Soft Computing

, Volume 23, Issue 21, pp 11247–11262 | Cite as

Revised convexity, normality and stability properties of the dynamical feedback fuzzy state space model (FFSSM) of insulin–glucose regulatory system in humans

  • Izaz Ullah KhanEmail author
  • Tahir Ahmad
  • Normah Maan
Methodologies and Application


This research tries to explore more important structural properties of the insulin–glucose regulatory system in humans. Consequently, an important theorem, namely “revised modified optimized defuzzified value theorem” for feedback systems is derived and then proved. Moreover, the properties concerning the convexity, normality and the bounded-input bounded-output stability of the induced solution of FFSSM are researched. The proposed theorems and lemmas are successfully implemented and verified for the insulin–glucose system in humans. The successful and promising results and proofs of the theorems of the relevant properties improve the credibility and reliability of the FFSSM model of the insulin–glucose regulatory system in humans.


Insulin–glucose regulations Feedback systems Fuzzy state space model (FSSM) Inverse modeling Dynamical systems Modern control theory 



We are thankful to the respectable editors and reviewers for their relevant, credible and useful reviews and suggestions. Thanks are due to COMSATS University Islamabad, Abbottabad Campus, Pakistan, and UTM Malaysia.


This study was funded by no agency/grant.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors. The data presented are obtained from the widely accepted published research Sturis (1991), Sturis et al. (1991) and Tolić et al. (2000).


  1. Abu Arqub O (2013) Series solution of fuzzy differential equations under strongly generalized differentiability. J Adv Res Appl Math 5:31–52. MathSciNetCrossRefGoogle Scholar
  2. Abu Arqub O (2017) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput Appl 28(7):1591–1610CrossRefGoogle Scholar
  3. Abu Arqub O, El-Ajou A, Momani S, Shawagfeh N (2013) Analytical solutions of fuzzy initial value problems by HAM. Appl Math Inf Sci 7:1903–1919. MathSciNetCrossRefGoogle Scholar
  4. Abu Arqub O, Momani S, Al-Mezel S, Kutbi M (2015) Existence, uniqueness, and characterization theorems for nonlinear fuzzy integrodifferential equations of Volterra type. Math Probl Eng. MathSciNetCrossRefzbMATHGoogle Scholar
  5. Abu Arqub O, Al-smadi MH, Momani SM, Hayat T (2016) Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput 20(8):3283–3302CrossRefzbMATHGoogle Scholar
  6. Abu Arqub O, Al-smadi MH, Momani SM, Hayat T (2017a) Application of reproducing kernel algorithms for solving second-order, two point fuzzy boundary value problems. Soft Comput 21(23):7191–7206CrossRefzbMATHGoogle Scholar
  7. Abu Arqub O, Momani S, Al-Mezel S, Kutbi M (2017b) A novel iterative numerical algorithm for the solutions of systems of fuzzy initial value problems. Appl Math Inf Sci 11(4):1059–1074CrossRefGoogle Scholar
  8. Ahmad T (1998) Mathematical and fuzzy modeling of interconnection in integrated circuits. Ph. D. thesis, Sheffield Hallam University, Sheffield, United KingdomGoogle Scholar
  9. Amin F, Fahmi A, Abdullah S, Ali A, Ahmed R, Ghanu F (2018a) Triangular cubic linguistic hesitant fuzzy aggregation operators and their application in group decision making. J Intell Fuzzy Syst 34:2401–2416CrossRefGoogle Scholar
  10. Amin F, Fahmi A, Abdullah S, Ali A, Ahmad KW (2018b) Some geometric operators with triangular cubic linguistic hesitant fuzzy number and their application in group decision-making. J Intell Fuzzy Syst 35(2):2485–2499CrossRefGoogle Scholar
  11. Amin F, Fahmi A, Abdullah S (2018c) Dealer using a new trapezoidal cubic hesitant fuzzy TOPSIS method and application to group decision-making program. Soft Comput. CrossRefzbMATHGoogle Scholar
  12. Aminu J, Ahmad T, Sulaiman S (2017) Representation of multi-connected system of Fuzzy State Space Modeling (FSSM) in potential method based on a network context. Malays J Fundam Appl Sci 13(4):711–716CrossRefGoogle Scholar
  13. Ang KH, Chong GCY, Li Y (2005) PID control system analysis, design, and technology. IEEE Trans Control Syst Technol 13(4):559–576CrossRefGoogle Scholar
  14. Aronoff L, Berkowitz K, Shreiner B, Want L (2004) Glucose metabolism and regulation: beyond insulin and glucagon. Diabetes Spectr 17(3):183–190CrossRefGoogle Scholar
  15. Ashaari A, Ahmad T, Shamsuddin M, Zenian S (2015) Fuzzy state space model for a pressurizer in a nuclear power plant. Malays J Fundam Appl Sci 11(2):57–61Google Scholar
  16. Bay JS (1999) Fundamentals of linear state space systems. WCB/McGraw Hill, New YorkGoogle Scholar
  17. Cao SG, Rees NW (1995) Identification of dynamic fuzzy system. Fuzzy Sets Syst 74:307–320CrossRefzbMATHGoogle Scholar
  18. Durbin J, Koopman S (2001) Time series analysis by state space methods. Oxford University Press, Oxford. ISBN 978-0-19-852354-3zbMATHGoogle Scholar
  19. Fahmi A, Abdullah S, Amin F, Nasir S, Asad A (2017a) Aggregation operators on triangular cubic fuzzy numbers and its application to multi-criteria decision making. J Intell Fuzzy Syst 33(6):3323–3337CrossRefGoogle Scholar
  20. Fahmi A, Abdullah S, Amin F, Ali A (2017b) Precursor selection for sol–gel synthesis of titanium carbide nano powders by a new cubic fuzzy multi-attribute group decision-making model. J Intell Fuzzy Syst. CrossRefGoogle Scholar
  21. Fahmi A, Amin F, Abdullah S, Asad A (2018a) Cubic fuzzy Einstein aggregation operators and its application to decision-making. Int J Syst Sci 49(11):2385–2397CrossRefMathSciNetGoogle Scholar
  22. Fahmi A, Abdullah S, Amin F, Khan MSA (2018b) Trapezoidal cubic fuzzy number Einstein hybrid weighted averaging operators and its application to decision making. Soft Comput. CrossRefzbMATHGoogle Scholar
  23. Fahmi A, Abdullah S, Amin F, Asad A, Rehaman K (2018c) Expected values of aggregation operators on cubic triangular fuzzy number and its application to multi-criteria decision making problems. Eng Math 2(1):1–11CrossRefGoogle Scholar
  24. Fahmi A, Abdullah S, Amin F, Ali A (2018d) Weighted Average Rating (War) method for solving group decision making problem using triangular cubic fuzzy hybrid aggregation (Tcfha). Punjab Univ J Math 50(1):23–34MathSciNetGoogle Scholar
  25. Fahmi A, Abdullah S, Amin F (2018e) Expected values of aggregation operators on cubic trapezoidal fuzzy number and its application to multi-criteria decision making problems. Eng Math 2(2):51–65Google Scholar
  26. Franklin GF, Powell JD, Emami-Naeini A (2018) Feedback control of dynamic systems, 8th edn. Pearson, HobokenzbMATHGoogle Scholar
  27. Gonz’alez R, Cipriano A (2016) An insulin infusion fuzzy controller with state estimation for artificial pancreas systems. Rev Iberoam Autom Inf Ind 13:393–402CrossRefGoogle Scholar
  28. Hangos KM, Lakner R, Gerzson M (2001) Intelligent control systems: an introduction with examples. Springer, New YorkzbMATHGoogle Scholar
  29. Hangos KM, Bokor J, Szederkenvi G (2004) Analysis and control of nonlinear process systems. Springer, New YorkGoogle Scholar
  30. Hurwitz A (1895) On the conditions under which an equation has only roots with negative real parts. Mathematische Annalen 46:273–284CrossRefMathSciNetGoogle Scholar
  31. Huu TP, Sone A, Miura N (2017) GA-optimized Fuzzy State Space Model of multi degree freedom structure under seismic excitation. In: ASME pressure vessels and piping conference.
  32. Keener J, Sneyd J (1998) Mathematical physiology. Springer, New York, pp 594–603CrossRefzbMATHGoogle Scholar
  33. Khan IU (2013) Feedback fuzzy state space modeling for solving inverse problems in a multivariable dynamical system. Ph.D. thesis, Department of Mathematics, Universiti Teknologi Malaysia, Skudai, Johor Bahru, MalaysiaGoogle Scholar
  34. Khan IU, Ahmad T, Normah M (2012a) On the structural, number theoretic and fuzzy relational properties of large multi-connected systems of feedback fuzzy state space models (FFSSM’s). J Appl Sci Res 8(2):1103–1113Google Scholar
  35. Khan IU, Ahmad T, Normah M (2012b) Feedback fuzzy state space modeling and optimal production planning for steam turbine of a combined cycle power generation plant. Res J Appl Sci 7(2):100–107Google Scholar
  36. Khan IU, Ahmad T, Maan N (2013) An inverse feedback fuzzy state space modeling (FFSSM) for insulin–glucose regulatory system in humans. Sci Res Essays 8(25):1570–1583Google Scholar
  37. Kim YW, Kim KH, Choi HJ, Lee DS (2005) Anti-diabetic activity of beta-glucans and their enzymatically hydrolyzed oligosaccharides from Agaricus blazei. Biotechnol Lett 27(7):483–487CrossRefGoogle Scholar
  38. Klir GJ, Yuan B (1995) Fuzzy sets and logic: theory and applications. PTR Prentice Hall, New JerseyzbMATHGoogle Scholar
  39. Maxwell JC (1868) On governors. Proc R Soc Lond 16: 270–283. JSTOR 112510Google Scholar
  40. Meszéna D, Lakatos E, Szederkényi G (2014) Sensitivity analysis and parameter estimation of a human blood glucose regulatory system model. In: 11th International Workshop on Computational Systems Biology, Costa da Caparica, Lisbon, PortugalGoogle Scholar
  41. Mosekilde E (1996) Topics in nonlinear dynamics. World Scientific, Singapore, pp 263–279zbMATHGoogle Scholar
  42. Mythreyi K, Subramanian SC, Kumar RK (2014) Nonlinear glucose–insulin control considering delays part II: control algorithm. Control Eng Pract 28:26–33CrossRefGoogle Scholar
  43. Nimri R, Phillip M (2014) Artificial pancreas: fuzzy logic and control of glycemia current opinion in endocrinology. Diabetes Obes 21(4):251–256Google Scholar
  44. Nise NS (2010) Control systems engineering, 6th edn. Wiley, London. ISBN 978-0-470-54756-4zbMATHGoogle Scholar
  45. Normah M (2005) Mathematical modeling of mass transfer in a multi-stage rotating disc contractor column. Ph.D. thesis, Department of Mathematics, UTM Skudai, MalaysiaGoogle Scholar
  46. Otto M (1970) The origins of feedback control. The Colonial Press, Inc., ClintonzbMATHGoogle Scholar
  47. Pearson DW, Dray G, Peton N (1997) On linear fuzzy dynamical systems. In: Proceedings of the 2nd International ICSC Symposium, 17–19 Sept 1997, Nimes, France, pp 203–209Google Scholar
  48. Polonsky KS, Given BD, Pugh W, Liciniopaixao J, Thompson JE, Karrison T, Rubenstein AH (1986) Calculation of the systemic delivery rate of insulin in normal man. J Clin Endocrinol Metab 63:113–118CrossRefGoogle Scholar
  49. Polonsky KS, Given BD, Van Cauter E (1988) Twenty-four-hour profiles and pulsatile patterns of insulin secretion in normal and obese subjects. J Clin Invest 81:442–448CrossRefGoogle Scholar
  50. Priyadharsini S, Nandhini TSS, Chitra K, Mathumathi A (2018) Stability analysis of detecting diabetics on blood glucose regulatory systems. Int J Sci Res Sci Technol (IJSRST) 4(2):594–600Google Scholar
  51. Radomski D, Glowacka J (2018) Sensitivity analysis of the insulin–glucose mathematical model. In: Information technology in biomedicine, pp 455–468.
  52. Razidah I (2005) Fuzzy state space modeling for solving inverse problems in dynamic systems. Ph.D. thesis, Department of Mathematics, Faculty of Science, UTM Skudai, MalaysiaGoogle Scholar
  53. Razidah I, Jusoff K, Ahmad T, Ahmad S, Ahmad RS (2009) Fuzzy state space model of multivariable control systems. Comput Inf Sci 2:19–25Google Scholar
  54. Rita M, Li K, Wing C (2016) Econometric analyses of international housing markets. Routledge, LondonGoogle Scholar
  55. Rizza RA, Mandarino LJ, Gerich JE (1981) Dose-response characteristics for effects of insulin on production and utilization of glucose in man. Am J Physiol 240:E630–E639Google Scholar
  56. Romere C, Duerrschmid C, Bournat L, Constable P, Jain M, Xia F, Saha PK, Del. Solar M, Zhu B, York B, Sarkar P, Rendon DA, Gaber MW, LeMaire SA, Coselli JS, Milewicz DM, Sutton VR, Butte NF, Moore DD, Chopra AR (2016) Asprosin, a fasting-induced glucogenic protein hormone. Cell 165(3):566–579. CrossRefGoogle Scholar
  57. Routh EJ (1977) A treatise on the stability of a given state of motion: particularly steady motion. Macmillan and Co., LondonGoogle Scholar
  58. Routh EJ, Fuller AT (1975) Stability of motion. Taylor & Francis, LondonGoogle Scholar
  59. Russell SJ, El-Khatib FH, Sinha M, Magyar KL, McKeon K, Goergen LG, Balliro C, Hillard MA, Nathan DM, Damiano ER (2014) Outpatient glycemic control with a bionic pancreas in type 1 diabetes. N Engl J Med 371(4):313–325CrossRefGoogle Scholar
  60. Saade JJ (1996) Mapping convex and normal fuzzy sets. Fuzzy Sets Syst 81:251–256CrossRefMathSciNetzbMATHGoogle Scholar
  61. Sankaranarayanan S, Kumar SA, Cameron F, Maahs D (2017) Model-based falsification of an artificial pancreas control system. ACM SIGBED Rev 14(2):24–33CrossRefGoogle Scholar
  62. Shabestari PS, Panahi S, Hatef B, Jafari S, Sprott JC (2018) A new chaotic model for glucose–insulin regulatory system. Chaos Solitons Fractals 112:44–51CrossRefMathSciNetGoogle Scholar
  63. Shapiro ET, Tillil H, Polonsky KS, Fang VS, Rubenstein AH, Van Cauter E (1988) Oscillations in insulin secretion during constant glucose infusion in normal man: relationship to changes in plasma glucose. J Clin Endocrinol Metab 67:307–314CrossRefGoogle Scholar
  64. Soman E (2009) Scienceray, regulation of glucose by insulin archived. July 16, 2011, at the Wayback Machine, May 4Google Scholar
  65. Stuart B (1992) A history of control engineering, 1930–1955. IET, p 48. ISBN 978-0-86341-299-8Google Scholar
  66. Sturis J (1991) Possible mechanisms underlying slow oscillations of human insulin secretions. Ph.D. dissertation, The Technical University of Denmark, Lyngby, DenmarkGoogle Scholar
  67. Sturis J, Polonsky KS, Mosekilde E, Van Cauter E (1991) Computer model for mechanisms underlying ultradian oscillations of insulin and glucose. Am J Physiol 260:E801–E809Google Scholar
  68. Syau YR (2000) Closed and convex fuzzy sets. Fuzzy Sets Syst 110:287–291CrossRefMathSciNetzbMATHGoogle Scholar
  69. Todorov Y, Terziyska M (2018) NEO-fuzzy neural networks for knowledge based modeling and control of complex dynamical systems. In: Practical issues of intelligent innovationsGoogle Scholar
  70. Todorov Y, Terziyska M, Petrov M (2017) State-space fuzzy-neural predictive control. In: Recent contributions in intelligent systems.
  71. Tolić IM, Mosekilde E, Sturis J (2000) Modeling the insulin–glucose feedback system: the significance of pulsatile insulin secretion. J Theor Biol 207:361–375CrossRefGoogle Scholar
  72. Trevitt S, Simpson S, Wood A (2016) Artificial pancreas device systems for the closed-loop control of Type 1 diabetes: what systems are in development? J Diabetes Sci Technol 10(3):714–723CrossRefGoogle Scholar
  73. Turksoy K, Quinn L, Littlejohn E, Cinar A (2014) Multivariable adaptive identification and control for artificial pancreas systems. IEEE Trans Biomed Eng 61(3):883–891CrossRefGoogle Scholar
  74. Vasilyev AS, Ushakoy AV (2015) Modeling of dynamic systems with modulation by means of Kronecker vector-matrix representation. Sci Tech J Inf Technol Mech Opt 15(5):839–848Google Scholar
  75. Verdonk CA, Rizza RA, Gerich JE (1981) Effects of plasma glucose concentration on glucose utilization and glucose clearance in normal man. Diabetes 30:535–537CrossRefGoogle Scholar
  76. Yang XM (1995) Some properties of convex fuzzy sets. Fuzzy Sets Syst 72:129–132CrossRefMathSciNetzbMATHGoogle Scholar
  77. Yang XM, Yang FM (2002) A property on convex fuzzy sets. Fuzzy Sets Syst 126:269–271CrossRefMathSciNetzbMATHGoogle Scholar
  78. Yang K, Jung YS, Song CH (2007) Hypoglycemic effects of ganoderma applanatum and collybia confluens exo-polymers in streptozotocin-induced diabetic rats. Phytother Res 21(11): 1066–1069. PMID 17600864Google Scholar
  79. Yu C, Ljung L, Verhaegen M (2018) Identification of structured state-space models. Automatica 90:54–61CrossRefMathSciNetzbMATHGoogle Scholar
  80. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353CrossRefzbMATHGoogle Scholar
  81. Zhang HN, Lin ZB (2004) Hypoglycemic effect of ganoderma lucidum polysaccharides. Acta Pharmacol Sin 25(2): 191–195. PMID 14769208Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCOMSATS University IslamabadAbbottabadPakistan
  2. 2.Department of Mathematical Sciences, Faculty of ScienceUniversiti Teknologi MalaysiaSkudaiMalaysia

Personalised recommendations