极大代数上线性系统的最小实现

研究极大代数上线性系统单输入单输出的最小实现问题. 给出了存在2维最小实现的充要条件,该条件是用无穷序列{gi}^∞0元素之间的关系描述的,因而容易判断;同时,用涂奉生提出的结构标准形和最小实现算法给出了2维最小实现的构造方法,从而完全解决了2维最小实现问题.作为以上结果的推论,指出了涂奉生猜想在维数小于等于2的情况下成立,并通过反例说明涂奉生猜想在大于2维的情况下不成立.

Minimal Realization in Linear System of Max-algebra

The minimal realization of a low dimensional SISO linear system in the max-algebra is studied. The necessary and sufficient condition for the existence of 2-dimensional minimal realization is given, which is described by the relation of elements of the infinite sequence and is easy to check. The method of constructing a 2-dimension minimal realization is presented through the structural standardization and arithmetic of minimal realization developed by Fengsheng Tu, by which the problem of 2-dimension minimal realization is solved completly. Consequently it is proved that the conjecture of Fengsheng Tu is right with the dimension of the minimal realization no more than 2. Finally, a counter-example shows that the conjecture of Fengsheng Tu dose not hold with the dimensions of minimal realization bigger than 2.