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July  2012, 8(3): 765-780. doi: 10.3934/jimo.2012.8.765

Identification for systems governed by nonlinear interval differential equations

Tayel Dabbous 1,

1. 

Dep. of Elec. Eng., Higher Technological Institute, Ramadan 10th City

Received  January 2012 Revised  February 2012 Published  June 2012

In this paper we consider the identification problem for a class of systems governed by nonlinear time varying interval differential equations having unknown (interval) parameters. Using the fact that system output posses lower and upper bounds, we have developed two sets of ordinary differential equations that represent the behaviour of lower and upper bounds. Based on these differential equations, the interval identification problem is converted into an equivalent identification problem in which the unknown parameters are real valued functions. Using variational arguments, we have developed the necessary conditions of optimality for the equivalent problem on the basis of which the unknown lower and upper parameters (and hence the interval parameters) can be determined. Finally, we present some numerical simulations to illustrate the effectivness of the proposed technique.
Keywords: variational calculus, identification, Interval systems, interval differential equations.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765
References:
[1]

N. U. Ahmed, "Elements of Finite-Dimensional and Control Theory," Pitman Monographs and Surveys in Pure and Applied Mathematics, 37, Longman Scientific & Technical, Harlow, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[2]

B. Bedregal, R. Trinade and A. Doria-Neto, Basic concepts of interval digital signal processing, World Academy of Eng. and Tech., 40 (2008), 66-70. Google Scholar

[3]

J. Chen, G. Wang and L. Sheih, Interval Kalman filtering, IEEE Trans. on Aerospace and Electr. Systems, (1997). Google Scholar

[4]

T. E. Dabbous, Adaptive control of nonlinear systems using fuzzy systems, Industrial and Management Optimization, 6 (2010), 861-880.  Google Scholar

[5]

M. Kieffor, O. Didrit, L. Jaulin and É. Walter, "Applied Interval Analysis. With Examples in Parameter and State Estimation Robust Control and Robotics," With 1 CD-ROM (UNIX, Sun Solaris), Springer-Verlag London, Ltd., London, 2001.  Google Scholar

[6]

R. Moore, "Methods and Applications of Interval Analysis," SIAM Studies in Applied Mathematics, 2, SIAM, Philadelphia, PA, 1979.  Google Scholar

[7]

E. Oppenheimer and A. Michel, Application of interval analysis techniques to linear systems. II. The interval matrix exponential function, IEEE Trans. on Ciruits and Systems, 35 (1988), 1230-1242.  Google Scholar

[8]

E. Oppenheimer and A. Michel, Application of interval analysis techniques to linear systems. III. Initial value problem, IEEE Trans. on Ciruits and Systems, 35 (1988), 1243-1256.  Google Scholar

[9]

E. Oppenheimer and A. Michel, Application of interval analysis techniques to linear systems. I. Fundamental results, IEEE Trans. on Ciruits and Systems, 35 (1988), 1129-1138.  Google Scholar

[10]

A. Rapaport, J. L. Gouze and M. Hadj-Sadok, Interval observers for uncertain biological systems, Ecological Modeling, 133 (2000), 45-56. Google Scholar

[11]

G. Schröder, Differentiation of interval functions, Proceedings of AMS, 36 (1972), 485-490.  Google Scholar

[12]

K. Shahiari and S. Tarasiewicz, Linear time varying systems: Model parameters characterization using interval analysis, Int. Journal of Math. and Comp. in Sim., 1 (2008), 54-62. Google Scholar

[13]

Ye. Smagina and I. Brewer, Using interval arethmetic for robust state feedback design, Systems and Control Letter, 46 (2002), 187-194.  Google Scholar

[14]

A. Stancu, V. Puig and J. Quevedo, Observers for interval systems using set and trajecrory-based approaches, 44th IEEE Conf. on Decision and Control, 1 (2005), 6567-6572. Google Scholar

[15]

A. Yeşildirek and F. L. Lewis, Feedback linearization using neural networks, Automatica J. IFAC, 31 (1995), 1659-1664.  Google Scholar

show all references

References:
[1]

N. U. Ahmed, "Elements of Finite-Dimensional and Control Theory," Pitman Monographs and Surveys in Pure and Applied Mathematics, 37, Longman Scientific & Technical, Harlow, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[2]

B. Bedregal, R. Trinade and A. Doria-Neto, Basic concepts of interval digital signal processing, World Academy of Eng. and Tech., 40 (2008), 66-70. Google Scholar

[3]

J. Chen, G. Wang and L. Sheih, Interval Kalman filtering, IEEE Trans. on Aerospace and Electr. Systems, (1997). Google Scholar

[4]

T. E. Dabbous, Adaptive control of nonlinear systems using fuzzy systems, Industrial and Management Optimization, 6 (2010), 861-880.  Google Scholar

[5]

M. Kieffor, O. Didrit, L. Jaulin and É. Walter, "Applied Interval Analysis. With Examples in Parameter and State Estimation Robust Control and Robotics," With 1 CD-ROM (UNIX, Sun Solaris), Springer-Verlag London, Ltd., London, 2001.  Google Scholar

[6]

R. Moore, "Methods and Applications of Interval Analysis," SIAM Studies in Applied Mathematics, 2, SIAM, Philadelphia, PA, 1979.  Google Scholar

[7]

E. Oppenheimer and A. Michel, Application of interval analysis techniques to linear systems. II. The interval matrix exponential function, IEEE Trans. on Ciruits and Systems, 35 (1988), 1230-1242.  Google Scholar

[8]

E. Oppenheimer and A. Michel, Application of interval analysis techniques to linear systems. III. Initial value problem, IEEE Trans. on Ciruits and Systems, 35 (1988), 1243-1256.  Google Scholar

[9]

E. Oppenheimer and A. Michel, Application of interval analysis techniques to linear systems. I. Fundamental results, IEEE Trans. on Ciruits and Systems, 35 (1988), 1129-1138.  Google Scholar

[10]

A. Rapaport, J. L. Gouze and M. Hadj-Sadok, Interval observers for uncertain biological systems, Ecological Modeling, 133 (2000), 45-56. Google Scholar

[11]

G. Schröder, Differentiation of interval functions, Proceedings of AMS, 36 (1972), 485-490.  Google Scholar

[12]

K. Shahiari and S. Tarasiewicz, Linear time varying systems: Model parameters characterization using interval analysis, Int. Journal of Math. and Comp. in Sim., 1 (2008), 54-62. Google Scholar

[13]

Ye. Smagina and I. Brewer, Using interval arethmetic for robust state feedback design, Systems and Control Letter, 46 (2002), 187-194.  Google Scholar

[14]

A. Stancu, V. Puig and J. Quevedo, Observers for interval systems using set and trajecrory-based approaches, 44th IEEE Conf. on Decision and Control, 1 (2005), 6567-6572. Google Scholar

[15]

A. Yeşildirek and F. L. Lewis, Feedback linearization using neural networks, Automatica J. IFAC, 31 (1995), 1659-1664.  Google Scholar

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