Thursday , March 28 2024

Design of a Proportional Observer Based on the
ARX-Laguerre Model

Hassene BEDOUI*, Tarek GARNA,
Kamel BEN OTHMAN, Hassani MESSAOUD

LARATSI, National Engineering School of Monastir,
University of Monastir, Monastir, Tunisia
hassenebedoui@gmail.com; {tarek.garna;kamel.benothman;hassani.messaoud}@enim.rnu.tn

* Corresponding author

Abstract: A new ARX-Laguerre representation is recently built to model the dynamics of complex processes [1, 2]. The ARX-Laguerre models have proven their ability to accurately suit the behavior of systems. In this work, the model is exploited to diagnose the system by detecting its defaults. In this paper we build a proportional observer based on the ARX-Laguerre model. Therefore, the designed observer exploits the inputs and outputs of the Laguerre-ARX model to reconstruct the Laguerre filter outputs. The observer gain is calculated to ensure a fast asymptotic convergence of the estimation error. A simulation example is achieved to illustrate the ability of the proposed approach to estimate the Laguerre filter outputs.

Keywords: Diagnosis, ARX-Laguerre, proportional observer, SISO system, LMI, FDI.

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CITE THIS PAPER AS:
Hassene BEDOUI, Tarek GARNA, Kamel BEN OTHMAN, Hassani MESSAOUD, Design of a Proportional Observer Based on theARX-Laguerre Model, Studies in Informatics and Control, ISSN 1220-1766, vol. 24 (4), pp. 471-476, 2015. 
https://doi.org/10.24846/v24i4y201511

Introduction

The dynamics of physical processes are often modeled by mathematical relations.These relations are generally differential equations or state representations dedicated exclusively to theoretical models, transfer functions or regressive representations (ARX, ARMAX, Arimax). Recently, Bouzraraet al [1,2] proposed a new representationable to model the dynamics of complex physical processes. The proposed representation called ARX-Laguerre achieves a significant complexity reduction compared to the linear ARX standard model. The principle of the ARX-Laguerre model is based on filtering the input and the output of theARX standard model by the orthogonal Laguerre functions. This new representation is useful in case of observers based diagnosis for state representation modeled systems. Generally, observers used for linear systems are often Luenberger-ones or with proportional gain [3]. In this work, we propose a proportional observer which can reconstructthe Laguerre filter outputs from the inputs and outputs of the ARX-Laguerre model.The reconstruction is achieved througha comparison between the outputs of the estimated Laguerre filters and the real outputs. The ARX-Laguerre based diagnosis is anew approach since the proposed model is recent [1,2].

This work is presented as follows: A theoretical study on the new ARX-Laguerre linear modeling is presented in Section 1, followed in the second section by developing its recursive representation used in the observer design. In the third section, we present the synthesis of proportional observer exploiting the ARX-Laguerre model.

This step is characterized by the development of the observer structure, the synthesis of gain matrices and Lyapunov ones. In addition, the conditions of existence of the observer are established. A work on improving the performance of the new observer is achieved in the same section. In the last section, the synthesis technique of the proportional observer is applied to an illustrative example showing the effectiveness of the developed method.

REFERENCES

  1. BOUZRARA, K., T. GARNA, H. MESSAOUD, J. RAGOT, Online Identification of the ARX Model Expansion on Laguerre Orthonormal Bases With Filters on Model Input and Output, International Journal of Control, vol. 86, 2013, pp. 369-385.
  2. BOUZRARA, K., T. GARNA, H. MESSAOUD, J. RAGOT, Decomposition of an ARX Model on Laguerre Orthonormal Bases, ISA Transactions, vol. 51, 2012, pp. 848-860.
  3. VAN SCHRICK, D., PI-Observer-based Reconstruction of Effect-variables and Construction of Characteristic Curves, at the 5-th Asian Control Conference, 2004, pp. 937-942.
  4. MALTI, Représentation de Systèmes Discrets sur la Base des Filtres Orthogonaux – Application à la Modélisation de Systèmes Dynamiques Multivariables, PhD thesis, Institut National Polytechnique de Loraine, France, 1999.
  5. BEDOUI, , K. BEN OTHMAN, Multi-observer for Uncertain Output Nonlinear Systems, in 8-th International Conference on Applied Mathematics, Simulation, Modelling (ASM ’14), Florence, Italy, 2014, pp. 354-359.
  6. BORNE, P., G. DAUPHIN-TANGUY, J. P. RICHARD, G. ROTELLA, I. ZAMBETTAKIS, Commande et Optimisation des Processus, Collection Méthodes et Techniques de l’Ingénieur, Editions Technip, 1990.
  7. HE, Y.-Y., X.-J. MA, Z.-Q. SUN, Analysis and Design of Fuzzy Controller and Fuzzy Observer, IEEE Transactions on Fuzzy Systems, vol. 6, 1998, pp. 41-51.
  8. IKEDA, T., K. TANAKA, H. O. WANG, Robust Stabilization of a Class of Uncertain Nonlinear Systems via Fuzzy Control: Quadratic Stabilizability, H∞ Control Theory, and Linear Matrix Inequalities, IEEE Transactions on Fuzzy Systems, vol. 4, 1996, pp. 1-13.
  9. CHEN, J., C. LOPEZ-TORIBIO, R. PATTON, Fuzzy Observers for Nonlinear Dynamic Systems Fault Diagnosis,” in Proceedings of the 37-th IEEE Conference on Decision and Control, 1998, pp. 84-89.
  10. BEDOUI, H., K. BEN OTHMAN, A. KHEDHER, Fault Detection and Isolation of Fuzzy System with Uncertain Parameters Using the Bounded Approach, WSEAS Transactions on Systems and Control, 9, art. 28, 2014, pp. 269-276.
  11. AITOUCHE, A., L. BELKOURA, A. JOUNI, S. C. OLTEANU, Embedded P.E.M. Fuel Cell Stack Nonlinear Observer by Means of a Takagi-Sugeno Approach, Studies in Informatics and Control, ISSN 1220-1766, vol. 24(1), 2015, pp. 61-70.
  12. OTSUKA, N., T. SOGA, Stabilizability Conditions for Switched Linear Systems with Constant Input via Switched Observer, Studies in Informatics and Control, ISSN 1220-1766, vol. 22(1) 2013, pp. 7-14.