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Genetic algorithms with filters for optimal control problems in fed-batch bioreactors

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Abstract

When using a genetic algorithm (GA) to solve optimal control problems that can arise in a fed-batch bioreactor, the most obvious direct approach is to rely on a finite dimensional discretization of the optimal control problem into a nonlinear programming problem. Usually only the control function is discretized, and the continuous control function is approximated by a series of piecewise constant functions. Even though the piecewise discretized controls that the GA produces for the optimal control problem may give good performances, the control policies often show very high activity and differ considerably from those obtained using a continuous optimization strategy. The present study introduces a few filters into a real-coded genetic algorithm as additional operators and investigates the smoothing capabilities of the filters employed. It is observed that inclusion of a filter significantly smoothens the optimal control profile and often encourages the convergence of the algorithm. The applicability of the technique is illustrated by solving two previously reported optimal control problems in fed-batch bioreactors that are known to have singular arcs.

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Abbreviations

A :

Expression defined by Eq. 20

f :

Nonlinear function of x and u, dimension (n × 1)

F :

Filtering radius

G :

Generation number

G max :

Maximum number of generations allowed

k :

Index used for interval number in Eqs. 10, 11 and 12

m :

Number of control variables

n :

Number of state variables

N :

Number of intervals into which the time horizon is divided

P :

Population size

PI:

Performance index

p m :

Mutation probability

Q :

Relative cost factor in Eq. 36

t :

Time (h)

u :

Scalar control variable

u :

Control variable vector, dimension (m×1)

u c :

Candidate for the control (Eqs. 4, 5, 6, 7, 8, 9, 10, 11 and 12)

u b :

Best control value (Eqs. 4, 5, 6, 7, 8, 9, 10, 11 and 12))

x :

State variable vector, dimension (n×1)

δ :

Tunable parameter in first-order filter (δ=0.8)

ϕ :

Smoothing parameter in shift and smoothing filter (ϕ=0.25)

Φ:

Performance index

FO:

First-order filter

GA:

Genetic algorithm

IDP:

Iterative dynamic programming

MF:

Median filter

RCGA:

Real-coded genetic algorithm

f:

Final

max:

Maximum

min:

Minimum

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Authors and Affiliations

  1. Department of Chemical Engineering, Indian Institute of Science, Bangalore, 560012, India

    Debasis Sarkar & Jayant M. Modak

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  1. Debasis Sarkar
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  2. Jayant M. Modak
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Correspondence to Jayant M. Modak.

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Sarkar, D., Modak, J.M. Genetic algorithms with filters for optimal control problems in fed-batch bioreactors. Bioprocess Biosyst Eng 26, 295–306 (2004). https://doi.org/10.1007/s00449-004-0366-0

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  • DOI: https://doi.org/10.1007/s00449-004-0366-0

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