# American Institute of Mathematical Sciences

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

## A unified parameter identification method for nonlinear time-delay systems

 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.
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. 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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
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