On zero semimodules of systems over semirings with applications to queueing systems

In this paper, a zero semimodule for a linear system over a semiring is introduced. Unlike a linear system over a field, an (A, B)-controlled invariant sub-semimodule V is not equivalent to an (A, B)-controlled invariant sub-semimodule of feedback type V-FB. The solvability condition for the disturbance decoupling problem (DDP) cannot be established by knowing the maximal (A, B)-controlled invariant sub-semimodule. In this paper, an extended zero semimodule is used to find a tight upper bound for the maximal (A, B)controlled invariant sub-semimodule of feedback type V-FB(*), if one exists. Thus a connection is established between the geometric control method and the frequency domain method. This connection implies a necessary condition for the solvability of DDP. For example systems over some special semirings, this tight upper bound is equal to V F; then we have a necessary and sufficient condition for the solvability of DDP. A queueing system, described by (Max,+)-algebra, is studied to illustrate the main results.