Dimension Reduction and Feedback Stabilization for Max-Plus Linear Systems and Applications in VLSI Array Processors

This paper investigates the dimension reduction and feedback stabilization of max-plus linear systems and applies them to control and optimize the very large scale integration (VLSI) array processors. We introduce the weakly similar relation between max-plus matrices and the pseudoequivalent relation between autonomous max-plus linear systems, and point out that two systems are pseudoequivalent if and only if their state matrices are weakly similar. The reduced system is defined by using the pseudoequivalence, whose dimension is determined by the row rank of the original state matrix. We focus on obtaining a reduced system which maintains the stability and retains the steady-state period. An algorithm of polynomial complexity is developed to find such a reduced system. The reduced system is then used to design a state feedback controller to stabilize a max-plus linear system. Finally, we use the VLSI array processors as an example to demonstrate how the presented methods work in practical applications.