Abstract
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.
Similar content being viewed by others
References
Bastin G, Dochain D, 1990. On-line Estimation and Adaptive Control of Bioreactors: a Volume in Process Measurement and Control. Elsevier, Amsterdam, the Netherlands. https://doi.org/10.1016/C2009-0-12088-3
Bastin G, Gevers MR, 1988. Stable adaptive observers for nonlinear time-varying systems. IEEE Trans Autom Contr, 33(7):650–658. https://doi.org/10.1109/9.1273
Besançon G, de León-Morales J, Huerta-Guevara O, 2006. On adaptive observers for state affine systems. Int J Contr, 79(6):581–591. https://doi.org/10.1080/00207170600552766
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. https://doi.org/10.14736/kyb-2018-1-0155
Diop S, Fliess M, 1991. Nonlinear observability, identifiability, and persistent trajectories. Proc 30th IEEE Conf on Decision and Control, p.714–719. https://doi.org/10.1109/CDC.1991.261405
Dochain D, 2008. Automatic Control of Bioprocesses. Wiley. https://www.wiley.com/en-mx/Automatic+Control+of+Bioprocesses-p-9781848210257
Farza M, M’Saad M, Maatoug T, et al., 2009. Adaptive observers for nonlinearly parameterized class of nonlinear systems. Automatica, 45(10):2292–2299. https://doi.org/10.1016/j.automatica.2009.06.008
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. https://doi.org/10.1016/j.automatica.2014.10.032
Gauthier JP, Hammouri H, Othman S, 1992. A simple observer for nonlinear systems applications to bioreactors. IEEE Trans Autom Contr, 37(6):875–880. https://doi.org/10.1109/9.256352
Hammouri H, Nadri M, 2013. An observer design for a class of implicit systems. Syst Contr Lett, 62(3):256–261. https://doi.org/10.1016/j.sysconle.2012.11.001
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. https://doi.org/10.1016/j.jfranklin.2010.03.004
Kreisselmeier G, 1977. Adaptive observers with exponential rate of convergence. IEEE Trans Autom Contr, 22(1):2–8. https://doi.org/10.1109/TAC.1977.1101401
Lafont F, Busvelle E, Gauthier JP, 2011. An adaptive high-gain observer for wastewater treatment systems. J Process Contr, 21(6):893–900. https://doi.org/10.1016/j.jprocont.2011.03.006
Liang XY, Zhang JF, Xia XH, 2008. Adaptive synchronization for generalized Lorenz systems. IEEE Trans Autom Contr, 53(7):1740–1746. https://doi.org/10.1109/TAC.2008.928318
Luders G, Narendra K, 1973. An adaptive observer and identifier for a linear system. IEEE Trans Autom Contr, 18(5):496–499. https://doi.org/10.1109/TAC.1973.1100369
Marino R, Tomei P, 1995a. Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. IEEE Trans Autom Contr, 40(7):1300–1304. https://doi.org/10.1109/9.400471
Marino R, Tomei P, 1995b. Nonlinear Control Design: Geometric, Adaptive, and Robust. Prentice Hall, Limited, London, UK.
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. https://doi.org/10.1007/s12555-010-0205-9
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. https://doi.org/10.1007/s00034-011-9320-y
Raghavan S, Hedrick JK, 1994. Observer design for a class of nonlinear systems. Int J Contr, 59(2):515–528. https://doi.org/10.1080/00207179408923090
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. https://doi.org/10.1049/iet-cta.2012.0318
Zhang Q, 2002. Adaptive observer for multiple-input-multiple-output (MIMO) linear time-varying systems. IEEE Trans Autom Contr, 47(3):525–529. https://doi.org/10.1109/9.989154
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partly supported by the Czech Science Foundation (No. GA19-05872S)
Contributors
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.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1631/FITEE.2000368
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1631/FITEE.2000368