Estimation of the State and the Unknown Inputs of a Multimodel with non Measurable Decision Variables

 

E. Maherzi¹, M. Besbes*¹, S. Zemmel² and A. Mami³

 

¹ High School of Technology and Computer Science, University of Carthage. 45 rue des entrepreneurs, Charguia 2, Tunis Carthage 2035, Tunisia. *mongi.besbes@gmail.com

² High School of Applied Sciences and Technologies of Gafsa, University of Gafsa.

³ Department of physics, Faculty of Sciences of Tunis, University of El Manar.

 

ABSTRACT

This paper treats the estimation of the state of a nonlinear system with unknown input. The nonlinear system is described by a multimodel with unknown function of activation but depending only on the state. The method of design of the multiobserver is described by using the second method of Lyapunov and their candidate functions. The sufficient obtained stability conditions are expressed in terms of Linear Matrix Inequalities (LMI) and are obtained first using the Lyapunov quadratic functions and secondly by using Lyapunov polyquadratic functions. This latter technique seems to be less conservative and less constraining than the first. Illustrative examples are presented in this paper.

Keywords: Discrete multimodel; multiobserver with unknown inputs; non measurable variables of decision; quadratic stabilization; polyquadratic stabilization; unknown input estimation; Linear Matrix Inequalities (LMI).

 

RESUMEN

Este artículo trata la estimación de estado de un sistema no lineal con una entrada desconocida. El sistema no lineal se describe por un multi-modelo con una función desconocida de activación, pero dependiendo sólo en su estado. El método de diseño del multi-observador se detalla mediante el segundo método de Lyapunov y sus funciones candidatos. Las condiciones de estabilidad obtenidos se expresan en términos de desigualdades matriciales lineales (LMI) y se obtienen de la utilización de las funciones cuadráticas de Lyapunov en un primer estudio y de las funciones poli cuadráticas de Lyapunov en un segundo estudio que aparece menos conservador y menos restrictivo que el primero. Múltiples ejemplos ilustrativos se presentan en este documento.

 

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References

[1] A. Akhenak et al., "State estimation of uncertain multiple model with unknown inputs", in 43rd IEEE Conference on Decision and Control, Atlantic, Paradise Island, Bahamas, vol. 4, 2004, pp. 3563-3568.         [ Links ]

[2] A. Akhenak et al., "Unknown input observer based-approach, application to secure communications", First Conference on Analysis and control of Chaotic Systems, volume 1, part 1, 2006.         [ Links ]

[3] A. Akhenak et al., "State estimation via Multiple observer with unknown input. Application to the three tank system", 5th IFAC Symposium on Fault Detection Supervision and Safety for Technical Processes, Safe process, pp. 245-251, Washington, USA, June 9-11 2003.         [ Links ]

[4] A. Jadbabaie. "A reduction in conservatism in stability and L2 gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities". In Proc. Of the IFAC, China, 1999. pp. 285-289.         [ Links ]

[5] D. Marx and B. Dept. "Unknown input observers for switched nonlinear discrete time descriptor systems", Automatica Control, IEEE Transactions on publication data. February 2008.         [ Links ]

[6] E. Maherzi et al., "Polyquadratic stabilization of a multi inputs multimodel with quantified commands". International Journal of mathematics and computer in simulation. Issue 4, Vol. 1, pp. 344-349, 2007.         [ Links ]

[7] J. Daafouz and J. Bernussou. "Parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties". Systems and Control Letters, 43/5:355 359, August 2001.         [ Links ]

[8] J.L. Mata-Machuca et al., "Chaotic Systems Synchronization Via High Order Observer Design", Journal of Applied Research and Technology (JART), Vol.9, pp 57, 68, 2011.         [ Links ]

[9] J. Moreno, "Quasi-Unknown input observers for linear systems", IEEE Conference on Decision and Control, pp. 732-737, 2001.         [ Links ]

[10] T.A. Johansen Et al., "On the interpretation and identification dynamic Takagi-Sugeno fuzzy models". IEEE Trans on Fuzzy Systems. Vol. 8.n. 3. pp.297-313.         [ Links ]

[11] K. Gasso, et al., "Structure identification in multiple model representation elimination and merging of local models", IEEE Conference on Decision and Control, Vol. 3, pp. 2992-2997, 2001.         [ Links ]

[12] K. Tanaka and M. Sugeno. "Stability analysis and design of fuzzy control systems". Fuzzy Sets and Systems, 45(2):135-156, 1992.         [ Links ]

[13] K. Tanaka et al., "Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based design", IEEE Trans. Fuzzy Systems, Vol. 6 (1), pp. 250-256, 1998.         [ Links ]

[14] L. Vandenberghe and S. Boyd, "Semi definite programming", SIAM Review 38 (1) (1996) 49-95.         [ Links ]

[15] M. Chadli et al., "State and unknown input estimation for discrete time multiple mode", Journal of the Franklin Institute 346 (2009) 593-610.         [ Links ]

[16] M. Chadli, and A. Elahajjaji, "Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties - comment on", Fuzzy Sets and Systems Journal 157 (9) (2006) 1276-1281.         [ Links ]

[17] M. Darouach et al., "Full-order observers for linear systems with unknown inputs", IEEE Transactions on Automatic Control 39 (3) (1994) 606-609.         [ Links ]

[18] P. Amann et al., "Identification of fuzzy relational models for fault detection", Control Engineering Practice, Vol. 9 (5), pp. 555-562, 2001.         [ Links ]

[19] P. Kudva et al., "Observers for linear systems with unknown inputs", IEEE Trans. on Automatic Control Vol. 25 (1), pp. 113-115, 1980.         [ Links ]

[20] R. Dixon, "Observer-based FDIA, application to an electromechanical positioning system", Control Engineering Practice, Vol. 12, pp. 1113-1125, 2004.         [ Links ]

[21] R.J. Patton et al., "Fuzzy observer for nonlinear dynamic systems fault diagnosis", IEEE Conference on Decision and Control, Vol. 1, pp. 84-89, 1998.         [ Links ]

[22] R. Acevedo-Gomez, et al., "State Variables Monitoring Using a Class Of Nonlinear Observer Based Estimaot, Applied To Continuous Bio-System", Journal of Applied Research and Technology (JART), Vol 6 n3, pp. 147-158.         [ Links ]

[23] S.K. Dassanake et al., "Using unknown input observers to detect and isolate sensor faults in a turbofan engine", Digital Avionics Systems Conferences, Vol. 7, pp. 6E51-6E57, 2000.         [ Links ]

[24] S.K. Chang, et al., "Design of general structured observers for linear systems with unknown inputs", Journal of the Franklin Institute 334 (2) (1997) 1025-1030.         [ Links ]

[25] Sami Zemmel et al., "Synthesis of a Robust Multiobserver for the Estimation of Unknown Inputs Using the Piecewise Quadratic Functions". American Journal of Applied Sciences 7 (9):1264-1276, 2010, ISSN 1546-9239.         [ Links ]

[26] T.A. Johansen and R. Babuska, "Multiobjective Identification of Takagi-Sugeno Fuzzy Models", IEEE Trans. on Fuzzy Systems, Vol. 11 (6), pp. 847-860, 2003.         [ Links ]

[27] T.A. Johansen. "Computation of Lyapunov function for smooth nonlinear systems using convex optimization". Automatica, 2000. no. 36, pp. 1617-1626.         [ Links ]

[28] T. Floquet, and J. Barbot, "A sliding mode approach of unknown input observers for linear systems", in: IEEE Conference on Decision and Control, vol. 2, 2004, pp.1724-1729.         [ Links ]