Optimal input design for parameter estimation for nonlinear dynamical systems with bounded-errors and application in aeronautic domain

Optimal input design for parameter estimation for nonlinear dynamical systems with bounded-errors and application in aeronautic domain

Qiaochu Li Carine Jauberthie Lilianne Denis-Vidal Zohra Cherfi Moussa Maïga 

Laboratoire Mathématiques Appliquées de Compiègne (LMAC) Université de Technologie de Compiègne EA 2222 — 15 Rue Roger Couttolenc, 60200 Compiègne, France

LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France

Laboratoire Roberval UMR7337 - Université de Technologie de Compiègne, CNRS 15 Rue Roger Couttolenc, 60200 Compiègne, France

Corresponding Author Email: 
qiaochu.li,lilianne.denis-vidal@utc.fr; cjaubert,mmaiga@laas.fr; zohra.cherfi@utc.fr
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30 April 2017
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This paper deals with optimal input design for parameter estimation in a bounded error context. Interval analysis will be served as a tool in this paper. In this context, the measurement noise and parameters are considered bounded. The model parameter estimation is realized by combining the valided numeric integration tools and set-membership operations. To improve the accuracy on parameters, the sensitivity analysis based optimal input design is thus proposed, more accurate estimation results have been obtained which is our main contribution. This process for obtaining optimal input design has been tested on an aircraft model. The estimation results on parameters by using optimal input have been compared with the case by using a non optimal input.


state estimation, parameter estimation, nonlinear system, bounded error, optimal input design, interval analysis

1. Introduction
2. Présentation du problème et cas d’étude
3. Estimation d’état et de paramètres en utilisant l’analyse par intervalles
4. Entrée optimale via le critère MIGMAG
5. Application
6. Conclusion

Bendtsen C., Stauning O. (1996, aug). FADBAD, a flexible C++ package for automatic differentiation. Technical Report no IMM–REP–1996–17. Lyngby, Denmark, Department of Mathematical Modelling, Technical University of Denmark.

Berz M., Hoffstätter G., Atter G. H. (1998). Computation and application of taylor polynomials with interval remainder bounds. Reliable Computing, vol. 4, p. 83–97.

Berz M., Makino K. (1998). Verified integration of odes and flows using differential algebraic methods on high-order taylor models. Reliable Computing, vol. 4, no 4, p. 361-369.

Chabert G. (2014). Ibex : C++ library for constraint processing over real numbers. https://github.com/ibex-team/ibex-lib/.

Chen R. (1975). Input design for aircraft parameter identification: using time optimal control formulation. In Methods for Aircraft State and Parameter Identification , AGARD-CP-172, paper 13.

Coton P., Bucharles A., Jauberthie C., Le Moing T., Planckaert L. et. (2001). Caire - identification des dérivées de stabilité dynamique. Rapport technique. ph.2. Rapport technique 1/05650, ONERA.

Dreyfus S. (1965). Dynamic programming and the calculus of variations. Santa monica, CA, RAND Corporation.

Eijgenraam P. (1981). The solution of initial value problems using interval arithmetic. Amsterdam, Stichting Mathematisch Centrum.

Jauberthie C. (2002). Méthodologies de planification d’expériences pour systèmes dynamiques. Université de Technologies de Compiègne.

Jauberthie C., Chanthery E. (2013). Optimal input design for a nonlinear dynamical uncertain aerospace system. In Proceedings of IFAC Symposium on Nonlinear Control Systems (NOLCOS), p. 469–474. Toulouse, France.

Jaulin L., Kieffer M., Didrit O., Waletr E. (2001). Applied interval analysis: with examples in parameter and state estimation, robust control and robotics (vol. 1). Springer.

Jaulin L., Walter E. (1993). Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica, vol. 29, p. 1053 - 1064.

Kieffer M.,Walter E. (2011). Guaranteed estimation of the parameters of nonlinear continuoustime models: Contributions of interval analysis. International Journal of Adaptive Control and Signal Processing, vol. 25, no 3, p. 191–207.

Lohner R. J. (1987). Enclosing the solutions of ordinary initial and boundary value problems. In E. W. Kaucher, U. W. Kulisch, C. Ullrich (Eds.), Computer arithmetic: Scientific computation and programming languages, p. 255–286. Stuttgart,Wiley-Teubner Series in Computer Science.

Maïga M., Ramdani N., Travé-Massuyès. (2014). A CSP versus a zonotope-based method for solving guard set intersection in nonlinear hybrid reachability. Mathematics in Computer Science, vol. 8, no 3-4.

Maïga M., Ramdani N., Travé-Massuyès L., Combastel C. (2015). A comprehensive method for reachability analysis of uncertain nonlinear hybrid systems. IEEE Transactions on Automatic Control, vol. PP, no 99, p. 1-1.

Moore R. E. (1966). Interval analysis. New Jersey, Prentice Hall, 1st edition.

Nazin S., Polyak B. (2005). Interval parameter estimation under model uncertainty. Mathematical and Computer Modelling of Dynamical Systems, vol. 11, no 2, p. 225–237.

Nedialkov N. S. (1999). Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. Rapport technique.

Nedialkov N. S., Jackson K. R., Corliss G. F. (1999). Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation, vol. 105, p. 21–68.

Nedialkov N. S., Jackson K. R., Pryce J. D. (2001). An effective high-order interval method for validating existence and uniqueness of the solution of an ivp for an ode. Reliable Computing, vol. 7, no 6, p. 449–465.

Raïssi T., Ramdani N., Candau Y. (2004). Set membership state and parameter estimation for systems described by nonlinear differential equations. Automatica, vol. 40, no 10, p. 1771–1777.

Ramdani N., Nedialkov N. S. (2011). Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques. Nonlinear Analysis: Hybrid Systems, vol. 5, no 2, p. 149 - 162.

Rihm R. (1992). Enclosing solutions with switching points in ordinary differential equations. In L. Atanassova, J. Herzberger (Eds.), Computer Arithmetic and Enclosure Methods, p. 419–425. Amsterdam, North–Holland.

Rihm R. (1994). Interval methods for initial value problems in odes. In Imacs-gamm international workshop on validated computations. Amsterdam, Elsevier.

Schweppe F. (1968). Recursive state estimation: unknown but bounded errors and system inputs. IEEE Transactions on Automatic Control, vol. 13, no 1, p. 22–28.

Schweppe F. (1973). Uncertain dynamic systems. New Jersey, Prentice-Hall: Englewood Cliffs.

Walter E., Kieffer M. (2007). Guaranteed nonlinear parameter estimation in knowledge-based models. Journal of Computational and Applied Mathematics, vol. 199, no 2, p. 277–285.