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On the computation of invariant sets for constrained nonlinear systems: an interval arithmetic approach. (English) Zbl 1086.93035

Summary: This paper deals with the computation of control invariant sets for constrained nonlinear systems. The proposed approach is based on the computation of an inner approximation of the one step set, that is, the set of states that can be steered to a given target set by an admissible control action. Based on this procedure, control invariant sets can be computed by recursion.
We present a method for the computation of the one-step set using interval arithmetic. The proposed specialized branch and bound algorithm provides an inner approximation with a given bound of the error; this makes it possible to achieve a trade off between accuracy of the computed set and computational burden. Furthermore an algorithm to approximate the one step set by an inner bounded polyhedron is also presented; this allows us to relax the complexity of the obtained set, and to make easier the recursion and storage of the sets.

MSC:

93C55 Discrete-time control/observation systems
93C57 Sampled-data control/observation systems
93B03 Attainable sets, reachability
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References:

[1] Bemporad, A., A predictive controller with artificial Lyapunov function for linear systems with input/state constraints, Automatica, 34, 1255-1260 (1998) · Zbl 0938.93524
[2] Bertsekas, D.; Rhodes, I., On the minmax reachability of target set and target tubes, Automatica, 7, 233-247 (1971) · Zbl 0215.21801
[3] Blanchini, F., Ultimate boundedness control for discrete-time uncertain systems via set-induced Lyapunov functions, IEEE Transactions on Automatic Control, 39, 428-433 (1994) · Zbl 0800.93754
[4] Blanchini, F., Set invariance in control, Automatica, 35, 1747-1767 (1999) · Zbl 0935.93005
[5] Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory (1994), SIAM: SIAM Philadelphia, PA
[6] Cannon, M.; Deshmukh, V.; Kouvaritakis, B., Nonlinear model predictive control with polytopic invariant sets, Automatica, 39, 1487-1494 (2003) · Zbl 1033.93022
[7] Cannon, M., Kouvaritakis, B., & Deshmukh, V. (2003). Enlargement of polytopic terminal region in NMPC by interpolation and partial invariance. In Proceedings of the ACC; Cannon, M., Kouvaritakis, B., & Deshmukh, V. (2003). Enlargement of polytopic terminal region in NMPC by interpolation and partial invariance. In Proceedings of the ACC · Zbl 1034.93023
[8] Chen, H.; Allgöwer, F., A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica, 34, 10, 1205-1218 (1998) · Zbl 0947.93013
[9] Chen, W., Ballance, D., & O’Reilly, J. (2001). Optimization of attraction domains of nonlinear MPC via LMI methods. In Proceedings of the ACC; Chen, W., Ballance, D., & O’Reilly, J. (2001). Optimization of attraction domains of nonlinear MPC via LMI methods. In Proceedings of the ACC
[10] Gilbert, E. G.; Tan, K., Linear systems with state and control constraints: The theory and application of maximal output admissible sets, IEEE Transactions on Automatic Control, 36, 1008-1020 (1991) · Zbl 0754.93030
[11] Hansen, E., Global optimization using interval analysis (1992), Marcel Dekker, Inc: Marcel Dekker, Inc New York · Zbl 0762.90069
[12] Jaulin, L.; Kieffer, M.; Didrit, O.; Walter, E., Applied interval analysis with examples in parameter and state estimation, robust control and robotics (2001), Springer: Springer Berlin · Zbl 1023.65037
[13] Johansen, T., Approximate explicit receding horizon control of constrained nonlinear systems, Automatica, 40, 293-300 (2004) · Zbl 1051.49018
[14] Kearfott, R., Rigorous global search: Continuous problems (1996), Kluwer: Kluwer Dordrecht, Netherlands · Zbl 0876.90082
[15] Kerrigan, E., 2000. Robust constraint satisfaction: Invariant sets and predictive control; Kerrigan, E., 2000. Robust constraint satisfaction: Invariant sets and predictive control
[16] Knueppel, O., Profil/bias—a fast interval library, Computing, 53, 3-4, 277-387 (1994) · Zbl 0808.65055
[17] Limon, D., Alamo, T., & Camacho, E. F. (2003). Robust MPC based on a contractive sequence of sets. In Proceedings of the CDC; Limon, D., Alamo, T., & Camacho, E. F. (2003). Robust MPC based on a contractive sequence of sets. In Proceedings of the CDC
[18] Limon, D., Alamo, T., & Camacho, E. F. (2005). Enlarging the domain of attraction of MPC controllers. Automatica; Limon, D., Alamo, T., & Camacho, E. F. (2005). Enlarging the domain of attraction of MPC controllers. Automatica · Zbl 1061.93045
[19] Mayne, D., Control of constrained dynamic systems, European Journal of Control, 7, 87-99 (2001) · Zbl 1293.93299
[20] Mayne, D. Q.; Schröeder, W. R., Robust time-optimal control of constrained linear systems, Automatica, 33, 12, 2103-2118 (1997) · Zbl 0910.93052
[21] Moore, R., Interval analysis (1966), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0176.13301
[22] Rump, S. (1999). Developments in reliable computing; Rump, S. (1999). Developments in reliable computing
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