Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets : a modal interval analysis approach
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 189-204.

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.

DOI : 10.1051/cocv:2008025
Classification : 65G40, 93B05, 93B51
Mots clés : nonlinearity, uncertainty, constraints, robust controllable set, quantified set inversion, minimax optimization, interval analysis, modal intervals
@article{COCV_2009__15_1_189_0,
     author = {Wan, Jian and Veh{\'\i}, Josep and Luo, Ningsu and Herrero, Pau},
     title = {Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets : a modal interval analysis approach},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {189--204},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {1},
     year = {2009},
     doi = {10.1051/cocv:2008025},
     mrnumber = {2488575},
     zbl = {1158.93010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008025/}
}
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
PY  - 2009
SP  - 189
EP  - 204
VL  - 15
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008025/
DO  - 10.1051/cocv:2008025
LA  - en
ID  - COCV_2009__15_1_189_0
ER  - 
%0 Journal Article
%A Wan, Jian
%A Vehí, Josep
%A Luo, Ningsu
%A Herrero, Pau
%T Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets : a modal interval analysis approach
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 189-204
%V 15
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008025/
%R 10.1051/cocv:2008025
%G en
%F COCV_2009__15_1_189_0
Wan, Jian; Vehí, Josep; Luo, Ningsu; Herrero, Pau. Control of constrained nonlinear uncertain discrete-time systems via robust controllable sets : a modal interval analysis approach. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 189-204. doi : 10.1051/cocv:2008025. http://www.numdam.org/articles/10.1051/cocv:2008025/

[1] F. Blanchini, Set invariance in control. Automatica 35 (1999) 1747-1767. | MR | Zbl

[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. Automatica 41 (2005) 1583-1589. | MR | Zbl

[3] M. Cannon, V. Deshmukh and B. Kouvaritakis, Nonlinear model predictive control with polytopic invariant sets. Automatica 39 (2003) 1487-1494. | MR | Zbl

[4] H. Chen and F. Allgower, A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34 (1998) 1205-1217. | MR | Zbl

[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. | MR | Zbl

[6] E. Hansen, Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992). | MR | Zbl

[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. | MR | Zbl

[8] L. Jaulin, M. Kieffer, O. Didrit and E. Walter, Applied Interval Analysis. Springer, London (2001). | MR | Zbl

[9] E. Kaucher, Interval analysis in the extended interval space IR, Comput. Suppl. 2. Springer, Heidelberg (1980) 33-49. | MR | Zbl

[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. | MR | Zbl

[12] W. Kühn, Rigorously computed orbits of dynamical systems without the wrapping effect. Computing 61 (1998) 47-67. | MR | Zbl

[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. Automatica 33 (1997) 2103-2118. | MR | Zbl

[15] R. Moore, Interval Analysis. Prentice Hall, Englewood Cliffs, NJ (1966). | MR | Zbl

[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. | MR | Zbl

[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 Control 64 (2003) 1844-1857. | MR

Cité par Sources :