Estimation of Sporulated Cell Concentration of Bacillus thuringiensis in a Batch Biochemical Reactor via Simple State Observers
Abstract
:1. Introduction
2. Benchmark Model
Plant Model for Batch Process Model Using Bacillus thuringiensis Production
3. Linear Model of the System in Equations (1a–d)–(2a–c)
Observability Properties of the System in Equations (1a–d)–(2a–c)
4. Nonlinear Observability Analysis
5. Observer Design
Proposed Observer Algorithm for B. thuringiensis Bioreactor
- (i)
- The first stage involves the proper definition of a model bioreactor, which involves process modeling and, hence, the general state–space dynamical model or the plant model.
- (ii)
- The observer must then be formulated in mathematical terms (Equation (14)).
6. Sketch of Proof of Proposition 2 (Equation (15))
7. Numerical Results and Simulation Research
7.1. Dynamics of Bacillus thuringiensis Production Culture in a Batch Bioreactor
7.2. Observability Properties
7.3. Observer Design: Application to Bioreactor System for B. thuringiensis Production
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Nonlinear Observability Analysis
References
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Observer Design (Type) | Microorganism | Bioprocesses | Reference |
---|---|---|---|
Kalman filter | Kluyveromyces marxianus yeast (β-galactosidase enzyme) | Biochemical reactor | [23] |
Extended Kalman filter (EKF) | Microalgae | Biotechnological processes (photobioreactors) | [24] |
Unscented Kalman filter (UKF) (state and parameter estimation) | Microalgae (biomass estimation) | Biotechnological processes (photobioreactors) | [25] |
Cubature Kalman filter (CKF) | Estimation for penicillin | Biotechnological processes | [26] |
Linear observer | Biological reactor | [27] | |
Nonlinear observer | Yeats | Fed-batch reactor for ethanol production | [28,29] |
Sliding-mode observer | δ-endotoxin production of Bacillus thuringiensis | Batch bioprocess | [30,31] |
Neural observer | Anaerobic digestion process | Anaerobic process for paper mills’ effluent treatment | [32] |
Adaptive observer | Estimation of the biomass concentration (Escherichia coli) | Sigma-Point Kalman filter | [33] |
Modeling Biotechnology: modeling and simulation | In bioprocess | Different bioprocesses | [34,35] |
Control of a bioreactor | Yeast fermentation | [11] |
Symbol | Description | Value | Units |
---|---|---|---|
Substrate concentration | |||
Dissolved oxygen | |||
Time | |||
Sporulated cells’ concentration | |||
Vegetative cells’ concentration | |||
Total cell concentration | |||
Specific growth rate | |||
Maximum specific growth rate | |||
Maintenance constant | |||
Kinetic constant representing the spore formation | |||
Cell death specific rate | |||
Saturation constant | |||
Growth yield | |||
Inlet volumetric flow rate of air | 1800 | ||
Volumetric oxygen transfer coefficient | |||
Maximum kinetic constant representing the spore formation | 0.5 | ||
Exponential parameter for spore formation | 1 | ||
Parameter for spore formation | 1 | ||
Maximum death cell specific rate | 0.1 | ||
Exponential parameter for death rate | 5 | ||
Parameter for death rate | 4.9 | ||
saturation concentration (OD concentration in equilibrium with the oxygen partial pressure of the gaseous phase) | 0.00759 | ||
Oxygen consumption constant by growth | |||
Oxygen consumption constant for maintenance | |||
Ventilation constant |
Assumption/Definition | Remark | Reference |
---|---|---|
A2. | It is a realistic assumption based on a mass conservation principle. | [58] |
A3. | Select the parameters g1 and g2 to obey the restriction. | [56,58] |
D. | Dynamic error. |
Definition No. | ||
---|---|---|
1 | ||
2 |
[g/L] | [g/L h] Slope of the Line/Coefficient of Determination ) | [g/L h] Slope of the Line/Coefficient of Determination ) | |
---|---|---|---|
Process lines (slope of the line in Figure 4) | Exponential growth (first stage, see Figure 4) 2,3 | Sporulation (second stage, see Figure 4) 2,3 | Reference |
Batch | In this work (see Figure 4) | ||
Batch | In this work (see Figure 4) | ||
Batch | In this work (see Figure 4) | ||
Batch | In this work (see Figure 4) | ||
Run 10 12 | [63] | ||
Run 11 12 | [63] | ||
Run 12 12 | [63] |
Measurement (Equation (5)) | (Equation (8)) | ||
---|---|---|---|
1 | 4 | ||
2 | 4 | ||
3 | 4 | ||
4 | 4 |
Test Code | Figure No. | System (Estimator) |
Parameter Disturbance Nominal Value 0.53 1/h |
Initial Conditions: | |||
---|---|---|---|---|---|---|---|
T1 | 6a,b | Model | |||||
ELO | |||||||
PO | |||||||
T2 | 7a,b | Model | |||||
ELO | |||||||
PO | |||||||
T3 | 8a,b | Model | |||||
ELO | |||||||
PO | |||||||
T4 | 9a,b | Model | |||||
ELO | |||||||
PO | |||||||
T5 | 10a–c | Model | |||||
ELO | |||||||
PO |
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Zárate-Castrejón, J.L.; López-Pérez, P.A.; López-López, M.; Núñez-Colín, C.A.; Veloz-García, R.A.; Mukhtar, H.; Peña-Caballero, V. Estimation of Sporulated Cell Concentration of Bacillus thuringiensis in a Batch Biochemical Reactor via Simple State Observers. Mathematics 2024, 12, 3996. https://doi.org/10.3390/math12243996
Zárate-Castrejón JL, López-Pérez PA, López-López M, Núñez-Colín CA, Veloz-García RA, Mukhtar H, Peña-Caballero V. Estimation of Sporulated Cell Concentration of Bacillus thuringiensis in a Batch Biochemical Reactor via Simple State Observers. Mathematics. 2024; 12(24):3996. https://doi.org/10.3390/math12243996
Chicago/Turabian StyleZárate-Castrejón, José Luis, Pablo A. López-Pérez, Milagros López-López, Carlos A. Núñez-Colín, Rafael A. Veloz-García, Hamid Mukhtar, and Vicente Peña-Caballero. 2024. "Estimation of Sporulated Cell Concentration of Bacillus thuringiensis in a Batch Biochemical Reactor via Simple State Observers" Mathematics 12, no. 24: 3996. https://doi.org/10.3390/math12243996
APA StyleZárate-Castrejón, J. L., López-Pérez, P. A., López-López, M., Núñez-Colín, C. A., Veloz-García, R. A., Mukhtar, H., & Peña-Caballero, V. (2024). Estimation of Sporulated Cell Concentration of Bacillus thuringiensis in a Batch Biochemical Reactor via Simple State Observers. Mathematics, 12(24), 3996. https://doi.org/10.3390/math12243996