Estimation of Bounded Model Uncertainties

Olivier Adrot; Jean-Marie Flaus; José Ragot

doi:10.20965/jrm.2006.p0661

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JRM Vol.18 No.5 pp. 661-671

doi: 10.20965/jrm.2006.p0661

(2006)

Paper:

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Estimation of Bounded Model Uncertainties

Olivier Adrot^, Jean-Marie Flaus^, and José Ragot^**

^*Laboratoire d’Automatique de Grenoble - UMR 5528, ENSIEG, BP 46, 38402 St Martin d’Hères Cedex, France

^**Centre de Recherche en Automatique de Nancy - UMR 7039, ENSEM, 2, avenue de la Forêt de Haye, 54516 Vandœuvre-lès-Nancy Cedex, France

Received:

February 3, 2006

Accepted:

July 10, 2006

Published:

October 20, 2006

Keywords:

uncertain model, interval analysis, estimation theory, robot control

Abstract

We identify parameters of a given input-output model so that estimated model output is consistent with the measured output of the system modeled. Parameter estimation based on a set-membership approach is a nonprobabilistic method for characterizing the uncertainty with which each model parameter is known. The model is consistent with data if the estimated output domain contains measured system output at each instant. Dynamic linear Multi-Input Multi-Output (MIMO) models are considered in this paper. Every equation error is bounded while model parameters fluctuate within a time-invariant domain represented by a zonotope. Our proposal helps find the characteristics of this domain, e.g., center, shape, size, by taking into account coupling between bounded variables of output equations to increase model accuracy.

Cite this article as:

O. Adrot, J. Flaus, and J. Ragot, “Estimation of Bounded Model Uncertainties,” J. Robot. Mechatron., Vol.18 No.5, pp. 661-671, 2006.

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References

[1] O. Adrot, “Diagnostic à base de modèles incertains utilisant l’analyse par intervalles: l’approche bornante,” Ph.D. of the “Institut National Polytechnique de Lorraine,” France, 2000.
[2] O. Adrot, D. Maquin, and J. Ragot, “Bounding approach to the fault detection of uncertain dynamic systems,” IFAC Safeprocess2000, 2000.
[3] O. Adrot, D. Maquin, and J. Ragot, “Diagnosis of an uncertain static system,” IEEE CDC’2000, 2000.
[4] O. Adrot, J. Ragot, and J.-M. Flaus, “Characterization of bounded uncertainties,” Complex Systems, Intelligence and Modern Technology Applications CSIMTA, Cherbourg, 2004.
[5] V. Cerone, “Parameter bounds for ARMAX models from records with bounded errors in variables,” International Journal of Control, Vol.57, No.1, pp. 225-235, 1993.
[6] T. Clément and S. Gentil, “Recursive membership set estimation for ARMAX models: an output-error approach,” 12^th IMACS World Congress, 1988.
[7] E. Fogel and Y. F. Huang, “On the value of information in system identification-bounded noise case,” Automatica, Vol.18, No.2, pp. 229-238, 1982.
[8] L. Jaulin and E. Walter, “Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis,” Mathematics and Computers in Simulation, No.35, pp. 123-137, 1993.
[9] M. Milanese and G. Belforte, “Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: linear families of models,” IEEE Transactions on Automatic Control, Vol.AC-27, No.2, pp. 408-413, 1982.
[10] M. Milanese, J. Norton, H. Piet-Lahanier, and E. Walter, “Bounding approaches to system identification,” Plenum Press, New York & London, 1996.
[11] S. H. Mo and J. P. Norton, “Fast and robust algorithm to compute exact polytope parameter bounds,” Math. and Comput. in Simul., Vol.32, pp. 481-493, 1990.
[12] R. E. Moore, “Methods and applications of interval analysis,” SIAM, Philadelphia, Pennsylvania, 1979.
[13] A. Neumaier, “Interval methods for systems of equations,” Cambridge University Press, Cambridge, 1990.
[14] H. Piet-Lahanier and E. Walter, “Exact recursive characterization of feasible parameter sets in the linear case,” Math. and Comput. in Simul., Vol.32, pp. 495-504, 1990.
[15] S. Ploix, O. Adrot, and J. Ragot, “Parameter Uncertainty Computation in static Linear Models,” IEEE CDC’99, 1999.
[16] S. Ploix, O. Adrot, and J. Ragot, “Bounding approach to the diagnosis of uncertain static systems,” IFAC Safeprocess2000, 2000.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] O. Adrot, “Diagnostic à base de modèles incertains utilisant l’analyse par intervalles: l’approche bornante,” Ph.D. of the “Institut National Polytechnique de Lorraine,” France, 2000.

[2] [2] O. Adrot, D. Maquin, and J. Ragot, “Bounding approach to the fault detection of uncertain dynamic systems,” IFAC Safeprocess2000, 2000.

[3] [3] O. Adrot, D. Maquin, and J. Ragot, “Diagnosis of an uncertain static system,” IEEE CDC’2000, 2000.

[4] [4] O. Adrot, J. Ragot, and J.-M. Flaus, “Characterization of bounded uncertainties,” Complex Systems, Intelligence and Modern Technology Applications CSIMTA, Cherbourg, 2004.

[5] [5] V. Cerone, “Parameter bounds for ARMAX models from records with bounded errors in variables,” International Journal of Control, Vol.57, No.1, pp. 225-235, 1993.

[6] [6] T. Clément and S. Gentil, “Recursive membership set estimation for ARMAX models: an output-error approach,” 12^th IMACS World Congress, 1988.

[7] [7] E. Fogel and Y. F. Huang, “On the value of information in system identification-bounded noise case,” Automatica, Vol.18, No.2, pp. 229-238, 1982.

[8] [8] L. Jaulin and E. Walter, “Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis,” Mathematics and Computers in Simulation, No.35, pp. 123-137, 1993.

[9] [9] M. Milanese and G. Belforte, “Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: linear families of models,” IEEE Transactions on Automatic Control, Vol.AC-27, No.2, pp. 408-413, 1982.

[10] [10] M. Milanese, J. Norton, H. Piet-Lahanier, and E. Walter, “Bounding approaches to system identification,” Plenum Press, New York & London, 1996.

[11] [11] S. H. Mo and J. P. Norton, “Fast and robust algorithm to compute exact polytope parameter bounds,” Math. and Comput. in Simul., Vol.32, pp. 481-493, 1990.

[12] [12] R. E. Moore, “Methods and applications of interval analysis,” SIAM, Philadelphia, Pennsylvania, 1979.

[13] [13] A. Neumaier, “Interval methods for systems of equations,” Cambridge University Press, Cambridge, 1990.

[14] [14] H. Piet-Lahanier and E. Walter, “Exact recursive characterization of feasible parameter sets in the linear case,” Math. and Comput. in Simul., Vol.32, pp. 495-504, 1990.

[15] [15] S. Ploix, O. Adrot, and J. Ragot, “Parameter Uncertainty Computation in static Linear Models,” IEEE CDC’99, 1999.

[16] [16] S. Ploix, O. Adrot, and J. Ragot, “Bounding approach to the diagnosis of uncertain static systems,” IFAC Safeprocess2000, 2000.

Estimation of Bounded Model Uncertainties

Olivier Adrot*, Jean-Marie Flaus*, and José Ragot**

Olivier Adrot^, Jean-Marie Flaus^, and José Ragot^**