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Robust finite time stability of fractional-order linear delayed systems with nonlinear perturbations

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Abstract

This letter is concerned with robust stability of fractional-order linear delayed system with nonlinear perturbations over finite time interval. By using inequality technique, two new sufficient conditions for the finite time stability for such systems with order α: 0 α ≤ 0.5 and 0.5 < α < 1 are presented, respectively. A numerical example is presented to demonstrate the validity and feasibility of the obtained results.

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References

  1. P. Dorato, “Short time stability in linear time-varying systems,” Proc. of IRE International Convention Record Part 4, pp. 83–87, 1961.

    Google Scholar 

  2. L. Weiss and E. F. Infante, “Finite time stability under perturbing forces and on product spaces,” IEEE Trans. on Automatic Control, vol. 1, pp. 54–59, February 1967.

    Article  MathSciNet  Google Scholar 

  3. F. Amato, C. Cosentino, and A. Merola, “Sufficient conditions for finite-time stability and stabilization of nonlinear quadratic systems,” IEEE Trans. on Automatic Control, vol. 55, no. 2, pp. 430–434, 2010.

    Article  MathSciNet  Google Scholar 

  4. K. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.

    MATH  Google Scholar 

  5. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

    MATH  Google Scholar 

  6. J. R. Wang, L. L. Lv, and Y. Zhou, “New concepts and results in stability of fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2530–2538, June 2012.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Ganesh, R. Sakthivel, N. I. Mahmudov, and S. M. Anthoni, “Approximate controllability of fractional integrodifferential evolution equations,” J. of Applied Mathematics, vol. 2013, 291816, 2013.

    Article  MathSciNet  Google Scholar 

  8. R. Sakthivel, R. Ganesh, Y. Ren, and S. M. Anthoni, “Approximate controllability of nonlinear fractional dynamical systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 18. no. 12, pp. 3498–3508, December 2013.

    Article  MathSciNet  Google Scholar 

  9. R. Sakthivel and Y. Ren, “Approximate controllability of fractional differential equations with state dependent delay,” Results in Mathematics, vol. 63, no. 3–4, pp. 949–963, June 2013.

    MATH  MathSciNet  Google Scholar 

  10. R. Sakthivel, N. I. Mahmudov, and J. J. Nieto, “Controllability for a class of fractional-order neutral evolution control systems,” Applied Mathematics and Computation, vol. 218, no. 20, pp. 10334–10340, June 2012.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Lu and Y. Chen, “Robust stability and stabilization of fractional-order interval systems with the fractional order α: the 0 < α < 1 case,” IEEE Trans. on Autom. Control, vol. 55, no. 1, pp. 152–158, 2011.

    Google Scholar 

  12. Y. Chen, H. Ahn, 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, April 2007.

    Article  MATH  MathSciNet  Google Scholar 

  13. K. Moornani and H. Mohammad, “On robust stability of linear time invariant fractional-order systems with real parametric uncertainties,” ISA Transactions, vol. 48, no. 4, pp. 484–490, 2009.

    Article  Google Scholar 

  14. Y. Lim, K. Oh, and H. Ahn, “Stability and stabilization of fractional-order linear systems subject to input saturation,” IEEE Trans. on Automatic Control, vol. 58, no.4, pp. 1062–1067, 2013.

    Article  MathSciNet  Google Scholar 

  15. Z. Jiao and Y. Zhong, “Robust stability for fractional-order systems with structured and unstructured uncertainties,” Computers Mathematics with Applications, vol. 64, no. 10, pp. 3258–3266, 2012.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. Deng, C. Li, and J. Lu, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, pp. 409–416, June 2007.

    Article  MATH  MathSciNet  Google Scholar 

  17. K. A. Moornani and M. Haeri, “On robust stability of LTI fractional-order delay systems of retarded and neutral type,” Automatica, vol. 46, no. 2, pp. 362–368, February 2011.

    Article  Google Scholar 

  18. S. J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi, and T. Abdeljawad, “Mittag-leffler stability theorem for fractional nonlinear systems with delay,” Abstract and Applied Analysis, vol. 2010, pp. 108651, 2010.

    Article  MathSciNet  Google Scholar 

  19. M. A. Pakzad, S. Pakzad, and M. A. Nekoui, “Stability analysis of time-delayed linear fractional-order systems,” Int. J. of Control Automation and Systems, vol. 11, no. 3, pp. 519–525, June 2011.

    Article  Google Scholar 

  20. X. Zhang, “Some results of linear fractional order time-delay system,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 407–411, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Lazarevic and A. Spasic, “Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach,” Mathematical and Computer Modelling, vol. 49, no. 3–4, pp. 475–481, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Lazarevic and D. Debeljkovic, “Finite time stability analysis of linear autonomous fractional order systems with delayed state,” Asian Journal of Control, vol. 7, no. 4, pp. 440–447, December 2005.

    Article  MathSciNet  Google Scholar 

  23. M. Lazarevic, “Finite time stability analysis of PDα fractional control of robotic time-delay systems,” Mechanics Research Communications, vol. 33, no. 2, pp. 269–279, March–April 2006.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. H. Delavari, D. Baleanu, and J. Sadati, “Stability analysis of Caputo fractional-order nonlinear systems revisited,” Nonlinear Dynamics, vol. 67, no. 4, pp. 2433–2439, March 2012.

    Article  MATH  MathSciNet  Google Scholar 

  26. L. P. Chen, Y. Chai, R. C. Wu, and J. Yang, “Stability and stabilization of a class of nonlinear fractional-order systems with Caputo derivative,” IEEE Trans. on Circuits and Systems II, vol. 59, no. 9, pp. 602–606, 2012.

    Article  Google Scholar 

  27. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality, Birkhauser, 2009.

    Book  Google Scholar 

  28. V. Lakshmikantham, “Theory of fractional functional differential equations” Nonlinear Analysis: TMA, vol. 69, no. 10, pp. 3337–3343, 2008.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Electrical Engineering and Automation, Hefei University of Technology, Hefei, 230009, China

    Liping Chen, Yigang He & Lisheng Yin

  2. School of Mathematics, Anhui University, Hefei, 230039, China

    Ranchao Wu

  3. School of Automation, Chongqing University, Chongqing, 400044, China

    Yi Chai

Authors
  1. Liping Chen
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  2. Yigang He
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  3. Ranchao Wu
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  4. Yi Chai
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  5. Lisheng Yin
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Corresponding author

Correspondence to Liping Chen.

Additional information

Recommended by Editor Ju Hyun Park.

This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar under Grant (No. 50925727), the National Natural Science Foundation of China (No. 61374135), the Fundamental Research Funds for the Central Universities (No. 2012HGCX0003), the Key Grant Project of Chinese Ministry of Education under Grant (No. 313018), the Natural Science Foundation of Anhui Province (No. 11040606 M12), the Universities Natural Science Foundation of Anhui Province (No. KJ2012A219) and the China Postdoctoral Science Foundation funded project (No. 2013M541823).

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Chen, L., He, Y., Wu, R. et al. Robust finite time stability of fractional-order linear delayed systems with nonlinear perturbations. Int. J. Control Autom. Syst. 12, 697–702 (2014). https://doi.org/10.1007/s12555-013-0436-7

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  • DOI: https://doi.org/10.1007/s12555-013-0436-7

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