Constant-gain nonlinear adaptive observers revisited: an application to chemostat systems


This study deals with constant-gain adaptive observers for nonlinear systems, for which relatively few solutions are available for some particular cases. We introduce an asymptotic observer of constant gain for nonlinear systems that have linear input. This allows the observer design to be formulated within the linear matrix inequality paradigm provided that a strictly positive real condition between the input disturbance and the output is fulfilled. The proposed observer is then applied to a large class of nonlinear chemostat dynamical systems that are widely used in the fermentation process, cell cultures, medicine, etc. In fact, under standard practical assumptions, the necessary change of the chemostat state coordinates exists, allowing use of the constant-gain observer. Finally, the developed theory is illustrated by estimating pollutant concentration in a Spirulina maxima wastewater treatment facility.


  1. Bastin G, Dochain D, 1990. On-line Estimation and Adaptive Control of Bioreactors: a Volume in Process Measurement and Control. Elsevier, Amsterdam, the Netherlands.

    Google Scholar 

  2. Bastin G, Gevers MR, 1988. Stable adaptive observers for nonlinear time-varying systems. IEEE Trans Autom Contr, 33(7):650–658.

    MathSciNet  Article  Google Scholar 

  3. Besançon G, de León-Morales J, Huerta-Guevara O, 2006. On adaptive observers for state affine systems. Int J Contr, 79(6):581–591.

    MathSciNet  Article  Google Scholar 

  4. Čelikovský S, Torres-Muñoz JA, Dominguez-Bocanegra AR, 2018. Adaptive high gain observer extension and its application to bioprocess monitoring. Kybernetika, 54(1): 155–174.

    MathSciNet  MATH  Google Scholar 

  5. Diop S, Fliess M, 1991. Nonlinear observability, identifiability, and persistent trajectories. Proc 30th IEEE Conf on Decision and Control, p.714–719.

  6. Dochain D, 2008. Automatic Control of Bioprocesses. Wiley.

  7. Farza M, M’Saad M, Maatoug T, et al., 2009. Adaptive observers for nonlinearly parameterized class of nonlinear systems. Automatica, 45(10):2292–2299.

    MathSciNet  Article  Google Scholar 

  8. Farza M, Bouraoui I, Ménard T, et al., 2014. Adaptive observers for a class of uniformly observable systems with nonlinear parametrization and sampled outputs. Automatica, 50(11):2951–2960.

    MathSciNet  Article  Google Scholar 

  9. Gauthier JP, Hammouri H, Othman S, 1992. A simple observer for nonlinear systems applications to bioreactors. IEEE Trans Autom Contr, 37(6):875–880.

    MathSciNet  Article  Google Scholar 

  10. Hammouri H, Nadri M, 2013. An observer design for a class of implicit systems. Syst Contr Lett, 62(3):256–261.

    MathSciNet  Article  Google Scholar 

  11. Karimi HR, Zapateiro M, Luo N, 2010. A linear matrix inequality approach to robust fault detection filter design of linear systems with mixed time-varying delays and nonlinear perturbations. J Franklin Inst, 347(6):957–973.

    MathSciNet  Article  Google Scholar 

  12. Kreisselmeier G, 1977. Adaptive observers with exponential rate of convergence. IEEE Trans Autom Contr, 22(1):2–8.

    MathSciNet  Article  Google Scholar 

  13. Lafont F, Busvelle E, Gauthier JP, 2011. An adaptive high-gain observer for wastewater treatment systems. J Process Contr, 21(6):893–900.

    Article  Google Scholar 

  14. Liang XY, Zhang JF, Xia XH, 2008. Adaptive synchronization for generalized Lorenz systems. IEEE Trans Autom Contr, 53(7):1740–1746.

    MathSciNet  Article  Google Scholar 

  15. Luders G, Narendra K, 1973. An adaptive observer and identifier for a linear system. IEEE Trans Autom Contr, 18(5):496–499.

    Article  Google Scholar 

  16. Marino R, Tomei P, 1995a. Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. IEEE Trans Autom Contr, 40(7):1300–1304.

    MathSciNet  Article  Google Scholar 

  17. Marino R, Tomei P, 1995b. Nonlinear Control Design: Geometric, Adaptive, and Robust. Prentice Hall, Limited, London, UK.

    Google Scholar 

  18. Mondal S, Chakraborty G, Bhattacharyy K, 2010. LMI approach to robust unknown input observer design for continuous systems with noise and uncertainties. Int J Contr Autom Syst, 8(2):210–219.

    Article  Google Scholar 

  19. Pourgholi M, Majd VJ, 2011. A nonlinear adaptive resilient observer design for a class of Lipschitz systems using LMI. Circ Syst Signal Process, 30(6):1401–1415.

    MathSciNet  Article  Google Scholar 

  20. Raghavan S, Hedrick JK, 1994. Observer design for a class of nonlinear systems. Int J Contr, 59(2):515–528.

    MathSciNet  Article  Google Scholar 

  21. Wu HS, 2013. A class of adaptive robust state observers with simpler structure for uncertain non-linear systems with time-varying delays. IET Contr Theory Appl, 7(2):218–227.

    MathSciNet  Article  Google Scholar 

  22. Zhang Q, 2002. Adaptive observer for multiple-input-multiple-output (MIMO) linear time-varying systems. IEEE Trans Autom Contr, 47(3):525–529.

    MathSciNet  Article  Google Scholar 

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Author information



Corresponding author

Correspondence to Sergej Čelikovský.

Additional information

This research was partly supported by the Czech Science Foundation (No. GA19-05872S)


Jorge A. TORRES and Sergej ČELIKOVSKÝ conceptualized the research. Arno SONCK developed the theoretical findings of the work. Arno SONCK and Alma R. DOMINGUEZ contributed to the design and simulation of the experiment results. Arno SONCK and Jorge A. TORRES drafted the manuscript. Sergej ČELIKOVSKÝ revised and finalized the paper.

Compliance with ethics guidelines

Jorge A. TORRES, Arno SONCK, Sergej ČELIKOVSKÝ, and Alma R. DOMINGUEZ declare that they have no conflict of interest.

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Torres, J.A., Sonck, A., Čelikovský, S. et al. Constant-gain nonlinear adaptive observers revisited: an application to chemostat systems. Front Inform Technol Electron Eng 22, 68–78 (2021).

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Key words

  • Nonlinear observers
  • Adaptive observers
  • Coordinate change
  • Chemostat
  • Pollutant observation

CLC number

  • TP273