Esmat Sadat Alaviyan Shahri and Saeed Balochian. Analysis of Fractional-order Linear Systems with Saturation Using Lyapunov s Second Method and Convex Optimization. International Journal of Automation and Computing, vol. 12, no. 4, pp. 440-447, 2015. https://doi.org/10.1007/s11633-014-0856-8
Citation: Esmat Sadat Alaviyan Shahri and Saeed Balochian. Analysis of Fractional-order Linear Systems with Saturation Using Lyapunov s Second Method and Convex Optimization. International Journal of Automation and Computing, vol. 12, no. 4, pp. 440-447, 2015. https://doi.org/10.1007/s11633-014-0856-8

Analysis of Fractional-order Linear Systems with Saturation Using Lyapunov s Second Method and Convex Optimization

doi: 10.1007/s11633-014-0856-8
  • Received Date: 2013-09-05
  • Rev Recd Date: 2014-04-02
  • Publish Date: 2015-08-01
  • In this paper, local stability and performance analysis of fractional-order linear systems with saturating elements are shown, which lead to less conservative information and data on the region of stability and the disturbance rejection. Then, a standard performance analysis and global stability by using Lyapunov s second method are addressed, and the introduction of Lyapunov s function candidate whose sub-level set provide stability region and performance with a restricted state space origin is also addressed. The results include both single and multiple saturation elements and can be extended to fractional-order linear systems with any nonlinear elements and nonlinear noise that satisfy Lipschitz condition. A noticeable application of these techniques is analysis of control fractional-order linear systems with saturation control inputs.

     

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  • [1]
    Y. Li, Y. Q. Chen, I. Podlubny. Mittag-leffler stability of fractional order nonlinear dynamic system. Automatica, vol. 45, no. 8, pp. 1965-1969, 2009.
    [2]
    J. C. Trigeassou, N. Maamri, J. Sabatier, A. Oustalou. A Lyapunov approach to the stability of fractional differential equations. Signal Processing, vol. 91, no. 3, pp. 437-445, 2011.
    [3]
    J. J. Sabatier, M. Moze, C. Farges. LMI stability conditions for fractional order systems. Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1594-1609, 2010.
    [4]
    Y. Li, Y. Q. Chen, I. Podlubny. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1810-1821, 2010.
    [5]
    Y. Q. Chen. Ubiquitous fractional order controls? In Proceedings of the Second IFAC Workshop on Fractional Derivatives and Applications, ISEP, Porto, Portugal, pp. 481-492, 2006.
    [6]
    S. Y. Xing, J. G. Lu. Robust stability and stabilization of fractional-order linear systems with nonlinear uncertain parameters: An LMI approach. Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1163-1169, 2009.
    [7]
    H. Delavari, D. Baleanu, J. Sadati. Stability analysis of caputo fractional-order nonlinear system revisited. Nonlinear Dynamics, vol. 67, no. 4, pp. 2433-2439, 2012.
    [8]
    H. Delavari, R. Ghaderi, A. Ranjbar, N. S. Momani. Fractional order controller for two-degree of freedom polar robot. In Proceedings of International Workshop on New Trends in Science and Technology, Ankara, Turkey, 2008.
    [9]
    C. Farges, J. Sabatier, M. Moze. Fractional order polytopic systems: Robust stability and stabilisation. Advances in Difference Equations, 2011. (Online first)
    [10]
    S. Balochian, A. K. Sedigh, A. Zare. Stabilization of multiinput hybrid fractional order systems with state delay. ISA Transactions, vol. 50, no. 1, pp. 21-27, 2011.
    [11]
    S. Balochian, A. K. Sedigh. Sufficient condition for stabilization of linear time invariant fractional order switched systems and variable structure control stabilizers. ISA Transactions, vol. 51, no. 1, pp. 65-73, 2012.
    [12]
    M. Y. Ongun, D. Arslan, R. Garrappa. Nonstandard finite difference schemes for a fractional order Brusselator system. Advance in Difference Equations, 2013. (Online first)
    [13]
    H. S. Ahn, Y. Q. Chen. Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica, vol. 44, no. 11, pp. 2985-2988, 2008.
    [14]
    I. Petras. Fractional-order Nonlinear Systems Modeling, Berlin and Heidelberg, Germany: Springer-Verlag, 2011.
    [15]
    M. O. Efe. Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct drive robot arm. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, no. 6, pp. 1561-1570, 2008.
    [16]
    M. Pourgholi, V. J. Majd. A nonlinear adaptive resilient observer design for a class of lipschitz systems using LMI. Journal of Circuits, Systems, and Signal Processing, vol. 30, no. 6, pp. 1401-1415, 2011.
    [17]
    E. Amini Boroujeni, H. R. Momeni. Observer based control of a class of nonlinear fractional order system using LMI. World Academy of Science, Engineering and Technology, vol. 61, pp. 779-782, 2012.
    [18]
    Y. Chen, B. M. Vinagre, D. Xue, V. Feliu. Fractional-Order Systems and Controls Fundamentals and Applications, London, UK: Springer-Verlag, 2010.
    [19]
    I. Petras, D. Bednarova. Control of fractional-order nonlinear system: A review. Acta Mechanica et Automatica, vol. 5, no. 2, pp. 96-100, 2011.
    [20]
    D. Baleanu, Z. B. Guven, J. A. T. Machado. New Trends in Nanotechnology and Fractional Calculus Applications, Netherlands: Springer, 2010.
    [21]
    M. D. Ortigueira. An introduction to the fractional continuous-time linear systems: The 21st century systems. IEEE Circuits and Systems Magazine, vol. 8, no. 3, pp. 19-26, 2008.
    [22]
    J. Sabatier, O. P. Agrawal, J. A. T. Machado. Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, London, UK: Springer, 2007.
    [23]
    B. Guo, D. Huang. Existence and stability of standing waves for nonlinear fractional Schrödinger equations. Journal of Mathematical Physics, vol. 53, no. 8, Article number 083702, 2012.
    [24]
    N. Laskin. Fractional quantum mechanics. Physical Review E, vol. 62, pp. 3135-3145, 2000.
    [25]
    R. Metzler, J. Klafter. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General, vol. 37, no. 31, pp.R161-R208, 2004.
    [26]
    S. Das. Functional Fractional Calculus, 2nd ed., Berlin Heidelberg, Germany: Springer-Verlag, pp. 1-220, 2011.
    [27]
    L. Stamova, G. Stamov. Lipschitz stability criteria for functional differential systems of fractional. Journal of Mathematical Physics, vol. 54, no. 4, Article number 043502, 2013.
    [28]
    R. Magin, M. D. Ortigueira, I. Podlubny, J. Trujillo. On the fractional signals and systems. Signal Processing, vol. 91, no. 3, pp. 350-371, 2011.
    [29]
    F. Liu, M. M. Meerschaert, S. Momani, N. N. Leonenko, W. Chen, O. P. Agrawal. Fractional differential equations. International Journal of Differential Equations, vol. 2010, Article number 215856, 2010.
    [30]
    L. G. Yuan, Q. G. Yang. Parameter identification and synchronization of fractional-order chaotic systems. Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 305-316, 2012.
    [31]
    J. J. E. Slotin, W. A. Li. Applied Nonlinear Control, Englewood Cliffs, New Jersey, USA: Prentice Hall, 1991.
    [32]
    H. Hindi, S. Boyd. Analysis of linear systems with saturation using convex optimization. In Proceedings of the 37th IEEE Conference on Decision and Control, IEEE, Florida, USA, pp. 903-908, 1998.
    [33]
    L. Ghaoui, S. Niculescu. Advances in Linear Matrix Inequality Method in Control, Philadelphia, USA: Society for Industrial and Applied Mathematics, 2000.
    [34]
    T. S. Hu, Z. L. Lin, B. M. Chen. An analysis and design method for linear system subject to actuator saturation and disturbance. Automatica, vol. 38, no. 2, pp. 351-359, 2002.c
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