Non-Fragile Adaptive Sliding-Mode Observer Design for a Class of Fractional-Order Pseudo-Linear Systems with State Delay
Subject Areas : electrical and computer engineeringمجيد پرويزيان 1 , خسرو خانداني 2 , وحيد جوهري مجد 3
1 -
2 - Arak University
3 -
Keywords: Von-fragile adaptive observer, fractional-order pseudo-linear systems, sliding mode, LMIs,
Abstract :
In recent years, fractional order systems and fractional order control have increasingly attracted the attention of researchers in various fields of science and engineering. On the other hand, numerous control approaches have been extended in order to be utilized in fractional order systems. Despite this fact, few research studies have been devoted to generalizing integer order observers to fractional order ones. Since the applications of fractional order systems are increasing, developing fractional order observers seems to be essential. In this paper the problem of non-fragile adaptive sliding mode observer design for a class of fractional-order nonlinear systems with time delay is addressed. First, the states of the fractional-order pseudo-linear time-delay system with matched nonlinearity are estimated employing the sliding mode control method. Then the state estimation problem of fractional order systems with mismatched nonlinearity has been investigated. The asymptotic stability of the estimation error dynamics is proven by employing the Lyapunov stability analysis method for fractional order systems. The sufficient stability conditions are derived in the form of Linear Matrix Inequalities (LMIs). Eventually, the effective performance of the proposed approach in this paper has been corroborated through simulation of a numerical example and also a case study of a fractional order economic system.
[1] M. Hu, Y. Li, S. Li, C. Fu, D. Qin, and Z. Li, "Lithium-ion battery modeling and parameter identification based on fractional theory," Energy, vol. 165, pt. B, pp. 153-163, Dec. 2018.
[2] X. Wang, T. Thu Giang Hoang, Z. Pan, and Y. Q. Chen, "Fractional-order modelling and control for two parallel PWM rectifiers," IFAC-PapersOnLine, vol. 51, no. 4, pp. 54-59, Feb. 2018.
[3] T. Wei and Y. S. Li, "Identifying a diffusion coefficient in a time-fractional diffusion equation," Mathematics and Computers in Simulation, vol. 151, pp. 77-95, Sept. 2018.
[4] M. Bonforte, Y. Sire, and J. L. Vazquez, "Optimal existence and uniqueness theory for the fractional heat equation," Nonlinear Analysis: Theory, Methods & Applications, vol. 153, pp. 142-168, Apr. 2017.
[5] J. D. Gabano, T. Poinot, and H. Kanoun, "Identification of a thermal system using continuous linear parameter-varying fractional modelling," IET Control Theory & Applications, vol. 5, no. 7, pp. 889-899, May 2011.
[6] F. C. Meral, T. J. Royston, and R. Magin, "Fractional calculus in viscoelasticity: an experimental study," Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 939-945, Apr. 2010.
[7] C. Zopf, S. E. Hoque, and M. Kaliske, "Comparison of approaches to model viscoelasticity based on fractional time derivatives," Computational Materials Science, vol. 98, pp. 287-296, Feb. 2015.
[8] Y. Long, B. Xu, D. Chen, and W. Ye, "Dynamic characteristics for a hydro-turbine governing system with viscoelastic materials described by fractional calculus," Applied Mathematical Modelling, vol. 58, pp. 128-139, Jun. 2018.
[9] C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, and J. H. T. Bates, "The role of fractional calculus in modeling biological phenomena: a review," Communications in Nonlinear Science and Numerical Simulation, vol. 51, pp. 141-159, Oct. 2017.
[10] 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, Aug. 2009.
[11] I. Kheirizad, A. A. Jalali, and K. Khandani, "Stabilisation of unstable FOPDT processes with a single zero by fractional-order controllers," International J. of Systems Science, vol. 44, no. 8, pp. 1533-1545, Aug. 2013.
[12] Y. Farid, V. J. Majd, and A. Ehsani-Seresht, "Fractional-order active fault-tolerant force-position controller design for the legged robots using saturated actuator with unknown bias and gain degradation," Mechanical Systems and Signal Processing, vol. 104, pp. 465-486, May 2018.
[13] S. P. Nangrani and S. S. Bhat, "Fractional order controller for controlling power system dynamic behavior," Asian J. of Control, vol. 20, no. 1, pp. 403-414, Jan. 2018.
[14] M. Netto and L. Nili, "A robust data-driven koopman kalman filter for power systems dynamic state estimation," IEEE Trans. on Power Systems, vol. 33, no. 6, pp. 7228-7237, Jun. 2018.
[15] J. Zhao, "Dynamic state estimation with model uncertainties using H∞ extended kalman filter," IEEE Trans. on Power Systems, vol. 33, no. 1, pp. 1099-1100, Mar. 2018.
[16] H. Bao and J. H. Oh, "Novel state estimation framework for humanoid robot," Robotics and Autonomous Systems, vol. 98, pp. 258-275, Dec. 2017.
[17] N. Ramdani, L. Trave-Massuyes, and C. Jauberthie, "Mode discernibility and bounded-error state estimation for nonlinear hybrid systems," Automatica, vol. 91, pp. 118-125, May 2018.
[18] M. Bettayeb and S. Djennoune, "A note on the controllabity and the observability of fractional dynamical systems," in IFAC Proc. Volumes, vol. 139, no. 11, pp. 493-498, Jan, 2006.
[19] S. Dadras and H. R. Momeni, "Fractional sliding mode observer design for a class of uncertain fractional order nonlinear systems," in Proc. of the 50th IEEE Conf. on Decision and Control and European Control Conf., CDC-ECC’11, pp. 6925-6930, Orlando, FL, USA, 12-15 Dec. 2011.
[20] Z. Belkhatir and T. M. Laleg-Kirati, "High-order sliding mode observer for fractional commensurate linear systems with unknown input," Automatica, vol. 82, pp. 209-217, Aug. 2017.
[21] 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 Trans., vol. 63, pp. 1-10, Jul. 2016.
[22] Y. H. Lan and Y. Zhou, "Non-fragile observer-based robust control for a class of fractional-order nonlinear systems," Systems & Control Letters, vol. 62, pp. 1143-1150, Dec. 2013.
[23] E. A. Boroujeni and H. R. Momeni, "Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems," Signal Processing, vol. 92, no. 10, pp. 2365-2370, Oct. 2012.
[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 J. of Automation and Computing, vol. 10, no. 4, pp. 296-302, Aug. 2013.
[25] E. A. Boroujeni and H. R. Momeni, "An iterative method to design optimal non-fragile H∞ observer for Lipschitz nonlinear fractional-order systems," Nonlinear Dynamics, vol. 80, no. 4, pp. 1801-1810, Jun. 2015.
[26] L. Liu, Z. Han, and W. Li, "H∞ non-fragile observer-based sliding mode control for uncertain time-delay systems," J. of the Franklin Institute, vol. 347, no. 2, pp. 567-576, Mar. 2010.
[27] Y. Liu, Y. Niu, and Y. Zou, "Non-fragile observer-based sliding mode control for a class of uncertain switched systems," J. of the Franklin Institute, vol. 351, no. 2, pp. 952-963, Feb. 2014.
[28] L. Gao, D. Wang, and Y. Wu, "Non-fragile observer-based sliding mode control for Markovian jump systems with mixed mode-dependent time delays and input nonlinearity," Applied Mathematics and Computation, vol. 229, pp. 374-395, Feb. 2014.
[29] Y. Kao, W. Li, and C. Wang, "Nonfragile observer‐based H∞ sliding mode control for Itô stochastic systems with Markovian switching," International J. of Robust and Nonlinear Control, vol. 24, no. 15, pp. 2035-2047, Oct. 2014.
[30] F. Zhong, H. Li, and S. Zhong, "State estimation based on fractional order sliding mode observer method for a class of uncertain fractional-order nonlinear systems," Signal Processing, vol. 127, pp. 168-184, Oct. 2016.
[31] I. Podlubny, Fractional Differential Equations, New York, Academic Press, 1999.
[32] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, Philadelphia: SIAM, 1994.
[33] S. Liu, W. Jiang, X. Li, and X. F. Zhou, "Lyapunov stability analysis of fractional nonlinear systems," Applied Mathematics Letters, vol. 51, pp. 13-19, Jan. 2016.