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

Jian Wan; Josep Vehí; Ningsu Luo; Pau Herrero

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 1, page 189-204
  • ISSN: 1292-8119

Abstract

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A general framework for computing robust controllable sets of constrained nonlinear uncertain discrete-time systems as well as controlling such complex systems based on the computed robust controllable sets is introduced in this paper. The addressed one-step control approach turns out to be a robust model predictive control scheme with feasible unit control horizon and contractive constraint. The solver of 1-dimensional quantified set inversion in modal interval analysis is extended to 2-dimensional cases for computing robust controllable sets off-line with a clear semantic interpretation, where both universal and existential quantifiers are concerned simultaneously. An interval-based solver of constrained minimax optimization is also proposed to compute one-step control inputs online in a reliable way, which guarantee to drive the system state contractively along the computed robust controllable sets to a selected terminal robust control invariant set.

How to cite

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Wan, Jian, et al. "Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets: a modal interval analysis approach." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 189-204. <http://eudml.org/doc/250563>.

@article{Wan2009,
abstract = { A general framework for computing robust controllable sets of constrained nonlinear uncertain discrete-time systems as well as controlling such complex systems based on the computed robust controllable sets is introduced in this paper. The addressed one-step control approach turns out to be a robust model predictive control scheme with feasible unit control horizon and contractive constraint. The solver of 1-dimensional quantified set inversion in modal interval analysis is extended to 2-dimensional cases for computing robust controllable sets off-line with a clear semantic interpretation, where both universal and existential quantifiers are concerned simultaneously. An interval-based solver of constrained minimax optimization is also proposed to compute one-step control inputs online in a reliable way, which guarantee to drive the system state contractively along the computed robust controllable sets to a selected terminal robust control invariant set. },
author = {Wan, Jian, Vehí, Josep, Luo, Ningsu, Herrero, Pau},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinearity; uncertainty; constraints; robust controllable set; quantified set inversion; minimax optimization; interval analysis; modal intervals; nonlinearity; robust controllable set; interval analysis},
language = {eng},
month = {1},
number = {1},
pages = {189-204},
publisher = {EDP Sciences},
title = {Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets: a modal interval analysis approach},
url = {http://eudml.org/doc/250563},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Wan, Jian
AU - Vehí, Josep
AU - Luo, Ningsu
AU - Herrero, Pau
TI - Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets: a modal interval analysis approach
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2009/1//
PB - EDP Sciences
VL - 15
IS - 1
SP - 189
EP - 204
AB - A general framework for computing robust controllable sets of constrained nonlinear uncertain discrete-time systems as well as controlling such complex systems based on the computed robust controllable sets is introduced in this paper. The addressed one-step control approach turns out to be a robust model predictive control scheme with feasible unit control horizon and contractive constraint. The solver of 1-dimensional quantified set inversion in modal interval analysis is extended to 2-dimensional cases for computing robust controllable sets off-line with a clear semantic interpretation, where both universal and existential quantifiers are concerned simultaneously. An interval-based solver of constrained minimax optimization is also proposed to compute one-step control inputs online in a reliable way, which guarantee to drive the system state contractively along the computed robust controllable sets to a selected terminal robust control invariant set.
LA - eng
KW - Nonlinearity; uncertainty; constraints; robust controllable set; quantified set inversion; minimax optimization; interval analysis; modal intervals; nonlinearity; robust controllable set; interval analysis
UR - http://eudml.org/doc/250563
ER -

References

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  1. F. Blanchini, Set invariance in control. Automatica35 (1999) 1747–1767.  Zbl0935.93005
  2. J.M. Bravo, D. Limon, T. Alamo and E.F. Camacho, On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach. Automatica41 (2005) 1583–1589.  Zbl1086.93035
  3. M. Cannon, V. Deshmukh and B. Kouvaritakis, Nonlinear model predictive control with polytopic invariant sets. Automatica39 (2003) 1487–1494.  Zbl1033.93022
  4. H. Chen and F. Allgower, A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica34 (1998) 1205–1217.  Zbl0947.93013
  5. E. Gardenes, M.A. Sainz, L. Jorba, R. Calm, R. Estela, H. Mielgo and A. Trepat, Modal intervals. Reliab. Comput.7 (2001) 77–111.  
  6. E. Hansen, Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992).  Zbl0762.90069
  7. P. Herrero, M.A. Sainz, J. Vehí and L. Jaulin, Quantified set inversion algorithm with applications to control. Reliab. Comput.11 (2005) 369–382.  Zbl1081.65519
  8. L. Jaulin, M. Kieffer, O. Didrit and E. Walter, Applied Interval Analysis. Springer, London (2001).  Zbl1023.65037
  9. E. Kaucher, Interval analysis in the extended interval space IR, Comput. Suppl.2. Springer, Heidelberg (1980) 33–49.  Zbl0419.65031
  10. E.C. Kerrigan, Robust Constraint Satisfaction: Invariant Sets and Predictive Control. Ph.D. thesis, University of Cambridge, USA (2000).  
  11. J. Klamaka, Controllability of nonlinear discrete systems. Internat. J. Appl. Math. Comput. Sci.12 (2002) 173–180.  
  12. W. Kühn, Rigorously computed orbits of dynamical systems without the wrapping effect. Computing61 (1998) 47–67.  Zbl0910.65052
  13. 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.  
  14. D.Q. Mayne and W.R. Schroeder, Robust time-optimal control of constrained linear systems. Automatica33 (1997) 2103–2118.  Zbl0910.93052
  15. R. Moore, Interval Analysis. Prentice Hall, Englewood Cliffs, NJ (1966).  Zbl0176.13301
  16. 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.  
  17. S.P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity. Reliab. Comput.8 (2002) 321–418.  Zbl1020.65029
  18. A.N. Sirotin and A.M. Formal'skii, Reachability and controllability of discrete-time systems under control actions bounded in magnitude and norm. Autom. Remote Control64 (2003) 1844–1857.  Zbl1209.93017

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