Skip to main content

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

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  • Abu Arqub O (2013) Series solution of fuzzy differential equations under strongly generalized differentiability. J Adv Res Appl Math 5:31–52. https://doi.org/10.5373/jaram.1447.051912

    MathSciNet  Article  Google Scholar 

  • Abu Arqub O (2017) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput Appl 28(7):1591–1610

    Article  Google Scholar 

  • 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. https://doi.org/10.12785/amis/070528

    MathSciNet  Article  Google Scholar 

  • 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. https://doi.org/10.1155/2015/835891

    MathSciNet  Article  MATH  Google Scholar 

  • 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):3283–3302

    Article  MATH  Google Scholar 

  • 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):7191–7206

    Article  MATH  Google Scholar 

  • 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):1059–1074

    Article  Google Scholar 

  • Ahmad T (1998) Mathematical and fuzzy modeling of interconnection in integrated circuits. Ph. D. thesis, Sheffield Hallam University, Sheffield, United Kingdom

  • 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 34:2401–2416

    Article  Google Scholar 

  • 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):2485–2499

    Article  Google Scholar 

  • 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. https://doi.org/10.1007/s00500-018-3476-3

    Article  MATH  Google Scholar 

  • 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):711–716

    Article  Google Scholar 

  • Ang KH, Chong GCY, Li Y (2005) PID control system analysis, design, and technology. IEEE Trans Control Syst Technol 13(4):559–576

    Article  Google Scholar 

  • Aronoff L, Berkowitz K, Shreiner B, Want L (2004) Glucose metabolism and regulation: beyond insulin and glucagon. Diabetes Spectr 17(3):183–190

    Article  Google Scholar 

  • 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):57–61

    Google Scholar 

  • Bay JS (1999) Fundamentals of linear state space systems. WCB/McGraw Hill, New York

    Google Scholar 

  • Cao SG, Rees NW (1995) Identification of dynamic fuzzy system. Fuzzy Sets Syst 74:307–320

    Article  MATH  Google Scholar 

  • Durbin J, Koopman S (2001) Time series analysis by state space methods. Oxford University Press, Oxford. ISBN 978-0-19-852354-3

    MATH  Google Scholar 

  • 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):3323–3337

    Article  Google Scholar 

  • 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. https://doi.org/10.1515/jisys-2017-0083

    Article  Google Scholar 

  • 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):2385–2397

    Article  MathSciNet  Google Scholar 

  • 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. https://doi.org/10.1007/s00500-018-3242-6

    Article  MATH  Google Scholar 

  • 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):1–11

    Article  Google Scholar 

  • 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):23–34

    MathSciNet  Google Scholar 

  • 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):51–65

    Google Scholar 

  • Franklin GF, Powell JD, Emami-Naeini A (2018) Feedback control of dynamic systems, 8th edn. Pearson, Hoboken

    MATH  Google Scholar 

  • Gonz’alez R, Cipriano A (2016) An insulin infusion fuzzy controller with state estimation for artificial pancreas systems. Rev Iberoam Autom Inf Ind 13:393–402

    Article  Google Scholar 

  • Hangos KM, Lakner R, Gerzson M (2001) Intelligent control systems: an introduction with examples. Springer, New York

    MATH  Google Scholar 

  • Hangos KM, Bokor J, Szederkenvi G (2004) Analysis and control of nonlinear process systems. Springer, New York

    Google Scholar 

  • Hurwitz A (1895) On the conditions under which an equation has only roots with negative real parts. Mathematische Annalen 46:273–284

    Article  MathSciNet  Google Scholar 

  • 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. https://doi.org/10.1115/pvp2017-65334

  • Keener J, Sneyd J (1998) Mathematical physiology. Springer, New York, pp 594–603

    Book  MATH  Google Scholar 

  • 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, Malaysia

  • 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):1103–1113

    Google Scholar 

  • 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):100–107

    Google Scholar 

  • 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):1570–1583

    Google Scholar 

  • 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):483–487

    Article  Google Scholar 

  • Klir GJ, Yuan B (1995) Fuzzy sets and logic: theory and applications. PTR Prentice Hall, New Jersey

    MATH  Google Scholar 

  • Maxwell JC (1868) On governors. Proc R Soc Lond 16: 270–283. JSTOR 112510

  • 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, Portugal

  • Mosekilde E (1996) Topics in nonlinear dynamics. World Scientific, Singapore, pp 263–279

    MATH  Google Scholar 

  • Mythreyi K, Subramanian SC, Kumar RK (2014) Nonlinear glucose–insulin control considering delays part II: control algorithm. Control Eng Pract 28:26–33

    Article  Google Scholar 

  • Nimri R, Phillip M (2014) Artificial pancreas: fuzzy logic and control of glycemia current opinion in endocrinology. Diabetes Obes 21(4):251–256

    Google Scholar 

  • Nise NS (2010) Control systems engineering, 6th edn. Wiley, London. ISBN 978-0-470-54756-4

    MATH  Google Scholar 

  • Normah M (2005) Mathematical modeling of mass transfer in a multi-stage rotating disc contractor column. Ph.D. thesis, Department of Mathematics, UTM Skudai, Malaysia

  • Otto M (1970) The origins of feedback control. The Colonial Press, Inc., Clinton

    MATH  Google Scholar 

  • 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 203–209

  • 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 63:113–118

    Article  Google Scholar 

  • 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 81:442–448

    Article  Google Scholar 

  • 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):594–600

    Google Scholar 

  • Radomski D, Glowacka J (2018) Sensitivity analysis of the insulin–glucose mathematical model. In: Information technology in biomedicine, pp 455–468. https://doi.org/10.1007/978-3-319-91211-0_40

  • 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, Malaysia

  • Razidah I, Jusoff K, Ahmad T, Ahmad S, Ahmad RS (2009) Fuzzy state space model of multivariable control systems. Comput Inf Sci 2:19–25

    Google Scholar 

  • Rita M, Li K, Wing C (2016) Econometric analyses of international housing markets. Routledge, London

    Google Scholar 

  • 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–E639

    Google Scholar 

  • 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. https://doi.org/10.1016/j.cell.2016.02.063

    Article  Google Scholar 

  • Routh EJ (1977) A treatise on the stability of a given state of motion: particularly steady motion. Macmillan and Co., London

    Google Scholar 

  • Routh EJ, Fuller AT (1975) Stability of motion. Taylor & Francis, London

    Google Scholar 

  • 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):313–325

    Article  Google Scholar 

  • Saade JJ (1996) Mapping convex and normal fuzzy sets. Fuzzy Sets Syst 81:251–256

    Article  MathSciNet  MATH  Google Scholar 

  • Sankaranarayanan S, Kumar SA, Cameron F, Maahs D (2017) Model-based falsification of an artificial pancreas control system. ACM SIGBED Rev 14(2):24–33

    Article  Google Scholar 

  • Shabestari PS, Panahi S, Hatef B, Jafari S, Sprott JC (2018) A new chaotic model for glucose–insulin regulatory system. Chaos Solitons Fractals 112:44–51

    Article  MathSciNet  Google Scholar 

  • 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 67:307–314

    Article  Google Scholar 

  • Soman E (2009) Scienceray, regulation of glucose by insulin archived. July 16, 2011, at the Wayback Machine, May 4

  • Stuart B (1992) A history of control engineering, 1930–1955. IET, p 48. ISBN 978-0-86341-299-8

  • Sturis J (1991) Possible mechanisms underlying slow oscillations of human insulin secretions. Ph.D. dissertation, The Technical University of Denmark, Lyngby, Denmark

  • 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–E809

    Google Scholar 

  • Syau YR (2000) Closed and convex fuzzy sets. Fuzzy Sets Syst 110:287–291

    Article  MathSciNet  MATH  Google Scholar 

  • Todorov Y, Terziyska M (2018) NEO-fuzzy neural networks for knowledge based modeling and control of complex dynamical systems. In: Practical issues of intelligent innovations

  • Todorov Y, Terziyska M, Petrov M (2017) State-space fuzzy-neural predictive control. In: Recent contributions in intelligent systems. https://doi.org/10.1007/978-3-319-41438-6_17

  • Tolić IM, Mosekilde E, Sturis J (2000) Modeling the insulin–glucose feedback system: the significance of pulsatile insulin secretion. J Theor Biol 207:361–375

    Article  Google Scholar 

  • 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):714–723

    Article  Google Scholar 

  • Turksoy K, Quinn L, Littlejohn E, Cinar A (2014) Multivariable adaptive identification and control for artificial pancreas systems. IEEE Trans Biomed Eng 61(3):883–891

    Article  Google Scholar 

  • 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):839–848

    Google Scholar 

  • Verdonk CA, Rizza RA, Gerich JE (1981) Effects of plasma glucose concentration on glucose utilization and glucose clearance in normal man. Diabetes 30:535–537

    Article  Google Scholar 

  • Yang XM (1995) Some properties of convex fuzzy sets. Fuzzy Sets Syst 72:129–132

    Article  MathSciNet  MATH  Google Scholar 

  • Yang XM, Yang FM (2002) A property on convex fuzzy sets. Fuzzy Sets Syst 126:269–271

    Article  MathSciNet  MATH  Google Scholar 

  • 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 17600864

  • Yu C, Ljung L, Verhaegen M (2018) Identification of structured state-space models. Automatica 90:54–61

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  • Zhang HN, Lin ZB (2004) Hypoglycemic effect of ganoderma lucidum polysaccharides. Acta Pharmacol Sin 25(2): 191–195. PMID 14769208

Download references

Acknowledgements

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.

Funding

This study was funded by no agency/grant.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Izaz Ullah Khan.

Ethics declarations

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).

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliation.

Communicated by V. Loia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khan, I.U., Ahmad, T. & Maan, N. Revised convexity, normality and stability properties of the dynamical feedback fuzzy state space model (FFSSM) of insulin–glucose regulatory system in humans. Soft Comput 23, 11247–11262 (2019). https://doi.org/10.1007/s00500-018-03682-w

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-03682-w

Keywords

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