Abstract
The phrase input multiplicities means that an input variable with more than one value produces the same output value as if there were a single input–single output process. With input multiplicities, the value of the process gain changes as the manipulated variable changes, and beyond a certain input value, the sign of the gain also changes. A conventional PI controller for processes with input multiplicities may give unstable, less economical, or oscillatory responses. In the present work, control problems of a continuous bioreactor exhibiting two input multiplicities in the dilution rate on productivity were experimentally analyzed. A regulatory problem for the evaluation of controllers was taken up, i.e. a step change was made in the feed substrate concentration from 20 to 25 g/l at steady state conduction at lower (0.09386 h−1) and higher (0.2278 h−1) dilution rates for the same productivity of 2.9 g/l h. The nonlinear PI controller gave a more stable and fast response at both input dilution rates. The linear PI controller designed for a lower input dilution rate was stable, with some oscillations at the lower dilution rate, but the response was unstable at a higher dilution rate due to the input multiplicity behaviour of the process. Thus, nonlinear PI controller performance was found to be superior to that of the linear controller, and earlier reported theoretical results have been validated by the present experimental work.
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Abbreviations
- D :
-
Dilution rate (h−1)
- K c :
-
Controller gain
- K i :
-
Substrate inhibition constant (g/l)
- K m :
-
Substrate saturation constant (g/l)
- P :
-
Product concentration (g/l)
- P m :
-
Product saturation constant (g/l)
- Q :
-
Product cell produced per unit time (g/h l)
- S :
-
Substrate concentration (g/l)
- S f :
-
Feed substrate concentration (g/l)
- S f,s :
-
Steady state feed substrate concentration (g/l)
- t :
-
Time (h)
- X :
-
Biomass concentration (g/l)
- Z :
-
Defined by Eq. 15
- α:
-
Product yield parameter (g/g)
- β:
-
Product yield parameter (h−1)
- μ:
-
Specific growth rate (h−1)
- μm :
-
Maximum specific growth rate (h−1)
- τI :
-
Integral time constant (h)
- ψ:
-
Cell mass yield (g/g)
- Δ:
-
Deviation variable from steady state
References
Koppel LB (1982) Input multiplicities in nonlinear multivariable control systems. AIChE J 28(6):935–944
Koppel LB (1983) Input multiplicities in process control. Chem Eng Edu (spring) 58–63:89–92
Koppel LB (1985) Conditions imposed by process static’s on multivariable process dynamics. AIChE J 31(1):70–75
Dash KS, Koppel LB (1989) Sudden destabilization of controlled chemical processes. Chem Engg Comm 84:129–157
Henson MA, Seborg DE (1992) Nonlinear control strategies for continuous fermentation. Chem Eng Sci 47(4):821–835
Reddy GP, Chidambaram M (1995) Nonlinear control of bioreactors with input multiplicities in dilution rate. Bioprocess Eng 12:151–155
Acknowledgments
The authors acknowledge the financial assistance provided by the All India Council for Technical Education (AICTE), New Delhi, India to carry out the present work.
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Kumar, S.V.S., Kumar, V.R. & Reddy, G.P. Nonlinear control of bioreactors with input multiplicities—an experimental work. Bioprocess Biosyst Eng 28, 45–53 (2005). https://doi.org/10.1007/s00449-005-0014-3
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DOI: https://doi.org/10.1007/s00449-005-0014-3