Abstract
An algorithm for computing the maximal controlled invariant set and the least restrictive controller for discrete time systems is proposed. We show how the algorithm can be encoded using quantifier elimination, which leads to a semi-decidability result for definable systems. For discrete time linear systems with all sets specified by linear inequalities, a more efficient implementation is proposed using linear programming and Fourier elimination. If in addition the system is in controllable canonical form, the input is scalar and unbounded, the disturbance is scalar and bounded and the initial set is a rectangle, then the problem is decidable.
Research supported by ONR under grant N00014-97-1-0946, by DARPA under contract F33615-98-C-3614, and by ARO under grant MURI DAAH04-96-1-0341.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
D.S. Arnon, G.E. Collins, and S. McCallum. Cylindrical algebraic decomposition I: the basic algorithm. SIAM Journal on Computing, 13(4):865–877, 1984.
T. Başar and G. J. Olsder. Dynamic Non-cooperative Game Theory. Academic Press, 2nd edition, 1995.
A. Bensoussan and J.L. Menaldi. Hybrid control and dynamic programming. Dynamics of Continuous, Discrete and Impulsive Systems, (3):395–442, 1997.
M.S. Branicky, V.S. Borkar, and S.K. Mitter. A unified framework for hybrid control: Model and optimal control theory. IEEE Transactions on Automatic Control, 43(1):31–45, 1998.
P.E. Caines and Y.J. Wei. Hierarchical hybrid control systems: A lattice theoretic formulation. IEEE Transactions on Automatic Control, 43(4):501–508, April 1998.
R.N. Cernikov. The solution to linear programming problems by elimination of unknowns. Soviet Mathematics Doklady, 2:1099–1103, 1961.
V. Chandru. Variable elimination in linear constraints. The Computer Journal, 36(5):463–472, 1993.
T. Dang and O. Maler. Reachability analysis via face lifting. In Hybrid Systems: Computation and Control, vol. 1386 of LNCS, pp. 96–109. Springer Verlag, 1998.
A. Dolzmann and T. Sturm. REDLOG: Computer algebra meets computer logic. ACM SIGSAM Bulletin, 31(2):2–9, 1997.
L.B.J. Fourier. Analyse des travaux de l’Academie Royale des Sciences, pendant l’annee 1824, Partie matematique. Histoire de l’Academie Royale des Sciences de l’Institut de France 7, 1827.
G. Grammel. Maximum principle for a hybrid system via singular perturbations. SIAM Journal of Control and Optimization, 37(4):1162–1175, 1999.
M.R. Greenstreet and I. Mitchell. Integrating projections. In Hybrid Systems: Computation and Control, vol. 1386 of LNCS, pp. 159–174. Springer Verlag, 1998.
M. Heymann, F. Lin, and G. Meyer. Control synthesis for a class of hybrid systems subject to configuration-based safety constraints. In Hybrid and Real Time Systems, vol. 1201 of LNCS, pp. 376–391. Springer Verlag, 1997.
C. Lassez and J.-L. Lassez. Quantifier elimination for conjunctions of linear constraints via a convex hull algorithm. In Symbolic and Numeric Computation for Artificial Intelligence, pages 103–122. Academic Press, 1992.
J. Lewin. Differential Games. Springer-Verlag, 1994.
J. Lygeros, C. Tomlin, and S. Sastry. Controllers for reachability specifications for hybrid systems. Automatica, pages 349–370, March 1999.
O. Maler, A. Pnueli, and J. Sifakis. On the synthesis of discrete controllers for timed systems. In Theoretical Aspects of Computer Science, vol. 900 of LNCS, pp. 229–242. Springer Verlag, 1995.
A. Nerode and W. Kohn. Multiple agent hybrid control architecture. In Hybrid Systems, vol. 736 of LNCS, pp. 297–316. Springer Verlag, New York, 1993.
G. Pappas, G. Lafferriere, and S. Sastry. Hierarchically consistent control systems. In IEEE Conference on Decision and Control, pages 4336–4341, December 1998.
B. Piccoli. Necessary conditions for hybrid optimization. In IEEE Conference on. Decision and Control, pages 410–415, December 7–10 1999.
P. J. G. Ramadge and W. M. Wonham. The control of discrete event systems. Proceedings of the IEEE, Vol.77(1):81–98, 1989.
A. Seidenberg. A new decision method for elementary algebra. Annals of Mathematics, 60:387–374, 1954.
H.J. Sussmann. A maximum principle for hybrid optimal control problems. In IEEE Conference on Decision and Control, pages 425–430, December 7–10 1999.
A. Tarski. A decision method for elementary algebra and geometry. University of California Press, 1951.
W. Thomas. On the synthesis of strategies in infinite games. In Ernst W. Mayr and Claude Puech, editors, Proceedings of STACS 95, vol. 900 of LNCS, pp. 1–13. Springer Verlag, Munich, 1995.
C. Tomlin, J. Lygeros, and S. Sastry. Computing controllers for nonlinear hybrid systems. In Hybrid Systems: Computation and Control, vol. 1569 of LNCS, pp. 238–255. Springer Verlag, 1999.
R. Vidal, S. Schaffert, J. Lygeros, and S. Sastry. Controlled invariance of discrete time systems. Technical Report UCB/ERL M99/65, Electronics Research Laboratory, University of California, Berkeley, 1999.
H. Wong-Toi. The synthesis of controllers for linear hybrid automata. In IEEE Conference on Decision and Control, pages 4607–4613, December 10–12 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Vidal, R., Schaffert, S., Lygeros, J., Sastry, S. (2000). Controlled Invariance of Discrete Time Systems. In: Lynch, N., Krogh, B.H. (eds) Hybrid Systems: Computation and Control. HSCC 2000. Lecture Notes in Computer Science, vol 1790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46430-1_36
Download citation
DOI: https://doi.org/10.1007/3-540-46430-1_36
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67259-3
Online ISBN: 978-3-540-46430-3
eBook Packages: Springer Book Archive