Mathematical model with time‐delay and delayed controller for a bioreactor
In this paper, a fractional Lotka–Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time‐delay τ$$ \tau $$ due to the dead‐time (non‐trivial) that the microbe need...
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| Vydáno v: | Mathematical methods in the applied sciences Ročník 46; číslo 1; s. 248 - 266 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Freiburg
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15.01.2023
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| ISSN: | 0170-4214, 1099-1476 |
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| Abstract | In this paper, a fractional Lotka–Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time‐delay
τ$$ \tau $$ due to the dead‐time (non‐trivial) that the microbe needed to metabolize the substrate. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay
τ$$ \tau $$ as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators, or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller
u(t)=kpx(t)$$ u(t)={k}_px(t) $$, which is shown to fail to stabilize the obtained model under the proposed analysis. Another suitable choice is the use of a delayed controller
u(t)=krx(t−h)$$ u(t)={k}_rx\left(t-h\right) $$ which successfully stabilizes the model even when it is unstable. The delay
h$$ h $$ in the feedback control is due to the dead‐time necessary to obtain the measurement of the biomass in the bioreactor by dry weight. Finally, the proposed theoretical results are corroborated through numerical simulations. |
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| AbstractList | In this paper, a fractional Lotka–Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time‐delay τ$$ \tau $$ due to the dead‐time (non‐trivial) that the microbe needed to metabolize the substrate. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay τ$$ \tau $$ as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators, or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller u(t)=kpx(t)$$ u(t)={k}_px(t) $$, which is shown to fail to stabilize the obtained model under the proposed analysis. Another suitable choice is the use of a delayed controller u(t)=krx(t−h)$$ u(t)={k}_rx\left(t-h\right) $$ which successfully stabilizes the model even when it is unstable. The delay h$$ h $$ in the feedback control is due to the dead‐time necessary to obtain the measurement of the biomass in the bioreactor by dry weight. Finally, the proposed theoretical results are corroborated through numerical simulations. In this paper, a fractional Lotka–Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time‐delay τ$$ \tau $$ due to the dead‐time (non‐trivial) that the microbe needed to metabolize the substrate. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay τ$$ \tau $$ as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators, or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller u(t)=kpx(t)$$ u(t)={k}_px(t) $$, which is shown to fail to stabilize the obtained model under the proposed analysis. Another suitable choice is the use of a delayed controller u(t)=krx(t−h)$$ u(t)={k}_rx\left(t-h\right) $$ which successfully stabilizes the model even when it is unstable. The delay h$$ h $$ in the feedback control is due to the dead‐time necessary to obtain the measurement of the biomass in the bioreactor by dry weight. Finally, the proposed theoretical results are corroborated through numerical simulations. In this paper, a fractional Lotka–Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis . The model contemplates a time‐delay due to the dead‐time (non‐trivial) that the microbe needed to metabolize the substrate. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators, or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller , which is shown to fail to stabilize the obtained model under the proposed analysis. Another suitable choice is the use of a delayed controller which successfully stabilizes the model even when it is unstable. The delay in the feedback control is due to the dead‐time necessary to obtain the measurement of the biomass in the bioreactor by dry weight. Finally, the proposed theoretical results are corroborated through numerical simulations. |
| Author | Itzá‐Ortiz, Benjamín A. Villafuerte‐Segura, Raúl Alvarado‐Santos, Eduardo López‐Pérez, Pablo A. |
| Author_xml | – sequence: 1 givenname: Raúl orcidid: 0000-0003-3904-5401 surname: Villafuerte‐Segura fullname: Villafuerte‐Segura, Raúl organization: Universidad Autónoma del Estado de Hidalgo – sequence: 2 givenname: Benjamín A. orcidid: 0000-0002-6189-3266 surname: Itzá‐Ortiz fullname: Itzá‐Ortiz, Benjamín A. email: itza@uaeh.edu.mx organization: Universidad Autónoma del Estado de Hidalgo – sequence: 3 givenname: Pablo A. surname: López‐Pérez fullname: López‐Pérez, Pablo A. organization: Universidad Autónoma del Estado de Hidalgo – sequence: 4 givenname: Eduardo orcidid: 0000-0002-1058-3794 surname: Alvarado‐Santos fullname: Alvarado‐Santos, Eduardo organization: Universidad Autónoma del Estado de Hidalgo |
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| SubjectTerms | bifurcation theory Biomass Bioreactors Controllers Delay delay control/observation systems Feedback control Hopf bifurcation Lotka–Volterra model mathematical modeling or simulation for problems pertaining to biology Mathematical models Norton–Simons–Massagué model Substrates Zymomonas mobilis |
| Title | Mathematical model with time‐delay and delayed controller for a bioreactor |
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