Abstract
A specification of the linear system theory over dioids is proposed for periodic systems. Using the conventional periodic system theory as a guideline, we study periodic systems for which the underlying algebraic structure is a dioid. The focus is on representations (impulse response and state model) associated with such systems, the properties of these representations as well as the state space realization.
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. Wiley.
Bittanti, S. 1996. Deterministic and stochastic linear periodic systems. In: Springer (ed.), Time Series and Linear Systems, pp. 141–182.
Blondel, V., and Portier, N. 1999. The minimal realization problem in max-plus algebra is NP-hard. In Notes de lectures des Journèes thèmatiques Algebres tropicales-ALAPEDES. Paris: ENS.
Bolzern, P., Colaneri, P., and Scatolini, R. 1986. Zeros of discrete-time linear periodic systems. IEEE Transactions on Automatic Control 31: 1057–1058.
Braker, H. 1993. Algorithms and applications in timed discrete event systems. Ph.D. thesis, Delft University of Technology.
Bru, R., Coll, C., Hernandez, V., and Sanchez, E. 1997. Geometrical conditions for the reachability and realizability of positive periodic discrete systems. Linear Algebra and its Applications 256: 109–124.
Cohen, G., Moller, P., Quadrat, J. P., and Viot, M. 1989. Algebraic tools for the performance evaluation of discrete event systems. IEEE Proceedings: Special issue on Discrete Event Systems 77(1).
Colaneri, P., and Kucera, V. 1997. The model matching problem for periodic discrete-time systems. IEEE Transactions on Automatic Control 42: 1472–1476.
Cottenceau, B., Hardouin, L., Boimond, J. L., and Ferrier, J. L. 1999. Synthesis of greatest linear feedback for TEG in dioids. IEEE Trans. on Automatic Control 44(6): 1258–1262.
Cottenceau, B., Hardouin, L., Boimond, J. L., and Ferrier, J. L. 2001. Model reference control for timed event graphs in dioids. (37): 1451–1458.
Cuninghame-Green, R. 1979. Minimax Algebra. Number 166 in Lecture notes in Economics and Mathematical Systems. Springer.
De Schutter, B. 2000. On the ultimate behavior of the sequence of consecutive powers of a matrix in the max-plus algebra. Linear Algebra and Its Applications 307(1–3): 103–117.
De Schutter, B., and De Moor, B. 1995. Minimal realization in the max algebra is an extended linear complementarity problem. Systems and Control Letters 25(2): 103–111.
Gaubert, S. 1992. Théorie des systèmes linéaires dans les dioïdes. Ph.D. thesis, Ecole des Mines de Paris.
Gaubert, S. 1994. On rational series in one variable over certain dioids. Tech. Rep. 2162, INRIA.
Kailath, T. 1980. Linear Systems. Prentice Hall.
Kamen, E. W. 1996. Fundamentals of linear time-varying systems. In W. S. Levine (ed.), The Control Handbook. IEEE Press and CRC Press.
Lahaye, S. 2000. Contribution á l'ètude des systèmes linèaires non stationnaires dans l'algèbre des dioïdes. Ph.D. thesis, ISTIA, Universitè d'Angers.
Lahaye, S., Boimond, J. L., and Hardouin, L. 1999a. Optimal control of (min, +) linear time-varying systems. In Proceedings of PNPM'99. Zaragoza.
Lahaye, S., Boimond, J. L., and Hardouin, L. 1999b. Timed event graphs with variable resources: performance evaluation, modeling in (min, +) algebra. JESA 33(8–9): 1015–1032.
Menguy, E., Boimond, J. L., Hardouin, L., and Ferrier, J. L. 2000. A first step towards adaptive control for linear systems in max algebra. J. Discrete Event Dynamic Systems 10(4): 347–367.
Misra, P. 1996. Time-invariant representation of discrete periodic systems. Automatica 32: 267–272.
Olsder, G. J. 1986. On the characteristic equation and minimal realizations for discrete event dynamic systems. In A. Bensoussan and J. Lions (eds.), Analysis and Optimization of Systems. Number 83 in Lecture notes in Control and Information Sciences, pp. 189–201. Springer.
Olsder, G. J., and Schutter, B. D. 1999. The minimal realization problem in the max-plus algebra. In V. Blondel, E. Sontag, M. Vidyasagar, and J. Willems (eds.), Open Problems in Mathematical Systems and Control Theory. London: Springer.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lahaye, S., Boimond, J.L. & Hardouin, L. Linear Periodic Systems Over Dioids. Discrete Event Dynamic Systems 14, 133–152 (2004). https://doi.org/10.1023/B:DISC.0000018568.03525.93
Issue Date:
DOI: https://doi.org/10.1023/B:DISC.0000018568.03525.93