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Periodic disturbance rejection for fractional-order dynamical systems

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Published/Copyright: May 23, 2015
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 18 Issue 3

Abstract

This paper proposes a simple fractional-order derivative controller, based on an adaptive orthogonal signals generator, which permits both to reconstruct an unknown multi-sinusoidal disturbance and cancel its effect on the system output. An interesting feature is that the disturbance is removed by the generated internal signals with no additional dynamics in the cancellation algorithm. An opportune choice of the fractional-order controller guarantees the closed-loop stability of the system if the location of the plant frequency response at the estimated frequencies belongs to a halfplane passing through the origin of the complex plane, i.e. no information about the order of the system to be controlled, the relative degree, the nature of its poles and zeros, is required. The case of multi-sinusoidal disturbance is also analyzed. Simulations are presented that highlight the performances of the proposed method.

Keywords : active noise and vibration control; orthogonal signals generator; fractional controllers; sinusoidal parameters estimation

References

[1] M. Bodson, S.C. Douglas, Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency. Automatica 33, No 12 (1997), 2213-2221.Search in Google Scholar

[2] R. Caponetto, G. Dongola, L. Fortuna, I. Petr´aˇs, Fractional Order Systems. World Scientific, Singapore etc. (2010).10.1142/7709Search in Google Scholar

[3] J. Chanderasekar, L. Liu, D. Patt, P.P. Friedmann, D.S. Bernstein, Adaptive harmonic steady-state control for disturbance rejection. IEEE Trans. on Contr. Syst. Tech. 14, No 6 (2006), 993-1007.Search in Google Scholar

[4] Y.Q. Chen, I. Petras, D. Xue, Fractional order control - a tutorial. In: American Control Conference (ACC), St. Louis, MO - USA (2009), 1397-1411.10.1109/ACC.2009.5160719Search in Google Scholar

[5] P. Chen, M. Sun, Q. Yan, X. Fang, Adaptive asymptotic rejection of unmatched general periodic disturbances in output-feedback nonlinear systems. IEEE Trans. on Aut. Contr. 57, No 4 (2012), 1056-1061.Search in Google Scholar

[6] S. Das, Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008).Search in Google Scholar

[7] Z. Ding, Asymptotic rejection of asymmetric periodic disturbances in output-feedback nonlinear systems. Automatica 43, No 3 (2007), 555-561.Search in Google Scholar

[8] D. Ding, Z. Wang, H. Dong, H. Shu, Distributed H∞ state estimation with stochastic parameters and nonlinearities through sensor networks: The finite-horizon case. Automatica 48, No 8 (2012), 1575-1585.Search in Google Scholar

[9] H. Dong, Z. Wang, H. Gao, Distributed H∞ filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks. IEEE Trans. on Ind. Electr. 60, No 10 (2013), 4665-4672.Search in Google Scholar

[10] G. Fedele, A. Ferrise, Non adaptive second order generalized integrator for identification of a biased sinusoidal signal. IEEE Trans. on Aut. Contr. 57, No 7 (2012), 1838-1842.Search in Google Scholar

[11] G. Fedele, A. Ferrise, Biased disturbance compensation with unknown frequency. IEEE Trans. on Aut. Contr. 58, No 12 (2013), 3207-3212.Search in Google Scholar

[12] G. Fedele, A. Ferrise, Periodic disturbance rejection with unknown frequency and unknown plant structure. J. of Frank. Inst. 351, No 2 (2014), 1074-1092.Search in Google Scholar

[13] G. Fedele, A. Ferrise, D. Frascino, Multi-sinusoidal signal estimation by an adaptive SOGI-filters bank. In: 15th IFAC Symp. on System Identification, Saint-Malo, France (2008), 402-407.10.3182/20090706-3-FR-2004.00066Search in Google Scholar

[14] B. Francis, W.Wonham, The internal model principle of control theory. Automatica 12, No 5 (1976), 457-464.[15] Z. Gao, Scaling and bandwidth-parameterization based controller tuning. In: American Control Conference (ACC), Denver, Colorado, USA (2003), 4989-4996.Search in Google Scholar

[16] Y. Jiang, S. Li, Rejection of nonharmonic disturbances in a class of nonlinear systems with nonlinear exosystems. Asian J. of Contr. 13, No 6 (2011), 858-867.Search in Google Scholar

[17] M. Karimi-Ghartemani, F. Merrikh-Bayat, Necessary and sufficient conditions for perfect command following and disturbance rejection in fractional order systems. In: 17th World Congress of the International Federation of Automatic Control (IFAC), Seoul, Korea (2008), 364-369.10.3182/20080706-5-KR-1001.00062Search in Google Scholar

[18] M. Li, D. Li, J. Wang, C. Zhao, Active disturbance rejection control for fractional-order system. ISA Trans. 52, No 3 (2013), 365-374.Search in Google Scholar

[19] Y. Luo, Y.Q. Chen, Y.G. Pi, Fractional Order Adaptive Feedforward Cancellation In: American Control Conference (ACC), San Francisco, CA (2011), 19-24.Search in Google Scholar

[20] C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractionalorder Systems and Controls. Springer (2010).10.1007/978-1-84996-335-0Search in Google Scholar

[21] C.A. Monje, B.M. Vinagre, V. Feliu, Y. Chen, Tuning and auto-tuning of fractional order controllers for industry applications. Contr. Eng. Pract. 16, No 7 (2008), 798-812.Search in Google Scholar

[22] A. Oustaloup, La Commande CRONE: Command Robuste d’Ordre Non Entier. Hermes (1991).Search in Google Scholar

[23] A. Oustaloup, F. Levron, B. Mathieu, F.M. Nanot, Frequency-band complex non-integer differentiator: Characterization and synthesis. IEEE Trans. on Circuits Syst. I: Fundam. Theory Appl. 47, No 1 (2000), 25-39.Search in Google Scholar

[24] F. Padula, A. Visioli, Tuning rules for optimal PID and fractional-order PID controllers. J. of Proc. Contr. 21, No 1 (2011), 69-81.Search in Google Scholar

[25] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego-Boston etc. (1999).Search in Google Scholar

[26] I. Podlubny, I. Petras, Analogue realizations of fractional-order controllers. Nonlin. Dynam. 29, No 1 (2002), 281-296.Search in Google Scholar

[27] A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements. Chaos, Solitons and Fractals 40, No 5 (2009), 2317-2328.Search in Google Scholar

[28] M.A. Rahimian, M.S. Tavazoei, Optimal tuning for fractional-order controllers: An integer-order approximating filter approach. J. of Dyn. Sys., Meas., Contr. 135, No 2 (2013), 021017/1-021017/11.10.1115/1.4023066Search in Google Scholar

[29] H. Ramezanian, S. Balochian, A. Zare, Design of optimal fractionalorder pid controllers using particle swarm optimization algorithm forautomatic voltage regulator (AVR) system. J. of Contr., Aut., Electr. Sys. 24, No 5 (2013), 601-611.Search in Google Scholar

[30] H.F. Raynauld, A. Zergainoh, State-space representation for fractionalorder controllers. Automatica 36, No 7 (2000), 1017-1021.Search in Google Scholar

[31] B. Shen, Z. Wang,H. Shu, G. Wei, Robust H∞ finite-horizon filtering with randomly occurred nonlinearities and quantization effects. Automatica 46, No 11 (2010), 1743-1751.Search in Google Scholar

[32] M.S. Tavazoei, A note on fractional-order derivatives of periodic functions. Automatica 46, No 5 (2010), 945-948.Search in Google Scholar

[33] A. Tepljakov, E. Petlenkov, J. Belikov, FOMCON: a MATLAB toolbox for fractional-order system identification and control. Int. Journal of Microelect. and Comp. Sci. 2, No 2 (2011), 51-62.Search in Google Scholar

[34] B.M. Vinagre, V. Feliu, Optimal fractional controllers for rational order systems: A special case of the Wiener-Hopf spectral factorization method. IEEE Trans. on Aut. Contr. 52, No 12 (2007), 2385-2389.Search in Google Scholar

[35] M. Zhang, H. Lan, W. Ser, Cross-updated active noise control system with online secondary path modeling. IEEE Trans. on Speech Audio Proc. 9, No 5 (2001), 598-602. Search in Google Scholar

Received: 2014-6-23
Published Online: 2015-5-23
Published in Print: 2015-6-1

© Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Fcaa Related News, Events And Books (Fcaa-Volume 18-3-2015)
  3. Decay solutions for a class of fractional differential variational inequalities
  4. A biomathematical view on the fractional dynamics of cellulose degradation
  5. The spreading property for a prey-predator reaction-diffusion system with fractional diffusion
  6. Fractional variation of Hölderian functions
  7. Periodic disturbance rejection for fractional-order dynamical systems
  8. Successive approximation: A survey on stable manifold of fractional differential systems
  9. When do fractional differential equations have solutions that are bounded by the Mittag--Leffler function ?
  10. On explicit stability conditions for a linear fractional difference system
  11. Fractional differential inclusions in the Almgren sense
  12. Time-optimal control of fractional-order linear systems
  13. Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain
  14. Nonexistence results for a class of evolution equations in the Heisenberg group
  15. High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)
  16. Dyadic nonlocal diffusions in metric measure spaces
  17. Fractional derivative anomalous diffusion equation modeling prime number distribution
  18. Time-fractional diffusion equation in the fractional Sobolev spaces
  19. Continuous time random walk models associated with distributed order diffusion equations
https://doi.org/10.1515/fca-2015-0037
Keywords for this article
active noise and vibration control; orthogonal signals generator; fractional controllers; sinusoidal parameters estimation

Articles in the same Issue

  1. Frontmatter
  2. Fcaa Related News, Events And Books (Fcaa-Volume 18-3-2015)
  3. Decay solutions for a class of fractional differential variational inequalities
  4. A biomathematical view on the fractional dynamics of cellulose degradation
  5. The spreading property for a prey-predator reaction-diffusion system with fractional diffusion
  6. Fractional variation of Hölderian functions
  7. Periodic disturbance rejection for fractional-order dynamical systems
  8. Successive approximation: A survey on stable manifold of fractional differential systems
  9. When do fractional differential equations have solutions that are bounded by the Mittag--Leffler function ?
  10. On explicit stability conditions for a linear fractional difference system
  11. Fractional differential inclusions in the Almgren sense
  12. Time-optimal control of fractional-order linear systems
  13. Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain
  14. Nonexistence results for a class of evolution equations in the Heisenberg group
  15. High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)
  16. Dyadic nonlocal diffusions in metric measure spaces
  17. Fractional derivative anomalous diffusion equation modeling prime number distribution
  18. Time-fractional diffusion equation in the fractional Sobolev spaces
  19. Continuous time random walk models associated with distributed order diffusion equations
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