A Frequency-Domain Approach for Max-Plus Linear Systems

This paper surveys recent investigations on a frequency-domain approach to study max-plus linear systems, which can be used to model queueing systems, communication networks, and manufacturing systems. The challenging problem for a well-developed frequency-domain theory of such systems is the lack of inverse operations. This paper proposes a frequency-domain approach by revisiting Kalman's original realization theory for max-plus linear systems. The main advantage of Kalman's theory is that the frequency-domain method and the state-variable method merge into a single framework. Moreover, it introduces the concepts of poles and zeros as semimodules instead of point poles and zeros, which cannot be traditionally defined without inverse operations. Moreover, the pole and zero semimodules can characterize the common Petri net components in the solutions to the model matching problem.