Research Article Open Access

Synthesis of a Robust Multiobserver for the Estimation of Unknown Inputs Using the Piecewise Quadratic Functions

Sami Zemmel1, Mongi Besbes2, Elyes Maherzi1 and Radhi Mhiri2
  • 1 , Afganistan
  • 2 ,
American Journal of Applied Sciences
Volume 7 No. 9, 2010, 1264-1276

DOI: https://doi.org/10.3844/ajassp.2010.1264.1276

Published On: 30 September 2010

How to Cite: Zemmel, S., Besbes, M., Maherzi, E. & Mhiri, R. (2010). 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. https://doi.org/10.3844/ajassp.2010.1264.1276

Abstract

Problem statement: The estimation of states and the unknown inputs of a nonlinear system described by a multimodel are done by a multiobserver. The stabilization of the multiobserver calls upon uses both quadratic and no quadratic functions of Lyapunov. Although the stabilization using the quadratic approach is interesting from the point of view implementation, the step showed its limits for the multimodel. However, the problem paused by the quadratic method lies in the obligation to satisfy several LMI with respect to the same Lyapunov matrix P, these results are shown very conservative. Approach: To reduce the conservatism of the quadratic approach we propose another approach which is exclusively based on Lyapunov piecewise quadratic functions. The conditions obtained by the stabilization of the multiobserver are expressed in term of matrix inequalities with constraints on the matrices rank. Results: The estimation of both states and unknown inputs of a multimodel using the quadratic approach per pieces leads to results less conservative than the quadratic approach. Academic examples illustrate the robustness of the piecewise quadratic approach. Conclusion: In this article we proposed new sufficient conditions of stability of a multiobserver able to the estimation of states and unknown inputs of a nonlinear system describes by a multimodel subjected to the influence of the unknown inputs. The study in was carried out by considering two approaches. The first approach is based on Lyapunov quadratic functions; it is significant to note the great difficulty in finding satisfying results by this approach for the multimodel systems. For this reason we proposed an approach based on piecewise quadratic functions which led to interesting results (proposition 1) and less conservative than the quadratic approach. The conditions suggested in this article concern both the multiobserver stabilization and the estimation of states and the unknown inputs of a multimodel with measurable variables of decision (μξ(k)). The synthesis of a multiobserver with no measurable variables of decision is not approached. This point can constitute an interesting prospect for this study.

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Keywords

  • Discrete multimodel
  • unknown input multiobserver
  • quadratic stabilization
  • piecewise quadratic functions
  • unknown input estimation