Estimation of initial functions for systems with delays from discrete measurements

Krzysztof Fujarewicz; Krzysztof Fujarewicz

doi:10.3934/mbe.2017011

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 165-178. doi: 10.3934/mbe.2017011

Previous Article Next Article

Estimation of initial functions for systems with delays from discrete measurements

Krzysztof Fujarewicz

Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

Received: 11 September 2015 Accepted: 12 July 2016 Published: 01 February 2017
MSC : Primary: 65M99, 65K05, 93E12; Secondary: 93C57

The work presents a gradient-based approach to estimation of initial functions of time delay elements appearing in models of dynamical systems. It is shown how to generate the gradient of the estimation objective function in the initial function space using adjoint sensitivity analysis. It is assumed that the system is continuous-time and described by ordinary differential equations with delays but the estimation is done based on discrete-time measurements of the signals appearing in the system. Results of gradient-based estimation of initial functions for exemplary models are presented and discussed.
- Delay differential equations,
- systems with delays,
- initial functions,
- parameter estimation,
- adjoint sensitivity analysis,
- sampled systems
Citation: Krzysztof Fujarewicz. Estimation of initial functions for systems with delays from discrete measurements[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 165-178. doi: 10.3934/mbe.2017011

Related Papers:

Abstract

The work presents a gradient-based approach to estimation of initial functions of time delay elements appearing in models of dynamical systems. It is shown how to generate the gradient of the estimation objective function in the initial function space using adjoint sensitivity analysis. It is assumed that the system is continuous-time and described by ordinary differential equations with delays but the estimation is done based on discrete-time measurements of the signals appearing in the system. Results of gradient-based estimation of initial functions for exemplary models are presented and discussed.

References

[1]	[ M. Anguelova,B. Wennberg, State elimination and identifiability of the delay parameter for nonlinear time-delay systems, Automatica, 44 (2008): 1373-1378.
[2]	[ C. T. H. Baker,E. I. Parmuzin, Identification of the initial function for nonlinear delay differential equations, Russ. J. Numer. Anal. Math. Modelling, 20 (2005): 45-66.
[3]	[ C. T. H. Baker,E. I. Parmuzin, Initial function estimation for scalar neutral delay differential equations, Russ. J. Numer. Anal. Math. Modelling, 23 (2008): 163-183.
[4]	[ L. Belkoura,J. P. Richard,M. Fliess, Parameters estimation of systems with delayed and structured entries, Automatica, 45 (2009): 1117-1125.
[5]	[ K. Fujarewicz,A. Galuszka, Generalized backpropagation through time for continuous time neural networks and discrete time measurements, Artificial Intelligence and Soft Computing -ICAISC 2004 (eds. L. Rutkowski, J. Siekmann, R. Tadeusiewicz and L. A. Zadeh), Lecture Notes in Computer Science, 3070 (2004): 190-196.
[6]	[ K. Fujarewicz,M. Kimmel,A. Swierniak, On fitting of mathematical models of cell signaling pathways using adjoint systems, Math. Biosci. Eng., 2 (2005): 527-534.
[7]	[ K. Fujarewicz,M. Kimmel,T. Lipniacki,A. Swierniak, Adjoint systems for models of cell signalling pathways and their application to parametr fitting, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 4 (2007): 322-335.
[8]	[ K. Fujarewicz,K. Lakomiec, Parameter estimation of systems with delays via structural sensitivity analysis, Discrete and Continuous Dynamical Systems -series B, 19 (2014): 2521-2533.
[9]	[ K. Fujarewicz,K. Lakomiec, Adjoint sensitivity analysis of a tumor growth model and its application to spatiotemporal radiotherapy optimization, Mathematical Biosciences and Engineering, 13 (2016): 1131-1142.
[10]	[ M. Jakubczak,K. Fujarewicz, Application of adjoint sensitivity analysis to parameter estimation of age-structured model of cell cycle, in Information Technologies in Medicine, (eds. E. Pietka, P. Badura, J. Kawa and W. Wieclawek), Advances in Intelligent Systems and Computing, 472 (2016): 123-131.
[11]	[ K. Ł akomiec, S. Kumala, R. Hancock, J. Rzeszowska-Wolny and K. Fujarewicz, Modeling the repair of DNA strand breaks caused by $γ$-radiation in a minichromosome, Physical Biology 11 (2014), 045003.
[12]	[ M. Liu,Q. G. Wang,B. Huang,C. C. Hang, Improved identification of continuous-time delay processes from piecewise step tests, Journal of Process Control, 17 (2007): 51-57.
[13]	[ R. Loxton,K. L. Teo,V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010): 2113-2119.
[14]	[ B. Ni,D. Xiao,S. L. Shah, Time delay estimation for MIMO dynamical systems with time-frequency domain analysis, Journal of Process Control, 20 (2010): 83-94.
[15]	[ B. Rakshit,A. R. Chowdhury,P. Saha, Parameter estimation of a delay dynamical system using synchronization inpresence of noise, Chaos, Solitons and Fractals, 32 (2007): 1278-1284.
[16]	[ J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003): 1667-1694.
[17]	[ Y. Tang,X. Guan, Parameter estimation of chaotic system with time-delay: A differential evolution approach, Chaos, Solitons and Fractals, 42 (2009): 3132-3139.
[18]	[ Y. Tang,X. Guan, Parameter estimation for time-delay chaotic systems by particle swarm optimization, Chaos, Solitons and Fractals, 40 (2009): 1391-1398.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)