Abstract
We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be performed either continuously or step-wise. We present and compare both approaches. Even in the case of two species in competition, which is not directly accessible with our approach due to lack of controllability, feedback control improves identifiability of the non-dominant growth rate.
References
Abed EH, Wang HO, Chen RC (1994) Stabilization of period doubling bifurcations and implications for control of chaos. Physica D 70:154–164
Aborhey S, Williamson D (1978) State amd parameter estimation of microbial growth process. Automatica 14:493–498
Allgower EL, Georg K (2003) Introduction to numerical continuation methods. Society for Industrial and Applied Mathematics, Boston
Antonelli R, Harmand J, Steyer JP, Astolfi A (2003) Set point regulation of an anaerobic digestion process with bounded output feedback. IEEE Trans Control Syst Technol 11:495–504
Astrom K, Wittenmark B (1994) Adaptive Control, 2nd edn. Prentice-Hall, New Jersey
Bastin G, Dochain D (1990) On-line estimation and adaptive control of bioreactors. Elsevier, Amsterdam
Benoît E, (1991) Dynamic bifurcations: proceedings of a conference held in Luminy, France, March 5–10, 1990, Lecture notes in mathematics, vol 1493. Springer, Berlin
Baltes M, Schneider R, Sturm C, Reuss M (1994) Optimal experimental design for parameter estimation in unstructured growth models. Biotechnol Progress 10(5):480–488
Boudjellaba H, Sari T (1999) Stability loss delay in harvesting competing populations. J Differ Equ 152:394–408
Busvelle E, Gauthier J-P (2003) On determining unknown functions in differential systems, with an application to biological reactors. ESAIM Control Optim Calc Var 9:509–522
Dochain D (2003) State and parameter estimation in chemical and biochemical processes : a survey. J Process Control 13(8):801–818
Dochain D, Bastin G (1984) Adaptive identification and control algorithms for non linear bacterial growth systems. Automatica 20(5):621–634
Dochain D, Bastin G (1986) On-line estimation of microbial growth rates. Automatica 22(6):705–711
Dochain D, Pauss A (1988) On-line estimation of specific growth rates : an illustrative case study. Can J Chem Eng 66(4):626–631
Dochain D, Perrier M (1997) Dynamical modelling, analysis, monitoring and control design for nonlinear bioprocesses. Adv Biochem Eng Biotechnol 56:147–197
Dochain D, Perrier M, Guay M (2010) Extremum seeking control and its application to process and reaction systems: a survey. Math Comput Simul 16(6):535–553
Guay M, Dochain D, Perrier M (2004) Adaptive extremum seeking control of stirred tank bioreactors. Automatica 40(5):881–888
Holmberg A (1982) On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities. Math Biosci 62:23–43
Holmberg A, Ranta J (1982) Procedures for parameter and state estimation of microbial growth process models. Automatica 18:181–193
Insperger T (2006) Act-and-wait concept for continuous-time control systems with feedback delay. IEEE Trans Control Syst Technol 14(5):974–977
Jadot F, Bastin G, Viel F (1999) Robust global stabilization of stirred tank reactors with saturated output-feedback. Eur J Control 5:361–371
Kapoor N, Daoutidis P (1999) An observer-based anti-windup scheme for non-linear systems with input constraints. Int J Control 9:18–29
Kovarova-Kova K, Egli T (1998) Growth kinetics of suspended microbial cells: from single-substrate-controlled growth to mixed-substrate kinetics. Microbiol Mol Biol Rev 62(3):646–666
Kurtz MJ, Henson MA (1997) Input–output linearizing control of constrained nonlinear processes. J Process Control 7(1):3–17
Lobry JR, Flandrois J-P (1991) Comparison of estimates of Monod’s growth model from the same data set. Binary 3:20–23
Lobry C, Rapaport A, Sari T (February 2009) Stability loss delay in the chemostat with a slowly varying washout rate. In: Proceedings of the MATHMOD international conference on mathematical modelling, Vienna (Austria)
O’Malley R (1991) Singular perturbation methods for ordinary differential equations. Springer, New York
Perrier M, Feyode Azevedo S, Ferreira E, Dochain D (2000) Tuning of observer-based estimators: theory and application to the on-line estimation of kinetic parameters. Control Eng Pract 8(4):377–388
Posten C, Munack A (1990) On-line application of parameter estimation accuracy to biotechnical processes. In: Proceedings of the 9th American control conference (ACC), San Diego (CA), pp 2181–2186
Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421–428
Rapaport A, Harmand J (2002) Robust regulation of a class of partially observed nonlinear continuous bioreactors. J Process Control 12:291–302
Robinson JA (1985) Determining microbial kinetic parameters using non-linear regression analysis. Adv Microb Ecol 8:61–114
Reddy GP, Chidambaram M (1994) Nonlinear control of bioreactors with input multiplicities. Bioproc Eng 11:97–100
Sieber J, Gonzalez-Buelga A, Neild SA, Wagg DJ, Krauskopf B (2008) Experimental continuation of periodic orbits through a fold. Phys Rev Lett 100:244101
Satishkumar B, Chidambaram M (1999) Control of unstable bioractor using fuzzy-tuned PI controller. Bioproc Eng 20:127–132
Schaum A, Alvarez J, Lopez-Arenas T (2012) Saturated PI control of continuous bioreactors with Haldane kinetics. Chem Eng Sci 68:520–529
Sieber J, Rapaport A, Rodrigues S, Desroches M (July 2012) A new method for the reconstruction of unknown non-monotonic growth function in the chemostat. In: Proceedings of the IEEE Mediterranean conference on control and automation, Barcelona (Spain)
Smith HL, Waltman P (1995) The theory of the chemostat. Cambridge University Press, Oxford
Vanrollenghem P, Keesman K (1996) Identification of biodegradation models under model and data uncertainty. Water Sci Technol 33(2):91–105
Veraart A, Faassen E, Dakos V, van Nes E, Lurling M, Scheffer M (2012) Recovery rates reflect distance to a tipping point in a living system. Nature 481:357–359
Versyck K, Claes J, Van Impe J (1997) Practical identification of unstructured growth kinetics by application of optimal experimental design. Biotechnol Progress 13(5):524–531
Versyck K, Claes J, Van Impe J (1998) Optimal experimental design for practical identification of unstructured growth models. Math Comput Simul 46(5-6):621–629
Acknowledgments
JS’ research is supported by the EPSRC Grant EP/J010820/1
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sieber, J., Rapaport, A., Rodrigues, S. et al. A method for the reconstruction of unknown non-monotonic growth functions in the chemostat. Bioprocess Biosyst Eng 36, 1497–1507 (2013). https://doi.org/10.1007/s00449-013-0912-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00449-013-0912-8