Forward Reachability Of A Class Of Autonomous Interval Max-plus Systems

In our life,Many problems can be translated into a Max-Plus algebraic model,such as production,communication,traffic,etc.Max-Plus linear system and interval Max-Plus system are two commonly used models.Reachability analysis is a fundamental problem in the areas of systems theory and of denpendability analysis.Forward reachability contains reach set from a set of initial conditions and reach tube.Scholars have already used the max-plus polyhedra and maxplus cone,or the difference-bound matrices to study the reachability of Max-Plus algebra and Max-Plus linear system.When analyzing the forward reachability of interval MaxPlus system,we use addition operation and maximum operation on the interval.But the interval operations are essentially computing the upper and lower bounds of the variable,which can cause deviation.Therefore,the previous methods for reachability can not be directly used to analyze the forward reachability of interval Max-Plus system,then what we should do is to improve the method to reduce the deviation.In practical problems,the parameter of each variable is either infinite or often changing randomly in a limited range.This article mainly researches a kind of state matrix,the element of which is either infinite or a closed subset in R,in a system.In this paper,we study the period of this kind of interval matrix to simplify the computation for reachability,and research forward reachability of the autonomous interval Max-Plus system,which is determined by this kind of interval matrix.This article mainly studies reach set.Firstly,we use the obtained conclusions to calculate the cycle of this kind of matrix,the period of all the elements of the quasi-diagonal matrix and of the quasi-diagonal interval matrix,respectively.And some rules of the power of the matrix are obtained.Secondly,in autonomous interval Max-Plus system,for the interval operations and forward reachability,we find a precise method to calculate reach set??take all the points,as well as a step.By them,we can turn the complicated set into a Simpler one.And in two-dimensional space,we get some conclusions about the reach set from three kinds of initial conditions.The paper also find a simpler way-using the key points of the border.Some other conclusions of the n-dimensional space are also obtained.Finally,we take some numerical examples to show the step and the operation process of the two methods.