This paper examines the properties of the greatest subsolution for linear equations in the max-plus algebra. The greatest subsolution is a relaxed solution of the linear equations, and gives a unified and reasonable solution whether there exists a strict solution or not. Accordingly, it forms part of a key algorithm for deriving a control law in the field of controller design, and some effective controllers based on the greatest subsolution have been proposed. However, there remain several issues to be discussed regarding the properties of the greatest subsolution. Hence, the main focus of this paper is on the following fundamental properties: 1) Formulation as an optimization problem, 2) Uniqueness of the greatest subsolution, 3) Necessary and sufficient condition for the correspondence of the greatest subsolution with the strict solution. These results could provide flexibility of the controller design based on the greatest subsolution, and facilitate the performance evaluation of the controller. Finally, the uniqueness of the strict solution of the linear equations is examined, and it is confirmed through illustrative examples.

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.

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.

A ripple-free dual-rate control system is designed for a single-input single-output dual-rate system, in which the sampling interval of a plant output is longer than the holding interval of a control input. The dual-rate system is converged to a multi-input single-output single-rate system using the lifting technique, and a control system is designed based on an error system using the steady-state variable. Because the proposed control law is designed so that the control input is constant in the steady state, the intersample output as well as the sampled output converges to the set-point without both steady-state error and intersample ripples when there is neither modeling nor disturbance. Furthermore, in the proposed method, a two-degree-of-freedom integral compensation is designed, and hence, the transient response is not deteriorated by the integral action because the integral action is canceled when there is neither modeling nor disturbance. Moreover, in the presence of the modeling error or disturbance, the integral compensation is revealed, and hence, the steady-state error is eliminated on both the intersample and sampled response.