SpringerLink
Log in

Static output feedback ℋ ∞ control for a fractional-order glucose-insulin system

  • Regular Papers
  • Control Theory
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

This paper presents the ℋ ∞ static output feedback control of nonlinear fractional-order systems. Based on the extended bounded real lemma, the ℋ ∞ control is formulated and sufficient conditions are derived in terms of linear matrix inequalities (LMIs) formulation by using the fractional Lyapunov direct method where the fractional-order α belongs to 0 < α < 1. The control approach is finally applied to the regulation of the glucose level in diabetes type 1 treatment. Therefore, it is attempted to incorporate fractional-order into the mathematical minimal model of glucose-insulin system dynamics and it is still an interesting challenge to show, how the order of a fractional differential system affects the dynamics of the system in the presence of meal disturbance. Numerical simulations are carried out to illustrate our proposed results and show that the nonlinear fractional-order glucose-insulin systems are, at least, as stable as their integer-order counterpart in the presence of exogenous glucose infusion or meal disturbance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, Singapore, 2001.

    Google Scholar 

  2. I. Podlubny, Fractional Differential Equations, Academic, New York, 1999.

    MATH  Google Scholar 

  3. A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006.

    MATH  Google Scholar 

  4. K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.

    MATH  Google Scholar 

  5. O. Heaviside, Electromagnetic Theory, 3rd ed., Chelsea Publishing Company, New York, 1971.

    Google Scholar 

  6. N. Engheta, “On fractional calculus and fractional multipoles in electromagnetism” IEEE Trans. Antennas and Propagation, vol. 44, no. 4, pp. 554–566, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Sun, A. Abdelwahad, and B. Onaral, “Linear approximation of transfer function with a pole of fractional order” IEEE Trans. Aut. Contr., vol. 29, no. 5, pp. 441–444, 1984.

    Article  MATH  Google Scholar 

  8. R. Bagley and R. Calico, “Fractional order state equations for the control of viscoelastically damped structures” J. Guidance, Contr. & Dynamics, vol. 14, no. 2, pp. 304–311, 1991.

    Article  Google Scholar 

  9. Y. Rossikhin and M. Shitikova, “Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system” Acta Mechanica, vol. 120, no. 1-4, pp. 109–125, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  10. I. N’Doye, H. Voos, and M. Darouach, “Observerbased approach for fractional-order chaotic synchronization and secure communication” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 3, no. 3, pp. 442–450, 2013.

    Article  MathSciNet  Google Scholar 

  11. J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, Oxford, 2003.

    MATH  Google Scholar 

  12. K. S. Cole, “Electric conductance of biological systems” Proc, of the Cold Spring Harbor Symposia on Quantitative Biology, New York, USA, 1993.

    Google Scholar 

  13. T. J. Anastasio, “The fractional-order dynamics of brainstem vestibulo-oculomotor neurons” Biological Cybernetics, vol. 72, no. 1, pp. 69–79, 1994.

    Article  Google Scholar 

  14. E. Ahmed, A. M. El-Sayed, and H. A. A. El-Saka, “Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models” J. of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 542–553, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Ahmed and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics” Physica A: Statistical Mechanics and its Applications, vol. 379, no. 2, pp. 607–614, 2007.

    Article  MathSciNet  Google Scholar 

  16. Y. S. Ding and H. P. Ye, “A fractional-order differential equation model of HIV infection of CD4+ t-cells” Mathematical and Computer Modeling, vol. 50, no. 3-4, pp. 386–392, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. P. Ye and Y. S. Ding, “Nonlinear dynamics and chaos in a fractional-order HIV model” Mathematical Problems in Engineering, vol. 2009, no. 378614, 2009.

    Google Scholar 

  18. M. G. Markakis, G. D. Mitsis, G. P. Papavassilopoulos, P. A. Ioannou, and V. Z. Marmarelis, “A switching control strategy for attenuation of blood glucose disturbances” Optimal Control Applications & Methods, vol. 32, no. 2, pp. 185–195, 2011.

    Article  MathSciNet  Google Scholar 

  19. F. Chee, A. V. Savkin, T. L. Fernando, and S. Nahavandi, “Optimal ℋ ∞ insulin injection control for blood glucose regulation in diabetics patients” IEEE Trans. on Biomedical Engineering, vol. 52, no. 10, pp. 1625–1631, 2005.

    Article  Google Scholar 

  20. M. Fisher, “A semi closed-loop algorithm for the control of blood glucose levels in diabetics” IEEE Trans. on Biomedical Engineering, vol. 38, no. 1, pp. 57–61, 1991.

    Article  Google Scholar 

  21. P. Kaveh and Y. B. Shtessel, “Blood glucose regulation using higher-order sliding mode control” Int. J. Robust & Nonlinear Contr., vol. 18, no. 4-5, pp. 557–569, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Chee and T. Fernando, Closed Loop Control of Blood Glucose, Springer, Berlin, 2007.

    Google Scholar 

  23. F. Chee, T. Fernando, A. Savkin, and V. Heeden, “Expert PID control system for blood glucose control in critically ill patients” IEEE Trans. on Information Technology in Biomedicine, vol. 7, no. 4, pp. 419–425, 2003.

    Article  Google Scholar 

  24. R. Parker, F. Doyle, and N. Peppas, “A modelbased algorithm for blood glucose control in type I diabetic patients” IEEE Trans. on Biomedical Engineering, vol. 46, no. 2, pp. 148–157, 1999.

    Article  Google Scholar 

  25. S. Lyunch and B. Bequette, “Model predictive control of blood glucose in type I diabetes using subcutaneous glucose measurements” Proc. IEEE American Contr. Conf., Anchorage, USA, pp. 4039–4040, 2002.

    Google Scholar 

  26. S. Faruque Ali and R. Padhi, “Optimal blood glucose regulation of diabetic patients using single network adaptive critics” Optimal Control Applications & Methods, vol. 32, no. 2, pp. 196–214, 2009.

    Google Scholar 

  27. I. N’Doye, H. Voos, M. Darouach, J. G. Schneider, and N. Knauf, “ℋ ∞ static output feedback stabilization of nonlinear fractional-order glucose-insulin system” Proc. IFAC Workshop on Fractional Differentiation and Its Application, Grenoble, France, 2013.

    Google Scholar 

  28. I. N’Doye, H. Voos, M. Darouach, J. G. Schneider, and N. Knauf, “Static output feedback stabilization of nonlinear fractional-order glucose-insulin system” Prof. of IEEE EMBS Conference on Biomedical Engineering and Sciences, Langkawi, Malaysia, 2012.

    Google Scholar 

  29. I. N’Doye, H. Voos, M. Darouach, J. G. Schneider, and N. Knauf, “An unknown input fractional-order observer design for fractional-order glucose-insulin system” Prof. of IEEE EMBS Conference on Biomedical Engineering and Sciences, Langkawi, Malaysia, 2012.

    Google Scholar 

  30. I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.

    MathSciNet  MATH  Google Scholar 

  31. C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications, Springer, Berlin, 2010.

    Book  Google Scholar 

  32. I. Petráš, “A note on the fractional-order volta system” Commun Nonlinear Sci. Numer. Simulat., vol. 15, no. 2, pp. 384–393, 2010.

    Article  MATH  Google Scholar 

  33. I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, Berlin, 2011.

    Book  Google Scholar 

  34. W. Deng, “Short memory principle and a predictorcorrector approach for fractional differential equations” Journal Computational Applied Mathematics, vol. 206, no. 1, pp. 174–188, 2007.

    Article  MATH  Google Scholar 

  35. L. Dorckák, “Numerical models for simulation of the fractional-order control systems” Tech. Rep. UEF-04-94, Institute of Experimental Physics, Academy of Sciences, Slovakia, 1994.

    Google Scholar 

  36. Y. Li, Y. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1810–1821, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Pourgholi and V. J. Majd, “A nonlinear adaptive resilient observer design for a class of Lipschitz systems using LMI” Circuits, Systems, and Signal Processing, vol. 30, no. 6, pp. 1401–1415, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  38. L. Fadiga, C. Farges, J. Sabatier, and M. Moze, “On computation of ℋ ∞ norm for commensurate fractional-order system” Proc. of Conference on Decision and Control and European Control Conference, Orlando, FL, USA, 2011.

    Google Scholar 

  39. J. Zhuang and Z. Yisheng, “State feedback ℋ ∞ optimal control for linear fractional-order systems” Proc. Chinese Control Conference, Bei**g, China, 2010.

    Google Scholar 

  40. J. Shen, J. Lam, and P. Li, “Reduced-order ℋ ∞ filtering for commensurate fractional-order systems” Proc. IEEE Conf. Decision & Contr., Frorence, Italy, 2013.

    Google Scholar 

  41. C. Crusius and A. Trofino-Neto, “Sufficient LMI conditions for output feedback control problems” IEEE Trans. Aut. Contr., vol. 44, no. 5, pp. 1053–1057, 1999.

    Article  MATH  Google Scholar 

  42. C. R. Bowden, R. N. Bergman, G. Toffolo, and C. Cobelli, “Minimal modeling, partition analysis, and identification of glucose disposal in animals and man” Proc. of International Conference on Cybernetics and Society, Cambridge, MA, pp. 129–135, 1980.

    Google Scholar 

  43. G. Toffolo, R. N. Bergman, D. T. Finegood, C. R. Bowden, and C. Cobelli, “Quantitative estimation of beta cell sensitivity to glucose in the intact organism: a minimal model of insulin kinetics in the dog” Diabetes, vol. 29, no. 12, pp. 979–990, 1980.

    Article  Google Scholar 

  44. A. De Gaetano and O. Arino, “Mathematical modeling of the intravenous glucose tolerance test” J. Math. Biol., vol. 40, no. 2, pp. 136–168, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Makroglou, J. Li, and Y. Kuang, “Mathematical models and software tools for glucose-insulin regulatory system and diabetes: an overview” Applied Numerical Mathematics, vol. 56, no. 3-4, pp. 559–573, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  46. C. Neatpisarnvanit and J. Boston, “Estimation of plasma insulin from plasma glucose” IEEE Trans. on Biomedical Engineering, vol. 49, no. 11, pp. 1253–1259, 2002.

    Article  Google Scholar 

  47. R. L. Ollerton, “Application of optimal control theory to diabetes mellitus” Int. J. Contr., vol. 50, no. 6, pp. 2503–2522, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  48. C. Farges, M. Moze, and J. Sabatier, “Pseudostate feedback stabilization of commensurate fractional order systems” Automatica, vol. 46, no. 10, pp. 1730–1734, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  49. E. V. Cauter, E. Shapiro, and H. Tillil, “Circadian modulation of glucose and insulin responses to meals: relationship to cortisol rhythm” American Journal of Physiology, vol. 262, no. 4, pp. 467–475, 1992.

    Google Scholar 

  50. A. G. Hernandez, L. Fridman, R. Leder, S. I. Andrade, C. R. Monsalve, Y. Shtessel, and A. Levant, “High-order sliding mode control for blood glucose regulation in the presence of uncertain dynamics” Proc. of 33rd Annual International Conference of the IEEE EMBS, Boston, Massachusetts, 2011.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology(KAUST), Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Ibrahima N’Doye

  2. Faculty of Science, Technology and Communication (FSTC), University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, Belval, L-1359, Luxembourg

    Holger Voos

  3. Research Center for Automatic Control of Nancy (CRAN UMR, 7039, CNRS), IUT de Longwy, University of Lorraine, 186 rue de Lorraine 54400, Cosnes et Romain, France

    Mohamed Darouach

  4. Luxembourg Centre for Systems Biomedicine, University of Luxembourg, 7, Avenue des hauts Fournaux, L-4362, Belval, Luxembourg

    Jochen G. Schneider

Authors
  1. Ibrahima N’Doye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Holger Voos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed Darouach
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jochen G. Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ibrahima N’Doye.

Additional information

Recommended by Associate Editor Guang-Hong Yang under the direction of Editor PooGyeon Park.

This present work is supported by the National Research Fund, Luxembourg and the European Commission (FP7-COFUND).

Ibrahima N’Doye received his Ph.D. degree in Automatic Control from the University Henri Poincaré of Nancy at the Research Center of Automatic Control (CRAN-CNRS, University of Lorraine), France and the University Hassan II Ain Chock, Casablanca, Morocco, in 2011. From 2012 to 2014, he was a Postdoc at sthe Faculty of Science, Technology and Communication, Research Unit of Engineering Sciences. Since 2014, he is a Postdoc at the King Abdullah University of Science and Technology (KAUST) in the Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE). His research interests are in the area estimation and control of fractional-order systems and nonlinear dynamic systems with applications in energy systems and biomedicine.

Holger Voos studied Electrical Engineering at the Saarland University and received the Doctoral Degree in Automatic Control from the Technical University of Kaiserslautern, Germany, in 2002. From 2000 to 2004, he was with Bodenseewerk Gerätetechnik GmbH, Germany, where he worked as a Systems Engineer in aerospace and robotics. From 2004 to 2010, he was a Professor at the University of Applied Sciences Ravensburg-Weingarten, Germany, and the head of the Mobile Robotics Lab there. Since 2010, he is a Professor at the University of Luxembourg in the Faculty of Science, Technology and Communication, Research Unit of Engineering Sciences. He is the head of the Automatic Control Research Group and also the head of the Automation Lab in the Interdisciplinary Centre of Security, Reliability and Trust (SnT) at the University of Luxembourg. His research interests are in the area of distributed and networked control, model predictive control and safe and secure automation systems with applications in mobile robotics, energy systems and biomedicine.

Mohamed Darouach graduated from “Ecole Mohammadia d’Ingénieurs”, Rabat, Morocco, in 1978, and received the Docteur Ingénieur and Doctor of Sciences degrees from Nancy University, France, in 1983 and 1986, respectively. From 1978 to 1986 he was Associate Professor and Professor of automatic control at “Ecole Hassania des Travaux Publics”, Casablanca, Morocco. Since 1987 he has been a Professor at University of Lorraine. He has been a Vice Director of the Research Center in Automatic Control of Nancy (CRAN UMR 7039, Nancy-University, CNRS) from 2005 to 2013. He obtained a degree Honoris Causa from the Technical University of IASI and since 2010 he is a member of the Scientific council of Luxembourg University. He held invited positions at the University of Alberta, Edmonton. His research interests span theoretical control, observers design, and control of large-scale uncertain systems with applications.

Jochen G. Schneider graduated at the Robert-Schumann-Gymnasium Saarlouis in 1989. From 1990 to 1997, he studied Human Medicine at University Hospital of the Saarland, Medical Faculty at Homburg/Saar. He obtained a full licensed to practice medicine in 2000. He received the Doctoral Degree (Thesis defense) and the Habilitation in Clinical Biochemistry in 2002 and 2010, respectively. Since 2010, he is the head Translational Medicine Group, Luxembourg Centre for Systems Biomedicine & Honorary Consultant at Saarland University Medical Center Homburg/Saar. His research interests are in the area of medical translational and experimental.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

N’Doye, I., Voos, H., Darouach, M. et al. Static output feedback ℋ ∞ control for a fractional-order glucose-insulin system. Int. J. Control Autom. Syst. 13, 798–807 (2015). https://doi.org/10.1007/s12555-013-9192-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-013-9192-y

Keywords

Search

Navigation