AUTHORS: Ramzi Ben Messaoud, Salah Hajji

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ABSTRACT: In this article, we propose a new nonlinear observer concept. The basic idea for our observer’s design is to use mean value theorem (MVT) and lsqnonlin algorithm to determine the estimation error (e = x−xˆ) and MVT parameters βi (between 0 and 1) . The stability study is carried out thanks to the quadratic function of Lyapunov. Two numerical examples are provided to show the performance of the proposed approach. The first studies a chaotic system with a linear term ( ˙x = f(x, u)) and the second deals with a linear system ( ˙x = Ax + f(x, u)).

KEYWORDS: lsqnonlin algorithm;Nonlinear observer; Nonlinear system; mean value theorem; State estimation

REFERENCES:

[1] Zeitz .M, The extended Luenberger observer for nonlinear systems, Systems & Control Letters 9 (1987) 149-156.

[2] Noboru .S, Branislav .R and Kouji .U, Nonlinear Luenberger observer design via invariant manifold computation. Proceedings of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 20141989; 27 :199–216.

[3] Chouaib .A, Vincent .A, Laurent .B and Pascal .D, State and Parameter Estimation: A Nonlinear Luenberger Observer Approach, IEEE Transactions on Automatic Control, Vol: 62, Issue: 2, Feb. 2017.

[4] Zhu .F, State estimation and unknown input reconstruction via both reduced-order and highorder sliding mode observers. Journal of Process Control 2012; 22 :296–302.

[5] Shtessel Y, Edwards C, Fridman L, Levant A. “Sliding Mode Control and Observation”. Birkhauser: New York, 2014. ¨

[6] Loza.F, Bejarano .J, Fridman .L. Unmatched uncertainties compensation based on high-order sliding mode observation. International Journal of Robust and Nonlinear Control 2013; 23 :754– 764.

[7] Cho .M, Rajamani .R, A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Transactions on Automatic Control 1997; 42 :534–537.

[8] Ekramian .M, Sheikholeslam .F, Hosseinnia .S, Yazdanpanah .M., Adaptive state observer for Lipschitz nonlinear systems. Systems Control Letter 2013; 62:319–323.

[9] Holland .JH, Outline for a logical theory of adaptive systems. J ACM. 1962;9(3):297314.

[10] Dalil .I, Benoit .x, Didier .M and Jose .R , Observer for Lipschitz nonlinear systems: mean value theorem and sector nonlinearity transformation , 2012 IEEE International Symposium on Intelligent Control (ISIC), 3-5 Oct. 2012, Croatia.

[11] Zemouche .A and Boutayeb .M, Observers for a class of Lipschitz systems with extension to Hinfinity performance analysis. Systems & Control Letters, 57:18–27, 2008.

[12] Gridsada .P, Rajesh .R, and Damrongrit .P, Nonlinear Observer for Bounded Jacobian Systems, With Applications to Automotive Slip Angle Estimation, IEEE Transactions on Automatic control, Vol. 56, No. 5, May 2011.

[13] Sahoo .P .K, Riedel .T, Mean Value Theorems and Functional Equations, World Scientific, 1999.

[14] Eriksson .K, Estep .D, and Johnson .C, Applied Mathematics: Body and Soul, Volume 3: Calculus in Several Dimensions, Springer-Verlag, 2004.

[15] . Kirkpatrick .S, Gelatt .C.D and Vecchi. M.P, Optimization by simulated annealing, Science 220(4598), 671680 (1983).

[16] Yanbin .Z, Jian .T, Ning .Z .S, A note on observer design for one-sided Lipschitz nonlinear systems, Systems and Control Letters, 59 (2010), 66-71.

[17] Ali .Z, Mohamed .B and Iulia .B .G, Observer Design for Nonlinear Systems: An Approach Based on the Differential Mean Value Theorem, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12- 15, 2005.

[18] Yuhua .X and Yuling .W, A new chaotic system without linear term and its impulsive synchronization, Optik - International Journal for Light and Electron Optics, Vol.125, Issue 11, June 2014, Pages 2526-2530.

[19] Spong .M, Modeling and control of elastic joint robots, ASME J. Dyn. Syst. Meas. Control 109, 310319 (1987)