A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME SYSTEMS USING HYBRID AUTOMATA

Nacim Meslem, Nacim Ramdani, Yves Candau

2008

Abstract

This work is about state estimation in the bounded error context for non linear continuous time systems. The main idea is to seek to estimate not an optimal value for the unknown state vector but the set of feasible values, thus to characterize simultaneously the value of the vector and its uncertainty. Our contribution resides in the use of comparison theorems for differential inequalities and the analysis of the monotonicity of the dynamical systems with respect to the uncertain variables. The uncertain dynamical system is then bracketted between two hybrid dynamical systems. We show how to obtain this systems and to use them for state estimation with a prediction-correction type observer. An example is given with bioreactors.

References

  1. Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., and Yovine, S. (1995). The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138:3-34.
  2. Chernousko, F. (2005). Ellipsoidal state estimation for dynamical systems. Nonlinear analysis.
  3. Chisci, L., Garulli, A., and Zappa, G. (1996). Recursive state bounding by parallelotopes. Automatica, 32:1049-1055.
  4. Combastel, C. (2005). A state bounding observer for uncertain nonlinear continuous-time systems based on zonotopes. In 44th IEEE Conference on decision and control and European control conference ECC 2005.
  5. Dochain, D. (2003). State and parameter estimation in chemical and biochemical processes: a tutorial. Journal of process control, 13:801-818.
  6. Hermann, R. (1963). On the accessibility problem in control theory. In in Int. Symp. on nonlinear differential equations and nonliear mechanics, pages 325-332, New York. Academic Press.
  7. Hermann, R. and J.Krener, A. (1977). Nonlinear controllability and observability. Transactions on Automatic Control, AC-22:728-740.
  8. Hirsch, M. and Smith, H. (2005). Monotone dynamical systems. In Canada, A., Drabek, P., and Fonda, A., editors, Handbook of Differential Equations, Ordinary Differential Equations, volume 2, chapter 4. Elsevier.
  9. Jaulin, L., Kieffer, M., Didrit, O., and Walter, E. (2001). Applied interval analysis: with examples in parameter and state estimation, robust control and robotics. Springer-Verlag, London.
  10. Kieffer, M. and Walter, E. (2006). Guaranteed nonlinear state estimation for continuous-time dynamical models from discrete-time measurements. In Proceedings 6th IFAC Symposium on Robust Control, Toulouse.
  11. Kieffer, M., Walter, E., and Simeonov, I. (2006). Guaranteed nonlinear parameter estimation for continuoustime dynamical models. In Proceedings 14th IFAC Symposium on System Identification, pages 843-848, Newcastle, Aus.
  12. Marcelli, C. and Rubbioni, P. (1997). A new extension of classical müller theorem. Nonlinear Analysis, Theory, Methods & Applications, 28(11):1759-1767.
  13. Moore, R. (1966). Interval analysis. Prentice-Hall, Englewood Cliffs.
  14. Müller, M. (1926). Uber das fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen. Math. Z., 26:619-645.
  15. Nedialkov, N. (1999). Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. PhD University of Toronto.
  16. Raïssi, T., Ramdani, N., and Candau, Y. (2004). Set membership state and parameter estimation for systems described by nonlinear differential equations. Automatica, 40(10):1771-1777.
  17. Ramdani, N., Meslem, N., Raïssi, T., and Candau, Y. (2006). Set-membership identification of continuoustime systems. In Proceedings 14th IFAC Symposium on System Identification, pages 446-451, Newcastle, Aus.
  18. Smith, H. (1995). Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems. Ams. providence, ri edition.
  19. Walter, W. (1997). Differential inequalities and maximum principles: Theory, new methods and applications. Nonlinear Analysis, Theory, Methods & Applications, 30(8):4695-4711.
Download


Paper Citation


in Harvard Style

Meslem N., Ramdani N. and Candau Y. (2008). A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME SYSTEMS USING HYBRID AUTOMATA . In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO, ISBN 978-989-8111-32-6, pages 32-37. DOI: 10.5220/0001490400320037


in Bibtex Style

@conference{icinco08,
author={Nacim Meslem and Nacim Ramdani and Yves Candau},
title={A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME SYSTEMS USING HYBRID AUTOMATA},
booktitle={Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO,},
year={2008},
pages={32-37},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0001490400320037},
isbn={978-989-8111-32-6},
}


in EndNote Style

TY - CONF
JO - Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO,
TI - A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME SYSTEMS USING HYBRID AUTOMATA
SN - 978-989-8111-32-6
AU - Meslem N.
AU - Ramdani N.
AU - Candau Y.
PY - 2008
SP - 32
EP - 37
DO - 10.5220/0001490400320037