This paper considers an optimal feedback control problem for a class of fed-batch fermentation processes. Our main contributions are as follows. Firstly, a dynamic optimization problem for fed-batch fermentation processes is modeled as an optimal control problem of switched dynamical systems, and a general state-feedback controller is designed for this dynamic optimization problem. Unlike the existing switched dynamical system optimal control problem, the state-dependent switching method is applied to design the switching rule, and the structure of this state-feedback controller is not restricted to a particular form. Then, this problem is transformed into a mixed-integer optimal control problem by introducing a discrete-valued function. Furthermore, each of these discrete variables is represented by using a set of 0-1 variables. By using a quadratic constraint, these 0-1 variables are relaxed such that they are continuous on the closed interval $ [0, 1] $. Accordingly, the original mixed-integer optimal control problem is transformed intoa nonlinear parameter optimization problem. Unlike the existing works, the constraint introduced for these 0-1 variables are at most quadratic. Thus, it does not increase the number of locally optimal solutions of the original problem. Next, an improved gradient-based algorithm is developed based on a novel search approach, and a large number of numerical experiments show that this novel search approach can effectively improve the convergence speed of this algorithm, when an iteration is trapped to a curved narrow valley bottom of the objective function. Finally, numerical results illustrate the effectiveness of this method developed by this paper.
Citation: Xiang Wu, Yuzhou Hou, Kanjian Zhang. Optimal feedback control for a class of fed-batch fermentation processes using switched dynamical system approach[J]. AIMS Mathematics, 2022, 7(5): 9206-9231. doi: 10.3934/math.2022510
This paper considers an optimal feedback control problem for a class of fed-batch fermentation processes. Our main contributions are as follows. Firstly, a dynamic optimization problem for fed-batch fermentation processes is modeled as an optimal control problem of switched dynamical systems, and a general state-feedback controller is designed for this dynamic optimization problem. Unlike the existing switched dynamical system optimal control problem, the state-dependent switching method is applied to design the switching rule, and the structure of this state-feedback controller is not restricted to a particular form. Then, this problem is transformed into a mixed-integer optimal control problem by introducing a discrete-valued function. Furthermore, each of these discrete variables is represented by using a set of 0-1 variables. By using a quadratic constraint, these 0-1 variables are relaxed such that they are continuous on the closed interval $ [0, 1] $. Accordingly, the original mixed-integer optimal control problem is transformed intoa nonlinear parameter optimization problem. Unlike the existing works, the constraint introduced for these 0-1 variables are at most quadratic. Thus, it does not increase the number of locally optimal solutions of the original problem. Next, an improved gradient-based algorithm is developed based on a novel search approach, and a large number of numerical experiments show that this novel search approach can effectively improve the convergence speed of this algorithm, when an iteration is trapped to a curved narrow valley bottom of the objective function. Finally, numerical results illustrate the effectiveness of this method developed by this paper.
| [1] |
X. Lin, Z. Su, Y. Yang, S. Zhang, The potential of ionic liquids in biopharmaceutical engineering, Chinese J. Chem. Eng., 30 (2021), 236–243. https://doi.org/10.1016/j.cjche.2020.11.015 doi: 10.1016/j.cjche.2020.11.015
|
| [2] |
J. Yang, W. Li, Q. Liu, H. Liu, Dissolution of antibiotics mycelium in ionic liquids: Performance and mechanism, Chinese J. Chem. Eng., 26 (2017), 252–258. https://doi.org/10.1016/j.cjche.2017.04.003 doi: 10.1016/j.cjche.2017.04.003
|
| [3] |
S. Basharat, Z. Huang, M. Gong, X. Lv, A. Ahmed, I. Hussain, et al., A review on current conventional and biotechnical approaches to enhance biosynthesis of steviol glycosides in Stevia rebaudiana, Chinese J. Chem. Eng., 30 (2021), 92–104. https://doi.org/10.1016/j.cjche.2020.10.018 doi: 10.1016/j.cjche.2020.10.018
|
| [4] |
Y. Yu, G. Li, W. Han, L. Zhu, T. Si, H. Wang, et al., An efficient preparation of porous polymeric microspheres by solvent evaporation in foam phase, Chinese J. Chem. Eng., 29 (2021), 409–416. https://doi.org/10.1016/j.cjche.2020.09.002 doi: 10.1016/j.cjche.2020.09.002
|
| [5] |
M. N. Pantano, M. C. Fernandez, M. E. Serrano, O. A. Ortiz, G. J. E. Scaglia, Tracking control of optimal profiles in a nonlinear fed-batch bioprocess under parametric uncertainty and process disturbances, Ind. Eng. Chem. Res., 57 (2018), 11130–11140. https://doi.org/10.1021/acs.iecr.8b01791 doi: 10.1021/acs.iecr.8b01791
|
| [6] |
D. Niu, L. Zhang, F. Wang, Modeling and parameter updating for nosiheptide fed-batch fermentation process, Ind. Eng. Chem. Res., 55 (2016), 8395–8402. https://doi.org/10.1021/acs.iecr.6b01245 doi: 10.1021/acs.iecr.6b01245
|
| [7] |
F. A. Gorrini, S. I. Biagiola, J. L. Figueroa, A. V. Wouwer, Simple observer-based feedback strategy for controlling fed-batch hybridoma cultures, Ind. Eng. Chem. Res., 56 (2017), 15072–15082. https://doi.org/10.1021/acs.iecr.7b03395 doi: 10.1021/acs.iecr.7b03395
|
| [8] |
A. Simorgh, A. Razminia, J. A. T. Machado, Optimal control of nonlinear fed-batch process using direct transcription method, Comput. Chem. Eng., 130 (2019), 106561. https://doi.org/10.1016/j.compchemeng.2019.106561 doi: 10.1016/j.compchemeng.2019.106561
|
| [9] |
L. Veiter, J. Kager, C. Herwig, Optimal process design space to ensure maximum viability and productivity in Penicillium chrysogenum pellets during fed-batch cultivations through morphological and physiological control, Microb. Cell Fact., 19 (2020), 33. https://doi.org/10.1186/s12934-020-1288-5 doi: 10.1186/s12934-020-1288-5
|
| [10] |
J. Schorsch, C. C. Castro, L. D. Couto, C. Nobre, M. Kinnaert, Optimal control for fermentative production of fructo-oligosaccharides in fed-batch bioreactor, J. Process Contr., 78 (2019), 124–138. https://doi.org/10.1016/j.jprocont.2019.03.004 doi: 10.1016/j.jprocont.2019.03.004
|
| [11] |
X. Wu, K. Zhang, M. Cheng, Optimal control of bioprocess systems using hybrid numerical optimization algorithms, Optimization, 67 (2018), 1287–1306. https://doi.org/10.1080/02331934.2018.1466299 doi: 10.1080/02331934.2018.1466299
|
| [12] |
F. Rossi, S. Copelli, A. Colombo, C. Pirola, F. Manenti, Online model-based optimization and control for the combined optimal operation and runaway prediction and prevention in (fed-)batch systems, Chem. Eng. Sci., 138 (2015), 760–771. https://doi.org/10.1016/j.ces.2015.09.006 doi: 10.1016/j.ces.2015.09.006
|
| [13] |
H. S. Shin, H. C. Lim, Optimal fed-batch operation for recombinant cells with segregational plasmid instability, Chem. Eng. Commun., 195 (2008), 1122–1143. https://doi.org/10.1080/00986440801943396 doi: 10.1080/00986440801943396
|
| [14] |
R. F. Rao, X. D. Li, Input-to-state stability in the meaning of switching for delayed feedback switched stochastic financial system, AIMS Mathematics, 6 (2021), 1040–1064. https://doi.org/10.3934/math.2021062 doi: 10.3934/math.2021062
|
| [15] |
X. Wu, K. Zhang, M. Cheng, Adaptive numerical approach for optimal control of a single train, J. Syst. Sci. Complex, 32 (2019), 1053–1071. https://doi.org/10.1007/s11424-018-7277-7 doi: 10.1007/s11424-018-7277-7
|
| [16] |
X. Wu, K. Zhang, M. Cheng, Optimal control of constrained switched systems and application to electrical vehicle energy management, Nonlinear Anal. Hybrid Syst., 30 (2018), 171–188. https://doi.org/10.1016/j.nahs.2018.05.006 doi: 10.1016/j.nahs.2018.05.006
|
| [17] |
X. Wu, K. Zhang, M. Cheng, A switched dynamical system approach towards the optimal control of chemical processes based on a gradient-based parallel optimization algorithm, Comput. Chem. Eng., 118 (2018), 180–194. https://doi.org/10.1016/j.compchemeng.2018.08.007 doi: 10.1016/j.compchemeng.2018.08.007
|
| [18] | D. Sahoo, G. Samanta, Impact of fear effect in a two prey-one predator system with switching behaviour in predation, Differ. Equ. Dyn. Syst., in press. https://doi.org/10.1007/s12591-021-00575-7 |
| [19] |
S. Saha, G. Samanta, Modelling of a two prey and one predator system with switching effect, Comput. Math. Biophys., 9 (2021), 90–113. https://doi.org/10.1515/cmb-2020-0120 doi: 10.1515/cmb-2020-0120
|
| [20] |
S. Saha, G. Samanta, Modeling of insect-pathogen dynamics with biological control, Math. Biol. Bioinform., 15 (2020), 268–294. https://doi.org/10.17537/2020.15.268 doi: 10.17537/2020.15.268
|
| [21] |
M. Huang, S. Liu, Y. Zhang, Mathematical modeling and analysis of biological control strategy of aphid population, AIMS Mathematics, 7 (2022), 6876–6897. https://doi.org/10.3934/math.2022382 doi: 10.3934/math.2022382
|
| [22] |
D. Liberzon, A. S. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag., 19 (1999), 59–70. https://doi.org/10.1109/37.793443 doi: 10.1109/37.793443
|
| [23] |
H. Lin, P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Auto. Control, 54 (2009), 308–322. https://doi.org/10.1109/.2005.1466986 doi: 10.1109/.2005.1466986
|
| [24] |
D. Yang, C. X. Huang, G. D. Zong, Finite-time H-infinity bumpless transfer control for switched systems: A state-dependent switching approach, Int. J. Robust Nonlinear Control, 30 (2020), 1417–1430. https://doi.org/10.1002/rnc.4825 doi: 10.1002/rnc.4825
|
| [25] |
X. Huo, L. Ma, X. D. Zhao, G. D. Zong, Event-triggered adaptive fuzzy output feedback control of MIMO switched nonlinear systems with average dwell time, Appl. Math. Comput., 365 (2020), 124665. https://doi.org/10.1016/j.amc.2019.124665 doi: 10.1016/j.amc.2019.124665
|
| [26] |
A. Kundu, A new condition for stability of switched linear systems under restricted minimum dwell time switching, Syst. Control Lett., 135 (2020), 104597. https://doi.org/10.1016/j.sysconle.2019.104597 doi: 10.1016/j.sysconle.2019.104597
|
| [27] |
A. A. Kahloul, A. Sakly, Hybrid approach for constrained optimal control of nonlinear switched systems, J. Control Auto. Electr. Syst., 31 (2020), 865–873. https://doi.org/10.1007/s40313-020-00586-9 doi: 10.1007/s40313-020-00586-9
|
| [28] |
X. Xu, X. Mao, Y. Li, H. Zhang, New result on robust stability of switched systems with all subsystems unstable, IET Control Theory Appl., 13 (2019), 2138–2145. https://doi.org/10.1049/iet-cta.2019.0018 doi: 10.1049/iet-cta.2019.0018
|
| [29] |
S. Sui, S. Tong, C. L. Chen, K. Sun, Fuzzy adaptive optimal control for nonlinear switched systems with actuator hysteresis, Int. J. Adapt. Control Signal Process., 33 (2019), 609–625. https://doi.org/10.1002/acs.2975 doi: 10.1002/acs.2975
|
| [30] |
P. Wang, J. Zhao, Stability and guaranteed cost analysis of switched positive systems with mode-dependent dwell time and sampling, IET Control Theory Appl., 14 (2020), 378–385. https://doi.org/10.1049/iet-cta.2019.0466 doi: 10.1049/iet-cta.2019.0466
|
| [31] |
X. Xu, P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Trans. Auto. Control, 49 (2004), 2–16. https://doi.org/10.1109/TAC.2003.821417 doi: 10.1109/TAC.2003.821417
|
| [32] |
X. Wu, K. Zhang, M. Cheng, Computational method for optimal control of switched systems with input and state constraints, Nonlinear Anal. Hybrid Syst., 26 (2017), 1–18. https://doi.org/10.1016/j.nahs.2017.04.001 doi: 10.1016/j.nahs.2017.04.001
|
| [33] |
H. Chen, W. Zhang, On weak topology for optimal control of switched nonlinear systems, Automatica, 81 (2017), 409–415. https://doi.org/10.1016/j.automatica.2017.03.039 doi: 10.1016/j.automatica.2017.03.039
|
| [34] |
X. Liu, S. Li, K. Zhang, Optimal control of switching time in switched stochastic systems with multi-switching times and different costs, Int. J. Control, 90 (2017), 1604–1611. https://doi.org/10.1080/00207179.2016.1214879 doi: 10.1080/00207179.2016.1214879
|
| [35] |
X. Wu, K. Zhang, C. Sun, Numerical algorithm for a class of constrained optimal control problems of switched systems, Numer. Algor., 67 (2014), 771–792. https://doi.org/10.1007/s11075-013-9822-8 doi: 10.1007/s11075-013-9822-8
|
| [36] |
H. G. Bock, C. Kirches, A. Meyer, A. Potschka, Numerical solution of optimal control problems with explicit and implicit switches, Optim. Meth. Softw., 33 (2018), 450–474. https://doi.org/10.1080/10556788.2018.1449843 doi: 10.1080/10556788.2018.1449843
|
| [37] |
X. Xu, P. J. Antsaklis, Optimal control of switched systems via non-linear optimization based on direct differentiations of value functions, Int. J. Control, 75 (2002), 1406–1426. https://doi.org/10.1080/0020717021000023825 doi: 10.1080/0020717021000023825
|
| [38] |
X. Wu, K. Zhang, C. Sun, Constrained optimal control of switched systems based on modified BFGS algorithm and filled function method, Int. J. Comput. Math., 91 (2014), 1713–1729. https://doi.org/10.1080/00207160.2013.859678 doi: 10.1080/00207160.2013.859678
|
| [39] |
X. Wu, K. Zhang, Three-dimensional trajectory design for horizontal well based on optimal switching algorithms, ISA Trans., 58 (2015), 348–356. https://doi.org/10.1016/j.isatra.2015.04.002 doi: 10.1016/j.isatra.2015.04.002
|
| [40] |
R. Vasudevan, H. Gonzalez, R. Bajcsy, S. S. Sastry, Consistent approximations for the optimal control of constrained switched systems-Part 1: A conceptual algorithm, SIAM J. Control Optim., 51 (2013), 4463–4483. https://doi.org/10.1137/120901490 doi: 10.1137/120901490
|
| [41] |
R. Vasudevan, H. Gonzalez, R. Bajcsy, S. S. Sastry, Consistent approximations for the optimal control of constrained switched systems-Part 2: An implementable algorithm, SIAM J. Control Optim., 51 (2013), 4484–4503. https://doi.org/10.1137/120901507 doi: 10.1137/120901507
|
| [42] |
X. Wu, K. Zhang, M. Cheng, X. Xin, A switched dynamical system approach towards the economic dispatch of renewable hybrid power systems, Int. J. Electr. Power Energy Syst., 103 (2018), 440–457. https://doi.org/10.1016/j.ijepes.2018.06.016 doi: 10.1016/j.ijepes.2018.06.016
|
| [43] |
T. Sardarmehni, A. Heydari, Sub-optimal scheduling in switched systems with continuous-time dynamics: A gradient descent approach, Neurocomputing, 285 (2018), 10–22. https://doi.org/10.1016/j.neucom.2018.01.003 doi: 10.1016/j.neucom.2018.01.003
|
| [44] |
H. R. Tabrizidooz, M. Pourbabaee, M. Hedayati, Optimal control of switched systems by a modified pseudo spectral method, Iran J. Math. Chem., 8 (2017), 161–173. https://doi.org/10.22052/ijmc.2017.44718 doi: 10.22052/ijmc.2017.44718
|
| [45] |
G. Wu, J. Sun, J. Chen, Optimal linear quadratic regulator of switched systems, IEEE Trans. Auto. Control, 64 (2019), 2898–2904. https://doi.org/10.1109/TAC.2018.2872204 doi: 10.1109/TAC.2018.2872204
|
| [46] |
A. Heydari, Optimal switching with minimum dwell time constraint, J. Franklin Inst., 354 (2017), 4498–4518. https://doi.org/10.1016/j.jfranklin.2017.04.015 doi: 10.1016/j.jfranklin.2017.04.015
|
| [47] |
T. M. Caldwell, T. D. Murphey, Switching mode generation and optimal estimation with application to skid-steering, Automatica, 47 (2011), 50–64. https://doi.org/10.1016/j.automatica.2010.10.010 doi: 10.1016/j.automatica.2010.10.010
|
| [48] |
X. Wu, K. Zhang, M. Cheng, Sensitivity analysis for an optimal control problem of chemical processes based on a smoothing cost penalty function approach, Chem. Eng. Res. Design., 146 (2019), 221–238. https://doi.org/10.1016/j.cherd.2019.04.011 doi: 10.1016/j.cherd.2019.04.011
|
| [49] |
X. Wu, Q. Liu, K. Zhang, X. Xin, Optimal-tuning of proportional-integral-derivative-like controller for constrained nonlinear systems and application to ship steering control, J. Franklin Inst., 355 (2018), 5667–5689. https://doi.org/10.1016/j.jfranklin.2018.06.017 doi: 10.1016/j.jfranklin.2018.06.017
|
| [50] |
K. Dehingia, H. K. Sarmah, K. Hosseini, K. Sadri, S. Salahshour, C. Park, An optimal control problem of immuno-chemotherapy in presence of gene therapy, AIMS Mathematics, 6 (2021), 11530–11549. https://doi.org/10.3934/math.2021669 doi: 10.3934/math.2021669
|
| [51] |
A. Das, K. Dehingia, H. K. Sharmah, C. Park, J. R. Lee, K. Sadri, et al., Optimal control of effector-tumor-normal cells dynamics in presence of adoptive immunotherapy, AIMS Mathematics, 6 (2021), 9813–9834. https://doi.org/10.3934/math.2021570 doi: 10.3934/math.2021570
|
| [52] |
A. Das, H. K. Sarmah, D. Bhattacharya, K. Dehingia, K. Hosseini, Combination of virotherapy and chemotherapy with optimal control for combating cancer, Math. Comput. Simulat., 194 (2022), 460–488. https://doi.org/10.1016/j.matcom.2021.12.004 doi: 10.1016/j.matcom.2021.12.004
|
| [53] |
B. J. Nath, H. K. Sarmah, H. Maurer, An optimal control strategy for antiretroviral treatment of HIV infection in presence of immunotherapy, Qual. Theory Dyn. Syst., 21 (2022), 30. https://doi.org/10.1007/s12346-022-00564-3 doi: 10.1007/s12346-022-00564-3
|
| [54] | N. U. Ahmed, Dynamic systems and control with applications, Singapore: World Scientific, 2006. https://doi.org/10.1142/6262 |
| [55] |
H. M. Saber, A. Ravindran, Nonlinear goal programming theory and practice: A survey, Comput. Oper. Res., 20 (1993), 275–291. https://doi.org/10.1016/0305-0548(93)90004-3 doi: 10.1016/0305-0548(93)90004-3
|
| [56] | K. L. Teo, C. Goh, K. Wong, A unified computational approach to optimal control problems, Essex: Longman Scientific $ & $ Technical, 1991. https://doi.org/10.1515/9783110883237.2763 |
| [57] | S. S. Rao, Engineering optimization: theory and practice, Chichester: John Wiley & Sons, 2019. https://doi.org/10.1002/9781119454816 |
| [58] |
Y. Mu, H. Teng, D. J. Zhang, W. Wang, Z. L. Xiu, Microbial production of 1, 3-propanediol by Klebsiella pneumoniae using crude glycerol from biodiesel preparations, Biotechnol Lett., 28 (2006), 1755–1759. https://doi.org/10.1007/s10529-006-9154-z doi: 10.1007/s10529-006-9154-z
|
| [59] |
A. Nikoobin, M. Moradi, Indirect solution of optimal control problems with state variable inequality constraints: finite difference approximation, Robotica, 35 (2017), 50–72. https://doi.org/10.1017/S0263574715000521 doi: 10.1017/S0263574715000521
|
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Xiang Wu, Yuzhou Hou, Kanjian Zhang. Optimal feedback control for a class of fed-batch fermentation processes using switched dynamical system approach[J]. AIMS Mathematics, 2022, 7(5): 9206-9231. doi: 10.3934/math.2022510
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