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Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets: a modal interval analysis approach

Published online by Cambridge University Press:  23 January 2009

Jian Wan,
Josep Vehí,
Ningsu Luo and
Pau Herrero
Jian Wan
Affiliation:
Institut d'Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, 17071 Girona, Spain. jwan@eia.udg.edu; vehi@eia.udg.edu; ningsu@eia.udg.edu; pherrero@eia.udg.edu
Josep Vehí
Affiliation:
Institut d'Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, 17071 Girona, Spain. jwan@eia.udg.edu; vehi@eia.udg.edu; ningsu@eia.udg.edu; pherrero@eia.udg.edu
Ningsu Luo
Affiliation:
Institut d'Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, 17071 Girona, Spain. jwan@eia.udg.edu; vehi@eia.udg.edu; ningsu@eia.udg.edu; pherrero@eia.udg.edu
Pau Herrero
Affiliation:
Institut d'Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, 17071 Girona, Spain. jwan@eia.udg.edu; vehi@eia.udg.edu; ningsu@eia.udg.edu; pherrero@eia.udg.edu
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Abstract

A general framework for computing robust controllable sets ofconstrained nonlinear uncertain discrete-time systems as well ascontrolling such complex systems based on the computed robustcontrollable sets is introduced in this paper. The addressedone-step control approach turns out to be a robust model predictivecontrol scheme with feasible unit control horizon and contractiveconstraint. The solver of 1-dimensional quantified set inversion inmodal interval analysis is extended to 2-dimensional cases forcomputing robust controllable sets off-line with a clear semanticinterpretation, where both universal and existential quantifiers areconcerned simultaneously. An interval-based solver of constrainedminimax optimization is also proposed to compute one-step controlinputs online in a reliable way, which guarantee to drive the systemstate contractively along the computed robust controllable sets to aselected terminal robust control invariant set.

Keywords

Nonlinearityuncertaintyconstraintsrobustcontrollable setquantified set inversionminimax optimizationinterval analysismodal intervals

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 1 , January 2009, pp. 189 - 204
Copyright
© EDP Sciences, SMAI, 2008

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References

Blanchini, F., Set invariance in control. Automatica 35 (1999) 1747–1767. CrossRef
Bravo, J.M., Limon, D., Alamo, T. and Camacho, E.F., On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach. Automatica 41 (2005) 1583–1589. CrossRef
Cannon, M., Deshmukh, V. and Kouvaritakis, B., Nonlinear model predictive control with polytopic invariant sets. Automatica 39 (2003) 1487–1494. CrossRef
Chen, H. and Allgower, F., A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34 (1998) 1205–1217. CrossRef
Gardenes, E., Sainz, M.A., Jorba, L., Calm, R., Estela, R., Mielgo, H. and Trepat, A., Modal intervals. Reliab. Comput. 7 (2001) 77–111. CrossRef
E. Hansen, Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992).
Herrero, P., Sainz, M.A., Vehí, J. and Jaulin, L., Quantified set inversion algorithm with applications to control. Reliab. Comput. 11 (2005) 369–382. CrossRef
L. Jaulin, M. Kieffer, O. Didrit and E. Walter, Applied Interval Analysis. Springer, London (2001).
E. Kaucher, Interval analysis in the extended interval space IR, Comput. Suppl. 2. Springer, Heidelberg (1980) 33–49.
E.C. Kerrigan, Robust Constraint Satisfaction: Invariant Sets and Predictive Control. Ph.D. thesis, University of Cambridge, USA (2000).
Klamaka, J., Controllability of nonlinear discrete systems. Internat. J. Appl. Math. Comput. Sci. 12 (2002) 173–180.
Kühn, W., Rigorously computed orbits of dynamical systems without the wrapping effect. Computing 61 (1998) 47–67. CrossRef
D. Limon, T. Alamo and E.F. Camacho, Robust MPC control based on a contractive sequence of sets, in Proc. 42nd IEEE Conf. Dec. Control (2003) 3706–3711.
Mayne, D.Q. and Schroeder, W.R., Robust time-optimal control of constrained linear systems. Automatica 33 (1997) 2103–2118. CrossRef
R. Moore, Interval Analysis. Prentice Hall, Englewood Cliffs, NJ (1966).
S.V. Rakovic, E.C. Kerrigan and D.Q. Mayne, Reachability computations for constrained discrete-time systems with state- and input-dependent disturbances, in Proc. 42nd IEEE Conf. Dec. Control (2003) 3905–3910.
Shary, S.P., A new technique in systems analysis under interval uncertainty and ambiguity. Reliab. Comput. 8 (2002) 321–418. CrossRef
Sirotin, A.N. and Formal'skii, A.M., Reachability and controllability of discrete-time systems under control actions bounded in magnitude and norm. Autom. Remote Control 64 (2003) 1844–1857. CrossRef