Towards geometric control of max-plus linear systems with applications to queueing networks

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. The max-plus linear systems have been studied for almost three decades, however, a well-established system theory on such specific systems is still an ongoing research. The geometric control theory in particular was proposed as the future direction for max-plus linear systems by Cohen et al. abstract realisation theory for traditional linear systems over fields to max-plus linear systems. The new generalised version of Kalman's abstract realisation theory not only provides a more concrete state space representation other than just a 'set-theoretic' representation for the canonical realisation of a transfer function, but also leads to the computational methods for the controlled invariant semimodules in the kernel and the equivalence kernel of the output map. These controlled invariant semimodules play key roles in the standard geometric control problems, such as disturbance decoupling problem and block decoupling problem. A queueing network is used to illustrate the main results in this article. 1. Introduction Traditional system theory focuses on linear time-invariant systems whose coefficients belong to a field. Recent applications in communication networks (Le Boudec and Thiran 2002), genetic regulatory networks (De Jong 2002) and queueing systems (Baccelli, Cohen, Olsder, and Quadrat 1992) require a new system theory for linear time-invariant systems with coefficients in a semiring. A semiring is understood as a set of objects without inverses with respect to the corresponding operations, for example, the max-plus algebra (Baccelli et al. 1992), the min-plus algebra (Le Boudec and Thiran 2002) and the Boolean semiring (Golan 1999). Especially, max-plus linear systems have been studied by researchers for the past three decades, for example, Another new research area for max-plus linear systems is to establish the geometric control theory (Wonham 1979) for systems over semirings as predicted in Cohen, Gaubert, …