0
Research Papers

Fractional-Order Nonlinear Disturbance Observer Based Control of Fractional-Order Systems

[+] Author and Article Information
Aldo Jonathan Muñoz-Vázquez

Mechatronic Engineering,
Polytechnic University of Victoria,
Ciudad Victoria 87138,
Tamaulipas, Mexico
e-mail: amunozv@upv.edu.mx

Vicente Parra-Vega

Robotics and Advanced Manufacturing,
Research Center for Advanced Studies,
Saltillo 25900,
Coahuila, Mexico
e-mail: vparra@cinvestav.edu.mx

Anand Sánchez-Orta

Robotics and Advanced Manufacturing,
Research Center for Advanced Studies,
Saltillo 25900,
Coahuila, Mexico
e-mail: anand.sanchez@cinvestav.edu.mx

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 31, 2018; final manuscript received April 30, 2018; published online May 30, 2018. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 13(7), 071007 (May 30, 2018) (8 pages) Paper No: CND-18-1047; doi: 10.1115/1.4040129 History: Received January 31, 2018; Revised April 30, 2018

The robust control for a class of disturbed fractional-order systems is presented in this paper. The proposed controller considers a dynamic observer to exactly compensate for matched disturbances in finite time, and a procedure to compensate for unmatched disturbances is then derived. The proposed disturbance observer is built upon continuous fractional sliding modes, producing a fractional-order reaching phase, leading to a continuous control signal, yet able to reject for some continuous but not necessarily differentiable disturbances. Numerical simulations and comparisons are presented to highlight the reliability of the proposed scheme.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Ohishi, K. , Nakao, M. , Ohnishi, K. , and Miyachi, K. , 1987, “ Microprocessor-Controlled DC Motor for Load-Insensitive Position Servo System,” IEEE Trans. Ind. Electron., 34(1), pp. 44–49. [CrossRef]
Li, S. , Yang, J. , Chen, W. H. , and Chen, X. , 2014, Disturbance Observer-Based Control: Methods and Applications, CRC Press, Boca Raton, FL.
Chen, W. H. , Yang, J. , Guo, L. , and Li, S. , 2016, “ Disturbance-Observer-Based Control and Related Methods—An Overview,” IEEE Trans. Ind. Electron., 63(2), pp. 1083–1095. [CrossRef]
Chen, W. H. , 2004, “ Disturbance Observer Based Control for Nonlinear Systems,” IEEE/ASME Trans. Mechatronics, 9(4), pp. 706–710. [CrossRef]
Iwasaki, M. , Shibata, T. , and Matsui, N. , 1999, “ Disturbance-Observer-Based Nonlinear Friction Compensation in Table Drive System,” IEEE/ASME Trans. Mechatronics, 4(1), pp. 3–8. [CrossRef]
Eom, K. S. , Suh, I. H. , Chung, W. K. , and Oh, S. R. , 1998, “ Disturbance Observer Based Force Control of Robot Manipulator Without Force Sensor,” IEEE International Conference on Robotics and Automation (ICRA), Leuven, Belgium, May 20, pp. 3012–3017.
Murakami, T. , Nakamura, R. , Yu, F. , and Ohnishi, K. , 1993, “ Force Sensorless Impedance Control by Disturbance Observer,” IEEE Power Conversion Conference, Yokohama, Japan, Apr. 19–21, pp. 352–357.
Katsura, S. , and Matsumto, Y. , 2007, “ Modeling of Force Sensing and Validation of Disturbance Observer for Force Control,” IEEE Trans. Ind. Electron., 54(1), pp. 530–538. [CrossRef]
Gupta, A. , and O'Malley, M. K. , 2011, “ Disturbance-Observer-Based Force Estimation for Haptic Feedback,” ASME J. Dyn. Syst. Meas. Control, 133(1), p. 014505. [CrossRef]
Wang, H. , and Chen, M. , 2016, “ Trajectory Tracking Control for an Indoor Quadrotor UAV Based on the Disturbance Observer,” Trans. Inst. Meas. Control, 38(6), pp. 675–692. [CrossRef]
Yang, J. , Li, S. , and Yu, X. , 2013, “ Sliding Mode Control for Systems With Mismatched Uncertainties Via Disturbance Observer,” IEEE Trans. Ind. Electron., 60(1), pp. 160–169. [CrossRef]
Mohammadi, A. , Tavakoli, M. , Marquez, H. J. , and Hashemzadeh, F. , 2013, “ Nonlinear Disturbance Observer Design for Robotic Manipulators,” Control Eng. Pract., 21(3), pp. 253–267. [CrossRef]
Liu, L. P. , Fu, Z. M. , and Song, X. N. , 2012, “ Sliding Mode Control With Disturbance Observer for a Class of Nonlinear Systems,” Int. J. Autom. Comput., 9(5), pp. 487–491. [CrossRef]
Chen, X. , Komada, S. , and Fukuda, T. , 2000, “ Design of a Nonlinear Disturbance Observer,” IEEE Trans. Ind. Electron., 47(2), pp. 429–437. [CrossRef]
Chen, W. H. , Ballance, D. J. , Gawthrop, P. J. , and O'Reilly, J. , 2000, “ A Nonlinear Disturbance Observer for Robotic Manipulators,” IEEE Trans. Ind. Electron., 47(4), pp. 932–938. [CrossRef]
Pashaei, S. , and Badamchizadeh, M. , 2016, “ A New Fractional-Order Sliding Mode Controller Via a Nonlinear Disturbance Observer for a Class of Dynamical Systems With Mismatched Disturbances,” ISA Trans., 63, pp. 39–48. [CrossRef] [PubMed]
Chen, M. , Shu-Yi, S. , Shi, P. , and Shi, Y. , 2017, “ Disturbance-Observer-Based Robust Synchronization Control for a Class of Fractional-Order Chaotic Systems,” IEEE Trans. Circuits Syst. II: Express Briefs, 64(4), pp. 417–421. [CrossRef]
Lin, C. , Chen, B. , Shi, P. , and Yu, J. P. , 2018, “ Necessary and Sufficient Conditions of Observer-Based Stabilization for a Class of Fractional-Order Descriptor Systems,” Syst. Control Lett., 112, pp. 31–35. [CrossRef]
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA. [PubMed] [PubMed]
Samko, S. , Kilbas, A. , and Marichev, O. , 1993, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, Switzerland.
Song, X. , Song, S. , and Tejado, I. , 2016, “ Fuzzy Adaptive Function Projective Combination Synchronization of a Class of Fractional-Order Chaotic and Hyperchaotic Systems,” Int. J. Innovative Comput., Inf. Control, 12(4), pp. 1317–1332. http://www.ijicic.org/ijicic-120420.pdf
Efe, M. Ö. , 2010, “ Fractional Order Sliding Mode Control With Reaching Law Approach,” Turk. J. Electr. Eng. Comput. Sci., 18(5), pp. 731–748.
Efe, M. Ö. , 2011, “ Integral Sliding Mode Control of a Quadrotor With Fractional Order Reaching Dynamics,” Trans. Inst. Meas. Control, 33(8), pp. 985–1003. [CrossRef]
Hu, W. , Ding, D. , and Wang, N. , 2016, “ Nonlinear Dynamic Analysis of a Simplest Fractional-Order Delayed Memristive Chaotic System,” ASME J. Comput. Nonlinear Dyn., 12(4), p. 041003. [CrossRef]
Alaviyan-Shahri, E. S. , Alfi, A. , and Tenreiro-Machado, J. A. , 2016, “ Stabilization of Fractional-Order Systems Subject to Saturation Element Using Fractional Dynamic Output Feedback Sliding Mode Control,” ASME J. Comput. Nonlinear Dyn., 12(3), p. 031014. [CrossRef]
Duc, T. M. , Hoa, N. V. , and Dao, T. P. , 2018, “ Adaptive Fuzzy Fractional-Order Nonsingular Terminal Sliding Mode Control for a Class of Second-Order Nonlinear Systems,” ASME J. Comput. Nonlinear Dyn., 13(3), p. 031004. [CrossRef]
Muñoz-Vázquez, A. J. , Parra-Veg, V. , Sánchez-Orta, A. , and Romero-Galván, G. , 2017, “ Output Feedback Finite-Time Stabilization of Systems Subject to Hölder Disturbances Via Continuous Fractional Sliding Modes,” Math. Probl. Eng., 2017, p. 3146231.
Muñoz-Vázquez, A. J. , Parra-Veg, V. , and Sánchez-Orta, A. , 2017, “ A Novel Continuous Fractional Sliding Mode Control,” Int. J. Syst. Sci., 48(13), pp. 2901–2908. [CrossRef]
Muñoz-Vázquez, A. J. , Parra-Vega, V. , Sánchez-Orta, A. , and Romero-Galván, G. , 2017, “ Finite-Time Disturbance Observer Via Continuous Fractional Sliding Modes,” Trans. Inst. Meas. Control, epub.
Pisano, A. , Rapaic, M. , Jelecic, Z. , and Usai, E. , 2010, “ Sliding Mode Control Approaches to the Robust Regulation of Linear Multivariable Fractional-Order Dynamics,” Int. J. Robust Nonlinear Control, 20(18), pp. 2045–2056. [CrossRef]
Pisano, A. , Rapaic, M. , Usai, E. , and Jelicic, Z. , 2012, “ Continuous Finite-Time Stabilization for Some Classes of Fractional Order Dynamics,” IEEE International Workshop on Variable Structure Systems (VSS), Mumbai, India, Jan. 12–14, pp. 16–21.
Jakovljević, B. , Pisano, A. , Rapaic, M. R. , and Usai, E. , 2015, “ On the Sliding-Mode Control of Fractional-Order Nonlinear Uncertain Dynamics,” Int. J. Robust Nonlinear Control, 26(4), pp. 782–798. [CrossRef]
Kamal, S. , Raman, A. , and Bandyopadhyay, B. , 2013, “ Finite-Time Stabilization of Fractional Order Uncertain Chain of Integrator: An Integral Sliding Mode Approach,” IEEE Trans. Autom. Control, 58(6), pp. 1597–1602. [CrossRef]
Petrás, I. , 2009, “ Stability of Fractional-Order Systems With Rational Orders: A Survey,” Fractional Calculus Appl. Anal., 12(3), pp. 269–298. http://www.math.bas.bg/complan/fcaa/volume12/fcaa123/IvoPetras_FCAA_12_3.pdf
Li, Y. , Chen, Y. Q. , and Podlubny, I. , 2009, “ Mittag-Leffler Stability of Fractional Order Nonlinear Dynamic Systems,” Automatica, 45(8), pp. 1965–1969. [CrossRef]
Li, Y. , Chen, Y. Q. , and Podlubny, I. , 2010, “ Stability of Fractional-Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–Leffler Stability,” Comput. Math. Appl., 59(5), pp. 1810–1821. [CrossRef]
Zhang, F. , Li, C. , and Chen, Y. Q. , 2011, “ Asymptotical Stability of Nonlinear Fractional Differential System With Caputo Derivative,” Int. J. Differ. Equations, 2011, p. 635165.
Aguila-Camacho, N. , Duarte-Mermoud, M. A. , and Gallegos, J. A. , 2014, “ Lyapunov Functions for Fractional Order Systems,” Commun. Nonlinear Sci. Numer. Simul., 19(9), pp. 951–2957. [CrossRef]
Duarte-Mermoud, M. A. , Aguila-Camacho, N. , Gallegos, J. A. , and Castro-Linares, R. , 2015, “ Using General Quadratic Lyapunov Functions to Prove Lyapunov Uniform Stability for Fractional Order Systems,” Commun. Nonlinear Sci. Numer. Simul., 22(1–3), pp. 650–659. [CrossRef]
Muñoz-Vázquez, A. J. , Parra-Vega, V. , Sánchez-Orta, A. , and Romero-Galván, G. , 2018, “ Quadratic Lyapunov Functions for Stability Analysis in Fractional-Order Systems With Not Necessarily Differentiable Solutions,” Syst. Control Lett., 116, pp. 15–19. [CrossRef]
Miller, K. S. , and Samko, S. , 1997, “ A Note on the Complete Monotonicity of the Generalized Mittag-Leffler Function,” Real Anal. Exch., 23(2), pp. 753–755. https://projecteuclid.org/euclid.rae/1337001380
Schiessel, H. , Metzler, R. , Blumen, A. , and Nonnenmacher, T. F. , 1995, “ Generalized Viscoelastic Models: Their Fractional Equations With Solutions,” J. Phys. A: Math. Gen., 28(23), pp. 6567–6584. [CrossRef]
Iftikhar, M. , Riu, D. , Druart, F. , Rosini, S. , Bultel, Y. , and Retière, N. , 2006, “ Dynamic Modeling of Proton Exchange Membrane Fuel Cell Using Non-Integer Derivatives,” J. Power Sources, 160(2), pp. 1170–1182. [CrossRef]
Magin, R. L. , 2010, “ Fractional Calculus Models of Complex Dynamics in Biological Tissues,” Comput. Math. Appl., 59(5), pp. 1586–1593. [CrossRef]
Humphrey, J. , Schuler, C. , and Rubinsky, B. , 1992, “ On the Use of the Weierstrass-Mandelbrot Function to Describe the Fractal Component of Turbulent Velocity,” Fluid Dyn. Res., 9(1–3), pp. 81–95. [CrossRef]
Machado, J. T. , 2013, “ Fractional Order Modelling of Dynamic Backlash,” Mechatronics, 23(7), pp. 741–745. [CrossRef]
Mandelbrot, B. B. , and Van Ness, J. W. , 1968, “ Fractional Brownian Motions, Fractional Noises and Applications,” SIAM Rev., 10(4), pp. 422–437. [CrossRef]
Carlson, G. , and Halijak, C. , 1964, “ Approximation of Fractional Capacitors (1/s)(1∕n) by a Regular Newton Process,” IEEE Trans. Circuit Theory, 11(2), pp. 210–213. [CrossRef]
Westerlund, S. , 1991, “ Dead Matter Has Memory!,” Phys. Scr., 43(2), pp. 174–179. [CrossRef]
Matignon, D. , 1998, “ Stability Properties for Generalized Fractional Differential Systems,” ESAIM: Proc., 5, pp. 145–158.
Utkin, V. , 1992, Sliding Modes in Control and Optimization, Springer-Verlag, Berlin. [CrossRef]
Royden, H. L. , and Fitzpatrick, P. , 1968, Real Analysis, Macmillan, New York.
Chen, Y. Q. , Petrás, I. , and Xue, D. , 2009, “ Fractional Order Control—A Tutorial,” IEEE American Control Conference (ACC), St. Louis, MO, June 10–12, pp. 1397–1411.
Oustaloup, A. , Mathieu, B. , and Lanusse, P. , 1995, “ The CRONE Control of Resonant Plants: Application to a Flexible Transmission,” Eur. J. Control, 1(2), pp. 113–121. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Nowhere differentiable disturbance

Grahic Jump Location
Fig. 2

Matched disturbance compensation. Fractional-order disturbance observer φ̂(t)=φ̂(tn)+γtnItαsign(σ(t)): (a) t versus φ̂(t), (b) t versus σ(t)=∫(φ−φ̂), and (c) t versus x(t).

Grahic Jump Location
Fig. 3

Matched disturbance compensation. Integer-order disturbance observer φ̂(t)=γ(σ(t)/‖σ(t)‖): (a) t versus φ̂(t), (b) t versus σ(t)=∫(φ−φ̂), and (c) t versus x(t).

Grahic Jump Location
Fig. 4

Matched disturbance compensation. Approximated integer-order disturbance observer φ̂(t)=γ(σ(t)/‖σ(t)‖+δ): (a) t versus φ̂(t), (b) t versus σ(t)=∫(φ−φ̂), and (c) t versus x(t).

Grahic Jump Location
Fig. 5

Unmatched disturbance. Disturbance estimation: (a) t versus φ1(t), (b) t versus φ̂1(t), and (c) t versus σ(t)=∫(φ1−φ̂1).

Grahic Jump Location
Fig. 6

Unmatched disturbance. Pseudostate and control signal. (a) t versus x̃2(t), (b) t versus u(t), and (c) t versus x1(t).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In