State geometric adjustability for interval max-plus linear systems

This article investigates the state geometric adjustability for interval max-plus linear systems, which means that the state vector sequence is transformed into a geometric vector sequence by using the state feedback control. It is pointed out that the geometric state vector sequence and its common ratio are closely related to the eigenvectors and eigenvalues of the special interval state matrix, respectively. Such an interval state matrix is determined by the eigen-robust interval matrix, which has a universal eigenvector relative to a universal eigenvalue. The state geometric adjustability is characterized by the solvability of interval max-plus linear equations, and a necessary and sufficient condition for the adjustability is given. A polynomial algorithm is provided to find the state feedback matrix. Several numerical examples and simulations are presented to demonstrate the results. At the same time, the proposed method is applied for the regulation of battery energy storage systems to optimize the start time of executing tasks for all processing units in each activity.