zbMATH — the first resource for mathematics

Finite-dimensional hybrid observer for delayed impulsive model of testosterone regulation. (English) Zbl 1394.92026
Summary: The paper deals with the model-based estimation of hormone concentrations that are inaccessible for direct measurement in the blood stream. Previous research demonstrated that the dynamics of nonbasal endocrine regulation can be closely captured by linear continuous models with time delays under a pulse-modulated feedback. The presence of continuous time delays is inevitable in such a model due to transport phenomena and the time necessary for an endocrine gland to produce a certain hormone quantity. Yet, thanks to the finite-dimensional reducibility of the linear time-delay part of the system, a finite-dimensional model can be used to reconstruct both the continuous and discrete states of the hybrid time-delay plant. A hybrid observer exploiting this possibility is suggested and analyzed by means of a discrete impulse-to-impulse mapping.
92C30 Physiology (general)
34K45 Functional-differential equations with impulses
93B05 Controllability
Full Text: DOI
[1] Lightman, S. L.; Conway-Campbell, B. L., The crucial role of pulsatile activity of the HPA axis for continuous dynamic equilibration, Nature Reviews Neuroscience, 11, 10, 710-718, (2010)
[2] Veldhuis, J. D., Recent insights into neuroendocrine mechanisms of aging of the human male hypothalamic-pituitary-gonadal axis, Journal of Andrology, 20, 1, 1-17, (1999)
[3] Churilov, A.; Medvedev, A.; Shepeljavyi, A., Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback, Automatica, 45, 1, 78-85, (2009) · Zbl 1154.93306
[4] Mattsson, P.; Medvedev, A., Modeling of testosterone regulation by pulse-modulated feedback: an experimental data study, Proceedings of the International Symposium on Computational Models for Life Sciences (CMLS ’13), AIP Publishing
[5] Goodwin, B. C.; Weber, G., Oscillatory behavior in enzymatic control processes, Advances of Enzime Regulation, 3, 425-438, (1965), Oxford, UK: Pergamon Press, Oxford, UK
[6] Goodwin, B. C., An entrainment model for timed enzyme syntheses in bacteria, Nature, 209, 5022, 479-481, (1966)
[7] Griffith, J. S., Mathematics of cellular control processes I. Negative feedback to one gene, Journal of Theoretical Biology, 20, 2, 202-208, (1968)
[8] Smith, W. R., Hypothalamic regulation of pituitary secretion of luteinizing hormone-II Feedback control of gonadotropin secretion, Bulletin of Mathematical Biology, 42, 1, 57-78, (1980) · Zbl 0423.92016
[9] Efimov, D. V.; Fradkov, A. L., Oscillatority conditions for nonlinear systems with delay, Journal of Applied Mathematics, 2007, (2007) · Zbl 1151.34054
[10] Haddad, W. M.; Chellaboina, V.; Nersesov, S. G., Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, (2006), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 1114.34001
[11] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations, (1989), Singapore: World Scientific, Singapore · Zbl 0719.34002
[12] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations, (1995), Singapore: World Scientific Publishing Company, Singapore · Zbl 0837.34003
[13] Yang, T., Impulsive Control Theory, (2001), Berlin, Germany: Springer-Verlag, Berlin, Germany
[14] Stamova, I., Stability Analysis of Impulsive Functional Differential Equations, (2009), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 1189.34001
[15] Evans, W. S.; Farhy, L. S.; Johnson, M. L.; Johnson, M. L.; Brand, L., Biomathematical modeling of pulsatile hormone secretion: a historical perspective, Methods in Enzymology: Computer methods, Volume A, 454, 345-366, (2009), Elsevier
[16] Churilov, A.; Medvedev, A.; Mattsson, P., Periodical solutions in a time-delay model of endocrine regulation by pulse-modulated feedback, Proceedings of the 51st IEEE Conference on Decision and Control
[17] Churilov, A.; Medvedev, A.; Mattsson, P., Finite-dimensional reducibility of time-delay systems under pulse-modulated feedback, Proceedings of the 52nd IEEE Conference on Decision and Control (CDC ’13)
[18] Walker, J. J.; Terry, J. R.; Tsaneva-Atanasova, K.; Armstrong, S. P.; McArdle, C. A.; Lightman, S. L., Encoding and decoding mechanisms of pulsatile hormone secretion, Journal of Neuroendocrinology, 22, 12, 1226-1238, (2010)
[19] Smith, W. R., Qualitative mathematical models of endocrine systems, American Journal of Physiology—Regulatory, Integrative and Comparative Physiology, 245, 4, R473-R477, (1983)
[20] Cartwright, M.; Husain, M., A model for the control of testosterone secretion, Journal of Theoretical Biology, 123, 2, 239-250, (1986)
[21] Das, P.; Roy, A. B.; Das, A., Stability and oscillations of a negative feedback delay model for the control of testosterone secretion, BioSystems, 32, 1, 61-69, (1994)
[22] Keenan, D. M.; Veldhuis, J. D., A biomathematical model of time-delayed feedback in the human male hypothalamic-pituitary-Leydig cell axis, American Journal of Physiology—Endocrinology and Metabolism, 275, 1, E157-E176, (1998)
[23] Ruan, S.; Wei, J., On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA Journal of Mathemathics Applied in Medicine and Biology, 18, 1, 41-52, (2001) · Zbl 0982.92008
[24] Mukhopadhyay, B.; Bhattacharyya, R., A delayed mathematical model for testosterone secretion with feedback control mechanism, International Journal of Mathematics and Mathematical Sciences, 2004, 3, 105-115, (2004) · Zbl 1066.92026
[25] Ren, H., Stability analysis of a simplified model for the control of testosterone secretion, Discrete and Continuous Dynamical Systems B. A Journal Bridging Mathematics and Sciences, 4, 3, 729-738, (2004) · Zbl 1114.92031
[26] Teel, A. R., Observer-based hybrid feedback: a local separation principle, Proceedings of the American Control Conference (ACC ’10)
[27] Cox, N.; Marconi, L.; Teel, A., High-gain observers and linear output regulation for hybrid exosystems, International Journal of Robust and Nonlinear Control, 24, 6, 1043-1063, (2014) · Zbl 1291.93051
[28] Alessandri, A.; Coletta, P., Design of luenberger observers for a class of hybrid linear systems, Hybrid Systems: Computation and Control: 4th International Workshop, HSCC 2001 Rome, Italy, March 28–30, 2001 Proceedings. Hybrid Systems: Computation and Control: 4th International Workshop, HSCC 2001 Rome, Italy, March 28–30, 2001 Proceedings, Lecture Notes in Computer Science, 2034, 7-18, (2001), Berlin, Germany: Springer, Berlin, Germany · Zbl 0991.93071
[29] Johnson, M. L.; Pipes, L.; Veldhuis, P. P.; Farhy, L. S.; Nass, R.; Thorner, M. O.; Evans, W. S.; Johnson, M. L.; Brand, L., AutoDecon: a robust numerical method for the quantification of pulsatile events, Methods in Enzymology: Computer Methods, Volume A, 454, 367-404, (2009), Elsevier
[30] Veldhuis, J. D.; Keenan, D. M.; Pincus, S. M., Motivations and methods for analyzing pulsatile hormone secretion, Endocrine Reviews, 29, 7, 823-864, (2008)
[31] Churilov, A.; Medvedev, A.; Shepeljavyi, A., A state observer for continuous oscillating systems under intrinsic pulse-modulated feedback, Automatica, 48, 6, 1117-1122, (2012) · Zbl 1244.93030
[32] Yamalova, D.; Churilov, A.; Medvedev, A., Hybrid state observer with modulated correction for periodic systems under intrinsic impulsive feedback, Proceedings of the 5th IFAC International Workshop on Periodic Control Systems (PSYCO ’13)
[33] Yamalova, D.; Churilov, A.; Medvedev, A., Hybrid state observer for time-delay systems under intrinsic impulsive feedback, Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS ’14)
[34] Raff, T.; Allgöwer, F., An impulsive observer that estimates the exact state of a linear continuous-time system in predetermined finite time, Proceedings of the Mediterranean Conference on Control & Automation (MED ’07), IEEE
[35] Raff, T.; Allgower, F., Observers with impulsive dynamical behavior for linear and nonlinear continuous-time systems, Proceedings of the 46th IEEE Conference on Decision and Control (CDC ’07)
[36] Chen, W.-H.; Li, D.-X.; Lu, X., Impulsive observers with variable update intervals for lipschitz nonlinear time-delay systems, International Journal of Systems Science, 44, 10, 1934-1947, (2013) · Zbl 1307.93075
[37] Chen, W.-H.; Yang, W.; Zheng, W. X., Adaptive impulsive observers for nonlinear systems: revisited, Automatica, 61, 232-240, (2015) · Zbl 1327.93093
[38] Yamalova, D.; Churilov, A.; Medvedev, A., State estimation in a delayed impulsive model of testosterone regulation by a finite-dimensional hybrid observer, Proceedings of the 14th European Control Conference (ECC ’15)
[39] Fargue, D. M., Réducibilité des systèmes héréditaires a des systèmes dynamiques, Comptes Rendus de l’Académie des Sciences, Paris, Série B, 277, 471-473, (1973)
[40] Fargue, D. M., Reductibilite des systemes hereditaires, International Journal of Non linear Mechanics, 9, 5, 331-338, (1974) · Zbl 0312.73049
[41] Murray, J. D., Mathematical Biology. I. An Introduction, (2002), New York, NY, USA: Springer, New York, NY, USA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.