权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

A Study of the System Theory via Controlled Path Integral

基于受控路径积分的系统论研究

基本信息

批准号：
03805020
负责人：
SHIMA Masasuke
金额：
$ 1.28万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1992
项目状态：
已结题

项目摘要

In this research project, we studied the possibility of system theory based on the controlled path integrals. The dynamical behavior of the system is described by a vector field X=X_0+X_1u^1+...+X_mu^m defined on a manifold M as a state space. u=(u^1,...,u^m) are control inputs to the system. The control u(t) given on [t_0,t_1] deter mines the state trajectory p(t). Assuming that a differential form omega is given and integrating omega(X) along the trajectory, we obtain the controlled path integral associated with omega and X. Outputs of a system, Lyapunov function and the performance index of the optimal control can be expressed by the controlled path integrals.Therefore, the control theory can be regarded as the study of the characteristics of controlled path integrals.Integration by parts yields finite term expressions of path integrals along the trajectory, which are useful in deriving the design methods along the trajectory such as the decoupling feedback control, the structure al … More gorithm and so on. Functional series expansion formulas of the Fliess type and the Volterra type are obtained by these expressions under the assumption of analyticity. The results was extended to the time-varying systems.A necessary and sufficient condition of the output controllability of the path integral is derived and expressed by the Lie derivatives of omega along the vector fields of the strong accessibility. If omega=dh,the condition accords with the previous results. It is also observed that the condition of the output invariance is derived more easily if we use the equality of the generalized Legendre-Clebsch condition, which is equivalent with the condition that a certain differential form is exact. This fact suggests the close relation of these notions. Using the controlled path integrals to express the system is nearly equivalent to describing the system with the cotangent bundle and the Hamiltonian system, where the symplectic structure and the Hamiltonian vector fields are useful in the study of stability. The notion of Hamiltonian control system is studied in detail and its equivalence and normal form are studied. We studied the control problem of the Berry phase, which is not integrable on the fiber space. The notion of the non-dissipative control is presented and its solvability condition is given for the cascade system. The approximate design methods of the nonlinear output regulation problem and the nonlinear almost model following problem are studied and partly solved. Less

在这个研究项目中，我们研究了基于受控路径积分的系统理论的可能性。系统的动力学行为用向量场X=X_0+X_1u^1+…来描述。+ x_m在流形m上定义为状态空间。u =(^ 1,…u^m)为系统的控制输入。在[t_0，t_1]上给出的控制u(t)决定了状态轨迹p(t)。假设给出一个微分形式，沿轨迹对(X)积分，得到与和X相关的可控路径积分。系统的输出、Lyapunov函数和最优控制的性能指标可以用可控路径积分来表示。因此，控制论可以看作是对被控路径积分特性的研究。分部积分法得到沿轨迹轨迹积分的有限项表达式，可用于解耦反馈控制、结构控制、多算法等沿轨迹设计方法的推导。利用这些表达式，在可解析性的假设下，得到了Fliess型和Volterra型的函数级数展开公式。将所得结果推广到时变系统。导出了路径积分输出可控性的一个充分必要条件，并用沿强可及性向量场的李导数表示。如果ω =dh，则条件与前面的结果一致。利用广义legende - clebsch条件的等式可以更容易地推导出输出不变性的条件，该条件等价于某一微分形式是精确的条件。这一事实表明了这两个概念的密切关系。用控制路径积分来表示系统几乎等同于用协切束和哈密顿系统来描述系统，其中辛结构和哈密顿向量场在稳定性研究中是有用的。详细研究了哈密顿控制系统的概念，研究了哈密顿控制系统的等价性和范式。研究了在光纤空间上不可积的Berry相位的控制问题。提出了非耗散控制的概念，并给出了串级系统的可解性条件。研究了非线性输出调节问题和非线性概模型跟随问题的近似设计方法，并得到了部分解决。少