Bioprocess and Biosystems Engineering

, Volume 27, Issue 5, pp 319–327 | Cite as

Nonlinear control for algae growth models in the chemostat

Original papers


This paper deals with output feedback control of phytoplanktonic algae growth models in the chemostat. The considered class of model is of variable yield type, meaning that the ratio between the environmental nutrient absorption rate and the cells’ growth rate varies, which is different from classical bioprocesses assumptions. On the basis of weak qualitative hypotheses on the analytical expressions of the involved biological phenomena (which guarantee robustness of the procedure toward modeling uncertainties) we propose a nonlinear controller and prove its ability to globally stabilize such processes. Finally, we illustrate our approach with numerical simulations and show its benefits for biological laboratory experiments, especially for ensuring persistence of the culture facing classical experimental problems.


Algae growth Chemostat Variable yield model Nonlinear control 


  1. 1.
    Bernard O (1995) Etude expérimentale et théorique de la croissance de Dunaliella tertiolecta soumise à une limitation variable de nitrate utilisation de la dynamique transitoire pour la conception et la validation de modèles. Université Paris VIGoogle Scholar
  2. 2.
    Bernard O, Gouzé JL (1995) Transient behavior of biological loop models with application to the Droop model. Math Biosci 127:19–43CrossRefGoogle Scholar
  3. 3.
    Chicone C (1999) Ordinary differential equations with applications texts in applied mathematics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  4. 4.
    Droop MR (1968)Vitamin B12 and marine ecology. IV. The kinetics of uptake growth and inhibition in Monochrysis Lutheri. J Mar Biol Assoc UK 48:689–733Google Scholar
  5. 5.
    Henson MA, Seborg DE (1997) Nonlinear process control. Prenctice Hall, Englewood CliffsGoogle Scholar
  6. 6.
    Khalil HK (1992) Nonlinear systems. Macmillan, New YorkGoogle Scholar
  7. 7.
    Lange K, Oyarzun FJ (1992) The attractiveness of the Droop equations. Math Biosci 111:61–278CrossRefGoogle Scholar
  8. 8.
    Luenberger DG (1979) Introduction to dynamic systems. Theory models and applications. Wiley, New YorkGoogle Scholar
  9. 9.
    Mailleret L (2003) Positive control for a class of nonlinear positive systems. In: Benvenutti L, De Santis A, Farina L (eds) Positive systems Lecture Notes in Control and Informations Sciences, vol 294. Springer, Berlin Heidelberg New York, pp 175–182Google Scholar
  10. 10.
    Mailleret L (2004) Stabilisation globale de systèmes dynamiques positifs mal connus. Applications en biologie. Université de Nice Sophia, AntipolisGoogle Scholar
  11. 11.
    Mailleret L, Bernard O, Steyer JP (2003) Robust regulation of anaerobic digestion. Processes Water Sci Technol 48(6):87–94Google Scholar
  12. 12.
    Mailleret L, Bernard O, Steyer JP (2004) Nonlinear adaptive control for bioreactors with unknown kinetics. Automatica 40(8):1379–1385CrossRefGoogle Scholar
  13. 13.
    Markus L (1956) Asymptotically autonomous differential systems. Annals of mathematics studies 36. Princeton University Press, New Jersey, pp 17–29Google Scholar
  14. 14.
    Monod J La technique de culture continue; théorie et applications. Annales de l’Institut Pasteur 79:390–401Google Scholar
  15. 15.
    Nijmeijer H, van der Schaft AJ (1991) Nonlinear dynamical control systems. Springer, Berlin Heidelberg New YorkGoogle Scholar
  16. 16.
    Oyarzun FJ, Lange K (1994) The attractiveness of the Droop equations. 2. Generic uptake and growth functions. Math Biosci 121:127–139CrossRefGoogle Scholar
  17. 17.
    Smith HL (1995) Monotone dynamical systems an introduction to the theory of competitive and cooperative systems. Mathematical surveys and monographs. American Mathematical SocietyGoogle Scholar
  18. 18.
    Smith HL, Waltman P (1995) The theory of the chemostat: dynamics of microbial competition. Cambridge University Press, CambridgeGoogle Scholar
  19. 19.
    Thieme HR (1992) Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations. J Math Biol 30:755–763CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Comore, InriaSophia-Antipolis cedexFrance

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