October  2014, 19(8): 2521-2533. doi: 10.3934/dcdsb.2014.19.2521

Parameter estimation of systems with delays via structural sensitivity analysis

1. 

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

Received  November 2013 Revised  May 2014 Published  August 2014

This article presents a method for sensitivity analysis of non-linear continuous-time models with delays and its application to parameter estimation. The method is universal and may be used for sensitivity analysis of any system given as a block diagram with arbitrary structure and any number of delays. The method gives sensitivity functions of model trajectories with respect to all model parameters, including delay times, and both forward and adjoint sensitivity analysis may be performed. Two examples application of the method are presented: identification of a Wiener model with delay and identification of a model of JAK-STAT cell signal transduction mechanism.
Citation: Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521
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.

[2]

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.

[3]

D. G. Cacuci, Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach, Journal Mathematical Physics, 22 (1981), 2794-2802. doi: 10.1063/1.525186.

[4]

J. B. Cruz (Ed.), Feedback Systems, McGraw-Hill, New York, 1972.

[5]

J. B. Cruz (Ed.), System Sensitivity Analysis, Benchmark Papers in Electrical Engineering and Computer Science, Dowden, Hutchinson and Ross, Inc., Stroudsburg, 1973.

[6]

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.

[7]

K. Fujarewicz and A. Galuszka, Generalized Backpropagation Through Time for Continuous Time Neural Networks and Discrete Time Measurements. in Artificial Intelligence and Soft Computing - ICAISC 2004 (eds. L. Rutkowski, J. Siekmann, R. Tadeusiewicz and L. A. Zadeh), Lecture Notes in Computer Science, 3070, Springer-Verlag, Berlin, 2004, 190-196.

[8]

K. Fujarewicz, M. Kimmel and A. Swierniak, On fitting of mathematical models of cell signaling pathways using adjoint systems, Mathematical Biosciences and Engineering, 2 (2005), 527-534. doi: 10.3934/mbe.2005.2.527.

[9]

K. Fujarewicz, Identification and suboptimal control of heat exchanger using generalized back propagation through time, Archives of Control Sciences, 10 (2000), 167-183.

[10]

F. Giri and E. W.Bai, eds, Block-oriented Nonlinear System Identification, Springer, 2010.

[11]

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.

[12]

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.

[13]

S. Mason, Feedback theory-Some properties of signal-flow graphs, Proc. IRE, 41 (1953), 1144-1156.

[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.

[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.

[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.

[17]

F. A. Rihan, Sensitivity analysis for dynamic systems with time-lags, Journal of Computational and Applied Mathematics, 151 (2003), 445-462. doi: 10.1016/S0377-0427(02)00659-3.

[18]

T. Swameye, G. Muller, J. Timmer, O. Sandra and U. Klingmuller, Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling, PNAS, 100 (2003), 1028-1033.

[19]

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.

[20]

Y. Tang and X. Guan, Parameter estimation for time-delay chaotic systems by particle swarm optimization, Chaos, Solitons and Fractals, 40 (2009), 1391-1398.

show all references

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.

[2]

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.

[3]

D. G. Cacuci, Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach, Journal Mathematical Physics, 22 (1981), 2794-2802. doi: 10.1063/1.525186.

[4]

J. B. Cruz (Ed.), Feedback Systems, McGraw-Hill, New York, 1972.

[5]

J. B. Cruz (Ed.), System Sensitivity Analysis, Benchmark Papers in Electrical Engineering and Computer Science, Dowden, Hutchinson and Ross, Inc., Stroudsburg, 1973.

[6]

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.

[7]

K. Fujarewicz and A. Galuszka, Generalized Backpropagation Through Time for Continuous Time Neural Networks and Discrete Time Measurements. in Artificial Intelligence and Soft Computing - ICAISC 2004 (eds. L. Rutkowski, J. Siekmann, R. Tadeusiewicz and L. A. Zadeh), Lecture Notes in Computer Science, 3070, Springer-Verlag, Berlin, 2004, 190-196.

[8]

K. Fujarewicz, M. Kimmel and A. Swierniak, On fitting of mathematical models of cell signaling pathways using adjoint systems, Mathematical Biosciences and Engineering, 2 (2005), 527-534. doi: 10.3934/mbe.2005.2.527.

[9]

K. Fujarewicz, Identification and suboptimal control of heat exchanger using generalized back propagation through time, Archives of Control Sciences, 10 (2000), 167-183.

[10]

F. Giri and E. W.Bai, eds, Block-oriented Nonlinear System Identification, Springer, 2010.

[11]

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.

[12]

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.

[13]

S. Mason, Feedback theory-Some properties of signal-flow graphs, Proc. IRE, 41 (1953), 1144-1156.

[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.

[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.

[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.

[17]

F. A. Rihan, Sensitivity analysis for dynamic systems with time-lags, Journal of Computational and Applied Mathematics, 151 (2003), 445-462. doi: 10.1016/S0377-0427(02)00659-3.

[18]

T. Swameye, G. Muller, J. Timmer, O. Sandra and U. Klingmuller, Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling, PNAS, 100 (2003), 1028-1033.

[19]

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.

[20]

Y. Tang and X. Guan, Parameter estimation for time-delay chaotic systems by particle swarm optimization, Chaos, Solitons and Fractals, 40 (2009), 1391-1398.

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