Max-plus matrix method and cycle time assignability and feedback stabilizability for min-max-plus systems

A variety of problems arising in nonlinear systems with timing constraints such as manufacturing plants, digital circuits, scheduling managements, etc., can be modeled as min-max-plus systems described by the expressions in which the operations minimum, maximum and addition appear. This paper applies the max-plus matrix method to analyze the cycle time assignability and feedback stabilizability of min-max-plus systems with min-max-plus inputs and max-plus outputs, which are nonlinear extensions of the systems studied in recent years. The max-plus projection matrix representation of closed-loop systems is introduced to establish some structural and quantitative relationships between reachability, observability, cycle time assignability and feedback stabilizability. The necessary and sufficient conditions for the cycle time assignability with respect to a state feedback and an output feedback, respectively, and the sufficient condition for the feedback stabilizability with respect to an output feedback are derived. Furthermore, one output feedback stabilization policy is designed so that the closed-loop systems take the maximal Lyapunov exponent as an eigenvalue. The max-plus matrix method based on max-plus algebra and directed graph is constructive and intuitive, and several numerical examples are given to illustrate this method.