for synchronization of space-clamped FitzHugh–Nagumo neurons. Nonlinear Dyn. 69, 2089–2096 (2012). doi:10.1007/s11071-012-0410-6

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by Higher Education Commission (HEC) of Pakistan by supporting Ph.D. studies of the first author through indigenous Ph.D. scholarship program (phase II, batch II, 2013).

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad, Pakistan

    Muhammad Siddique & Muhammad Rehan

Authors
  1. Muhammad Siddique
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Rehan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Rehan.

Appendices

Appendix 1: Proof of Theorem 2

Using (10)–(12), CCAS observers (22)–(23) and systems (1)–(2) reveal the error systems as

$$\begin{aligned} \dot{e}_\mathrm{m} (t)= & {} Ae_\mathrm{m} (t)+\left[ {f(x_\mathrm{m} (t))-f(\hat{{x}}_\mathrm{m} (t))} \right] \nonumber \\&+\,Bg(x_\mathrm{m} (t))\theta _\mathrm{m} -Bg(\hat{{x}}_\mathrm{m} (t))\widehat{\theta }_\mathrm{m} (t) \nonumber \\&-\,L_\mathrm{m} Ce_\mathrm{m} (t)+\frac{1}{2}{} { BF}e_\mathrm{o} (t), \end{aligned}$$
(36)
$$\begin{aligned} \dot{e}_\mathrm{s} (t)= & {} Ae_\mathrm{s} (t)+\left[ {f(x_\mathrm{s} (t))-f(\hat{{x}}_\mathrm{s} (t))} \right] \nonumber \\&+\,Bg(x_\mathrm{s} (t))\theta _\mathrm{s} -Bg(\hat{{x}}_\mathrm{s} (t))\widehat{\theta }_\mathrm{s} (t) \nonumber \\&-\,L_\mathrm{s} Ce_\mathrm{s} (t)+\frac{1}{2}{} { BF}e_\mathrm{o} (t),\end{aligned}$$
(37)
$$\begin{aligned} \dot{e}_\mathrm{o} (t)= & {} Ae_\mathrm{o} (t)+\left[ {f(\hat{{x}}_\mathrm{m} (t))-f(\hat{{x}}_\mathrm{s} (t))} \right] \nonumber \\&+\,Bg(\hat{{x}}_\mathrm{m} (t))\hat{{\theta }}_\mathrm{m}(t)-Bg(\hat{{x}}_\mathrm{s} (t))\hat{{\theta }}_\mathrm{s} (t) \nonumber \\&+\,L_\mathrm{m} Ce_\mathrm{m} (t)-L_\mathrm{s} Ce_\mathrm{s} (t)-{ BF}e_\mathrm{o} (t)-Bu_g.\nonumber \\ \end{aligned}$$
(38)

Applying \(\tilde{\theta }_\mathrm{m} (t)=\theta _\mathrm{m} -\hat{{\theta }}_\mathrm{m} (t)\) and \(g_\mathrm{m} (x_\mathrm{m} (t))=Bg(x_\mathrm{m} (t))\theta _\mathrm{m} \) and, further, employing the mathematical fact

$$\begin{aligned}&Bg(x_\mathrm{m} (t))\theta _\mathrm{m} -Bg(\hat{{x}}_\mathrm{m} (t))\hat{{\theta }}_\mathrm{m} (t)=g_\mathrm{m} (x_\mathrm{m} (t))\\&\quad -\,g_\mathrm{m} (\hat{{x}}_\mathrm{m} (t))+Bg(\hat{{x}}_\mathrm{m} (t))\tilde{\theta }_\mathrm{m} (t), \end{aligned}$$

we obtain

$$\begin{aligned} \dot{e}_\mathrm{m} (t)= & {} (A-L_\mathrm{m} C)e_\mathrm{m} (t)+\left[ {f(x_\mathrm{m} (t))-f(\hat{{x}}_\mathrm{m} (t))} \right] \nonumber \\&+\,\frac{1}{2}{} { BF}e_\mathrm{o} (t) +g_\mathrm{m} (x_\mathrm{m} (t))-g_\mathrm{m} (\hat{{x}}_\mathrm{m} (t)) \nonumber \\&+\,Bg(\hat{{x}}_\mathrm{m} (t))\widetilde{\theta }_\mathrm{m} (t). \end{aligned}$$
(39)

Similarly, it is implicit to obtain

$$\begin{aligned} \dot{e}_\mathrm{s} (t)= & {} (A-L_\mathrm{s} C)e_\mathrm{s} (t)+\left[ {f(x_\mathrm{s} (t))-f(\hat{{x}}_\mathrm{s} (t))} \right] \nonumber \\&+\,\frac{1}{2}{} { BF}e_\mathrm{o} (t) +g(x_\mathrm{s} (t))-g(\hat{{x}}_\mathrm{s} (t))\nonumber \\&+\,Bg(\hat{{x}}_\mathrm{s} (t))\widetilde{\theta }_\mathrm{s} (t). \end{aligned}$$
(40)

Using \(u_g =g(\hat{{x}}_\mathrm{m} (t))\hat{{\theta }}_\mathrm{m} -g\left( {\hat{{x}}_\mathrm{s} (t)} \right) \hat{{\theta }}_\mathrm{s} \), we have

$$\begin{aligned} \dot{e}_\mathrm{o} (t)= & {} (A - { BF})e_\mathrm{o} (t) + \left[ {f(\hat{x}_\mathrm{m} (t)) - f(\hat{x}_\mathrm{s} (t))} \right] \nonumber \\&+\, L_\mathrm{m} Ce_\mathrm{m} (t) - L_\mathrm{s} Ce_\mathrm{s} (t) \end{aligned}$$
(41)

Consider the Lyapunov function

$$\begin{aligned} V(t)= & {} e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m} e_\mathrm{m} (t)+e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} e_\mathrm{s} (t)+e_\mathrm{o}^\mathrm{T} (t)P_\mathrm{o} e_\mathrm{o} (t) \nonumber \\&+\,\tilde{\theta }_\mathrm{m}^\mathrm{T} (t)\Theta _\mathrm{m}^{-1} \tilde{\theta }_\mathrm{m} (t)+\tilde{\theta }_\mathrm{s}^\mathrm{T} (t)\Theta _\mathrm{s}^{-1} \tilde{\theta }_\mathrm{s} (t). \end{aligned}$$
(42)

Its time derivative along (39)–(41), by employing \(B^\mathrm{T}P_\mathrm{m} -R_\mathrm{m} C=0\) and \(B^\mathrm{T}P_\mathrm{s} -R_\mathrm{s} C=0\), is given by

$$\begin{aligned} \dot{V}(t)= & {} e_\mathrm{m}^\mathrm{T} (t)(A^\mathrm{T} - C^\mathrm{T} L_\mathrm{m}^\mathrm{T} )P_\mathrm{m} e_\mathrm{m} (t) + \left[ f(x_\mathrm{m} (t))\right. \nonumber \\&\left. - f(\hat{x}_\mathrm{m} (t)) \right] ^\mathrm{T} P_\mathrm{m}e_\mathrm{m} (t) \nonumber \\&+\, 0.5e_\mathrm{o}^\mathrm{T} (t)F^\mathrm{T} B^\mathrm{T} P_\mathrm{m} e_\mathrm{m} (t) + \left[ g_\mathrm{m} (x_\mathrm{m} (t))\right. \nonumber \\&\left. -\, g_\mathrm{m} (\hat{x}_\mathrm{m} (t)) \right] ^\mathrm{T}P_\mathrm{m} e_\mathrm{m} (t)\nonumber \\&+\, \widetilde{\theta }_\mathrm{m}^\mathrm{T} (t)g^\mathrm{T}(\hat{x}_\mathrm{m} (t))R_\mathrm{m} (y_\mathrm{m} (t)\nonumber \\&-\, C\hat{x}_\mathrm{m} (t)) + e_\mathrm{m}^\mathrm{T}(t)P_\mathrm{m} \left( {A - L_\mathrm{m} C} \right) e_\mathrm{m} (t)\nonumber \\&+\,e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m} \left[ {f(x_\mathrm{m} (t)) - f(\hat{x}_\mathrm{m} (t))}\right] \nonumber \\&+\, 0.5e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m} { BF}e_\mathrm{o} (t) \nonumber \\&+\, e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m} \left[ {g_\mathrm{m} (x_\mathrm{m} (t)) - g_\mathrm{m}(\hat{x}_\mathrm{m} (t))}\right] \nonumber \\&+\, (y_\mathrm{m} (t) - C\hat{x}_\mathrm{m} (t))^\mathrm{T} R_\mathrm{m}^\mathrm{T} g(\hat{x}_\mathrm{m}(t))\tilde{\theta }_\mathrm{m} (t) \nonumber \\&+\,e_\mathrm{s}^\mathrm{T} (t)(A^\mathrm{T} -C^\mathrm{T} L_\mathrm{s}^\mathrm{T} )P_\mathrm{s} e_\mathrm{s} (t) + \left[ f(x_\mathrm{m} (t)) \right. \nonumber \\ \end{aligned}$$
(43)
$$\begin{aligned}&\left. -\,f(\hat{x}_\mathrm{m} (t)) \right] ^\mathrm{T} P_\mathrm{s} e_\mathrm{s} (t)\nonumber \\&+\,0.5e_\mathrm{o}^\mathrm{T} (t)F^\mathrm{T} B^\mathrm{T} P_\mathrm{s} e_\mathrm{s} (t) + \,\left[ g_\mathrm{s} (x_\mathrm{s}(t))\right. \nonumber \\&\left. -\, g_\mathrm{s} (\hat{x}_\mathrm{s} (t)) \right] ^\mathrm{T} P_\mathrm{s} e_\mathrm{s} (t) \nonumber \\&+\,\tilde{\theta }_\mathrm{s}^\mathrm{T} (t)g^\mathrm{T} (\hat{x}_\mathrm{s} (t))R_\mathrm{s} (y_\mathrm{s} (t)\nonumber \\&-\,C\hat{x}_\mathrm{s} (t)) + \,e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} (A - L_\mathrm{s} C)e_\mathrm{s} (t) \nonumber \\&+\, e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} \left[ {f(x_\mathrm{s} (t)) - f(\hat{x}_\mathrm{s} (t))}\right] \nonumber \\&+\, 0.5e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} { BF}e_\mathrm{o} (t)\nonumber \\&+\, e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} \left[ {g_\mathrm{s} (x_\mathrm{s} (t)) - g_\mathrm{s} (\hat{x}_\mathrm{s} (t))} \right] + (y_\mathrm{m} (t)\nonumber \\&-\, C\hat{x}_\mathrm{m} (t))^\mathrm{T}R_\mathrm{s}^\mathrm{T} g(\hat{x}_\mathrm{s} (t))\tilde{\theta }_\mathrm{s} (t)\nonumber \\&+\,e_\mathrm{o}^\mathrm{T} (t)(A^\mathrm{T} - F^\mathrm{T} B^\mathrm{T} )P_\mathrm{o} e_\mathrm{o} (t)+\left[ f(\hat{x}_\mathrm{m} (t))\right. \nonumber \\&\left. -\, f(\hat{x}_\mathrm{s} (t)) \right] ^\mathrm{T}P_\mathrm{o} e_\mathrm{o} (t) \nonumber \\&+\, e_\mathrm{m}^\mathrm{T} (t)C^\mathrm{T} L_\mathrm{m}^\mathrm{T} P_\mathrm{o} e_\mathrm{o}(t) - e_\mathrm{s}^\mathrm{T} (t)C^\mathrm{T} L_\mathrm{s}^\mathrm{T} P_\mathrm{o} e_\mathrm{o} (t)\nonumber \\&+\, e_\mathrm{o}^\mathrm{T} (t)P_\mathrm{o} (A - { BF})e_\mathrm{o} (t) \end{aligned}$$

Using Assumption 1 for positive scaling factors \(\beta _1 \) and \(\beta _2 \), we have

$$\begin{aligned}&-\,\beta _1 \left[ {g_\mathrm{m} (x_\mathrm{m} (t))-g(\hat{{x}}_\mathrm{m} (t))} \right] ^\mathrm{T}\left[ {g_\mathrm{m} (x_\mathrm{m} (t))-g_\mathrm{m} (\hat{{x}}_\mathrm{m} (t))} \right] \nonumber \\&\quad +\,\beta _1 L_{g\mathrm{m}}^2 e_\mathrm{m}^\mathrm{T} \left( t \right) e_\mathrm{m} \left( t \right) >0,\\&-\,\beta _2 \left[ {g_\mathrm{s} (x_\mathrm{s} (t))-g_\mathrm{s} (\hat{{x}}_\mathrm{s} (t))} \right] ^\mathrm{T}\left[ {g_\mathrm{s} (x_\mathrm{s} (t))-g_\mathrm{s} (\hat{{x}}_\mathrm{s} (t))} \right] \nonumber \\&\quad +\,\beta _2 L_{g\mathrm{s}}^2 e_\mathrm{s}^\mathrm{T} \left( t \right) e_\mathrm{s} \left( t \right) >0, \end{aligned}$$

Employing the above inequalities, using (43), \(\dot{\tilde{\theta }}_\mathrm{m} (t)=-\dot{\hat{{\theta }}}_\mathrm{m} (t)\) and \(\dot{\tilde{\theta }}_\mathrm{s} (t)=-\dot{\hat{{\theta }}}_\mathrm{s} (t)\) and incorporating the adaptation laws (24)–(25) under Assumption 2, it implies that

$$\begin{aligned} \dot{V}\left( t \right)\le & {} e_\mathrm{m}^\mathrm{T} (t)(A^\mathrm{T} - C^\mathrm{T} L_\mathrm{m}^\mathrm{T} )P_\mathrm{m} e_\mathrm{m} (t)\\&+\, \left[ {f(x_\mathrm{m} (t)) - f(\hat{x}_\mathrm{m} (t))} \right] ^\mathrm{T} P_\mathrm{m} e_\mathrm{m} (t) \\&+\, 0.5e_\mathrm{o}^\mathrm{T} (t)F^\mathrm{T} B^\mathrm{T} P_\mathrm{m} e_\mathrm{m} (t) + \left[ g_\mathrm{m} (x_\mathrm{m} (t))\right. \nonumber \\&\left. -\, g(\hat{x}_\mathrm{m} (t)) \right] ^\mathrm{T} P_\mathrm{m} e_\mathrm{m} (t)\\&+\, e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m} (A- L_\mathrm{m} C)e_\mathrm{m} (t)\\&+\,e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m} \left[ {f(x_\mathrm{m}(t)) - f(\hat{x}_\mathrm{m} (t))} \right] \\&+\,0.5e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m}{} { BF}e_\mathrm{o} (t) + e_\mathrm{m}^\mathrm{T} (t)P_\mathrm{m} \left[ g(x_\mathrm{m} (t),\theta _\mathrm{m} )\right. \\&\left. -\,g(\hat{x}_\mathrm{m} (t),\theta _\mathrm{m} ) \right] \\&+\, e_\mathrm{s}^\mathrm{T}(t)(A^\mathrm{T} - C^\mathrm{T} L_\mathrm{s}^\mathrm{T} )P_\mathrm{s} e_\mathrm{s} (t) + \left[ f(x_\mathrm{m} (t))\right. \\&\left. -\, f(\hat{x}_\mathrm{m} (t)) \right] ^\mathrm{T} P_\mathrm{s} e_\mathrm{s} (t) \\&+\,0.5e_\mathrm{o}^\mathrm{T} (t)F^\mathrm{T} B^\mathrm{T} P_\mathrm{s} e_\mathrm{s} (t) + \,\left[ g(x_\mathrm{s} (t),\theta _\mathrm{s} )\right. \nonumber \\&\left. -\, g(\hat{x}_\mathrm{s} (t),\theta _\mathrm{s} ) \right] ^\mathrm{T}P_\mathrm{s} e_\mathrm{s} (t) \\ \end{aligned}$$
$$\begin{aligned}&+\,e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} (A - L_\mathrm{s} C)e_\mathrm{s} (t)+ e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s}\\&\times \,\left[ {f(x_\mathrm{s} (t)) - f(\hat{x}_\mathrm{s} (t))} \right] \\&+\, 0.5e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} { BF}e_\mathrm{o} (t)\\&+\,e_\mathrm{s}^\mathrm{T} (t)P_\mathrm{s} \left[ {g(x_\mathrm{s} (t),\theta _\mathrm{s} ) - g(\hat{x}_\mathrm{s} (t),\theta _\mathrm{s} )} \right] \\&+\, e_\mathrm{o}^\mathrm{T}(t)(A^\mathrm{T} - F^\mathrm{T} B^\mathrm{T} )P_\mathrm{o} e_\mathrm{o} (t)\\&+\,\left[ {f(\hat{x}_\mathrm{m} (t)) - f(\hat{x}_\mathrm{s} (t))} \right] ^\mathrm{T} P_\mathrm{o} e_\mathrm{o}(t) \\&+\, e_\mathrm{m}^\mathrm{T} (t)C^\mathrm{T} L_\mathrm{m}^\mathrm{T} P_\mathrm{o} e_\mathrm{o} (t) \\&-\,e_\mathrm{s}^\mathrm{T} (t)C^\mathrm{T} L_\mathrm{s}^\mathrm{T} P_\mathrm{o} e_\mathrm{o} (t)\\&+\,e_\mathrm{o}^\mathrm{T} (t)P_\mathrm{o} (A - { BF})e_\mathrm{o} (t)\\&+\, e_\mathrm{o}^\mathrm{T} (t)P_\mathrm{o} \left[ f(\hat{x}_\mathrm{m} (t) - f(\hat{x}_\mathrm{s} (t) \right] \\&+\, e_\mathrm{o}^\mathrm{T}(t)P_\mathrm{o} L_\mathrm{m} Ce_\mathrm{m} (t) - e_\mathrm{o}^\mathrm{T} (t)P_\mathrm{o} L_\mathrm{s} Ce_\mathrm{s} (t) \\ \end{aligned}$$
$$\begin{aligned}&-\, \alpha _{1} \left[ {f(x_\mathrm{m} (t))-f(\hat{x}_\mathrm{m} (t))} \right] ^\mathrm{T}\\&\times \,\left[ {f(x_\mathrm{m} (t)){-}f(\hat{x}_\mathrm{m} (t))} \right] + \alpha _{1} L_{f}^{2} e_\mathrm{m}^\mathrm{T} (t)e_\mathrm{m} (t) \\&-\, \alpha _{2} \left[ {f(x_\mathrm{s} (t)){-}f(\hat{x}_\mathrm{s} (t))} \right] ^\mathrm{T} \left[ f(x_\mathrm{s}(t))\right. \\&\left. -\,f(\hat{x}_\mathrm{s} (t)) \right] + \alpha _{2} L_{f}^{2} e_\mathrm{s}^\mathrm{T}(t)e_\mathrm{s} (t)\\&-\, \alpha _{3} \left[ {f(\hat{x}_\mathrm{m} (t)){-}f(\hat{x}_\mathrm{s} (t))} \right] ^\mathrm{T}\\&\times \,\left[ {f(\hat{x}_\mathrm{m} (t)){-}f(\hat{x}_\mathrm{s} (t))} \right] \\&+\, \alpha _{3} L_{f}^{2} e_\mathrm{o}^\mathrm{T} (t)e_\mathrm{o} (t) \\&-\, \beta _{1} \left[ g_\mathrm{m} (x_\mathrm{m} (t))\right. \\&\left. -\,g(\hat{x}_\mathrm{m} (t)) \right] ^\mathrm{T} \left[ {g_\mathrm{m} (x_\mathrm{m} (t)){-}g_\mathrm{m} (\hat{x}_\mathrm{m} (t))} \right] \nonumber \\&+\, \beta _{1} L_{{gm}}^{2} e_\mathrm{m}^\mathrm{T} \left( t \right) e_\mathrm{m} \left( t \right) \\&-\, \beta _{2} \left[ {g_\mathrm{s} (x_\mathrm{s} (t)){-}g_\mathrm{s} (\hat{x}_\mathrm{s} (t))} \right] ^\mathrm{T}\\&\times \left[ {g_\mathrm{s} (x_\mathrm{s} (t)){-}g_\mathrm{s} (\hat{x}_\mathrm{s} (t))} \right] + \beta _{2} L_{{gs}}^{2} e_\mathrm{s}^\mathrm{T} \left( t \right) e_\mathrm{s} \left( t \right) , \end{aligned}$$

which further reveals

$$\begin{aligned}&\dot{V}\left( t \right) \le E_{2}^\mathrm{T} (t)\Omega _{2} E_{2} (t)\\&E_2^\mathrm{T} (t)=\left[ {e_\mathrm{m}^\mathrm{T} (t)}\quad {e_\mathrm{s}^\mathrm{T} (t)}\quad {e_\mathrm{o}^\mathrm{T} (t)}\quad {\Delta f_{\mathrm{m}\hat{{m}}}^\mathrm{T} (t)}\quad {\Delta f_{\mathrm{s}\hat{{s}}}^\mathrm{T} (t)}\right. \nonumber \\&\left. \quad {\Delta f_{\hat{{m}}\hat{{s}}}^\mathrm{T} (t)}\quad {\Delta g_{\mathrm{m}\hat{{m}}}^\mathrm{T} (t)}\quad {\Delta g_{\mathrm{s}\hat{{s}}}^\mathrm{T} (t)} \right] ,\nonumber \\&\Delta g_{\mathrm{m}\hat{{m}}} (t)=\left[ {g_\mathrm{m} (x_\mathrm{m} (t))-g_\mathrm{m} (\hat{{x}}_\mathrm{m} (t))} \right] ,\nonumber \\&\Delta g_{\mathrm{s}\hat{{s}}} (t)=\left[ {g_\mathrm{s} (x_\mathrm{s} (t))-g_\mathrm{s} (\hat{{x}}_\mathrm{s} (t))} \right] .\nonumber \end{aligned}$$
(44)

If (26) is satisfied, the above inequality (44) implies \(\dot{V}(t) < 0 \). Hence, the errors \(e_\mathrm{m} (t), e_\mathrm{s} (t)\) and \(e_\mathrm{o} (t)\) converge to the origin, which entails synchronization of the master and the slave chaotic oscillators. \(\square \)

Appendix 2: Proof of Theorem 4

Applying the congruence transformation, that is, by pre- and post- multiplying (33) by \(\mathrm{diag} (I_n , I_n ,I_n ,\beta _1 I_n ,\alpha _1 I_n ,)\), where \(\beta _1 =1/{\eta _1 \bar{{\beta }}_1 }\) and \(\alpha _1 =1/{\eta _1 \bar{{\alpha }}_1 }\) for an appropriate number \(\eta _1 \), the resultant matrix inequality

$$\begin{aligned} \left[ {{\begin{array}{ccccc} {\Theta _1 }&{} {\tilde{P}_\mathrm{m} }&{} {\tilde{P}_\mathrm{m} }&{} {\frac{L_{g\mathrm{m}} }{\eta _1 }}&{} {\frac{L_f }{\eta _1 }} \\ {*}&{} {-\bar{{\alpha }}_1 I_n }&{} 0&{} 0&{} 0 \\ {*}&{} {*}&{} {-\bar{{\beta }}_1 I_n }&{} 0&{} 0 \\ {*}&{} {*}&{} {*}&{} {-\frac{1}{\eta _1 }\beta _1 I_n }&{} 0 \\ {*}&{} {*}&{} {*}&{} {*}&{} {-\frac{1}{\eta _1 }\alpha _1 I_n } \\ \end{array} }} \right] <0 \end{aligned}$$
(45)

is obtained. Employing Schur complement obtains

$$\begin{aligned} \zeta _{_1 }= & {} \left[ {{\begin{array}{ccc} {\Delta _1 }&{} {\tilde{P}_\mathrm{m} }&{} {\tilde{P}_\mathrm{m} } \\ {*}&{} {-\bar{{\alpha }}_1 I_n }&{} 0 \\ {*}&{} {*}&{} {-\bar{{\beta }}_1 I_n } \\ \end{array} }} \right] <0,\\ \Delta _1= & {} A^\mathrm{T}\tilde{P}_\mathrm{m} +\tilde{P}_\mathrm{m} A-C^\mathrm{T}H_1^\mathrm{T}\nonumber \\&-\,H_1 C+\bar{{\alpha }}_1 L_f^2 +\bar{{\beta }}_1 L_{g\mathrm{m}}^2.\nonumber \end{aligned}$$
(46)

Similarly, by using \(\beta _2 =1/{\eta _2 \bar{{\beta }}_2 }\), and \(\alpha _2 =1/{\eta _2 \bar{{\alpha }}_2 }\) for a scalar \(\eta _2 \), and following the same procedure as above, the matrix inequality (34) can be modified as

$$\begin{aligned} \zeta _{_2 }= & {} \left[ {{\begin{array}{ccc} {\Delta _2 }&{} {\tilde{P}_\mathrm{s} }&{} {\tilde{P}_\mathrm{s} } \\ {*}&{} {-\bar{{\alpha }}_2 I_n }&{} 0 \\ {*}&{} {*}&{} {-\bar{{\beta }}_2 I_n } \\ \end{array} }} \right] <0,\\ \Delta _2= & {} A^\mathrm{T}\tilde{P}_\mathrm{s} +\tilde{P}_\mathrm{s} A-C^\mathrm{T}H_2^\mathrm{T} -H_2 C+\bar{{\alpha }}_2 L_f^2 +\bar{{\beta }}_2 L_{g\mathrm{m}}^2.\nonumber \end{aligned}$$
(47)

By application of congruence transformation to (35) using \(\mathrm{diag}(I_n ,I_n ,\alpha _3 I_n ,\alpha _3 I_n ,)\), where \(\alpha _3 =1/{\bar{{\alpha }}_3 }\), the resultant inequality is obtained as

$$\begin{aligned} \left[ {{\begin{array}{cccc} {\Theta _3 }&{} {\bar{{P}}_\mathrm{o} }&{} 0&{} {L_f } \\ {*}&{} {-2\bar{{P}}_\mathrm{o} }&{} {I_n }&{} 0 \\ {*}&{} {*}&{} {-\alpha _3 I_n }&{} 0 \\ {*}&{} {*}&{} {*}&{} {-\alpha _3 I_n } \\ \end{array} }} \right] <0. \end{aligned}$$

By applying Schur complement, we achieve

$$\begin{aligned} \left[ {{\begin{array}{cc} {\Theta _3 +\bar{{\alpha }}_3 L_f^2 }&{} {\bar{{P}}_\mathrm{o} } \\ {*}&{} {-2\bar{{P}}_\mathrm{o} +\bar{{\alpha }}_3 I_n } \\ \end{array} }} \right] <0. \end{aligned}$$

Since we have

$$\begin{aligned}&\frac{1}{\bar{{\alpha }}_3 }\bar{{P}}_\mathrm{o} \bar{{P}}_\mathrm{o} -2\bar{{P}}_\mathrm{o} +\bar{{\alpha }}_3 I_n =(\bar{{P}}_\mathrm{o} -\bar{{\alpha }}_3 I_n )(\bar{{\alpha }}_3 I_n )^{-1} \\&(\bar{{P}}_\mathrm{o}-\bar{{\alpha }}_3 I_n )\ge 0,-\frac{1}{\bar{{\alpha }}_3 }\bar{{P}}_\mathrm{o} \bar{{P}}_\mathrm{o} \le -2\bar{{P}}_\mathrm{o} +\bar{{\alpha }}_3 I_n. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} \left[ {{\begin{array}{cc} {\Theta _3 +\bar{{\alpha }}_3 L_f^2 }&{} {\bar{{P}}_\mathrm{o} } \\ {*}&{} {-\frac{1}{\bar{{\alpha }}_3 }\bar{{P}}_\mathrm{o} \bar{{P}}_\mathrm{o} } \\ \end{array} }} \right] <0. \end{aligned}$$

By considering \(H_3 =F\bar{{P}}_\mathrm{o} \) and applying congruence transformation by \(\mathrm{diag}(P_\mathrm{o} , P_\mathrm{o} )\), it results into

$$\begin{aligned} \zeta _{3 }= & {} \left[ {{\begin{array}{cc} {\Delta _3 }&{} {P_\mathrm{o} } \\ {*}&{} {-\frac{1}{\bar{{\alpha }}_3 }I_n } \\ \end{array} }} \right] <0,\\ \Delta _3= & {} A^\mathrm{T}_\mathrm{o} P_\mathrm{o} +P_\mathrm{o} A-F^\mathrm{T}B^\mathrm{T}P_\mathrm{o} -P_\mathrm{o} { BF}+\bar{{\alpha }}_3 L_f^2.\nonumber \end{aligned}$$
(48)

By lumping together the linear matrix inequalities (46), (47), and (48) and, further, using \(H_1 =\tilde{P}_\mathrm{m} L_\mathrm{m} \) and \(H_2 =\tilde{P}_\mathrm{s} L_\mathrm{s} \), it produces

$$\begin{aligned}&\left[ {{\begin{array}{ccc} {\eta _1 \zeta _1 }&{} {\zeta _4 }&{} {\zeta _5 } \\ {*}&{} {\eta _2 \zeta _2 }&{} {\zeta _6 } \\ {*}&{} {*}&{} {\zeta _3 } \\ \end{array} }} \right] <0,\\&\zeta _{_4 } =\left[ {{\begin{array}{ccc} 0&{} 0&{} 0 \\ 0&{} 0&{} 0 \\ 0&{} 0&{} 0 \\ \end{array} }} \right] , \zeta _{_5 } =\left[ {{\begin{array}{cc} {0.5P_\mathrm{m} { BF}+C^\mathrm{T}L_\mathrm{m}^\mathrm{T} P_\mathrm{o} }&{} 0 \\ 0&{} 0 \\ 0&{} 0 \\ \end{array} }} \right] ,\nonumber \\&\zeta _{_6 } =\left[ {{\begin{array}{cc} {0.5P_\mathrm{s} { BF}-C^\mathrm{T}L_\mathrm{s}^\mathrm{T} P_\mathrm{o} }&{} 0 \\ 0&{} 0 \\ 0&{} 0 \\ \end{array} }} \right] .\nonumber \end{aligned}$$
(49)

We can regenerate the matrix inequality (26), by pre- and post-multiplying (49) by \( [\mathbb {I}_1^\mathrm{T} , \mathbb {I}_4^\mathrm{T} , \mathbb {I}_7^\mathrm{T} , \mathbb {I}_2^\mathrm{T} , \mathbb {I}_5^\mathrm{T} , \mathbb {I}_8^\mathrm{T} , \mathbb {I}_3^\mathrm{T} , \mathbb {I}_6^\mathrm{T} ]^\mathrm{T}\) and its transpose, respectively, where \(\mathbb {I}_{\alpha } \) is the matrix generated by replacing the ith \(0_{n\times n} \) with \(I_n \) in \(0_{n\times 8n} \) matrix (for example \(\mathbb {I}_2 = [0_{n\times n} , I_n, 0_{n\times n} , 0_{n\times n} , 0_{n\times n}, 0_{n\times n},0_{n\times n} ,0_{n\times n} ])\) and substituting \(\tilde{P}_\mathrm{m} =\eta _1 ^{-1}P_\mathrm{m} \),\(\tilde{P}_\mathrm{s} \,{=}\,\eta _1 ^{-1}P_\mathrm{s} \), \(\bar{{P}}_\mathrm{o} \,{=}\,P_\mathrm{o}^{-1} \), \(\bar{{\alpha }}_1 \,{=}\,1/{(\eta _1 \alpha _1 )}\), \(\bar{{\alpha }}_2 =1/{(\eta _2 \alpha _2 )}\), \(\bar{{\alpha }}_3 \,{=}\,\alpha _3 ^{-1}\), \(\bar{{\beta }}_1 \,{=}\,1/{(\eta _1 \beta _1 })\), and \(\bar{{\beta }}_2 \,{=}\,1/{(\eta _2 \beta _2 })\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Siddique, M., Rehan, M. A concept of coupled chaotic synchronous observers for nonlinear and adaptive observers-based chaos synchronization. Nonlinear Dyn 84, 2251–2272 (2016). https://doi.org/10.1007/s11071-016-2643-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2643-2

Keywords

Search

Navigation