T. A. Kharkovskaia, A. S. Kremlev, D. M. Sabirova, D. V. Efimov, R. Tarek

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The method of an interval observer design for nonlinear systems with parametric uncertainties is considered. The interval observer synthesis problem for systems with varying parameters consists in the following. If there is the uncertainty restraint for the state values of the system, limiting the initial conditions of the system and the set of admissible values for the vector of unknown parameters and inputs, the interval existence condition for the estimations of the system state variables, containing the actual state at a given time, needs to be held valid over the whole considered time segment as well. Conditions of the interval observers design for the considered class of systems are shown. They are: limitation of the input and state, the existence of a majorizing function defining the uncertainty vector for the system, Lipschitz continuity or finiteness of this function, the existence of an observer gain with the suitable Lyapunov matrix. The main condition for design of such a device is cooperativity of the interval estimation error dynamics. An individual observer gain matrix selection problem is considered. In order to ensure the property of cooperativity for interval estimation error dynamics, a static transformation of coordinates is proposed. The proposed algorithm is demonstrated by computer modeling of the biological reactor. Possible applications of these interval estimation systems are the spheres of robust control, where the presence of various types of uncertainties in the system dynamics is assumed, biotechnology and environmental systems and processes, mechatronics and robotics, etc.

Keywords: interval estimation, observer, nonlinear systems, parametric-varying systems, parametric uncertainty, bioreactor

1.        Bastin G., Van Impe J.F. Nonlinear and adaptive control in biotechnology: a tutorial. European Journal of Control, 1995, vol. 1, no. 1, pp. 37–53.
2.        Ajbar A., Alhumaizi K. Dynamics of the Chemostat: A Bifurcation Theory Approach. CRC Press, 2011, 368 p.
3.        Fossas E., Ros R.M., Fabregat J. Sliding mode control in a bioreactor model. Journal of Mathematical Chemistry, 2001, vol. 30, no. 2, pp. 203–218. doi: 10.1023/A:1017927804488
4.        Gordeeva Yu.L., Gordeev L.S. Matematicheskaya model' nepreryvnogo protsessa v bioreaktore s retsiklom substrata i biomassy [Mathematical model for continuous process in bioreactor with substrate and biomass recycle]. Vestnik of Astrakhan State Technical University. Series: Management, Computer Science and Informatics,2013, no. 2,
pp. 9–18.
5.        Mazenc F., Niculescu S.I., Bernard O. Interval observers for linear systems with delay. Proc. of the 48th IEEE conference on decision and control. Shanghai, China, 2009, pp. 1860–1865. doi: 10.1109/CDC.2009.5399818
6.        Moisan M., Bernard O., Gouze J.-L. Near optimal interval observers bundle for uncertain bioreactors. Automatica, 2009, vol. 45, no. 1, pp. 291–295. doi: 10.1016/j.automatica.2008.07.006
7.        Bernard O., Gouze J.-L. Closed loop observers bundle for uncertain biotechnological models. Journal of Process Control, 2004, vol. 14, no. 7, pp. 765–774. doi: 10.1016/j.jprocont.2003.12.006
8.        Moisan M., Bernard O. Interval observers for non monotone systems. Application to bioprocess models. IFAC Proceedings Volumes (IFAC-PapersOnline), 2005, vol. 16, pp. 43–48.
9.        Jaulin L. Nonlinear bounded-error state estimation of continuous time systems. Automatica, 2002, vol. 38, no. 6, pp. 1079–1082. doi: 10.1016/S0005-1098(01)00284-9
10.     Kiefer M., Walter E. Guaranteed nonlinear state estimator for cooperative systems. Numerical Algorithms, 2004, vol. 37, no. 1-4, pp. 187–198. doi: 10.1023/B:NUMA.0000049466.96588.a6
11.     Raissi T., Videau G., Zolghadri A. Interval observers design for consistency checks of nonlinear continuous-time systems. Automatica, 2010, vol. 46, no. 3, pp. 518–527. doi: 10.1016/j.automatica.2009.12.005
12.     Raissi T., Efimov D., Zolghadri A. Interval state estimation for a class of nonlinear systems. IEEE Transactions on Automatic Control, 2012, vol. 57, no. 1, art. no. 5983407, pp. 260–265. doi: 10.1109/TAC.2011.2164820
13.     Efimov D., Fridman L.M., Raissi T., Zolghadri A., Seydou R. Interval estimation for LPV systems applying high order sliding mode techniques. Automatica, 2012, vol. 48, no. 9, pp. 2365–2371. doi: 10.1016/j.automatica.2012.06.073
14.     Mazenc F., Bernard O. Interval observers for linear time-invariant systems with disturbances. Automatica, 2011, vol. 47, no. 1, pp. 140–147. doi: 10.1016/j.automatica.2010.10.019
15.     Chebotarev S.G., Kremlev A.S. Sintez interval'nogo nablyudatelya dlya lineinoi sistemy s peremennymi parametrami [Synthesis of interval observer for linear system with variable parameters]. Izv. vuzov. Priborostroenie, 2013, vol. 56, no. 4, pp. 42–46.
16.     Chebotarev S., Efimov D., Raïssi T., Zolghadri A. On interval observer design for a class of continuous-time LPV systems. IFAC Proceedings Volumes (IFAC-PapersOnline), 2013, vol. 9, part 1, pp. 68–73. doi: 10.3182/20130904-3-FR-2041.00068
17.     Chebotarev S., Kremlev A. Analysis conditions on interval observer synthesis for linear systems with variable parameters. 18th International Conference on Methods and Models in Automation and Robotics, MMAR 2013. Międzyzdroje, Poland, 2013, art. no. 6669939, pp. 390–392.
18.     Chebotarev S.G., Kremlev A.S. Analiz lineinykh sistem s peremennymi parametrami dlya sinteza interval'nykh nablyudatelei [Analysis of linear systems with varuable parameters for interval observer synthesis]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2012, no. 6 (82), pp. 50–53.
19.     Smith H.L. Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Providence: AMS, 1995, vol. 41, 174 p.
20.     Efimov D.V., Kremlev A.S., Kharkovskay T.A., Chebotarev S.G. Postroenie sistemy interval'nogo otsenivaniya dlya modeli regulyatsii gormona testosterona [Analysis and synthesis of complex systems interval estimation system design for testosterone hormone dynamic model]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2013, no. 6 (88), pp. 56–60.
21.     Efimov D., Raïssi T., Chebotarev S., Zolghadri A. Interval state observer for nonlinear time varying systems. Automatica, 2013, vol. 49, no. 1, pp. 200–205. doi: 10.1016/j.automatica.2012.07.004
22.     Efimov D., Raïssi T., Chebotarev S., Zolghadri A. On set-membership observer design for a class of periodical time-varying systems. Proceedings of the IEEE Conference on Decision and Control, 2012, art. no. 6426474, pp. 6767–6772. doi: 10.1109/CDC.2012.6426474

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