“Nonlinear Observer Design for a Nonlinear Cable/String FEM Model using Contraction Theory”

Authors: Yilmaz Türkyilmaz and Olav Egeland,
Affiliation: NTNU, Centre for Ships and Ocean Structures and NTNU, Department of Engineering Cybernetics
Reference: 2004, Vol 25, No 3, pp. 159-172.

Keywords: Cables, nonlinear models, finite element method, observers, nonlinear analysis

Abstract: Contraction theory is a recently developed nonlinear analysis tool which may be useful for solving a variety of nonlinear control problems. In this paper, using Contraction theory, a nonlinear observer is designed for a general nonlinear cable/ string FEM (Finite Element Method) model. The cable model is presented in the form of partial differential equations (PDE). Galerkin´s method is then applied to obtain a set of ordinary differential equations such that the cable model is approximated by a FEM model. Based on the FEM model, a nonlinear observer is designed to estimate the cable configuration. It is shown that the estimated configuration converges exponentially to the actual configuration. Numerical results and simulations are shown to be in agreement with the theoretical results.

PDF PDF (1205 Kb)        DOI: 10.4173/mic.2004.3.2

[1] BAICU, C.F, RAHN, C. D. DAWSON, D.M. (1999). Exponentially stabilizing boundary control of string-mass systems, Journal of Vibration and Control, 5, pp. 491-502 doi:10.1177/107754639900500309
[2] CANBOLAT, H., DAWSON, D., NACGARKATTI, S. P. COSTIC, B. (1998). Boundary control for a general class of string models, Proceedings of the American Control Conference. Pennsylvania.
[3] DEMETRIOU, M. A. (2001). Natural observers for second order lumped and distributed parameter systems using parameter-dependent Lyapunov functions, Proceedings of the American Control Conference, Arlington, VA.
[4] JOUFFROY, J., FOSSEN. T.I. SLOTINE, J.-J. E. (2004). Methodological remarks on contraction theory, exponentially stability and Lyapunov functions, Submitted to Automatica.
[5] KRISTIANSEN, D. (2000). Modeling of Cylinder Gyroscopes and Observer Design for Nonlinear Oscillatons, PhD thesis. Dept. of Engineering Cybernetics, Norwegian Univ. of Sci. and Tech.
[6] LOHMILLER, W. SLOTINE, J.-J. E. (1998). On contraction analysis for non-linear systems, Automatica, 34(6), pp. 683-696 doi:10.1016/S0005-1098(98)00019-3
[7] MORGÜL., Ö. (1994). A dynamic control law ibr the wave equation, Automatica, 30, pp. 1785-1792 doi:10.1016/0005-1098(94)90083-3
[8] QU, Z. (2000). Robust and adaptive boundary control of a stretched string, Proceedings of the American Control Conkrence, Illinois.
[9] SHAHRUZ, S.M. NARASIMHA, C. A. (1997). Suppression of vibration in tretched strings by the boundary control, Journal of Sound and Vibration, 204, pp. 835-840 doi:10.1006/jsvi.1997.0985
[10] ZIENKIEWICZ, O.C. TAYLOR, R. L. (2000). The Finite Element Method, 5th edn, Butterworth-Heinemann, Oxford, UK.

  title={{Nonlinear Observer Design for a Nonlinear Cable/String FEM Model using Contraction Theory}},
  author={Türkyilmaz, Yilmaz and Egeland, Olav},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}