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Optimal customer behavior in observable and unobservable discrete-time queues
Robust parameter estimation for constrained time-delay systems with inexact measurements
1. | School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China |
2. | School of Computer Science and Technology, Shandong Technology and Business University, Yantai 264005, China |
3. | School of Electrical Engineering, Computing, and Mathematical Sciences, Curtin University, Perth 6845, Australia |
4. | Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China |
In this paper, we consider estimation problems involving constrained nonlinear systems with the unknown time-delays and unknown system parameters. These unknown quantities are to be estimated such that a least-squares error function between the system output and a set of noisy measurements is minimized subject to the characteristic time constraints specifying the restrictions. We first present the classical estimation formulation, where the expectation of the error function is regarded as the cost function. Then, in order to obtain robust estimates against the noises in measurements, we propose a robust estimation formulation, in which the cost function is the variance of the error function and an additional constraint indicates an allowable sacrifice from the optimal expectation value of the classical estimation problem. For these two estimation problems, we derive the gradients of the corresponding cost and constraint functions with respect to time-delays and system parameters by solving some auxiliary time-delay systems backward in time. On this basis, we develop gradient-based optimization algorithms to determine the optimal time-delays and system parameters. Finally, we consider two example problems, including a parameter estimation problem in microbial batch fermentation process, to illustrate the effectiveness and applicability of our proposed algorithms.
References:
[1] |
H. T. Banks, J. A. Burns and E. M. Cliff,
Parameter estimation and identification for systems with delay, SIAM J. Control Optim., 19 (1981), 791-828.
doi: 10.1137/0319051. |
[2] |
Y. Bard,
Comparison of gradient methods for the solution of nonlinear parameter estimation problems, SIAM J. Numer. Anal., 7 (1970), 157-186.
doi: 10.1137/0707011. |
[3] |
Q. Q. Chai, R. Loxton, K. L. Teo and C. H. Yang,
A unified parameter identification method for nonlinear time-delay systems, J. Ind. Manag. Optim., 9 (2013), 471-486.
doi: 10.3934/jimo.2013.9.471. |
[4] |
D. Debeljković, Time-Delay Systems, InTech, 2011. Google Scholar |
[5] |
S. Diop, I. Kolmanovsky, P. E. Moraal and M. V. Nieuwstadt, Preserving stability/performance when facing an unknown time-delay, Control Eng. Pract., 9 (2001), 1319-1325. Google Scholar |
[6] |
P. J. Gawthrop and M. T. Nihtilä,
Identification of time-delays using a polynomial identification method, Syst. Control Lett., 5 (1985), 267-271.
doi: 10.1016/0167-6911(85)90020-9. |
[7] |
Q. Lin, R. Loxton, C. Xu and K. L. Teo,
Parameter estimation for nonlinear time-delay systems with noisy output measurements, Automatica J. IFAC, 60 (2015), 48-56.
doi: 10.1016/j.automatica.2015.06.028. |
[8] |
Q. Lin, R. Loxton and K. L. Teo,
The control parameterization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10 (2014), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[9] |
C. Y. Liu and Z. H. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer Optimization and Its Applications, 97. Springer, Heidelberg, Tsinghua University Press, Beijing, 2014.
doi: 10.1007/978-3-662-43793-3.![]() ![]() |
[10] |
C. Y. Liu, Z. H. Gong, E. Feng and H. C. Yin,
Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, J. Ind. Manag. Optim., 5 (2009), 835-850.
doi: 10.3934/jimo.2009.5.835. |
[11] |
C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta,
Robust multi-objective optimal switching control arising in $1, 3$-propanediol microbial fed-batch process, Nonlinear Anal. Hybrid Syst., 25 (2017), 1-20.
doi: 10.1016/j.nahs.2017.01.006. |
[12] |
C. Y. Liu, Z. H. Gong and K. L. Teo,
Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Appl. Math. Model., 53 (2018), 353-368.
doi: 10.1016/j.apm.2017.09.007. |
[13] |
C. Y. Liu, Z. H. Gong, H. W. J. Lee and K. L. Teo,
Robust bi-objective optimal control of $1, 3$-propanediol microbial batch production process, J. Process Contr., 78 (2019), 170-182.
doi: 10.1016/j.jprocont.2018.10.001. |
[14] |
C. Y. Liu, R. Loxton and K. L. Teo,
Optimal parameter selection for nonlinear multistage systems with time-delays, Comput. Optim. Appl., 59 (2014), 285-306.
doi: 10.1007/s10589-013-9632-x. |
[15] |
R. Loxton, K. L. Teo and V. Rehbock,
An optimization approach to state-delay identification, IEEE Trans. Aut. Control, 55 (2010), 2113-2119.
doi: 10.1109/TAC.2010.2050710. |
[16] |
R. B. Martin,
Optimal control drug scheduling of cancer chemotherapy, Automatica J. IFAC, 28 (1992), 1113-1123.
doi: 10.1016/0005-1098(92)90054-J. |
[17] |
P. Mendes and D. Kell,
Non-linear optimization of biochemical pathways: Applications to metabolic engineering and parameter estimation, Bioinformatics, 14 (1998), 869-883.
doi: 10.1093/bioinformatics/14.10.869. |
[18] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. |
[19] |
F. Pan, R. C. Han and D. M. Feng, An identification method of time-varying delay based on genetic algorithm, in Proceedings of the Second International Conference on Machine Learning and Cybernetics, (2003), 781–783. Google Scholar |
[20] |
X. M. Ren, A. B. Rad, P. T. Chan and W. L. Lo,
Online identification of continuous-time systems with unknown time-delay, IEEE Trans. Aut. Control, 50 (2005), 1418-1422.
doi: 10.1109/TAC.2005.854640. |
[21] |
J. P. Richard,
Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.
doi: 10.1016/S0005-1098(03)00167-5. |
[22] |
K. Schittkowski, A Fortran Implementation of a Sequential Quadratic Programming Algorithm with Distributed and Non-monotone Line Search-User's Guide, University of Bayreuth, Bayreuth, 2007. Google Scholar |
[23] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, New York-Heidelberg, 1980. |
[24] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[25] |
L. Wang, Q. Lin, R. Loxton, K. L. Teo and G. Cheng,
Optimal 1, 3-propanediol production: Exploring the trade-off between process yield and feeding rate variation, J. Process Contr., 32 (2015), 1-9.
doi: 10.1016/j.jprocont.2015.04.011. |
[26] |
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, J. Glob. Optim., 54 (2012), 307-323.
doi: 10.1007/s10898-012-9863-x. |
[27] |
Z. L. Xiu, B.-H. Song, L.-H. Sun and A.-P. Zeng,
Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a two-stage fermentation process, Biochem. Eng. J., 11 (2002), 101-109.
doi: 10.1016/S1369-703X(02)00033-5. |
[28] |
Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical modelling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol, J. Dalian Univ. Tech., 40 (2000), 428-433. Google Scholar |
[29] |
J. L. Yuan, X. Zhang, X. Zhu, E. Feng, H. C. Yin and Z. L. Xiu,
Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture, Nonlinear Anal. Hybrid Syst., 15 (2015), 112-131.
doi: 10.1016/j.nahs.2014.08.004. |
[30] |
J. L. Yuan, X. Zhu, X. Zhang, H. C. Yin, E. Feng and Z. L. Xiu,
Robust identification of enzymatic nonlinear dynamical systems for 1, 3-propanediol transport mechanisms in microbial batch culture, Appl. Math. Comput., 232 (2014), 150-163.
doi: 10.1016/j.amc.2014.01.027. |
show all references
References:
[1] |
H. T. Banks, J. A. Burns and E. M. Cliff,
Parameter estimation and identification for systems with delay, SIAM J. Control Optim., 19 (1981), 791-828.
doi: 10.1137/0319051. |
[2] |
Y. Bard,
Comparison of gradient methods for the solution of nonlinear parameter estimation problems, SIAM J. Numer. Anal., 7 (1970), 157-186.
doi: 10.1137/0707011. |
[3] |
Q. Q. Chai, R. Loxton, K. L. Teo and C. H. Yang,
A unified parameter identification method for nonlinear time-delay systems, J. Ind. Manag. Optim., 9 (2013), 471-486.
doi: 10.3934/jimo.2013.9.471. |
[4] |
D. Debeljković, Time-Delay Systems, InTech, 2011. Google Scholar |
[5] |
S. Diop, I. Kolmanovsky, P. E. Moraal and M. V. Nieuwstadt, Preserving stability/performance when facing an unknown time-delay, Control Eng. Pract., 9 (2001), 1319-1325. Google Scholar |
[6] |
P. J. Gawthrop and M. T. Nihtilä,
Identification of time-delays using a polynomial identification method, Syst. Control Lett., 5 (1985), 267-271.
doi: 10.1016/0167-6911(85)90020-9. |
[7] |
Q. Lin, R. Loxton, C. Xu and K. L. Teo,
Parameter estimation for nonlinear time-delay systems with noisy output measurements, Automatica J. IFAC, 60 (2015), 48-56.
doi: 10.1016/j.automatica.2015.06.028. |
[8] |
Q. Lin, R. Loxton and K. L. Teo,
The control parameterization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10 (2014), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[9] |
C. Y. Liu and Z. H. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer Optimization and Its Applications, 97. Springer, Heidelberg, Tsinghua University Press, Beijing, 2014.
doi: 10.1007/978-3-662-43793-3.![]() ![]() |
[10] |
C. Y. Liu, Z. H. Gong, E. Feng and H. C. Yin,
Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, J. Ind. Manag. Optim., 5 (2009), 835-850.
doi: 10.3934/jimo.2009.5.835. |
[11] |
C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta,
Robust multi-objective optimal switching control arising in $1, 3$-propanediol microbial fed-batch process, Nonlinear Anal. Hybrid Syst., 25 (2017), 1-20.
doi: 10.1016/j.nahs.2017.01.006. |
[12] |
C. Y. Liu, Z. H. Gong and K. L. Teo,
Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Appl. Math. Model., 53 (2018), 353-368.
doi: 10.1016/j.apm.2017.09.007. |
[13] |
C. Y. Liu, Z. H. Gong, H. W. J. Lee and K. L. Teo,
Robust bi-objective optimal control of $1, 3$-propanediol microbial batch production process, J. Process Contr., 78 (2019), 170-182.
doi: 10.1016/j.jprocont.2018.10.001. |
[14] |
C. Y. Liu, R. Loxton and K. L. Teo,
Optimal parameter selection for nonlinear multistage systems with time-delays, Comput. Optim. Appl., 59 (2014), 285-306.
doi: 10.1007/s10589-013-9632-x. |
[15] |
R. Loxton, K. L. Teo and V. Rehbock,
An optimization approach to state-delay identification, IEEE Trans. Aut. Control, 55 (2010), 2113-2119.
doi: 10.1109/TAC.2010.2050710. |
[16] |
R. B. Martin,
Optimal control drug scheduling of cancer chemotherapy, Automatica J. IFAC, 28 (1992), 1113-1123.
doi: 10.1016/0005-1098(92)90054-J. |
[17] |
P. Mendes and D. Kell,
Non-linear optimization of biochemical pathways: Applications to metabolic engineering and parameter estimation, Bioinformatics, 14 (1998), 869-883.
doi: 10.1093/bioinformatics/14.10.869. |
[18] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. |
[19] |
F. Pan, R. C. Han and D. M. Feng, An identification method of time-varying delay based on genetic algorithm, in Proceedings of the Second International Conference on Machine Learning and Cybernetics, (2003), 781–783. Google Scholar |
[20] |
X. M. Ren, A. B. Rad, P. T. Chan and W. L. Lo,
Online identification of continuous-time systems with unknown time-delay, IEEE Trans. Aut. Control, 50 (2005), 1418-1422.
doi: 10.1109/TAC.2005.854640. |
[21] |
J. P. Richard,
Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.
doi: 10.1016/S0005-1098(03)00167-5. |
[22] |
K. Schittkowski, A Fortran Implementation of a Sequential Quadratic Programming Algorithm with Distributed and Non-monotone Line Search-User's Guide, University of Bayreuth, Bayreuth, 2007. Google Scholar |
[23] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, New York-Heidelberg, 1980. |
[24] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[25] |
L. Wang, Q. Lin, R. Loxton, K. L. Teo and G. Cheng,
Optimal 1, 3-propanediol production: Exploring the trade-off between process yield and feeding rate variation, J. Process Contr., 32 (2015), 1-9.
doi: 10.1016/j.jprocont.2015.04.011. |
[26] |
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, J. Glob. Optim., 54 (2012), 307-323.
doi: 10.1007/s10898-012-9863-x. |
[27] |
Z. L. Xiu, B.-H. Song, L.-H. Sun and A.-P. Zeng,
Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a two-stage fermentation process, Biochem. Eng. J., 11 (2002), 101-109.
doi: 10.1016/S1369-703X(02)00033-5. |
[28] |
Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical modelling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol, J. Dalian Univ. Tech., 40 (2000), 428-433. Google Scholar |
[29] |
J. L. Yuan, X. Zhang, X. Zhu, E. Feng, H. C. Yin and Z. L. Xiu,
Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture, Nonlinear Anal. Hybrid Syst., 15 (2015), 112-131.
doi: 10.1016/j.nahs.2014.08.004. |
[30] |
J. L. Yuan, X. Zhu, X. Zhang, H. C. Yin, E. Feng and Z. L. Xiu,
Robust identification of enzymatic nonlinear dynamical systems for 1, 3-propanediol transport mechanisms in microbial batch culture, Appl. Math. Comput., 232 (2014), 150-163.
doi: 10.1016/j.amc.2014.01.027. |





0 | 0.434658 | 0.713674 | 3.068431 | 0.690517 |
0.001 | 0.414390 | 0.764191 | 2.986242 | 0.809873 |
0.004 | 0.408673 | 0.790692 | 2.930212 | 0.845580 |
0.007 | 0.396182 | 0.804993 | 2.830212 | 0.898728 |
0.01 | 0.365811 | 0.819147 | 2.805867 | 0.914341 |
0.04 | 0.319229 | 0.839143 | 2.761121 | 0.939468 |
0.07 | 0.306182 | 0.896250 | 2.748475 | 0.943645 |
0.1 | 0.283349 | 0.942031 | 2.536738 | 0.975285 |
0.17 | 0.253673 | 0.951192 | 2.245683 | 1.077120 |
0.178 | 0.249934 | 0.954031 | 2.231923 | 1.073528 |
0 | 0.434658 | 0.713674 | 3.068431 | 0.690517 |
0.001 | 0.414390 | 0.764191 | 2.986242 | 0.809873 |
0.004 | 0.408673 | 0.790692 | 2.930212 | 0.845580 |
0.007 | 0.396182 | 0.804993 | 2.830212 | 0.898728 |
0.01 | 0.365811 | 0.819147 | 2.805867 | 0.914341 |
0.04 | 0.319229 | 0.839143 | 2.761121 | 0.939468 |
0.07 | 0.306182 | 0.896250 | 2.748475 | 0.943645 |
0.1 | 0.283349 | 0.942031 | 2.536738 | 0.975285 |
0.17 | 0.253673 | 0.951192 | 2.245683 | 1.077120 |
0.178 | 0.249934 | 0.954031 | 2.231923 | 1.073528 |
Parameters | Lower bounds | Upper bounds | Optimal values | |||
0.0010 | 1.0000 | 0.0863 | 0.0743 | 0.0700 | 0.0648 | |
0.1000 | 2.0000 | 1.5346 | 1.5366 | 1.6385 | 1.9040 | |
10.000 | 500.00 | 287.91 | 289.79 | 331.63 | 442.56 | |
1.0000 | 5.2545 | 7.5042 | 10.759 | 15.629 | ||
0.0001 | 2.0000 | 0.0083 | 0.0092 | 1.0325 | 0.0124 | |
1.0000 | 20.000 | 13.598 | 15.787 | 16.587 | 17.001 | |
10.000 | 200.00 | 131.91 | 136.49 | 138.26 | 139.31 |
Parameters | Lower bounds | Upper bounds | Optimal values | |||
0.0010 | 1.0000 | 0.0863 | 0.0743 | 0.0700 | 0.0648 | |
0.1000 | 2.0000 | 1.5346 | 1.5366 | 1.6385 | 1.9040 | |
10.000 | 500.00 | 287.91 | 289.79 | 331.63 | 442.56 | |
1.0000 | 5.2545 | 7.5042 | 10.759 | 15.629 | ||
0.0001 | 2.0000 | 0.0083 | 0.0092 | 1.0325 | 0.0124 | |
1.0000 | 20.000 | 13.598 | 15.787 | 16.587 | 17.001 | |
10.000 | 200.00 | 131.91 | 136.49 | 138.26 | 139.31 |
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