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Optimal control of continuous fermentation processes

Decomposition methods for start-up, steady-state and minimum time problems

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Abstract

Three layer control structure is proposed for optimal control of continuous fermentation processes. The start-up optimization problems are solved as a first step for optimization layer building. A steady state optimization problem is solved by a decomposition method using prediction principle. A discrete minimum time optimal control problem with state delay is formulated and a decomposition method, based on an augmented Lagrange's function is proposed to solve it. The problem is decomposed in time domain by a new coordinating vector. The obtained algorithms are used for minimum time optimal control calculation of Baker's Yeast fermentation process.

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Abbreviations

x(t) g/l:

biomass concentration

s(t) g/l:

limiting substrate concentration

x 0 g/l:

inlet biomass concentration

s 0(t) g/l:

inlet substrate concentration

D(t) h−1 :

dilution rate

μ(t) h−1 :

specific growth rate

Y g/g:

yield coefficient

ɛ(t) h−1 :

specific limiting substrate consumption rate

k D h−1 :

disappearing constant

w 1, w 2 :

known constant or piece-wise disturbances

μ m h−1 :

maximum specific growth rate

k s g/l:

Michaelis-Menten's parameter

τ h:

time delay

x 0, s 0 g/l:

initial concentrations

¯x, ¯s, ¯D :

optimal steady state value

V min , V max , v=x,s,d,Δt :

bounds of variables

Δt h:

sampling period

K :

number of steps in the optimization horison

Js, J d :

performance indexes

L s :

Lagrange's function

L d :

Lagrange's functional

λ 0 :

weighting coefficient for the amount of the limiting substrate throwing out of the fermentor

λ 1, λ2 :

dual variables of Lagrange's function

\(\alpha _s^1 ,{\text{ }}\alpha _{\lambda _1 }^1 ,{\text{ }}\alpha _{\lambda _2 }^1 \) :

steps in steady state coordination procedure

\(\varepsilon _s ,{\text{ }}\varepsilon _{\lambda _1 } ,{\text{ }}\varepsilon _{\lambda _2 } \) :

errors values for steady state coordination process

λ v , v=x, s :

conjugate variables of Lagrange's functional

μ v , v=x,s :

penalty coefficients of augmented Lagrange's functional

ϱ v , v=x, s :

interconnections of the time

e v , v=x,s, D, λ x , λ s :

gradients of Lagrange's functional

j, l :

indexes of calculation procedures

ɛ :

values of errors in calculations

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Authors and Affiliations

  1. Institute of Informatics-BAS, Acad. G. Bonchev Str. bl. 2, 1113, Sofia, Bulgaria

    R. G. Tsoneva

  2. Central Laboratory of Bioinstrumentation and Automation-BAS, Acad. G. Bonchev Str. bl. 105, 1113, Sofia, Bulgaria

    T. D. Patarinska

Authors
  1. R. G. Tsoneva
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  2. T. D. Patarinska
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Additional information

The researches was supported by National Scientific Research Foundation under grants No NITN428/94 and No NITN440/94

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Tsoneva, R.G., Patarinska, T.D. Optimal control of continuous fermentation processes. Bioprocess Engineering 13, 189–196 (1995). https://doi.org/10.1007/BF00367253

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  • DOI: https://doi.org/10.1007/BF00367253

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