Kybernetika 54 no. 1, 155-174, 2018

Adaptive high gain observer extension and its application to bioprocess monitoring

Sergej Čelikovský, Jorge Antonio Torres-Muñoz and Alma Rosa Dominguez-BocanegraDOI: 10.14736/kyb-2018-1-0155


The adaptive version of the high gain observer for the strictly triangular systems subjected to constant unknown disturbances is proposed here. The adaptive feature is necessary due to the fact that the unknown disturbance enters in a way that cannot be suppressed by the high gain technique. The developed observers are then applied to a culture of microorganism in a bioreactor, namely, to the model of the continuous culture of Spirulina maxima. It is a common practice that just the biomass (or substrate) concentration is directly measured as the output of the process for monitoring and control purposes. This paper thereby shows both by theoretical analysis and numerical simulation that the adaptive high-gain observers offer a realistic option of online software sensors for substrate estimation.


nonlinear systems, adaptive observers, bioprocess


93C95, 90C46


  1. M. Abbaszadeh and H. J. Marquez: A generalized framework for robust nonlinear Hinfty filtering of Lipschitz descriptor systems with parametric and nonlinear uncertainties. Automatica 48 (2012), 5, 894-900.   DOI:10.1016/j.automatica.2012.02.033
  2. P. Agrawal and H. Lim: Analyses of various control schemes for continuous bioreactors. Advances in Biochemical Engineering/ Biotechnology 30 (1984), 61-90.   DOI:10.1007/bfb0006380
  3. G. Bastin and M. Gevers: Stable adaptive observers for nonlinear time-varying systems. IEEE Trans. Automat. Control 33 (1988), 7, 650-658.   DOI:10.1109/9.1273
  4. F. Bejarano and L. Fridman: High order sliding mode observer for linear systems with unbounded unknown inputs. Int. J. Control 83 (2010), 1920-1929.   DOI:10.1080/00207179.2010.501386
  5. G. Besancon, J. de Leon-Morales and O. Huerta-Guevara: On adaptive observers for state affine systems. Int. J. Control 79 (2006), 6, 581-591.   DOI:10.1080/00207170600552766
  6. A. R. Bocanegra-Domínguez et al.: Estudio teórico práctico de la remoción de contaminantes presentes en el río de Los Remedios, Estado de México. Tecnología y Ciencias del Agua 24.2 (2009), 81-91. (In Spanish)   CrossRef
  7. G. Bornard, F. Celle-Couenne and G. Gilles: Observability and Observers. In: Nonlinear Systems - T.1, 'Modeling and Estimation'. Chapman and Hall, London 1995, pp. 173-216.   DOI:10.1007/978-1-4615-2047-4_6
  8. R. O. Canizares and A. R. Domínguez: Growth of Spirulina maxima on swine waste. Bioresource Technol. 45 (1993), 1, 73-75.   DOI:10.1016/0960-8524(93)90148-5
  9. S. Diop and M. Fliess: Nonlinear observability, identifiability, and persistent trajectories. In: Proc. 30th IEEE Conference on Decision and Control 1 (1991), pp. 714-719.   DOI:10.1109/cdc.1991.261405
  10. D. Efimov and L. Fridman: Global sliding-mode observer with adjusted gains for locally Lipschitz systems. Automatica 47 (2011), 3, 565-570.   DOI:10.1016/j.automatica.2010.12.003
  11. M. Farza, I. Bouraou, T. Menard, R. Abdennou and M. M'Saad: Adaptive observers for a class of uniformly observable systems with nonlinear parametrization and sampled outputs. Automatica 50 (2014), 2951-2960.   DOI:10.1016/j.automatica.2014.10.032
  12. M. Farza, M. M'saad, T. Maatou and M. Kamoun: Adaptive observers for nonlinearly parameterized class of nonlinear systems. Automatica 45 (2009), 2292-2299.   DOI:10.1016/j.automatica.2009.06.008
  13. L. Fridman, Y. Shtessel, C. Edwards and X. Yan: Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems. Int. J. Robust Nonlinear Control 18 (2008), 399-412.   DOI:10.1002/rnc.1198
  14. J. P. Gauthier, H. Hammouri and S. Othman: A simple observer for nonlinear systems applications to bioreactors. IEEE Trans. Automat. Control 37 (1991), 875-880.   DOI:10.1109/9.256352
  15. L. Gerd and K. Narendra: An adaptive observer and identifier for a linear system. IEEE Trans. Automat. Control 18 (1973), 5, 496-499.   DOI:10.1109/tac.1973.1100369
  16. L. Guoping and D Ho: Full-order and reduced-order observers for Lipschitz descriptor systems: the unified LMI approach. IEEE Trans. Circuits Systems II: Express Briefs 53 (2006), 7, 563-567.   DOI:10.1109/tcsii.2006.875332
  17. H. Hammouri and M. Nadri: An observer design for a class of implicit systems. Systems Control Lett. 62 (2013), 3, 256-261.   DOI:10.1016/j.sysconle.2012.11.001
  18. R. Hermann and A. Krener: Nonlinear controllability and observability. IEEE Trans. Automat. Control 22 (1977), 5, 728-740.   DOI:10.1109/tac.1977.1101601
  19. K. Hamid-Reza, M. Zapateiro and N. Luo: 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 (2010), 6, 957-973.   DOI:10.1016/j.jfranklin.2010.03.004
  20. H. R. Karimi, M. Zapateiro, N. and Luo: 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 (2010), 6, 957-973.   DOI:10.1016/j.jfranklin.2010.03.004
  21. H. Khalil: Nonlinear Systems. Third edition. Prentice Hall, Englewood Cliffs, NJ 2002.   CrossRef
  22. M. J. Khosrowjerdi: Mixed H2/Hinfty approach to fault-tolerant controller design for Lipschitz non-linear systems. IET Control Theory A. 5 (2011), 2, 299-307.   DOI:10.1049/iet-cta.2009.0556
  23. G. Kreisselmeier: Adaptive observers with exponential rate of convergence. IEEE Trans. Automat. Control 22 (1977), 1, 2-8.   DOI:10.1109/tac.1977.1101401
  24. F. Lafon, E. Busvelle and J. P. Gauthier: An adaptive high-gain observer for wastewater treatment systems. Journal Process Control 21 (2011), 893-900.   DOI:10.1016/j.jprocont.2011.03.006
  25. X. Liang, Z. Jiangfeng and X. Xiaohua: Adaptive synchronization for generalized Lorenz systems. IEEE Trans. Automat. Control 53 (2008), 7, 1740-1746.   DOI:10.1109/tac.2008.928318
  26. F. Mairet et al.: Modelling neutral lipid production by the microalga Isochrysis aff. galbana under nitrogen limitation. Bioresource Technol. 102.1 (2011), 142-149.   DOI:10.1016/j.biortech.2010.06.138
  27. R. Marino and P. Tomei: Nonlinear Control Design. Geometric, Adaptive and Robust Approach. Prentice Hall, Englewood Cliffs, NJ 1995.   CrossRef
  28. R. Marino and P. Tomei: Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. IEEE Trans. Automat. Control 40 (1995), 7, 1300-1304.   DOI:10.1109/9.400471
  29. S. Raghavan and J. Hedrick: Observer design for a class of nonlinear systems. Int. J. Control 59 (1994), 2, 515-528.   DOI:10.1080/00207179408923090
  30. R. Rajamani: Observers for Lipschitz nonlinear systems. IEEE Trans. Automat. Control 43 (1998), 3, 397-401.   DOI:10.1109/9.661604
  31. A. Rodríguez-Mata, J. Torres-Muñoz, A. R. Domínguez, D. Hernandez-Villagran and S. Čelikovský: Nonlinear high-gain observers with integral action: Application to bioreactors. In: Proc. 8th International Conference on Electrical Engineering, Computing Science and Automatic Control, Cancun 2011, pp. 444-449.   DOI:10.1109/iceee.2011.6106611
  32. J. Sanchez-Torres, G. Loukianov, J. Moreno and S. V. Drakunov: An equivalent control based sliding mode observer using high order uniform robust sliding operators. In: Proc. American Control Conference, Montreal 2012, pp. 6160-6165.   CrossRef
  33. L. Travieso, E. Sánchez and R. Bora: Evaluation of laboratory and full-scale microalgae pond for tertiary treament of piggery wastes. Enviromental Technol. 25 (2004), 565-576.   DOI:10.1080/09593330.2004.9619347
  34. H. Wu: A class of adaptive robust state observers with simpler structure for uncertain non linear systems with time varying delays. IET Control Theory Appl. 7 (2013), 218-222.   DOI:10.1049/iet-cta.2012.0318
  35. L. Yong-Hong and Y. Zhou: Non-fragile observer-based robust control for a class of fractional-order nonlinear systems. Systems Control Lett. 62 (2013), 12, 1143-1150.   DOI:10.1016/j.sysconle.2013.09.007
  36. Q. Zhang: Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems. IEEE Trans. Automat. Control 47 (2002), 3, 525-529.   DOI:10.1109/9.989154
  37. A. Zemouche, M. Boutayeb, T. Maatoug and M. Kamoun: On LMI conditions to design observers for Lipschitz nonlinear systems. Automatica 49 (2013), 585-591.   DOI:10.1016/j.automatica.2012.11.029