# American Institute of Mathematical Sciences

February  2017, 14(1): 165-178. doi: 10.3934/mbe.2017011

## Estimation of initial functions for systems with delays from discrete measurements

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

Received  September 11, 2015 Accepted  July 12, 2016 Published  October 2016

Fund Project: The author is supported by the Polish National Science Centre under grant DEC-2013/11/B/ST7/01713

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.

Citation: Krzysztof Fujarewicz. Estimation of initial functions for systems with delays from discrete measurements. Mathematical Biosciences & Engineering, 2017, 14 (1) : 165-178. doi: 10.3934/mbe.2017011
##### References:
 [1] M. Anguelova and B. Wennberg, State elimination and identifiability of the delay parameter for nonlinear time-delay systems, Automatica, 44 (2008), 1373-1378. doi: 10.1016/j.automatica.2007.10.013. Google Scholar [2] C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for nonlinear delay differential equations, Russ. J. Numer. Anal. Math. Modelling, 20 (2005), 45-66. doi: 10.1515/1569398053270831. Google Scholar [3] C. T. H. Baker and E. I. Parmuzin, Initial function estimation for scalar neutral delay differential equations, Russ. J. Numer. Anal. Math. Modelling, 23 (2008), 163-183. doi: 10.1515/RJNAMM.2008.010. Google Scholar [4] L. Belkoura, J. P. Richard and M. Fliess, Parameters estimation of systems with delayed and structured entries, Automatica, 45 (2009), 1117-1125. doi: 10.1016/j.automatica.2008.12.026. Google Scholar [5] K. Fujarewicz and 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. Google Scholar [6] K. Fujarewicz, M. Kimmel and A. Swierniak, On fitting of mathematical models of cell signaling pathways using adjoint systems, Math. Biosci. Eng., 2 (2005), 527-534. doi: 10.3934/mbe.2005.2.527. Google Scholar [7] K. Fujarewicz, M. Kimmel, T. Lipniacki and 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. Google Scholar [8] K. Fujarewicz and K. Lakomiec, Parameter estimation of systems with delays via structural sensitivity analysis, Discrete and Continuous Dynamical Systems -series B, 19 (2014), 2521-2533. doi: 10.3934/dcdsb.2014.19.2521. Google Scholar [9] K. Fujarewicz and 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. Google Scholar [10] M. Jakubczak and 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. Google Scholar [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.Google Scholar [12] M. Liu, Q. G. Wang, B. Huang and C. C. Hang, Improved identification of continuous-time delay processes from piecewise step tests, Journal of Process Control, 17 (2007), 51-57. doi: 10.1016/j.jprocont.2006.08.002. Google Scholar [13] R. Loxton, K. L. Teo and V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119. doi: 10.1109/TAC.2010.2050710. Google Scholar [14] B. Ni, D. Xiao and S. L. Shah, Time delay estimation for MIMO dynamical systems with time-frequency domain analysis, Journal of Process Control, 20 (2010), 83-94. doi: 10.1016/j.jprocont.2009.10.002. Google Scholar [15] B. Rakshit, A. R. Chowdhury and P. Saha, Parameter estimation of a delay dynamical system using synchronization inpresence of noise, Chaos, Solitons and Fractals, 32 (2007), 1278-1284. Google Scholar [16] J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5. Google Scholar [17] Y. Tang and X. Guan, Parameter estimation of chaotic system with time-delay: A differential evolution approach, Chaos, Solitons and Fractals, 42 (2009), 3132-3139. doi: 10.1016/j.chaos.2009.04.045. Google Scholar [18] Y. Tang and X. Guan, Parameter estimation for time-delay chaotic systems by particle swarm optimization, Chaos, Solitons and Fractals, 40 (2009), 1391-1398. doi: 10.1016/j.chaos.2007.09.055. Google Scholar

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##### References:
 [1] M. Anguelova and B. Wennberg, State elimination and identifiability of the delay parameter for nonlinear time-delay systems, Automatica, 44 (2008), 1373-1378. doi: 10.1016/j.automatica.2007.10.013. Google Scholar [2] C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for nonlinear delay differential equations, Russ. J. Numer. Anal. Math. Modelling, 20 (2005), 45-66. doi: 10.1515/1569398053270831. Google Scholar [3] C. T. H. Baker and E. I. Parmuzin, Initial function estimation for scalar neutral delay differential equations, Russ. J. Numer. Anal. Math. Modelling, 23 (2008), 163-183. doi: 10.1515/RJNAMM.2008.010. Google Scholar [4] L. Belkoura, J. P. Richard and M. Fliess, Parameters estimation of systems with delayed and structured entries, Automatica, 45 (2009), 1117-1125. doi: 10.1016/j.automatica.2008.12.026. Google Scholar [5] K. Fujarewicz and 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. Google Scholar [6] K. Fujarewicz, M. Kimmel and A. Swierniak, On fitting of mathematical models of cell signaling pathways using adjoint systems, Math. Biosci. Eng., 2 (2005), 527-534. doi: 10.3934/mbe.2005.2.527. Google Scholar [7] K. Fujarewicz, M. Kimmel, T. Lipniacki and 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. Google Scholar [8] K. Fujarewicz and K. Lakomiec, Parameter estimation of systems with delays via structural sensitivity analysis, Discrete and Continuous Dynamical Systems -series B, 19 (2014), 2521-2533. doi: 10.3934/dcdsb.2014.19.2521. Google Scholar [9] K. Fujarewicz and 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. Google Scholar [10] M. Jakubczak and 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. Google Scholar [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.Google Scholar [12] M. Liu, Q. G. Wang, B. Huang and C. C. Hang, Improved identification of continuous-time delay processes from piecewise step tests, Journal of Process Control, 17 (2007), 51-57. doi: 10.1016/j.jprocont.2006.08.002. Google Scholar [13] R. Loxton, K. L. Teo and V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119. doi: 10.1109/TAC.2010.2050710. Google Scholar [14] B. Ni, D. Xiao and S. L. Shah, Time delay estimation for MIMO dynamical systems with time-frequency domain analysis, Journal of Process Control, 20 (2010), 83-94. doi: 10.1016/j.jprocont.2009.10.002. Google Scholar [15] B. Rakshit, A. R. Chowdhury and P. Saha, Parameter estimation of a delay dynamical system using synchronization inpresence of noise, Chaos, Solitons and Fractals, 32 (2007), 1278-1284. Google Scholar [16] J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5. Google Scholar [17] Y. Tang and X. Guan, Parameter estimation of chaotic system with time-delay: A differential evolution approach, Chaos, Solitons and Fractals, 42 (2009), 3132-3139. doi: 10.1016/j.chaos.2009.04.045. Google Scholar [18] Y. Tang and X. Guan, Parameter estimation for time-delay chaotic systems by particle swarm optimization, Chaos, Solitons and Fractals, 40 (2009), 1391-1398. doi: 10.1016/j.chaos.2007.09.055. Google Scholar
Model of the dynamical system with one isolated discrete delay element
Alternative structural representation of the delay element with additional input signal and zero initial condition
The sensitivity model (a) and the adjoint model (b) for one delay element presented in Fig. 2
The extended model
The system adjoint to the extended model presented in Fig. 4
Block diagram of the model A, used in Examples 1-4
Results of the numerical example 1; (a) － true and estimated initial function $\psi(t)$, (b) － output signal $y(t)$ of the model and the plant, note they are nearly indistinguishable due to very small prediction error, (c) － objective function value, (d) － prediction error i.e. difference between output signals of the plant and the model
Results of the numerical example 2
Results of the numerical example 3
Results of the numerical example 4
Block diagram of the model B, used in Example 5
Results of the numerical example 5
Block diagram of the model C, used in Example 6
Results of the numerical example 6
The adjoint system for the model A generating two signals: $\beta(t)$ which is a reversed in time gradient $\nabla_{\psi(t)}J$ and $\gamma(t)$ which integrated over time interval $(0,t_f)$ is equal to the gradient $\nabla_{\tau}J$
Comparison of six numerical examples
 Example Model Number of delays Sampling time Initial function(s) Delay time(s) Results 1 A (Fig. 6) 1 0+ Estimated Known Fig. 8 2 A (Fig. 6) 1 0+ Estimated Estimated Fig. 9 3 A (Fig. 6) 1 0.1 Estimated Estimated Fig. 10 4 A (Fig. 6) 1 0.1 Fixed (=0) Estimated Fig. 11 5 B (Fig. 12) 2 0+ Estimated Known Fig. 13 6 C (Fig. 14) 2 0+ Estimated Known Fig. 15
 Example Model Number of delays Sampling time Initial function(s) Delay time(s) Results 1 A (Fig. 6) 1 0+ Estimated Known Fig. 8 2 A (Fig. 6) 1 0+ Estimated Estimated Fig. 9 3 A (Fig. 6) 1 0.1 Estimated Estimated Fig. 10 4 A (Fig. 6) 1 0.1 Fixed (=0) Estimated Fig. 11 5 B (Fig. 12) 2 0+ Estimated Known Fig. 13 6 C (Fig. 14) 2 0+ Estimated Known Fig. 15
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