Skip to main content
SpringerLink
Log in

A Nonlinear Optimal Control Approach of Insulin Infusion for Blood Glucose Levels Regulation

  • Original Paper
  • Published:
Intelligent Industrial Systems

Abstract

The article proposes a nonlinear optimal control method for the administered infusion of insulin, aiming at regulation of patients’ blood glucose levels. The dynamic model of blood glucose concentration which receives as control input the insulin’s infusion rate, undergoes approximate linearization through Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization takes place round a temporary equilibrium which is updated at each iteration of the control method and which is defined by the last value of the system’s state vector and the last value of the control inputs vector exerted on it. For the linearized model, a robust H-infinity feedback controller is designed. To find the feedback control gain of the controller, an algebraic Riccati equation is solved at each iteration of the control algorithm. Through Lyapunov stability analysis it is first proven that the control loop satisfies the H-infinity tracking performance criterion, which signifies elevated robustness to model uncertainty and external perturbations. Moreover, the global asymptotic stability of the control loop is proven. The proposed control method can contribute to the treatment of diabetes patients and to blood glucose levels regulation in patients of intensive care units.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Leelarathna, L., English, S.W., Thabit, H., Caldwell, K., Allen, J.M., Kumareswaran, K., Wilinska, M.E., Nodele, M., Mangat, J., Evans, M.L., Bernstein, R., Havorka, R.: Feasibility of fully automated closed-loop glucose control using continuous subcutaneous glucose measurements in critical illness: a randomized control trial. J. Crit. Care 17, 1 (2013)

    Google Scholar 

  2. Wang, Q., Molenaar, P., Harsh, S., Freeman, K., Xie, J., Gold, C., Rovine, M., Ulbrecht, J.: Personalized state-space modeling of glucose dynamics for type 1 diabetes using continuously monitored glucose, insulin dose, and meal intake an extended kalman filter approach. J. Diabetes Sci. Technol. 8(2), 331345 (2014)

    Article  Google Scholar 

  3. Kalfon, P.: Automatisation du contrôle strict de la glycémie en réanimation (Computer protocols for tight glycaemic control in critically ill patients). Réanimation, 18, 532–537 (2009)

  4. Bridges, B.C., Preissig, C.M., Maher, K.O., Rigby, M.R.: Continuous glucose monitors prove highly accurate in critically ill children. J. Crit. Care 14, 1 (2010)

    Google Scholar 

  5. Chase, J., Shaw, G.M., Wong, X.W., Lotz, T., Lin, J., Hann, C.E.: Model-based Glycaemic Control in Critical Care—a review of the state of the possible. Biomed. Signal Process. Control 1(1), 321 (2006)

    Article  Google Scholar 

  6. Chee, F., Fernando, T., Vernon van Heerden, P.: Closed-loop glucose control in critically ill patients using continuous glucose monitoring system (CGMS) in real time. IEEE Trans. Inf. Technol. Biomed. 7(1), 43–53 (2003)

    Article  Google Scholar 

  7. Lunze, K., Singh, T., Walter, M., Brendel, M.D., Leonhardt, S.: Blood glucose control algorithms for type 1 diabetic patients: a methodological review. Biomed. Signal Process. Control 8(2m), 107–119 (2011)

    Google Scholar 

  8. Bhattacharjee, A., Sengupta, A., Sutradhar, A.: Nonparametric modelling of glucose-insulin process in IDDM patient using Hammerstein-Wiener model. In: 2010 11th Interantional. Conference on Control, Automation, Robotics and Vision, Singapore, (December 2010)

  9. Ting, C.W., Chek, C.: A novel blood glucose regulation using TSK-FCMAC: A Fuzzy CMAC based on the zero-ordered TSK fuzzy inference scheme. IEEE Trans. Neural Netw. 20(5), 856–871 (2009)

    Article  Google Scholar 

  10. Makroglou, A., Li, J., Kuang, Y.: Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview. Appl. Numer. Math. 56, 559–573 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chee, F., Savkin, A.V., Fernando, T.L., Nahavandi, S.: Optimal \(H_{\infty }\) insulin injection control for blood glucose regulation in diabetic patients. IEEE Trans. Biomed. Eng. 52(10), 1625–1631 (2005)

    Article  Google Scholar 

  12. Al-Helal, Z., Rehbock, V., Loxton, R.: Modellling and optimal control of blood glucose in the human body. J. Ind. Manag. Optim. 11(4), 1149–1164 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ridha, T.M., Moog, C., Delaleau, E., Fliess, M., Join, C.: A variable reference trajectory for model-free glycemia regulation. In: SIAM Conference on Control & its Applications (SIAM CT15), Jul 2015, Paris, France (2015)

  14. Liu, W., Tang, F.: Modeling a simplified regulatory system of blood glucose at molecular levels. J. Theor. Biol. 252, 608–620 (2008)

    Article  MathSciNet  Google Scholar 

  15. Marheineke, N., Cibis, T.M., Schiessl, S., Pielmeier, U.: Optimal control of glucose balance in ICU patients based on GlucoSafe model. J. Math. Ind. 4(3), 1–13 (2014)

    MathSciNet  Google Scholar 

  16. Boiroux, D., Finan, D.A., Jrgensen, J.B., Poulsen, N.K., Madsen, H.: Optimal Insulin Administration for People with Type 1 Diabetes. In: 9th International Symposium on Dynamics and Control of Process Systems (DYCOPS 2010), Leuven, Belgium, (July 2010)

  17. Kovacs, L., Szaloy, P., Benyo, B., Chase, J.G.: Optimal tight glycaemic control supported by differential geometric methods. In: IFBME Proceedings of the 5th European Medical and Biological Engineering Conference vol. 37, pp. 351–354 (2011)

  18. Kovacs, L., Szolay, P., Alamassy, Z., Benyo, Z., Bankai, L.: Quasi in-silico validations of a nonlinear LPV model-based robust glucose control algorirthm for Type I diabetes. In: 18th IFAC World Congress. Milano, Italy, (Sept. 2011)

  19. Kovacs, L., Kucsur, B., Bokor, J, Benyo, Z.: LPV fault detection of glucose-insulin system. In: IEEE MED 2006, 14th Mediterranean Conference on Control and Automation, Ancona, Italy (June 2006)

  20. Palumbo, P., Pepe, P., Panunzi, S., de Gaetano, A.: Glucose control by subcutaneous insulin administration: a DDE modelling approach. In: 18th IFAC World Control Conference, Milano, Italy, September, (2011)

  21. Palumbo, P., Pizzichelli, G., Panunzi, S., Pepe, P., de Gaetano, A.: Tests on virtual patient for an observer-based closed-loop of plasma glycaemia. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, Florida, (Dec. 2011)

  22. Rigatos, G., Rigatou, E., Zervos, N.: A nonlinear H-infinity approach to optimal control of the depth of anaesthesia. In: 12th International Conference of Computational Methods in Sciences and Engineering (ICCMSE 2016), Athems, Greece, (March 2016)

  23. Rigatos, G.G., Tzafestas, S.G.: Extended Kalman filtering for fuzzy modelling and multi-sensor fusion. Math. Comput. Model. Dyn. Syst. 13, 251–266 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Basseville, M., Nikiforov, I.: Detection of Abrupt Changes: Theory and Applications. Prentice-Hall, Upper Saddle River (1993)

    Google Scholar 

  25. Rigatos, G., Zhang, Q.: Fuzzy model validation using the local statistical approach. Fuzzy Sets Syst. 60(7), 882–904 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rigatos, G.G.: Modelling and Control for Intelligent Industrial Systems: Adaptive Algorithms in Robotcs and Industrial engineering. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  27. Rigatos, G.: Nonlinear Control and Filtering Using Differential Flatness Approaches: Applications to Electromechanicsl Systems. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  28. Rigatos, G.: Intelligent Renewable Energy Systems: Modelling and Control. Springer, New York (2017)

    Google Scholar 

  29. Toussaint, G.J., Basar, T., Bullo, F.: \(H_{\infty }\) optimal tracking control techniques for nonlinear underactuated systems. In: Proceedings of the IEEE CDC 2000, 39th IEEE Conference on Decision and Control. Sydney Australia (Dec. 2000)

  30. Lublin, L., Athans, M.: An experimental comparison of and designs for interferometer testbed. In: Francis, B., Tannenbaum, A. (eds.) Lectures Notes in Control and Information Sciences: Feedback Control, Nonlinear Systems and Complexity, pp. 150–177. Springer, London (1995)

  31. Doyle, J.C., Glover, K., Khargonekar, P.P., Francis, B.A.: State-space solutions to standard \(H_2\) and \(H_{\infty }\) control problems. IEEE Trans. Autom. Control 34, 831–847 (1989)

    Article  MATH  Google Scholar 

  32. Gibbs, B.P.: Advanced Kalman Filtering, Least Squares and Modelling, A practical handbook. Wiley, New York (2011)

  33. Simon, D.: A game theory approach to constrained minimax state estimation. IEEE Trans. Signal Process. 54(2), 405–412 (2006)

    Article  Google Scholar 

  34. Kovacs, L., Kulesar, B., Gyorgy, A., Benyo, Z.: Robust servo control of a novel type I diabetic model. Optim Control Appl. Methods 32, 215–238 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Magni, L., Raimondo, D., Bassi, L., Dalla Man, C., De Nicolao, G., Kovatchev, B., Cobelli, C.: Model-predictive control of typeI diabetes. J. Diabetes Sci. Technol. 1, 804–812 (2007)

    Article  Google Scholar 

  36. Chase, J., Shaw, G., Wang, X., Lotz, T., Lin, J., Hann, C.: Model-based glycaemia control in critical care: a review of the state of the possible. Biomed. Signal Process. Control 1, 3–21 (2006)

    Article  Google Scholar 

  37. Eren-Oruklu, M., Cinar, A., Quino, L., Smith, D.: Adaptive control strategy for regulation of blood-glucose levels in patients with type I diabetes. J. Process Control 19, 1333–1346 (2009)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras, Greece

    G. Rigatos

  2. Department of Industrial Engineering, University of Salerno, 84084, Fisciano, Italy

    P. Siano

  3. Department of Physics, Ural Federal University, 620002, Yekaterinesburg, Russia

    A. Melkikh

Authors
  1. G. Rigatos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Siano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Melkikh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Rigatos.

Appendix: Differential Flatness of the Blood Glucose–Insulin Infusion Model

Appendix: Differential Flatness of the Blood Glucose–Insulin Infusion Model

By exploiting the differential flatness properties of the blood glucose–insulin infusion model the problem of setpoints seletion in the system’s nonlinear optimal control can be easily solved. Actually, one can define reference setpoint only for the flat output of the system (blood glucose concentration) and can find the reference values for the rest of the state variables of the models through the differential relations connecting them to the flat output. It holds that:

The flat output of the system is \(y=x_1\) that is the blood glucose concentration. Next from the first row of the state-space model of the system, given in Eq. (44)

$$\begin{aligned} \begin{aligned} \dot{x}_1&=-{p_G}{x_1}-{S_I}[x_1+G_E]{x_2 \over {1+a_G{x_2}}} \\ \dot{x}_2&=-{n_c}{x_2}+{{n_I} \over V_a}[x_3-x_2] \\ \dot{x}_3&=-{n_k}{x_3}-{{{n_L}{x_3}} \over {1+{a_I}{x_3}}}-{n_I \over V_p}[x_3-x_2]+{1 \over V_p}u \end{aligned} \end{aligned}$$
(44)

one solves with respect to \(x_2\). This gives

$$\begin{aligned} x_2={{(\dot{x}_1+{p_G}{x_1})} \over {[-{a_G}(\dot{x}_1+{p_G}{x_1})-{S_I}(x_1+G_E)]}} \end{aligned}$$
(45)

From the second row of the state-space model of the system, given in Eq. (44), one solves with respect to \(x_3\). This gives

$$\begin{aligned} x_3={{V_a \over n_I}}\left[ \dot{x}_2+{n_c}{x_2}+{{n_I \over V_a}{x_2}}\right] \end{aligned}$$
(46)

Finally, from the third row of the state-space model of the system, given in Eq. (44), one solves with respect to the control input u. This gives

$$\begin{aligned} u={V_p}\left[ \dot{x}_3+{n_k}{x_3}+{{{n_L}{x_3}} \over {1+{a_I}{x_3}}}+{n_I \over V_p}[x_3-x_2]\right] \end{aligned}$$
(47)

Since all state variables and the control inputs of the blood glucose–insulin infusion model can be written as differential functions of its flat output (y concentration of blood glucose) it can be confirmed that the model is a differentially flat one.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rigatos, G., Siano, P. & Melkikh, A. A Nonlinear Optimal Control Approach of Insulin Infusion for Blood Glucose Levels Regulation. Intell Ind Syst 3, 91–102 (2017). https://doi.org/10.1007/s40903-016-0063-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40903-016-0063-8

Keywords

Search

Navigation