从矩阵半张量积到逻辑控制系统  

From semi-tensor product of matrices to logical control systems

作　　者：程代展[1,2] 齐洪胜[1] 刘挺[1] 王元华[2] 

机构地区：[1]中国科学院数学与系统科学研究院系统控制重点实验室,北京100190 [2]山东大学控制科学与工程学院,济南250061

出　　处：《中国科学：数学》2016年第10期1401-1424,共24页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:61273013和61333001)资助项目

摘　　要：利用矩阵半张量积建立起来的逻辑系统控制理论已经成为控制论中不可忽视的一个新方向.本文的目的是对该理论从基本数学工具和分析方法到主要结果作一个综合性的介绍.首先介绍矩阵半张量积,并对其作了一个深入的探讨和评述.接着说明如何利用矩阵半张量积将逻辑表达式转化为代数表达式.继而较详细地介绍逻辑动态系统及其控制理论近年得到的一些主要成果,包括逻辑动态系统的拓扑结构,逻辑控制系统的能控性和能观性、稳定性与镇定、干扰解耦、最优控制、逻辑系统与逻辑控制系统的辨识.进而对逻辑系统从经典布尔逻辑到k值逻辑、多值逻辑和混合逻辑的推广作了比较细致的推演.这些结果构成了这一新学科的理论基础.此外,对相关概念、记号以及基本方法与工具作了统一的规范处理.然后,对目前的学科进展情形与其应用/潜在应用作了一个较为全面的整体介绍.对逻辑动态(控制)系统存在的未解问题与进一步的研究方向给出一个带有启发性的推介.最后,对学科发展的大方向提出一个个人的预测:逻辑控制系统与动力学控制系统的结合可望产生"信息-控制系统",它就是钱学森预言的"理论控制论",也是美国学科发展报告提出的下一代控制理论.The logical control system theory（LCST） based on semi-tensor product（STP） of matrices has become a new direction in control field. The purpose of this paper is to give a comprehensive introduction to the logical control system, from the basic mathematical tools and analysis methods to the main results. First we introduce the STP of matrices. Meanwhile, a digging discussion is made to reveal some of its insides. Then we explain how to use STP to convert a logic expression into algebraic form. Then an itemized introduction to the main achievements in analysing and control design of LCSs in last few years is presented, which consists of the topological structure of logical systems, controllability, observability, and stabilization, the stability of disturbance decoupling, optimal control, identification of LCSs. Next, the extension from classical Boolean logic to k-valued logic, to multi-valued logic, and to mixed-valued logic is explained. All these results form the theoretical foundation of this new discipline. In addition, the concepts, notations, and the fundamental method/tool have been uniformly normalized. Then some current developments of LCST and its applications or potential applications have been discussed. Some open problems and possible research directions have also been presented. Finally, a personal prediction about the development of control theory is given as： The combination of logic control systems and dynamic control systems is expected to generate a ＂information-control system＂, it is what Tsien Hsueshen predicted ＂theorical control theory＂, and is also the next generation of control theory proposed by American discipline development report.

关 键 词：矩阵半张量积 逻辑动态(控制)系统 代数状态空间表示 泛逻辑系统 信息物理系统 

分 类 号：O151.21[理学—数学]