American Institute of Mathematical Sciences

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January  2021, 17(1): 317-337. doi: 10.3934/jimo.2019113

Robust parameter estimation for constrained time-delay systems with inexact measurements

Chongyang Liu 1, , Meijia Han 2, , Zhaohua Gong 1,, and  Kok Lay Teo 3,4,

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

* Corresponding author: Zhaohua Gong

Received  January 2019 Revised  April 2019 Published  September 2019

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.

Keywords: Nonlinear time-delay system, parameter estimation, characteristic time constraint, gradient computation, numerical optimization.
Mathematics Subject Classification: Primary: 49M37; Secondary: 39B72.
Citation: Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Reference output trajectory (black line) and corresponding perturbed sample data (blue stars) for Example 1
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Figure 2.  Output trajectories corresponding to the optimal time-delay parameter estimates in TABLE 1 for Examplek 1
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Figure 3.  Means and variances of the least-squares errors for 100,000 output realization according to (48) for Examplek 1
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Figure 4.  Error mean variations with respect to $ \epsilon $ for 100,000 output realization according to (49) for Examplek 1
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Figure 5.  State trajectories corresponding to the optimal parameters in TABLE 2 for Examplek 2
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Figure 6.  Mean and variance of the least-squares errors for 100,000 output realizations generated according to (59) for Examplek 2
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Figure 7.  Error mean variation of the least-squares error for 100,000 output realizations generated according to (61) for Examplek 2
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Table 1.  Optimal time-delay and parameter estimates for Examplek 1
$ \beta $ $ \alpha_1^* $ $ \alpha_2^* $ $ \zeta_1^* $ $ \zeta_2^* $
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
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$ \beta $ $ \alpha_1^* $ $ \alpha_2^* $ $ \zeta_1^* $ $ \zeta_2^* $
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
Table 2.  Bounds, optimal time-delay and parameter estimates for Examplek 2
Parameters Lower bounds Upper bounds Optimal values
$ \beta=0 $ $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.3 $
$ \alpha $ 0.0010 1.0000 0.0863 0.0743 0.0700 0.0648
$ \Delta_1 $ 0.1000 2.0000 1.5346 1.5366 1.6385 1.9040
$ k_1 $ 10.000 500.00 287.91 289.79 331.63 442.56
$ m_2 $ 1.0000 $ 20.000 $ 5.2545 7.5042 10.759 15.629
$ Y_2 $ 0.0001 2.0000 0.0083 0.0092 1.0325 0.0124
$ m_3 $ 1.0000 20.000 13.598 15.787 16.587 17.001
$ Y_3 $ 10.000 200.00 131.91 136.49 138.26 139.31
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Parameters Lower bounds Upper bounds Optimal values
$ \beta=0 $ $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.3 $
$ \alpha $ 0.0010 1.0000 0.0863 0.0743 0.0700 0.0648
$ \Delta_1 $ 0.1000 2.0000 1.5346 1.5366 1.6385 1.9040
$ k_1 $ 10.000 500.00 287.91 289.79 331.63 442.56
$ m_2 $ 1.0000 $ 20.000 $ 5.2545 7.5042 10.759 15.629
$ Y_2 $ 0.0001 2.0000 0.0083 0.0092 1.0325 0.0124
$ m_3 $ 1.0000 20.000 13.598 15.787 16.587 17.001
$ Y_3 $ 10.000 200.00 131.91 136.49 138.26 139.31
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