A Study of the System Theory via Controlled Path Integral

在本研究计画中，我们探讨了以受控路径积分为基础的系统理论的可能性。系统的动力学行为可用向量场X=X_0+X_1u^1+.+ X_mu^m定义在作为状态空间的流形M上。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）确定状态轨迹p（t）。假定给出一个微分形式ω，并沿轨迹沿着积分ω（X），我们得到与ω和X有关的受控路径积分。系统的输出、李雅普诺夫函数和最优控制的性能指标都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。分部积分得到了沿着轨迹的路径积分的有限项表达式，这些表达式有助于推导出沿着轨迹的设计方法，例如解耦反馈控制、结构Al ...更多信息 在解析性假设下，利用这些表达式得到了Fliess型和沃尔泰拉型函数级数展开式.将所得结果推广到时变系统，得到了路径积分输出能控的一个充要条件，并用Ω沿着强可达向量场的Lie导数表示.当ω =dh时，条件与前面的结果一致.我们还观察到，如果我们使用广义Legendre-Clebsch条件的等式，则更容易导出输出不变性的条件，这与某个微分形式是精确的条件是等价的。这一事实表明了这些概念之间的密切关系。用受控路径积分来表示系统几乎相当于用余切丛和汉密尔顿系统来描述系统，其中辛结构和汉密尔顿向量场在稳定性研究中很有用。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型。研究了在纤维空间上不可积的Berry相的控制问题。提出了串级系统无耗散控制的概念，给出了无耗散控制可解的条件。研究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，并部分解决了这两个问题。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