Abstract
In this paper we show that finding solutions of a system of multivariate polynomial equalities and inequalities in the max algebra is equivalent to solving an Extended Linear Complementarity Problem. This allows us to find all solutions of such a system of multivariate polynomial equalities and inequalities and provides a geometrical insight in the structure of the solution set. We also demonstrate that this enables us to solve many important problems in the max algebra and the max-min-plus algebra such as matrix decompositions, construction of matrices with a given characteristic polynomial, state space transformations and the (minimal) state space realization problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baccelli, F., Cohen, G., Olsder, G. J., and Quadrat, J. P. 1992.Synchronization and Linearity. New York: John Wiley & Sons.
Braker, J. G. 1993. An extended algorithm for performance evaluation of timed event graphs.Proc. 2nd European Control Conference, Groningen, The Netherlands, pp. 524–529.
Cottle, R. W., Pang, J. S., and Stone, R. E. 1992.The Linear Complementarity Problem. Boston: Academic Press.
Cuninghame-Green, R. A. 1979.Minimax Algebra. Berlin: Springer-Verlag.
De Schutter, B. and De Moor, B. 1993a. The extended linear complementarity problem, ESAT/SISTA, Katholieke Universiteit Leuven, Leuven, Belgium, Tech. Rep. 93-69. Submitted for publication.
De Schutter, B. and De Moor, B. 1994a. The characteristic equation and minimal state space realization of SISO systems in the max algebra.Proc. 11th Int. Conf. on Analysis and Optimization of Systems, Sophia-Antipolis, France, pp. 273–282.
De Schutter, B. and De Moor, B. 1994b. Minimal realization in the max algebra, ESAT/SISTA, Katholieke Universiteit Leuven, Leuven, Belgium, Tech. Rep. 94-29.
De Schutter, B. and De Moor, B. 1995. Minimal realization in the max algebra is an extended linear complementarity problem,Systems & Control Letters, 25:2, 103–111.
Gaubert, S. 1992.Théorie des Systèmes Linéaires dans les Dioïdes. PhD thesis, Ecole Nationale Supérieure des Mines de Paris, France.
Moller, P. 1986. Théorème de Cayley-Hamilton dans les dioïdes et application à l'étude des systèmes à événements discrets.Proc. 7th Int. Conf. on Analysis and Optimization of Systems, Antibes, France, pp. 215–226.
Motzkin, T. S., Raiffa, H., Thompson, G. L., and Thrall, R. M. 1953. The double description method. InContributions to the theory of games, no. 28 in Annals of Math. Studies. Princeton: Princeton University Press, pp. 51–73.
Olsder, G. J. 1991. Eigenvalues of dynamic max-min systems.Discrete Event Dynamic Systems: Theory and Applications 1: 177–207.
Author information
Authors and Affiliations
Additional information
Research assistant with the N.F.W.O. (Belgian National Fund for Scientific Research).
Senior research associate with the N.F.W.O.
Rights and permissions
About this article
Cite this article
de Schutter, B., de Moor, B. A method to find all solutions of a system of multivariate polynomial equalities and inequalities in the max algebra. Discrete Event Dyn Syst 6, 115–138 (1996). https://doi.org/10.1007/BF01797235
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01797235