access icon free State and output feedback boundary control of time fractional PDE–fractional ODE cascades with space-dependent diffusivity

This study considers the boundary control problem of a time fractional partial differential equation (PDE) – fractional ordinary differential equation (ODE) cascaded system with spatially varying diffusivity and Dirichlet connection. Using the backstepping transformation, a full-state feedback law is obtained. Meanwhile, the well-posedness of kernel functions in this transformation is established theoretically. Subsequently, a Mittag-Leffler convergent boundary observer is derived, which composes with the proposed state feedback law to enable stabilisation by output feedback. With the designed state and output feedback controllers, the Mittag-Leffler stability of the closed-loop system is then proved by the fractional Lyapunov method. Finally, the results of numerical simulations are provided for fractional cascaded plants when neither kernel ODE nor kernel PDE has an explicit solution.

Inspec keywords: closed loop systems; feedback; stability; observers; convergence of numerical methods; state feedback; Lyapunov methods; control nonlinearities; diffusion; nonlinear control systems; differential equations; partial differential equations; control system synthesis

Other keywords: Mittag-Leffler convergent boundary observer; boundary control problem; space-dependent diffusivity; Mittag-Leffler stability; designed state; fractional cascaded plants; fractional ordinary differential backstepping transformation; kernel ODE nor kernel PDE; fractional Lyapunov method; full-state feedback law; output feedback; time fractional PDE–fractional ODE cascades; time fractional partial differential equation

Subjects: Stability in control theory; Differential equations (numerical analysis); Numerical approximation and analysis; Control system analysis and synthesis methods; Mathematical analysis; Nonlinear control systems; Function theory, analysis

References

    1. 1)
      • 14. Rasheed, A., Anwar, M.S.: ‘Simulations of variable concentration aspects in a fractional nonlinear viscoelastic fluid flow’, Commun. Nonlinear Sci. Numer. Simul., 2018, 65, pp. 216230.
    2. 2)
      • 7. Krstic, M.: ‘Compensating actuator and sensor dynamics governed by diffusion PDEs’, Syst. Control Lett., 2009, 58, (5), pp. 372377.
    3. 3)
      • 40. Imran, M.A., Khan, I., Ahmad, M., et al: ‘Heat and mass transport of differential type fluid with non-integer order time-fractional Caputo derivatives’, J. Mol. Liq., 2017, 229, pp. 6775.
    4. 4)
      • 46. Smyshlyaev, A., Krstic, M.: ‘Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations’, IEEE Trans. Autom. Control, 2004, 49, (12), pp. 21852202.
    5. 5)
      • 47. Ablowitz, M.J., Kruskal, M.D., Ladik, J.: ‘Solitary wave collisions’, SIAM J. Appl. Math., 1979, 36, (3), pp. 428437.
    6. 6)
      • 32. Daftardar-Gejji, V., Babakhani, A.: ‘Analysis of a system of fractional differential equations’, J. Math. Anal. Appl., 2004, 293, (2), pp. 511522.
    7. 7)
      • 9. Sun, H., Chen, W., Li, C., et al: ‘Fractional differential models for anomalous diffusion’, Physica A, 2010, 389, (14), pp. 27192724.
    8. 8)
      • 21. Zhou, H.C., Lv, C., Guo, B.Z., et al: ‘Mittag-leffler stabilization for an unstable time-fractional anomalous diffusion equation with boundary control matched disturbance’, Int. J. Robust Nonlinear Control, 2019, 29, (13), pp. 43844401.
    9. 9)
      • 24. Wang, J., Xiong, X., Zhang, Y.: ‘Extending synchronization scheme to chaotic fractional-order chen systems’, Physica A, 2006, 370, (2), pp. 279285.
    10. 10)
      • 19. Chen, J., Cui, B., Chen, Y.: ‘Backstepping-based boundary control design for a fractional reaction diffusion system with a space-dependent diffusion coefficient’, ISA Trans., 2018, 80, pp. 203211.
    11. 11)
      • 27. Matignon, D.: ‘Stability results for fractional differential equations with applications to control processing’, Comput. Eng. Syst. Appl., 1996, 2, pp. 963968.
    12. 12)
      • 25. Podlubny, I.: ‘Fractional differential equations’ (Academic Press, San Diego, CA, 1999).
    13. 13)
      • 35. Ge, F., Chen, Y.: ‘Observer design for semilinear time fractional diffusion systems with spatially varying parameters’. Proc. of Int. Conf. on Fractional Differention and its Applications (ICFDA), Amman, Jordan, 2018, Available at http://dx.doi.org/10.2139/ssrn.3281639.
    14. 14)
      • 15. Meerschaert, M.M., Tadjeran, C.: ‘Finite difference approximations for fractional advection–dispersion flow equations’, J. Comput. Appl. Math., 2004, 172, (1), pp. 6577.
    15. 15)
      • 16. Ge, F., Chen, Y., Kou, C.: ‘Boundary feedback stabilisation for the time fractional-order anomalous diffusion system’, IET Control Theory Appl., 2016, 10, (11), pp. 12501257.
    16. 16)
      • 39. Miller, K.S., Samko, S.G.: ‘Completely monotonic functions’, Integral Transform. Spec. Funct., 2001, 12, (4), pp. 389402.
    17. 17)
      • 17. Chen, J., Zhuang, B., Chen, Y., et al: ‘Backstepping-based boundary feedback control for a fractional reaction diffusion system with mixed or robin boundary conditions’, IET Control Theory Appl., 2017, 11, (17), pp. 29642976.
    18. 18)
      • 8. Podlubny, I.: ‘Fractional-order systems and PIλDμ-controllers’, IEEE Trans. Autom. Control, 1999, 44, (1), pp. 208214.
    19. 19)
      • 1. d'Andréa Novel, B., Boustany, F., Conrad, F., et al: ‘Feedback stabilization of a hybrid PDE-ODE system: application to an overhead crane’, Math. Control Signal Syst., 1994, 7, (1), pp. 122.
    20. 20)
      • 4. Diagne, M., Bekiaris-Liberis, N., Otto, A., et al: ‘Control of transport PDE/nonlinear ODE cascades with state-dependent propagation speed’, IEEE Trans. Autom. Control, 2017, 62, (12), pp. 62786293.
    21. 21)
      • 13. Xu, Q., Xu, Y.: ‘Extremely low order time-fractional differential equation and application in combustion process’, Commun. Nonlinear Sci. Numer. Simul., 2018, 64, pp. 135148.
    22. 22)
      • 30. Tang, S., Xie, C.: ‘Stabilization for a coupled PDE-ODE control system’, J. Franklin Inst., 2011, 348, (8), pp. 21422155.
    23. 23)
      • 36. Luchko, Y.: ‘Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation’, Comput. Math. Appl., 2010, 59, (5), pp. 17661772.
    24. 24)
      • 29. Tang, S., Xie, C.: ‘State and output feedback boundary control for a coupled PDE-ODE system’, Syst, & Control Lett., 2011, 60, (8), pp. 540545.
    25. 25)
      • 31. Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: ‘‘Theory and applications of fractional differential equations’, vol. 204 (Elsevier Science Limited, Amsterdam, Netherlands, 2006).
    26. 26)
      • 22. Li, Y., Chen, Y., Podlubny, I.: ‘Stability of fractional-order nonlinear dynamic systems: lYapunov direct method and generalized mittag–leffler stability’, Comput. Math. Appl., 2010, 59, (5), pp. 18101821.
    27. 27)
      • 20. Chen, J., Cui, B., Chen, Y.: ‘Observer-based output feedback control for a boundary controlled fractional reaction diffusion system with spatially-varying diffusivity’, IET Control Theory Appl., 2018, 12, (11), pp. 15611572.
    28. 28)
      • 23. Duan, J.S., Chaolu, T., Rach, R.: ‘Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach–adomian–meyers modified decomposition method’, Appl. Math. Comput., 2012, 218, (17), pp. 83708392.
    29. 29)
      • 18. Cherstvy, A.G., Chechkin, A.V., Metzler, R.: ‘Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity’, Soft Mat., 2014, 10, (10), pp. 15911601.
    30. 30)
      • 45. Li, H., Cao, J., Li, C.: ‘High-order approximation to caputo derivatives and caputo-type advection-diffusion equations (III)’, J. Comput. Appl. Math., 2016, 299, (3), pp. 159175.
    31. 31)
      • 28. Baleanu, D., Machado, J.A.T., Luo, A.C.: ‘Fractional dynamics and control’ (Springer Science & Business Media, New York, USA, 2012).
    32. 32)
      • 3. Hasan, A., Aamo, O.M., Krstic, M.: ‘Boundary observer design for hyperbolic PDE-ODE cascade systems’, Automatica, 2016, 68, pp. 7586.
    33. 33)
      • 5. Krstic, M., Smyshlyaev, A.: ‘Boundary control of PDEs: a course on backstepping designs’ (Society for Industrial and Applied Mathematics, Philadelphia, USA, 2008).
    34. 34)
      • 26. Matignon, D., d'Andréa-Novel, B.: ‘Some results on controllability and observability of finite-dimensional fractional differential systems’, Comput. Eng. Syst. Appl., 1996, 2, pp. 952956.
    35. 35)
      • 6. Krstic, M., Smyshlyaev, A.: ‘Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays’, Syst. Control Lett., 2008, 57, (9), pp. 750758.
    36. 36)
      • 2. Zhou, H.C., Guo, B.Z., Wu, Z.H.: ‘Output feedback stabilisation for a cascaded wave PDE-ODE system subject to boundary control matched disturbance’, Int. J. Control, 2016, 89, (12), pp. 23962405.
    37. 37)
      • 38. Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: ‘Lyapunov functions for fractional order systems’, Commun. Nonlinear Sci. Numer. Simul., 2014, 19, (9), pp. 29512957.
    38. 38)
      • 33. Meurer, T., Kugi, A.: ‘Tracking control for boundary controlled parabolic PDEs with varying parameters: combining backstepping and differential flatness’, Automatica, 2009, 45, (5), pp. 11821194.
    39. 39)
      • 44. Li, C., Chen, A.: ‘Numerical methods for fractional partial differential equations’, Int. J. Comput. Math., 2018, 95, (6-7), pp. 10481099.
    40. 40)
      • 11. Pagnini, G.: ‘Nonlinear time-fractional differential equations in combustion science’, Fract. Calculus Appl. Anal., 2011, 14, (1), pp. 8093.
    41. 41)
      • 43. Luchko, Y., Mainardi, F., Povstenko, Y.: ‘Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation’, Comput. Math. Appl., 2013, 66, (5), pp. 774784.
    42. 42)
      • 10. Povstenko, Y.: ‘Fractional heat conduction in an infinite medium with a spherical inclusion’, Entropy, 2013, 15, (10), pp. 41224133.
    43. 43)
      • 42. Povstenko, Y.Z.: ‘Fractional heat conduction equation and associated thermal stress’, J. Therm. Stresses, 2004, 28, (1), pp. 83102.
    44. 44)
      • 12. Lü, Q., Zuazua, E.: ‘On the lack of controllability of fractional in time ODE and PDE’, Math. Control Signal Syst., 2016, 28, (2), pp. 121.
    45. 45)
      • 34. Ge, F., Meurer, T., Chen, Y.: ‘Mittag-leffler convergent backstepping observers for coupled semilinear subdiffusion systems with spatially varying parameters’, Syst. Control Lett., 2018, 122, pp. 8692.
    46. 46)
      • 37. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., et al: ‘Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems’, Commun. Nonlinear Sci. Numer. Simul., 2015, 22, (1-3), pp. 650659.
    47. 47)
      • 41. Sutulo, S., Soares, C.G.: ‘On applicability of mathematical models based on fractional calculus to ship dynamics’. IFAC Conf. on Control Applications in Marine systems, Rostock-Warnemunde, Germany, 2010, pp. 190195.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2019.1015
Loading

Related content

content/journals/10.1049/iet-cta.2019.1015
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading