Cycle time assignment of min-max systems

A variety of problems in computer networks, digital circuits, communication networks, manufacturing plants, etc., can be modelled as discrete event systems with maximum and minimum constraints. Systems with mixed constraints are non-linear and are called min-max systems. The cycle time vector of such a system arises as a performance measure for discrete event systems and provides the appropriate non-linear generalization of the spectral radius. This paper gives a complete account of the cycle time assignment by the state feedback for min-max systems. We describe some new definitions and results about such assignment which generalize the initial earlier works and shed new light on aspects of linear control theory. For an arbitrary min-max system, by introducing the concept of colouring graph and constructing the total condensation and its matrix representation, we give the canonical structure form. In order to design the state feedback system in which the internal structure property is unchanged and the cycle time can be assigned, we introduce and characterize the assignability, uniform state feedback and unmerged assignment for min-max systems. We present an algorithm for the unmerged assignment and illustrate our algorithm by means of an example.