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Full-order and Reduced-order Observer Design for One-sided Lipschitz Nonlinear Fractional Order Systems with Unknown Input

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  • Control Theory and Applications
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Abstract

This paper studies the problem of designing the unknown input observers (UIOs) for fractional order one-sided Lipchitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using the matrix generalized inverse approach, sufficient conditions for asymptotic stability of the observer error dynamic systems are presented, which guarantee the existence of the full-order and reduced-order UIOs. All the conditions are obtained in terms of linear matrix inequality (LMI). Furthermore, we show that the obtained results can be applied to a fractional order electrical circuit with the unknown input signal. Two examples are given to demonstrate the applicability of the proposed approach.

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References

  1. L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991.

    Article  Google Scholar 

  2. L. Chen, B. Basu, and D. McCabe, “Fractional order models for system identification of thermal dynamics of buildings,” Energy and Buildings, vol. 133, pp. 381–388, 2016.

    Article  Google Scholar 

  3. M. Ichise, Y. Nagayanagi, and T. Kojima, “An analog simulation of non-integer order transfer functions for analysis of electrode processes,” Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, vol. 33, no. 2, pp. 253–265, 1971.

    Article  Google Scholar 

  4. E. McAdams, A. Lackermeier, J. McLaughlin, D. Macken, and J. Jossinet, “The linear and non-linear electrical properties of the electrode-electrolyte interface,” Biosensors and Bioelectronics, vol. 10, no. 1, pp. 67–74, 1995.

    Article  Google Scholar 

  5. I. Podlubny, Fractional Differential Equations, Elsevier, 1998.

    MATH  Google Scholar 

  6. N. Aguila-Camacho, M. A. Duarte-Mermoud, and J. A. Gallegos, “Lyapunov functions for fractional order systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 9, pp. 2951–2957, 2014.

    Article  MathSciNet  Google Scholar 

  7. J. A. Gallegos and M. A. Duarte-Mermoud, “On the Lyapunov theory for fractional order systems,” Applied Mathematics and Computation, vol. 287, pp. 161–170, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  8. X.-J. Wen, Z.-M. Wu, and J.-G. Lu, “Stability analysis of a class of nonlinear fractional-order systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 55, no. 11, pp. 1178–1182, 2008.

    Article  Google Scholar 

  9. X. Yang, C. Li, T. Huang, and Q. Song, “Mittag-leffler stability analysis of nonlinear fractional-order systems with impulses,” Applied Mathematics and Computation, vol. 293, pp. 416–422, 2017.

    Article  MathSciNet  Google Scholar 

  10. S. Marir, M. Chadli, and D. Bouagada, “A novel approach of admissibility for singular linear continuous-time fractional-order systems,” International Journal of Control, Automation and Systems, vol. 15, no. 2, pp. 959–964, 2017.

    Article  MATH  Google Scholar 

  11. X. Zhang and Y. Chen, “Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional ordera: the 0< α <1 case,” ISA Transactions, 2017.

    Google Scholar 

  12. J.-G. Lu and Y.-Q. Chen, “Robust stability and stabilization of fractional-order interval systems with the fractional order a: The 0 < α < 1 case,” IEEE Transactions on Automatic Control, vol. 55, no. 1, pp. 152–158, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  13. H.-S. Ahn, Y. Chen, and I. Podlubny, “Robust stability test of a class of linear time-invariant interval fractional-order system using lyapunov inequality,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 27–34, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Li, Y. Chen, and I. Podlubny, “Mittag-leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  15. J.-C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, “A Lyapunov approach to the stability of fractional differential equations,” Signal Processing, vol. 91, no. 3, pp. 437–445, 2011.

    Article  MATH  Google Scholar 

  16. Y. Wei, J. Qiu, and H. R. Karimi, “Quantized filtering for continuous-time markovian jump systems with deficient mode information,” Asian Journal of Control, vol. 17, no. 5, pp. 1914–1923, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Tang, H. Gao, and J. Kurths, “Robust H ∞ self-triggered control of networked systems under packet dropouts,” IEEE Transactions on cybernetics, vol. 46, no. 12, pp. 3294–3305, 2016.

    Article  Google Scholar 

  18. S. Ahmad, M. Rehan, and K.-S. Hong, “Observer-based robust control of one-sided lipschitz nonlinear systems,” ISA Transactions, vol. 65, pp. 230–240, 2016.

    Article  Google Scholar 

  19. M.-J. Hu, Y.-W. Wang, and J.-W. Xiao, “Positive observer design for linear impulsive positive systems with interval uncertainties and time delay,” International Journal of Control, Automation and Systems, vol. 15, no. 3, pp. 1032–1039, 2017.

    Article  MathSciNet  Google Scholar 

  20. G. Liu, Y.-Y. Cao, and X.-H. Chang, “Fault detection observer design for fuzzy systems with local nonlinear models via fuzzy lyapunov function,” International Journal of Control, Automation and Systems, vol. 15, no. 5, pp. 2233–2242, 2017.

    Article  Google Scholar 

  21. G.-D. Hu, “Observers for one-sided lipschitz non-linear systems,” IMA Journal of Mathematical Control and Information, vol. 23, no. 4, pp. 395–401, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Xu, G.-D. Hu, and Y. Zhao, “Reduced-order observer design for one-sided lipschitz non-linear systems,” IMA Journal of Mathematical Control and Information, vol. 26, no. 3, pp. 299–317, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  23. C. Li, J. Wang, and J. Lu, “Observer-based robust stabilisation of a class of non-linear fractional-order uncertain systems: an linear matrix inequalitie approach,” IET Control Theory & Applications, vol. 6, no. 18, pp. 2757–2764, 2012.

    Article  MathSciNet  Google Scholar 

  24. Y.-H. Lan, W.-J. Li, Y. Zhou, and Y.-P. Luo, “Non-fragile observer design for fractional-order one-sided lipschitz nonlinear systems,” International Journal of Automation and Computing, vol. 10, no. 4, pp. 296–302, 2013.

    Article  Google Scholar 

  25. Y.-H. Lan, L.-L. Wang, L. Ding, and Y. Zhou, “Fullorder and reduced-order observer design for a class of fractional-order nonlinear systems,” Asian Journal of Control, vol. 18, no. 4, pp. 1467–1477, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Jmal, O. Naifar, A. Ben Makhlouf, N. Derbel, and M. A. Hammami, “On observer design for nonlinear caputo fractional-order systems,” Asian Journal of Control, 2018.

    Google Scholar 

  27. M. Darouach and L. Boutat-Baddas, “Observers for a class of nonlinear singular systems,” IEEE Transactions on Automatic Control, vol. 53, no. 11, pp. 2627–2633, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Trinh, T. D. Tran, and T. Fernando, “Disturbance decoupled observers for systems with unknown inputs,” IEEE Transactions on Automatic control, vol. 53, no. 10, pp. 2397–2402, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  29. W. Zhang, H. Su, H. Wang, and Z. Han, “Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using riccati equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4968–4977, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  30. W. Zhang, H. Su, F. Zhu, and G. M. Azar, “Unknown input observer design for one-sided lipschitz nonlinear systems,” Nonlinear Dynamics, vol. 79, no. 2, pp. 1469–1479, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Tian, J. Wang, and S. Ma, “Observer design for onesided lipschitz nonlinear continuous-time singular systems with unknown input,” in Control and Decision Conference (CCDC), 2016 Chinese, IEEE, 2016, pp. 1764–1769.

    Google Scholar 

  32. R. Zhang, D. Zeng, S. Zhong, and Y. Yu, “Event-triggered sampling control for stability and stabilization of memristive neural networks with communication delays,” Applied Mathematics and Computation, vol. 310, pp. 57–74, 2017.

    Article  MathSciNet  Google Scholar 

  33. Y. Wei, J. Qiu, H. R. Karimi, and W. Ji, “A novel memory filtering design for semi-markovian jump time-delay systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2017.

    Google Scholar 

  34. J. Qiu, H. R. Karimi et al., “Fuzzy-affine-model-based memory filter design of nonlinear systems with timevarying delay,” IEEE Transactions on Fuzzy Systems, 2017.

    Google Scholar 

  35. I. N’Doye, M. Darouach, H. Voos, and M. Zasadzinski, “Design of unknown input fractional-order observers for fractional-order systems,” International Journal of Applied Mathematics and Computer Science, vol. 23, no. 3, pp. 491–500, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Kong, M. Saif, and B. Liu, “Observer design for a class of nonlinear fractional-order systems with unknown input,” Journal of the Franklin Institute, vol. 354, no. 13, pp. 5503–5518, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  37. N. Djeghali, S. Djennoune, M. Bettayeb, M. Ghanes, and J.-P. Barbot, “Observation and sliding mode observer for nonlinear fractional-order system with unknown input,” ISA Transactions, vol. 63, pp. 1–10, 2016.

    Article  Google Scholar 

  38. H. Su, G. Chen, X. Wang, and Z. Lin, “Adaptive secondorder consensus of networked mobile agents with nonlinear dynamics,” Automatica, vol. 47, no. 2, pp. 368–375, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Song and S. He, “Robust finite-time H ∞ control for onesided lipschitz nonlinear systems via state feedback and output feedback,” Journal of the Franklin Institute, vol. 352, no. 8, pp. 3250–3266, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Kaczorek, “Positivity and reachability of fractional electrical circuits,” Acta Mechanica et Automatica, vol. 5, no. 2, pp. 42–51, 2011.

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Tao Zhan, Jiaming Tian & Shuping Ma

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  1. Tao Zhan
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  2. Jiaming Tian
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  3. Shuping Ma
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Corresponding author

Correspondence to Shuping Ma.

Additional information

Recommended by Associate EditorYang Tang under the direction of Editor Hamid Reza Karimi. This work is supported by National Natural Science Foundation of China (61473173), Major International (Regional) Joint Research Project of the National Natural Science Foundation of China (NSFC)(61320106011).

Tao Zhan is curretly a Ph.D. Candidate in School of Mathematics, Shandong University, Jinan, P. R. China. She is an exchange Ph.D student from 2017 to 2018 in Department of Applied Mathematics, University of Waterloo, Canada. Her research interests include nonlinear fractional order systems, singular system and impulsive control.

Jiaming Tian received the B.Sc. degree in information and computing science from the College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, China, in June 2014. Since September 2014, he has been working toward the Ph.D. degree on analysis and synthesis for singular Markov jump systems, in the School of Mathematics, Shandong University, Jinan, China. His research interests include stability analysis, nonlinear control and observer design for singular Markov jump systems.

Shuping Ma was born in Rizhao, China in 1970. She received her B.S. degree in Mathematics from Shandong University, China, in 1992, and, her M.S. and Ph.D. degrees in Mathematics and System Science of Shandong University, China, in 1997 and 2000, respectively. She joined School of Mathematics at Shandong University in 2000, where she is currently a professor. Her research interests include singular systems, time-delay systems, Markov jump systems, robust control and sliding mode control.

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Zhan, T., Tian, J. & Ma, S. Full-order and Reduced-order Observer Design for One-sided Lipschitz Nonlinear Fractional Order Systems with Unknown Input. Int. J. Control Autom. Syst. 16, 2146–2156 (2018). https://doi.org/10.1007/s12555-017-0684-z

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  • DOI: https://doi.org/10.1007/s12555-017-0684-z

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