American Institute of Mathematical Sciences

Advanced
April  2013, 9(2): 471-486. doi: 10.3934/jimo.2013.9.471

A unified parameter identification method for nonlinear time-delay systems

Qinqin Chai 1, , Ryan Loxton 2, , Kok Lay Teo 3, and  Chunhua Yang 4,

1. 

School of Information Science & Engineering, Central South University, Changsha, China
2. 

Department of Mathematics and Statistics, Curtin University, Perth 6845
3. 

Department of Mathematics and Statistics, Curtin University, Perth, W.A. 6845
4. 

School of Information Science and Engineering, Central South University, Changsha, 410083

Received  June 2012 Revised  October 2012 Published  February 2013

This paper deals with the problem of identifying unknown time-delays and model parameters in a general nonlinear time-delay system. We propose a unified computational approach that involves solving a dynamic optimization problem, whose cost function measures the discrepancy between predicted and observed system output, to determine optimal values for the unknown quantities. Our main contribution is to show that the partial derivatives of this cost function can be computed by solving a set of auxiliary time-delay systems. On this basis, the parameter identification problem can be solved using existing gradient-based optimization techniques. We conclude the paper with two numerical simulations.
Keywords: nonlinear optimization., Time-delay system, parameter identification.
Mathematics Subject Classification: Primary: 93B30; Secondary: 90C30, 90C90, 37N4.
Citation: Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471
References:
[1]

N. U. Ahmed, "Dynamic Systems and Control with Applications,", World Scientific Publishing Co. Pte. Ltd., (2006).   Google Scholar

[2]

L. Belkoura, J.-P. Richard and M. Fliess, Parameters estimation of systems with delayed and structured entries,, Automatica J. IFAC, 45 (2009), 1117.  doi: 10.1016/j.automatica.2008.12.026.  Google Scholar

[3]

Q. Q. Chai, C. H. Yang, K. L. Teo and W. H. Gui, Time-delayed optimal control of an industrial-scale evaporation process sodium aluminate solution,, Control Engineering Practice, 20 (2012), 618.   Google Scholar

[4]

R. Datko, Two examples of ill-posedness with respect to time delays revisited,, IEEE Transactions on Automatic Control, 42 (1997), 511.  doi: 10.1109/9.566660.  Google Scholar

[5]

L. Denis-Vidal, C. Jauberthie and G. Joly-Blanchard, Identifiability of a nonlinear delayed-differential aerospace model,, IEEE Transactions on Automatic Control, 51 (2006), 154.  doi: 10.1109/TAC.2005.861700.  Google Scholar

[6]

S. Diop, I. Kolmanovsky, P. E. Moraal and M. V. Nieuwstadt, Preserving stability/performance when facing an unknown time-delay,, Control Engineering Practice, 9 (2001), 1319.   Google Scholar

[7]

S. V. Drakunov, W. Perruquetti, J. P. Richard and L. Belkoura, Delay identification in time-delay systems using variable structure observers,, Annual Reviews in Control, 30 (2006), 143.   Google Scholar

[8]

P. J. Gawthrop and M. T. Nihtilä, Identification of time delays using a polynomial identification method,, Systems and Control Letters, 5 (1985), 267.  doi: 10.1016/0167-6911(85)90020-9.  Google Scholar

[9]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints,, Optimal Control Applications and Methods, 30 (2009), 341.  doi: 10.1002/oca.843.  Google Scholar

[10]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems,, Dynamics of Continuous, 18 (2011), 59.   Google Scholar

[11]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems,, Pacific Journal of Optimization, 7 (2011), 63.   Google Scholar

[12]

X. Liu, Constrained control of positive systems with delays,, IEEE Transactions on Automatic Control, 54 (2009), 1596.  doi: 10.1109/TAC.2009.2017961.  Google Scholar

[13]

R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints,, Automatica, 44 (2008), 2923.  doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[14]

R. Loxton, K. L. Teo and V. Rehbock, An optimization approach to state-delay identification,, IEEE Transactions on Automatic Control, 55 (2010), 2113.  doi: 10.1109/TAC.2010.2050710.  Google Scholar

[15]

R. Loxton, K. L. Teo and V. Rehbock, Robust suboptimal control of nonlinear systems,, Applied Mathematics and Computation, 217 (2011), 6566.  doi: 10.1016/j.amc.2011.01.039.  Google Scholar

[16]

D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", $3^{rd}$ edition, 116 (2008).   Google Scholar

[17]

R. B. Martin, Optimal control drug scheduling of cancer chemotherapy,, Automatica J. IFAC, 28 (1992), 1113.  doi: 10.1016/0005-1098(92)90054-J.  Google Scholar

[18]

F. Pan, R. C. Han and D. M. Feng, "An identification method of time-varying delay based on genetic algorithm,", in Proceedings of the 2003 International Conference on Machine Learning and Cybernetics, (2003), 781.   Google Scholar

[19]

C. Pignotti, A note on stabilization of locally damped wave equations with time delay,, Systems and Control Letters, 61 (2012), 92.  doi: 10.1016/j.sysconle.2011.09.016.  Google Scholar

[20]

J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems,, Automatica J. IFAC, 39 (2003), 1667.  doi: 10.1016/S0005-1098(03)00167-5.  Google Scholar

[21]

R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace, Optimal control of innate immune response,, Optimal Control Applications and Methods, 23 (2002), 91.  doi: 10.1002/oca.704.  Google Scholar

[22]

L. Wang, W. Gui, K. L. Teo, R. Loxton and C. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.  doi: 10.3934/jimo.2009.5.705.  Google Scholar

[23]

L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Optimal control problems arising in the zinc sulphate electrolyte purification process,, Journal of Global Optimization, 54 (2012), 307.  doi: 10.1007/s10898-012-9863-x.  Google Scholar

[24]

F. Y. Wang and Q. Yu, Optimal protein separations with time lags in control functions,, Journal of Process Control, 4 (1994), 135.   Google Scholar

[25]

K. H. Wong, L. S. Jennings and F. Benyah, The control parametrization enhancing transform for constrained time-delayed optimal control problems,, ANZIAM Journal, 43 ().   Google Scholar

[26]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso and C. R. Mirasso, Permutation-information-theory approach to unveil delay dynamics from time-series analysis,, Physical Review E, 82 (2010).  doi: 10.1103/PhysRevE.82.046212.  Google Scholar

show all references

References:
[1]

N. U. Ahmed, "Dynamic Systems and Control with Applications,", World Scientific Publishing Co. Pte. Ltd., (2006).   Google Scholar

[2]

L. Belkoura, J.-P. Richard and M. Fliess, Parameters estimation of systems with delayed and structured entries,, Automatica J. IFAC, 45 (2009), 1117.  doi: 10.1016/j.automatica.2008.12.026.  Google Scholar

[3]

Q. Q. Chai, C. H. Yang, K. L. Teo and W. H. Gui, Time-delayed optimal control of an industrial-scale evaporation process sodium aluminate solution,, Control Engineering Practice, 20 (2012), 618.   Google Scholar

[4]

R. Datko, Two examples of ill-posedness with respect to time delays revisited,, IEEE Transactions on Automatic Control, 42 (1997), 511.  doi: 10.1109/9.566660.  Google Scholar

[5]

L. Denis-Vidal, C. Jauberthie and G. Joly-Blanchard, Identifiability of a nonlinear delayed-differential aerospace model,, IEEE Transactions on Automatic Control, 51 (2006), 154.  doi: 10.1109/TAC.2005.861700.  Google Scholar

[6]

S. Diop, I. Kolmanovsky, P. E. Moraal and M. V. Nieuwstadt, Preserving stability/performance when facing an unknown time-delay,, Control Engineering Practice, 9 (2001), 1319.   Google Scholar

[7]

S. V. Drakunov, W. Perruquetti, J. P. Richard and L. Belkoura, Delay identification in time-delay systems using variable structure observers,, Annual Reviews in Control, 30 (2006), 143.   Google Scholar

[8]

P. J. Gawthrop and M. T. Nihtilä, Identification of time delays using a polynomial identification method,, Systems and Control Letters, 5 (1985), 267.  doi: 10.1016/0167-6911(85)90020-9.  Google Scholar

[9]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints,, Optimal Control Applications and Methods, 30 (2009), 341.  doi: 10.1002/oca.843.  Google Scholar

[10]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems,, Dynamics of Continuous, 18 (2011), 59.   Google Scholar

[11]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems,, Pacific Journal of Optimization, 7 (2011), 63.   Google Scholar

[12]

X. Liu, Constrained control of positive systems with delays,, IEEE Transactions on Automatic Control, 54 (2009), 1596.  doi: 10.1109/TAC.2009.2017961.  Google Scholar

[13]

R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints,, Automatica, 44 (2008), 2923.  doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[14]

R. Loxton, K. L. Teo and V. Rehbock, An optimization approach to state-delay identification,, IEEE Transactions on Automatic Control, 55 (2010), 2113.  doi: 10.1109/TAC.2010.2050710.  Google Scholar

[15]

R. Loxton, K. L. Teo and V. Rehbock, Robust suboptimal control of nonlinear systems,, Applied Mathematics and Computation, 217 (2011), 6566.  doi: 10.1016/j.amc.2011.01.039.  Google Scholar

[16]

D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", $3^{rd}$ edition, 116 (2008).   Google Scholar

[17]

R. B. Martin, Optimal control drug scheduling of cancer chemotherapy,, Automatica J. IFAC, 28 (1992), 1113.  doi: 10.1016/0005-1098(92)90054-J.  Google Scholar

[18]

F. Pan, R. C. Han and D. M. Feng, "An identification method of time-varying delay based on genetic algorithm,", in Proceedings of the 2003 International Conference on Machine Learning and Cybernetics, (2003), 781.   Google Scholar

[19]

C. Pignotti, A note on stabilization of locally damped wave equations with time delay,, Systems and Control Letters, 61 (2012), 92.  doi: 10.1016/j.sysconle.2011.09.016.  Google Scholar

[20]

J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems,, Automatica J. IFAC, 39 (2003), 1667.  doi: 10.1016/S0005-1098(03)00167-5.  Google Scholar

[21]

R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace, Optimal control of innate immune response,, Optimal Control Applications and Methods, 23 (2002), 91.  doi: 10.1002/oca.704.  Google Scholar

[22]

L. Wang, W. Gui, K. L. Teo, R. Loxton and C. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.  doi: 10.3934/jimo.2009.5.705.  Google Scholar

[23]

L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Optimal control problems arising in the zinc sulphate electrolyte purification process,, Journal of Global Optimization, 54 (2012), 307.  doi: 10.1007/s10898-012-9863-x.  Google Scholar

[24]

F. Y. Wang and Q. Yu, Optimal protein separations with time lags in control functions,, Journal of Process Control, 4 (1994), 135.   Google Scholar

[25]

K. H. Wong, L. S. Jennings and F. Benyah, The control parametrization enhancing transform for constrained time-delayed optimal control problems,, ANZIAM Journal, 43 ().   Google Scholar

[26]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso and C. R. Mirasso, Permutation-information-theory approach to unveil delay dynamics from time-series analysis,, Physical Review E, 82 (2010).  doi: 10.1103/PhysRevE.82.046212.  Google Scholar

[1]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206
[2]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[3]

Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008
[4]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[5]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194
[6]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030
[7]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[8]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454
[9]

Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems & Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024
[10]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[11]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931
[12]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027
[13]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[14]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[15]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[16]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[17]

Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075
[18]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[19]

Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493
[20]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (27)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences