A method for efficiently representing the state equation in a class of max-plus linear systems is proposed. We introduce a construct referred to as 'cell' in which the list of possible longest paths is stored. By imposing interval constraints on the system parameters, we can reduce the complexity of the state equation. The proposed method would be useful in scheduling applications for systems with adjustable system parameters.

This letter extends the existent MPL (Max-Plus Linear) state-space representation and proposes a new form that can account for both capacity and order constraints. It is often essential to consider these factors when applying the MPL approach to scheduling problems for production or transportation systems. The derived form is a type of augmented state-representation and can contribute to obtaining the earliest start and completion times for processes in installed facilities.

We develop an algorithm for a controller design method for Max-Plus Linear (MPL) systems with selective parameters. Since the conventional algorithm we proposed requires high computational load when the prediction horizon is large, two methods for reducing the calculation time are proposed. One is based upon the branch-and-bound method, and the other is to reuse the optimal solution. The effectiveness of these two methods is confirmed through numerical simulation.

We propose Max-Plus Linear (MPL) systems with selective parameters that can describe a certain class of Timed Petri nets (TPN). In this class, selector and joint places are incorporated with Single-Input and Single-Output Timed Event Graph (SISO TEG) subnets. We confirm that the proposed controller effectively works taking into account practical constraints through a numerical example.