Studies on the Controlled Path Integrals-I. Structure Algorithm

Abstract

First, it is shown that the output of the control system with statespace description can be expressed by the integral of completely integrable differential 1-form along the controlled path of the system, which is named as the “controlled path integral”.
Adopting the controlled path integrals as the outputs, we construct the “Structure Algorithm” for the nonlinear system defined on a differentiable manifold M with a vector field X=X₀+u¹X₁+…+u^mX_m, control inputs uⁱ (i=1, …, m), and outputs Y^k=∫ω^k (X) dt (k=1, …, l), where the differential 1-forms ω^k are not necessarily completely integrable. Coefficients of linear dependence of differential forms are assumed to be C^∞-functions defined on M. If the coefficients are constant around a point, we can derive a feedback control law which linearizes the input output relations of the system in an appropriate neighborhood of the point. Moreover, in order to calculate the feedback law, we can dispense with the local coordinates system, if the global coordinates describing the dynamic behaviour of the system are available.
Since the performance index of the optimal control can be also expressed by the controlled path integral, we recommend the controlled path integrals of differential forms and more general tensors for the study of control systems.