Bonnabel, Silvère (2007) Invariant asymptotic observers: theory and examples. PhD thesis Mathématiques et Automatique, CAS- Centre Automatique et Systèmes, ENSMP p.178.

Full text available as: - These_bonnabel.pdf ( 3679 Kb )

Licence: Copyright

Abstract

This thesis is about the construction of non-linear estimators of the asymptotic

observers type. We will first build an observer in order to estimate the concentrations in

reactive in a polymerization reactor of TOTAL. Then we will pursue theoretical questions

about the use of symmetries for the design of non-linear estimators.

The chemical reactor we worked on is a high pressure polymerization reactor which

produces plastics that are polymers made of two or three monomers. The estimation of

the concentrations in each reactive in the several zones of the reactor is based on a model

of the reaction. The model consists of a mass balance, an energy balance, and the use of a

chemical kinetics model. Thanks to the equations of the model, and to the measurements

of temperature and flows, we give a real-time estimation of the concentrations. It is a nonlinear

estimator, and the convergence is based on contraction properties. This estimator

was implemented and validated on the industrial unit.

The convergence of the estimator is independent of the choice of the units with which

the balances are written. We wondered if it always possible, when one builds an observer

for the concentrations of the Luenberger or extended Kalman filter type, to write correction

terms which do not depend on the units. We thus considered a more academic example : a

chemical exothermic reactor, for which the temperature and flows are measured, and the

chemical kinetics is of order one. We want to estimate the concentrations, and we want the

convergence properties to be independent of the physical units. This study showed that an

approach based on symmetries could suggest non-linear correction terms, and some change

of variables which help when it comes to studying global convergence of observers of the

Luenberger of extended Kalman filter type.

Then we developed a general method in order to write correction terms which systematically

preserve the symmetries of the system. The principal theoretical contribution of

the thesis is to give a precise method to write all the correction terms which preserve the

symmetries. The notion of the error between the true state and the estimated state is reexplored

via the notion of invariant state-error. The invariant state error dynamics has very

interesting properties. In particular it is independent of the trajectory for a left-invariant

trajectory on a Lie group. We apply the theory of invariants observers to mainly three

examples, a chemical reactor for which we build a non-linear globally convergent observer,

an example of a non-holonomic car for which we build an almost globally convergent observer, and an example of inertial navigation assisted by velocity measurements for which

we get the local convergence around any trajectory and such that the global behavior of

the error does not depend neither on the trajectory nor on the inputs.

Although the theory deals with the cases where the dimension of the symmetry group

is smaller than the dimension of the state, it seems very natural to use a similar method

to deal with the general case. This is the topic of the last part of the thesis where we look

at four examples. The synthesis of reduced observer for a class of lagrangian systems such

that all the positions are measured : the transformation group consists of all the change of

coordinates on the configuration space. The models of the Saint-Venant type on which the

Shallow-water model are based, and which are used in oceanography : the dimension of the

state space is infinite since the models use partial differential equations. The data fusion

in inertial navigation for which the measurement is an image, and thus is dimension of the

output is infinite. Finally, the parametric estimation of a two-states quantum system for

which the dimension of the group is bigger than the dimension of the state.

References

[1] N. Aghannan. Contrôle de Réacteurs de Polymérisation, observateur et invariance.

PhD thesis, Ecole des Mines de Paris, November 2003.

[2] N. Aghannan and P. Rouchon. On invariant asymptotic observers. In Proceedings

of the 41st IEEE Conference on Decision and Control, volume 2, pages 1479– 1484,

2002.

[3] N. Aghannan and P. Rouchon. An intrinsic observer for a class of lagrangian systems.

IEEE AC, 48(6) :936–945, 2003.

[4] V. Andrieu and L. Praly. On the existence of a kazantzis-kravaris/luenberger observer.

SIAM Journal on Control and Optimization, 45 :432–446, 2006.

[5] R. Aris and N.R. Amundson. An analysis of chemical reactor stability and controli,

ii,iii. Chem. Engng. Sci., 7 :121–155, 1958.

[6] V. Arnold. Méthodes Mathématiques de la Mécanique Classique. Mir Moscou, 1976.

[7] Didier Auroux. Etude de différentes méthodes d’assimilation de données pour l’environnement.

PhD thesis, Université de Nice Sophia-Antipolis, 2003.

[8] P. Becker and M. Busch. Modeling of ethylene copolymerizations with acrylate monomers.

Macromol. Theory Simul., 7 :435–446, 1998.

[9] S. Bonnabel. Invariant extended kalman filter. In Conference on Decision and Control

2007 (CDC07), 2007.

[10] S. Bonnabel, Ph. Martin, and P. Rouchon. Groupe de lie et observateur non-linéaire.

In CIFA 2006 (Conference Internationale Francophone d’Automatique), Bordeaux,

France., June 2006.

[11] S. Bonnabel, Ph. Martin, and P. Rouchon. A non-linear symmetry-preserving observer

for velocity-aided inertial navigation. In American Control Conference (ACC06),

pages 2910–2914, June 2006.

[12] S. Bonnabel, Ph. Martin, and P. Rouchon. Symmetry-preserving observers.

http ://arxiv.org/abs/math.OC/0612193, Accepted for publication in IEEE AC, Dec

2006.

[13] S. Bonnabel, M. Mirrahimi, and P. Rouchon. Observer-based hamiltonian identification

for quantum systems. Submitted, 2007.

[14] S. Bonnabel and P. Rouchon. Control and Observer Design for Nonlinear Finite and

Infinite Dimensional Systems, chapter On Invariant Observers, pages 53–66. Number

322 in Lecture Notes in Control and Information Sciences. Springer, 2005.

[15] R.W. Brockett. Remarks on finite dimensional nonlinear estimation. Asterique, 75-

76 :47–55, 1980.

[16] M. Buback and T. Dröge. High-pressure free-radical copolymerization of ethene and

methyl acrylate. Macromol. Chem. Phys., 198 :3627–3638, 1997.

[17] M. Buback, T. Dröge, A. Van Herk, and F.-O Mähling. High-pressure free-radical

copolymerization of ethene and n-butyl acrylate. Macromol. Chem. Phys., 197 :4119,

1996.

[18] M. Buback and J.Schweer. Chain-length dependence of free-radical rate coefficients,

termination rate coefficient in pure ethylene polymerization. Macromol. Chem., Rapid

commun., 9 :699–704, 1988.

[19] M. Buback and H. Lendle. The chemically initiated high pressure polymerization of

ethylene. Macromol. Chem., 184 :193–206, 1983.

[20] C. Cohen-Tannoudji, B. Diu, and F. Laloë. Mécanique Quantique, volume I& II.

Hermann, Paris, 1977.

[21] G. Creamer. Spacecraft attitude determination using gyros and quaternion measurements.

Journal of Astronautical Sciences, 44(3) :357–371, 1996.

[22] F. Fagnani and J. Willems. Representations of symmetric linear dynamical systems.

SIAM J. Control and Optim., 31 :1267–1293, 1993.

[23] J.P. Gauthier and I. Kupka. Deterministic Observation Theory and Applications.

Cambridge University Press, 2001.

[24] J.W. Grizzle and S.I. Marcus. The structure of nonlinear systems possessing symmetries.

IEEE Trans. Automat. Control, 30 :248–258, 1985.

[25] G. E. Ham. Copolymerization, High polymers Vol XVIII. Interscience publishers, New

York, 1964.

[26] T Hamel and R. Mahony. Attitude estimation on so(3) based on direct inertial measurements.

In International Conference on Robotics and Automation, ICRA2006, 2006.

[27] H. Hammouri and JP. Gauthier. Global time-varying linearization up to output injection.

SIAM J. Control Optim., 30 :1295–1310, 1992.

[28] S. Haroche and J.M. Raimond. Exploring the Quantum : Atoms, Cavities and Photons.

Oxford University Press, 2006.

[29] S. Jiang and M. Ghil. Tracking nonlinear solutions with simulated altimetric data in

a shallow water model. Journal of physical oceanography, 27 :72–95, 1997.

[30] R. Kalman and R. Bucy. New results in linear filtering and prediction theory. Basic

Eng., Trans. ASME, Ser. D,, 83(3) :95–108, 1961.

[31] A. Krener. Algebraic and Geometric Methods in Nonlinear Control Theory, chapter

The intrinsic geometry of dynamic observations, pages 77–87. D.Reidel Publishing

Company, 1986.

[32] Schwartz L. Opérateur invariants par rotation. fonctions métaharmoniques. In Séminaire

Schwarz, tome 2,exp 9, pages 1–5, 1954.

[33] L. Landau and E. Lifshitz. Mechanics. Mir, Moscow, 4th edition, 1982.

[34] W. Lohmiler and J.J.E. Slotine. On metric analysis and observers for nonlinear systems.

Automatica, 34(6) :683–696, 1998.

[35] D. Luenberger. An introduction to observers. IEEE Transaction on Automatic

Control, AC-16(6) :596-602, 16(6) :596–602, 1971.

[36] F. Stein Luft G. and Maximilian Dorn. The free-radical terpolymerisation of ethylene,

methyl acrylate and vinyl acetate at high pressure. Die Angewandte Makromolekulare

Chemie, 211 :131–140, 1993.

[37] R. Mahony, T. Hamel, and J-M Pflimlin. Complimentary filter design on the special

orthogonal group so(3). In Proceedings of the IEEE Conference on Decision and

Control, CDC05, Seville, 2005.

[38] D. H. S. Maithripala, W. P. Dayawansa, and J. M. BERG. Intrinsec observer-based

stabilization for simple mechanical systems on lie groups. SIAM J. Control and Optim.,

44 :1691–1711, 2005.

[39] S.I. Marcus. Algebraic and geometric methods in nonlinear filtering. SIAM J. Control

Optimization, 22 :817–844, 1984.

[40] Ph. Martin, P. Rouchon, and J. Rudolph. Invariant tracking. ESAIM : Control,

Optimisation and Calculus of Variations, 10 :1–13, 2004.

[41] T. McKenna. Polymer Reaction Engineering. cours de l’ENSPM, 2003.

[42] M. Mirrahimi and P. Rouchon. Mesure continue d’un ensemble statistique de systèmes

quantiques. In CIFA 2006 (Conférence Internationale Francophone d’Automatique),

Bordeaux, France, June 2006.

[43] M. Mirrahimi and P. Rouchon. Observer-based hamiltonian identification for quantum

systems. arXiv :math-ph/0703024v1, 2007.

[44] P. J. Olver. Equivalence, Invariants and Symmetry. Cambridge University Press,

1995.

[45] P. J. Olver. Classical Invariant Theory. Cambridge University Press, 1999.

[46] W. Respondek and I.A. Tall. Nonlinearizable single-input control systems do not

admit stationary symmetries. Systems and Control Letters, 46 :1–16, 2002.

[47] A.J. van der Schaft. Symmetries in optimal control. SIAM J. Control Optim., 25 :245–

259, 1987.

[48] Y.K. Song and J.W. Grizzle. The extended kalman filter as a local asymptotic observer.

Estimation and Control, 5 :59–78, 1995.

[49] M.W. Spong and F. Bullo. Controlled symmetries and passive walking,. IEEE Trans.

Automat. Control, 50 :1025–1031, 2005.

[50] Malcom P. Stevens. Polymer Chemistry, An introduction 3rd Ed. Oxford University

Press, 1999.

[51] J.J. Stoker. Differential Geometry. Wiley-Interscience, 1969.

[52] N.G Van Kampen. Stochastic process in physics and chemistry. Elsevier, North

Holland Personal Library, 1992.

[53] N.G. Van Kampen. Stochastic processes in physics and chemistry. Elsevier, 1992.

[54] J. Villermaux. Génie de la réaction chimique : conception et fonctionnement des

réacteurs. 2e éd.revue et augmentée edition, 1993.

[55] N. K. Read Zhang, S.X. and W.H. Ray. Runaway phenomena in low-density polyethylene

autoclave reactors. AIChE, 42 :10, 1996.

Statistiques de consultation

Repository Staff Only: edit this item