【作者】 陈磊；

【导师】 张国山；

【作者基本信息】 天津大学 ， 控制科学与工程， 2016， 硕士

【摘要】 对称系统是一类具有特殊结构的系统,有着广泛的应用背景,如电力系统,黏性材料系统,化学反应和空间结构系统等。目前,人们在对称系统理论领域已取得了很多成果。广义系统是一类更一般化系统,大量地出现在许多实际的系统模型中,是许多实际动态系统的自然表达,如经济系统,化学系统和微电子机械系统等。因此,广义系统也得到了国内外学者们的广泛关注,并取得了长足进展。本文针对当前对称系统、广义系统理论的研究现状,重点研究了通过反馈使非对称的正常系统和广义系统转化为对称系统的问题。首先,研究了矩阵方程的A~TX±X~T=B求解问题。提出了利用奇异值分解的方法求解矩阵方程A~TX±X~T=B。首先,给出了矩阵方程A~TX±X~T=B有解的充分必要条件,且给出了有解时解的一般表达式;然后,给出了矩阵A分别为实矩阵、行满秩矩阵、列满秩矩阵和非奇异矩阵时,此矩阵方程有解的充分必要条件和有解时解的一般表达式。然后,研究了基于状态反馈的非对称系统的对称化问题。首先,研究了通过状态反馈使非对称正常系统转化为对称系统的问题,给出了存在满足条件的状态反馈的充分必要条件和此时状态反馈控制律的参数化表达式,并给出了反馈后闭环系统稳定的充分必要条件。然后研究了通过状态反馈和变化矩阵使非对称广义系统转化为对称系统的问题,给出了存在满足条件的状态反馈控制和变换矩阵的充分必要条件和此时状态反馈控制律和变换矩阵的参数化表达式,且给出了反馈后闭环系统容许的充分必要条件。最后,研究了基于输出反馈和动态补偿的非对称系统的对称化问题。首先,提出了两种方法解决通过输出反馈使非对称系统转化为对称系统的问题,分别给出了存在满足条件的输出反馈控制的充分必要条件和此时输出反馈控制律的参数化表达式,且对于第一种方法给出了反馈后闭环系统稳定的充分必要条件。然后,通过将动态补偿控制等效为输出反馈控制,得到了通过动态补偿控制使非对称系统转化为对称系统的相应结论。更多还原

【Abstract】 Symmetric systems is a class of systems which have special structure and comprehensive application background,such as electrical and power networks,viscoelastic materials,chemical reaction,structural systems and so on.At present,there are many achievements in theory of symmetric systems.Descriptor systems are more general systems.They appear in many practical systems,which are natural mathematical representation for those systems.Examples of such systems include economical systems,chemical systems and micro-electro-mechanical systems.Therefore descriptor systems have received considerable attentions of scholars and made great progress.Based on the current research situation of symmetric system and descriptor system,this thesis focuses on the problem of the problem,that an asymmetric general or descriptor system is transformed into a state-space symmetric system through feedback control.First,the problem of solution to matrix equation A~TX±X~T=B is discussed.We propose a method to solve the matrix equation A~TX±X~T=B through singular value decomposition.Firstly,the necessary and sufficient condition for the existence of the solution to the matrix equation A~TX±X~T=B and its solution expression is given.Secondly,the necessary and sufficient condition for the existence of the solution to the matrix equation and its solution expression is given when matrix A is real,row full rank,column full rank and nonsingular respectively.Second,the problem that the normal and descriptor asymmetric systems are symmetrized through state feedback is investigated.Firstly,the problem that the normal asymmetric systems are transformed into state-space symmetric systems through state feedback control is concerned.The necessary and sufficient condition for the existence of state feedback control which meets the condition is given,and the expressions of state feedback control is obtained.The necessary and sufficient condition for stability of closed-loop systems is also given.Secondly,the problem that the descriptor asymmetric systems are transformed into state-space symmetric systems through state feedback control and transformation matrix is discussed.The necessary and sufficient condition for the existence of state feedback control and transformation matrix which meet the condition is given.And the expressions of state feedback control and transformation matrix are obtained.The necessary and sufficient condition for admissibility of closed-loop systems is also given.Third,the problem of that the asymmetric systems are symmetrized through output feedback and dynamic compensator is studied.Firstly,two methods to solve the problem that the asymmetric systems are symmetrized through output feedback control are given.The necessary and sufficient condition for the existence of output feedback control which meets the condition is given with the two methods,and the expressions of output feedback control is obtained.For the first method,the necessary and sufficient condition for stability of closed-loop systems is also given.Afterwards,dynamic feedback control is equivalent to output feedback control.Corresponding conclusions about that the asymmetric systems are symmetrized through dynamic compensation are obtained.更多还原