A Study of the System Theory via Controlled Path Integral

在本研究项目中，我们研究了基于受控路径积分的系统理论的可能性。定义在流形M上的向量场X=X_0+X_1u^1+…+X_u^m作为状态空间来描述系统的动力学行为。U=(u^1，...，u^m)是系统的控制输入。在[t_0，t_1]上给出的控制u(T)决定状态轨迹p(T)。假设给定一种微分形式欧米伽，并沿轨迹积分欧米伽(X)，我们得到了与欧米伽和X有关的受控路径积分。系统的输出、李亚普诺夫函数和最优控制的性能指标可以用受控路径积分来表示。因此，控制理论可以看作是对受控路径积分特性的研究。分部积分得到了沿轨迹的路径积分的有限项表达式，这对于沿轨迹的解耦反馈控制、结构…等设计方法的推导是有用的更多的算法等等。在解析性的假设下，利用这些表达式得到了Fliess型和Volterra型的泛函级数展开公式。将结果推广到时变系统，得到了路径积分输出能控的充要条件，并用强可达向量场上欧米伽的Lie导数表示。如果omega=dh，则条件与先前的结果一致。利用广义Legendre-Clebsch条件的等价性，可以更容易地推导出输出不变性的条件，而广义Legendre-Clebsch条件等价于某个微分形式是精确的。这一事实表明了这些概念之间的密切联系。用受控路径积分来表示系统几乎等同于用余切丛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，并研究了它的等价性和规范形。研究了在纤维空间中不可积的Berry相的控制问题。提出了非耗散控制的概念，并给出了该系统的可解性条件。研究并部分解决了非线性输出调节问题和非线性几乎模型跟踪问题的近似设计方法。较少

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