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

Abstract

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 on-going research. The geometric control theory in particular was proposed as the future direction for max-plus linear systems by Cohen et al. [Cohen, G., Gaubert, S. and Quadrat, J.P. (1999), ‘Max-plus Algebra and System Theory: Where we are and Where to Go Now’, Annual Reviews in Control, 23, 207--219]. This article generalises R.E. Kalman's 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.

Keywords:

• max-plus algebra
• (A, B)-controlled invariance
• disturbance decoupling

Acknowledgements

The author expresses her gratitude to her former advisor Prof. Michael K. Sain, who passed away in 2009, whose work was supported by the Frank M. Freimann Chair in Electrical Engineering, Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA. This work was supported by 2010--2011 Seed Grants for Transitional and Exploratory Projects (STEP) at Southern Illinois University Edwardsville.