一类自治的区间极大—加系统的向前可达性

发布时间：2018-06-01 21:38

本文选题：区间矩阵 + 周期　；参考：《河北师范大学》2017年硕士论文

【摘要】：在生产制造、通讯、运输等很多方面,我们都可以利用极大-加代数模型来解决相关问题,其中极大-加线性系统和区间极大-加系统是常用的两个模型.可达性分析是系统理论和可靠性分析等领域中的基本问题,向前可达性包括从某个状态开始继续向前发展的可达集和可达管.学者们已经利用极大-加多面体和极大-加锥体,或利用差分有界矩阵研究了极大-加代数和极大-加线性系统的可达性.而研究区间极大-加系统的向前可达性,用到了区间的加法和取极大两种运算.这两种运算的本质是变量上下界的运算,会带来误差.因此,以往研究可达性的方法不能直接用于解决区间极大-加系统中向前可达性的问题,需要改善运算方法以减小误差.而实际问题中,各变量的参数或无限制或经常在某一个有限区间内随机变动.本文主要研究这样一类矩阵,其元素或者为无限、或者为实数集上的闭区间.研究该类区间矩阵的周期,以简化区间极大-加系统的向前可达性的计算,并研究由这类区间矩阵确定的、自治的区间极大-加系统的向前可达性,着重研究其可达集.首先,利用已有结论计算这类区间矩阵的周期,进而研究一类准对角矩阵和准对角区间矩阵的所有元素的周期,得到了关于幂矩阵的一些性质.其次,在自治的区间极大-加系统中,针对区间运算和向前可达性的特点,本文找到了一个准确计算可达集的方法全部取点法,以及计算可达集的步骤.由此可以把比较复杂的初始状态集,转化为更为方便计算的初始状态集.在二维自治的区间极大-加系统中,本文得到了三类初始状态的系统可达集的计算规律,并给出了一个二维情况下更为简单的方法——局部取点法.进而还研究了9)维情况时可达集的一些计算方法.最后,通过数值例子展示了计算可达集的步骤和两种计算方法的运算过程.
[Abstract]:In many aspects, such as manufacturing, communication, transportation and so on, we can use the max-plus algebraic model to solve the related problems, in which the max-plus linear system and the interval max-additive system are two commonly used models. Reachability analysis is a fundamental problem in the field of system theory and reliability analysis. Forward reachability includes reachability sets and reachability tubes that continue to evolve from a certain state. Scholars have studied the reachability of Max-plus Algebra and Max-plus Linear Systems by using the maximal-plus polyhedron and the maximum-plus cone or by using the difference bounded matrix. In order to study the forward reachability of interval-maximum-additive systems, two operations, the addition of interval and the maximization of interval, are used. The essence of these two operations is the operation of upper and lower bounds of variables, which will bring errors. Therefore, the previous research methods of reachability can not be directly used to solve the problem of forward reachability in interval-maximum-additive systems, so it is necessary to improve the operation method to reduce the error. However, in practical problems, the parameters of each variable change randomly in a finite interval. In this paper, we study a class of matrices whose elements are either infinite or closed intervals on the set of real numbers. The period of the interval matrix is studied to simplify the calculation of the forward reachability of the interval maximal additive system, and the forward reachability of the autonomous interval maximum additive system determined by this kind of interval matrix is studied, with emphasis on its reachability set. First, the period of this kind of interval matrix is calculated by using the existing results, and then the period of all elements of a class of quasi-diagonal matrices and quasi-diagonal interval matrices is studied, and some properties of the power matrix are obtained. Secondly, according to the characteristics of interval operation and forward reachability in autonomous interval Max-Additive system, an accurate method of calculating reachability set is found in this paper, and the steps of calculating reachability set are given. Thus, the more complex initial state set can be transformed into a more computable initial state set. In this paper, we obtain the calculation law of reachability set of three kinds of initial state systems in the interval maximum additive system of two dimensional autonomy, and give a simpler method in two dimensional case, the local point taking method. Furthermore, some calculation methods for the reachable set in the case of 9) dimension are studied. Finally, numerical examples are given to show the steps of computing reachability set and the operation process of two methods.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.2

【相似文献】