# American Institute of Mathematical Sciences

January  2022, 18(1): 693-712. doi: 10.3934/jimo.2021159

## Parameter optimal identification and dynamic behavior analysis of nonlinear model for the solution purification process of zinc hydrometallurgy

 1 College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, 730050, China 2 College of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China 3 College of Electrical and Information Engineering, Key Laboratory of Gansu Advanced Control for Industrial Processes, National Demonstration Center for Experimental Electrical and Control Engineering Education, Lanzhou University of Technology, Lanzhou, 730050, China

* Corresponding author: Aimin An, Ph.D Professor; Email: anaiminll@163.com; ORCID:0000-0003-3607-6536

Received  March 2021 Revised  June 2021 Published  January 2022 Early access  September 2021

Impurity removal is a momentous part of zinc hydrometallurgy process, and the quality of products and the stability of the whole process are affected directly by its control effect. The application of dynamic model is of great significance to the prediction of key indexes and the optimization of process control. In this paper, considering the complex coupling relationship of stage II purification process, a hybrid modeling method of mechanism modeling and parameter identification modeling was proposed on the basis of not changing the actual production process of lead-zinc smeltery. Firstly, the overall nonlinear dynamic mechanism model was established, and then the deviation between the theoretical value and the actual detected outlet ion concentration was taken as the objective function to establish the parameter identification optimization model. Since the built model is nonlinear, it may pose implementation problems. On the premise of deriving the gradient vector and Hessian matrix of the objective function with respect to the parameter vector, an optimization algorithm based on the steepest descent method and Newton method is proposed. Finally, using the historical production data of a lead-zinc smeltery in China, the model parameters were accurately inversed. An intensive simulation validation and analysis of the dynamic characteristics about the whole model shows the accuracy and the potential of the model, also in the perspective of practical implementation, which provides the basis for the optimal control of system output and the guidance for the optimal control of zinc powder addition.

Citation: Qianqian Wang, Minan Tang, Aimin An, Jiawei Lu, Yingying Zhao. Parameter optimal identification and dynamic behavior analysis of nonlinear model for the solution purification process of zinc hydrometallurgy. Journal of Industrial and Management Optimization, 2022, 18 (1) : 693-712. doi: 10.3934/jimo.2021159
##### References:
 [1] N. Chen, J. Q. Zhou, W. H. Gui, C. H. Yang and J. Y. Dai, Two-layer optimal control for goethite iron precipitation process, Control Theory & Applications, 37 (2020), 222–228. [2] W. H. Gui and C. H. Yang, Intelligent Modeling, Control and Optimization of Complex Non-Ferrous Metallurgy Production Process, Science Press, Beijing, 2010. [3] W. H. Gui, C. H. Yang, X. F. Chen and Y. L. Wang, Modeling and optimization problems and challenges arising in nonferrous metallurgical processes, Acta Automatica Sinica, 39 (2013), 197-207. [4] J. Han, C. H. Yang, X. J. Zhou and W. H. Gui, Dynamic multi-objective optimization arising in iron precipitation of zinc hydrometallurgy, Hydrometallurgy, 173 (2017), 134-148.  doi: 10.1016/j.hydromet.2017.08.007. [5] X. F. Liu, B. Z. Zhou, B. Y. Xiao and G. P. Cai, Inertia parameter identification of an unknown captured space target, Aircraft Engineering and Aerospace Technology, 91 (2019), 1147-1155. [6] B. Sun, W. H. Gui, T. B. Wu, Y. L. Wang and C. H. Yang, An integrated prediction model of cobalt ion concentration based on oxidation-reduction potential, Hydrometallurgy, 140 (2013), 102-110.  doi: 10.1016/j.hydromet.2013.09.015. [7] B. Sun, W. H. Gui, Y. L. Wang, C. H. Yang and M. F. He, A gradient optimization scheme for solution purification process, Control Engineering Practice, 44 (2015), 89-103.  doi: 10.1016/j.conengprac.2015.07.008. [8] B. Sun, W. H. Gui, C. H. Yang, Y. L. Wang and M. F. He, Online estimation of impurity ion concentration in solution purification process, IFAC-PapersOnLine, 49 (2016), 178-183. [9] B. Sun, B. Zhang, C. H. Yang and W. H. Gui, Discussion on modeling and optimal control of nonferrous metallurgical purification process, Acta Automatica Sinica, 43 (2017), 880-892. [10] M. S. Sang, Y. Ding, M. L. Bao, Y. T. Fang and B. B. Lu, Propagation dynamics model considering the characteristics of 2019-nCoV and prevention-control measurements, System Engineering-Theory & Practice, 41 (2021), 124–133. doi: 10.12011/SETP2020-0911. [11] B. Sun, M. F. He, Y. L. Wang, W. H. Gui, C. H. Yang and Q. M. Zhu, A data-driven optimal control approach for solution purification process, Journal of Process Control, 68 (2018), 171-185.  doi: 10.1016/j.jprocont.2018.06.005. [12] L. Y. Wang, Modelling and Optimization Method Based on Control Parameterization in Purification Process of Zinc Hydrometallurgy, Ph.D thesis, Central South University in Changsha, 2009. [13] L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, J. Ind. Manag. Optim., 5 (2009), 705-718.  doi: 10.3934/jimo.2009.5.705. [14] T. B. Wu, C. H. Yang, Y. G. Li, H. Q. Zhu and W. H. Gui, Fuzzy operational-pattern based operating parameters collaborative optimization of cobalt removal process with arsenic salt, Acta Automatica Sinica, 40 (2014), 1690-1698. [15] X. Y. Wang, X. J. Zhou and C. H. Yang, Chance constrained optimization for copper removal process under uncertainty in zinc hydrometallurgy, CIESC Journal, 71 (2020), 1226-1233. [16] S. W. Xie, Y. F. Xie and F. B. Li, Optimal setting and control for iron removal process based on adaptive neural network soft-sensort, IEEE Transactions on Systems Man & Cybernetics Systems, 99 (2018), 1–13. [17] S. W. Xie, Y. F. Xie and H. Ying, A hybrid control strategy for real-time control of the iron removal process of the zinc hydrometallurgy plants, IEEE Transactions on Industrial Informatics, 14 (2018), 5278-5288.  doi: 10.1109/TII.2018.2815659. [18] S. W. Xie, Y. F. Xie, W. H. Gui and C. H. Yang, Weighted-coupling CSTR modeling and model predictive control with parameter adaptive correction for the goethite process, Journal of Process Control, 68 (2018), 254-267.  doi: 10.1016/j.jprocont.2018.05.006. [19] F. B. Zhou, C. G. Li, H. Q. Zhu and Y. G. Li, Determination of trace ions of cobalt and copper by UV-vis spectrometry in purification process of zinc hydrometallurgy, Optik, 184 (2019), 227-233.  doi: 10.1016/j.ijleo.2019.03.056. [20] H. Q. Zhu, C. H. Yang and W. H. Gui, Soft sensing method of ionic concentration in cobalt removal purification process based on WA and SVM, Computer Engineering and Applications, 47 (2011), 212-214. [21] Z. G. Zhang, Study on Industrial Application Hydrometallurgy of Zinc Purification System to High Impurity Raw Materials, Master thesis, Lanzhou University of Technology in Lanzhou, 2020.

show all references

##### References:
 [1] N. Chen, J. Q. Zhou, W. H. Gui, C. H. Yang and J. Y. Dai, Two-layer optimal control for goethite iron precipitation process, Control Theory & Applications, 37 (2020), 222–228. [2] W. H. Gui and C. H. Yang, Intelligent Modeling, Control and Optimization of Complex Non-Ferrous Metallurgy Production Process, Science Press, Beijing, 2010. [3] W. H. Gui, C. H. Yang, X. F. Chen and Y. L. Wang, Modeling and optimization problems and challenges arising in nonferrous metallurgical processes, Acta Automatica Sinica, 39 (2013), 197-207. [4] J. Han, C. H. Yang, X. J. Zhou and W. H. Gui, Dynamic multi-objective optimization arising in iron precipitation of zinc hydrometallurgy, Hydrometallurgy, 173 (2017), 134-148.  doi: 10.1016/j.hydromet.2017.08.007. [5] X. F. Liu, B. Z. Zhou, B. Y. Xiao and G. P. Cai, Inertia parameter identification of an unknown captured space target, Aircraft Engineering and Aerospace Technology, 91 (2019), 1147-1155. [6] B. Sun, W. H. Gui, T. B. Wu, Y. L. Wang and C. H. Yang, An integrated prediction model of cobalt ion concentration based on oxidation-reduction potential, Hydrometallurgy, 140 (2013), 102-110.  doi: 10.1016/j.hydromet.2013.09.015. [7] B. Sun, W. H. Gui, Y. L. Wang, C. H. Yang and M. F. He, A gradient optimization scheme for solution purification process, Control Engineering Practice, 44 (2015), 89-103.  doi: 10.1016/j.conengprac.2015.07.008. [8] B. Sun, W. H. Gui, C. H. Yang, Y. L. Wang and M. F. He, Online estimation of impurity ion concentration in solution purification process, IFAC-PapersOnLine, 49 (2016), 178-183. [9] B. Sun, B. Zhang, C. H. Yang and W. H. Gui, Discussion on modeling and optimal control of nonferrous metallurgical purification process, Acta Automatica Sinica, 43 (2017), 880-892. [10] M. S. Sang, Y. Ding, M. L. Bao, Y. T. Fang and B. B. Lu, Propagation dynamics model considering the characteristics of 2019-nCoV and prevention-control measurements, System Engineering-Theory & Practice, 41 (2021), 124–133. doi: 10.12011/SETP2020-0911. [11] B. Sun, M. F. He, Y. L. Wang, W. H. Gui, C. H. Yang and Q. M. Zhu, A data-driven optimal control approach for solution purification process, Journal of Process Control, 68 (2018), 171-185.  doi: 10.1016/j.jprocont.2018.06.005. [12] L. Y. Wang, Modelling and Optimization Method Based on Control Parameterization in Purification Process of Zinc Hydrometallurgy, Ph.D thesis, Central South University in Changsha, 2009. [13] L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, J. Ind. Manag. Optim., 5 (2009), 705-718.  doi: 10.3934/jimo.2009.5.705. [14] T. B. Wu, C. H. Yang, Y. G. Li, H. Q. Zhu and W. H. Gui, Fuzzy operational-pattern based operating parameters collaborative optimization of cobalt removal process with arsenic salt, Acta Automatica Sinica, 40 (2014), 1690-1698. [15] X. Y. Wang, X. J. Zhou and C. H. Yang, Chance constrained optimization for copper removal process under uncertainty in zinc hydrometallurgy, CIESC Journal, 71 (2020), 1226-1233. [16] S. W. Xie, Y. F. Xie and F. B. Li, Optimal setting and control for iron removal process based on adaptive neural network soft-sensort, IEEE Transactions on Systems Man & Cybernetics Systems, 99 (2018), 1–13. [17] S. W. Xie, Y. F. Xie and H. Ying, A hybrid control strategy for real-time control of the iron removal process of the zinc hydrometallurgy plants, IEEE Transactions on Industrial Informatics, 14 (2018), 5278-5288.  doi: 10.1109/TII.2018.2815659. [18] S. W. Xie, Y. F. Xie, W. H. Gui and C. H. Yang, Weighted-coupling CSTR modeling and model predictive control with parameter adaptive correction for the goethite process, Journal of Process Control, 68 (2018), 254-267.  doi: 10.1016/j.jprocont.2018.05.006. [19] F. B. Zhou, C. G. Li, H. Q. Zhu and Y. G. Li, Determination of trace ions of cobalt and copper by UV-vis spectrometry in purification process of zinc hydrometallurgy, Optik, 184 (2019), 227-233.  doi: 10.1016/j.ijleo.2019.03.056. [20] H. Q. Zhu, C. H. Yang and W. H. Gui, Soft sensing method of ionic concentration in cobalt removal purification process based on WA and SVM, Computer Engineering and Applications, 47 (2011), 212-214. [21] Z. G. Zhang, Study on Industrial Application Hydrometallurgy of Zinc Purification System to High Impurity Raw Materials, Master thesis, Lanzhou University of Technology in Lanzhou, 2020.
Flow chart of roasting, leaching and purification process of a lead-zinc smeltery
Temperature profiles of stage II solution purification in a lead-zinc smeltery
The equivalent CSTR model
Flow chart of solution algorithm for parameter identification model
Stage II purification reaction tank of zinc hydrometallurgy in a lead-zinc smeltery
Deviation function profile and parameter increment profile in solving process.(a)Deviation function profile; (b)Parameter increment profile
Nonlinear dynamic system model of the stage II purification process
Response profiles of outlet ion concentration.(a)Response profile of cobalt ion concentration; (b)Response profile of cadmium ion concentration
Variation profiles of outlet ion concentration when ${u_{\rm{b}}}$ is constant and ${u_{\rm{a}}}$ is variable.(a)Cobalt ion concentration at the outlet; (b)Cadmium ion concentration at the outlet
Variation profiles of outlet ion concentration when ${u_{\rm{a}}}$ is constant and ${u_{\rm{b}}}$ is variable.(a)Cobalt ion concentration at the outlet; (b)Cadmium ion concentration at the outlet
Variation profiles of outlet ion concentration when ${u_{\rm{a}}}$ and ${u_{\rm{b}}}$ change
The influence of inlet flow $Q$ on impurity ions concentration.(a) Cobalt ion concentration at the outlet; (b)Cadmium ion concentration at the outlet
Model test results.(a)Comparison of cobalt ion concentration at the outlet; (b) Comparison of cadmium ion concentration at the outlet
Values of relevant parameters in the reaction process
 Parameter Symbol Value Solution flow rate/(${{\rm{m}}^3}/{\rm{h}}$) $Q$ 160 Volume of single reaction tank/${{\rm{m}}^3}$ ${V_p}$ 108 Volume utilization of reaction tank/$\%$ $\backslash$ 80 Area coefficient/(${{\rm{m}}^2}/{\rm{kg}}$) $p$ 174
 Parameter Symbol Value Solution flow rate/(${{\rm{m}}^3}/{\rm{h}}$) $Q$ 160 Volume of single reaction tank/${{\rm{m}}^3}$ ${V_p}$ 108 Volume utilization of reaction tank/$\%$ $\backslash$ 80 Area coefficient/(${{\rm{m}}^2}/{\rm{kg}}$) $p$ 174
Characteristics of sample data
 Data type Symbol Average value Maximum value Minimum value Inlet cobalt ion concentration/(${\rm{mg/L}}$) ${x_{{\rm{a0}}}}$ 35.251 51.190 13.382 Inlet cadmium ion concentration/(${\rm{mg/L}}$) ${x_{{\rm{b0}}}}$ 298.278 433.148 113.229 Outlet cobalt ion concentration/(${\rm{mg/L}}$) ${\bar x_{\rm{a}}}$ 0.439 0.945 0.257 Outlet cadmium ion concentration/(${\rm{mg/L}}$) ${\bar x_b}$ 15.772 58.943 1.449
 Data type Symbol Average value Maximum value Minimum value Inlet cobalt ion concentration/(${\rm{mg/L}}$) ${x_{{\rm{a0}}}}$ 35.251 51.190 13.382 Inlet cadmium ion concentration/(${\rm{mg/L}}$) ${x_{{\rm{b0}}}}$ 298.278 433.148 113.229 Outlet cobalt ion concentration/(${\rm{mg/L}}$) ${\bar x_{\rm{a}}}$ 0.439 0.945 0.257 Outlet cadmium ion concentration/(${\rm{mg/L}}$) ${\bar x_b}$ 15.772 58.943 1.449
Inlet ion concentration values
 Inlet ion Symbol Value Cobalt ion/(${\rm{g/L}}$) ${x_{{\rm{a0}}}}$ 0.035 Cadmium ion/(${\rm{g/L}}$) ${x_{{\rm{b0}}}}$ 0.298
 Inlet ion Symbol Value Cobalt ion/(${\rm{g/L}}$) ${x_{{\rm{a0}}}}$ 0.035 Cadmium ion/(${\rm{g/L}}$) ${x_{{\rm{b0}}}}$ 0.298
Model error results
 Error Model in this paper Original model [8] Reformulated model [8] Maximum error (Cobalt ion) /${\rm{\% }}$ 31.12 49.10 32.27 Maximum error (Cadmium ion)/${\rm{\% }}$ 24.40 $\backslash$ $\backslash$ Average error (Cobalt ion)/${\rm{\% }}$ 11.05 13.81 11.90 Average error (Cadmium ion)/${\rm{\% }}$ 10.85 $\backslash$ $\backslash$
 Error Model in this paper Original model [8] Reformulated model [8] Maximum error (Cobalt ion) /${\rm{\% }}$ 31.12 49.10 32.27 Maximum error (Cadmium ion)/${\rm{\% }}$ 24.40 $\backslash$ $\backslash$ Average error (Cobalt ion)/${\rm{\% }}$ 11.05 13.81 11.90 Average error (Cadmium ion)/${\rm{\% }}$ 10.85 $\backslash$ $\backslash$
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