Authors

Lalitesh Kumar, Sukhwinder Singh Dhillon

Abstract

This article focuses on the design of tracking control for chaotic fractional order systems subjected to perturbations in a port-Hamiltonian framework. The fractional order systems of general form are modeled into port-controlled Hamiltonian form. Then, the extended results on the dissipativity, energy balance, and passivity of the fractional order systems are proved and presented in this paper. The port-controlled Hamiltonian form of the fractional order systems are proved to be asymptotically stable via energy balancing concept. Furthermore, a tracking controller is designed for the fractional order port-controlled Hamiltonian form by utilizing the matching conditions of the port-Hamiltonian systems. Stability of the system is established and analyzed explicitly for the closed-loop system with the help of direct Lyapunov method. Finally, an application example is solved with simulation results and discussions to prove the effectiveness of the propounded control design approach.

Keywords

Port-controlled Hamiltonian systems; Fractional order systems; Tracking control; Energy balancing; Passivity

Citation

BibTeX

@article{Kumar_2023,
  title={{Tracking control design for fractional order systems: A passivity-based port-Hamiltonian framework}},
  volume={138},
  ISSN={0019-0578},
  DOI={10.1016/j.isatra.2023.03.024},
  journal={ISA Transactions},
  publisher={Elsevier BV},
  author={Kumar, Lalitesh and Dhillon, Sukhwinder Singh},
  year={2023},
  pages={1--9}
}

Download the bib file

References

  • Maschke, Port-controlled Hamiltonian systems: Modelling origins and systemtheoretic properties. (1993)
  • Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
  • Donaire, A. et al. Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations. IEEE Transactions on Automatic Control vol. 61 1051–1056 (2016) – 10.1109/tac.2015.2458091
  • Hoang, N. H. & Ydstie, B. E. Integration of inventory control into the port‐Hamiltonian framework for dissipative stabilization of chemical reactors. Asian Journal of Control vol. 24 2490–2504 (2022) – 10.1002/asjc.2668
  • Mehra, R., Satpute, S. G., Kazi, F. & Singh, N. M. Control of a class of underactuated mechanical systems obviating matching conditions. Automatica vol. 86 98–103 (2017) – 10.1016/j.automatica.2017.07.033
  • Harandi, M. R. J. & Taghirad, H. D. Solution of matching equations of IDA-PBC by Pfaffian differential equations. International Journal of Control vol. 95 3368–3378 (2021) – 10.1080/00207179.2021.1972345
  • Fujimoto, K., Sakurama, K. & Sugie, T. Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations. Automatica vol. 39 2059–2069 (2003) – 10.1016/j.automatica.2003.07.005
  • Pang, S. et al. Toward Stabilization of Constant Power Loads Using IDA-PBC for Cascaded LC Filter DC/DC Converters. IEEE Journal of Emerging and Selected Topics in Power Electronics vol. 9 1302–1314 (2021) – 10.1109/jestpe.2019.2945331
  • Yang, S., Hu, C., Yu, J. & Jiang, H. Exponential Stability of Fractional-Order Impulsive Control Systems With Applications in Synchronization. IEEE Transactions on Cybernetics vol. 50 3157–3168 (2020) – 10.1109/tcyb.2019.2906497
  • Yin, C., Zhong, S. & Chen, W. Design of sliding mode controller for a class of fractional-order chaotic systems. Communications in Nonlinear Science and Numerical Simulation vol. 17 356–366 (2012) – 10.1016/j.cnsns.2011.04.024
  • Li, Y., Chen, Y. & Podlubny, I. Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica vol. 45 1965–1969 (2009) – 10.1016/j.automatica.2009.04.003
  • Li, Y., Chen, Y. & Podlubny, I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Computers & Mathematics with Applications vol. 59 1810–1821 (2010) – 10.1016/j.camwa.2009.08.019
  • Yu, J., Hu, H., Zhou, S. & Lin, X. Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems. Automatica vol. 49 1798–1803 (2013) – 10.1016/j.automatica.2013.02.041
  • Hu, J.-B., Lu, G.-P., Zhang, S.-B. & Zhao, L.-D. Lyapunov stability theorem about fractional system without and with delay. Communications in Nonlinear Science and Numerical Simulation vol. 20 905–913 (2015) – 10.1016/j.cnsns.2014.05.013
  • Aguila-Camacho, N., Duarte-Mermoud, M. A. & Gallegos, J. A. Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation vol. 19 2951–2957 (2014) – 10.1016/j.cnsns.2014.01.022
  • Duc, T. M. & Van Hoa, N. Stabilization of impulsive fractional-order dynamic systems involving the Caputo fractional derivative of variable-order via a linear feedback controller. Chaos, Solitons & Fractals vol. 153 111525 (2021) – 10.1016/j.chaos.2021.111525
  • Baleanu, D., Wu, G. & Zeng, S. Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos, Solitons & Fractals vol. 102 99–105 (2017) – 10.1016/j.chaos.2017.02.007
  • Ren, J. & Zhai, C. Stability analysis for generalized fractional differential systems and applications. Chaos, Solitons & Fractals vol. 139 110009 (2020) – 10.1016/j.chaos.2020.110009
  • Abed-Elhameed, Mittag–Leffler stability, control, and synchronization for chaotic generalized fractional-order systems. Adv Continuous Discret Model (2022)
  • Dadras, S. & Momeni, H. R. Passivity-based fractional-order integral sliding-mode control design for uncertain fractional-order nonlinear systems. Mechatronics vol. 23 880–887 (2013) – 10.1016/j.mechatronics.2013.05.009
  • Wang, Passivity-based control for fractional order unified chaotic system. (2014)
  • Kuntanapreeda, S. Adaptive control of fractional-order unified chaotic systems using a passivity-based control approach. Nonlinear Dynamics vol. 84 2505–2515 (2016) – 10.1007/s11071-016-2661-0
  • Azghandi, M. A., Barakati, S. M. & Yazdani, A. Passivity-Based Design of a Fractional-Order Virtual Capacitor for Active Damping of Multiparalleled Grid-Connected Current-Source Inverters. IEEE Transactions on Power Electronics vol. 37 7809–7818 (2022) – 10.1109/tpel.2022.3148242
  • Rakhshan, M., Gupta, V. & Goodwine, B. On Passivity of Fractional Order Systems. SIAM Journal on Control and Optimization vol. 57 1378–1389 (2019) – 10.1137/17m1126230
  • Chen, Robust passivity and feedback passification of a class of uncertain fractional-order linear systems. Internat J Systems Sci (2019)
  • Monje, (2010)
  • Liang, S., Wu, R. & Chen, L. Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays. Neurocomputing vol. 168 618–625 (2015) – 10.1016/j.neucom.2015.05.063
  • Liu, Y.-C., Dao, P. N. & Zhao, K. Y. On Robust Control of Nonlinear Teleoperators Under Dynamic Uncertainties With Variable Time Delays and Without Relative Velocity. IEEE Transactions on Industrial Informatics vol. 16 1272–1280 (2020) – 10.1109/tii.2019.2950617
  • Dao, P. N., Nguyen, V. T. & Liu, Y. Finite‐time convergence for bilateral teleoperation systems with disturbance and time‐varying delays. IET Control Theory & Applications vol. 15 1736–1748 (2021) – 10.1049/cth2.12155