A Study of the System Theory via Controlled Path Integral
Project/Area Number 
03805020

Research Category 
GrantinAid for General Scientific Research (C)

Allocation Type  Singleyear Grants 
Research Field 
機械力学・制御工学

Research Institution  HOKKAIDO UNIVERSITY 
Principal Investigator 
SHIMA Masasuke Hokkaido University, Fac. of Eng., Professor, 工学部, 教授 (10029457)

CoInvestigator(Kenkyūbuntansha) 
YAMASHITA Yuu Hokkaido University, Fac. of Eng., Assistant Professor, 工学部, 助手 (90210426)
ISURUGI Yoshihisa Hokkaido University, Fac. of Eng., Associate Professor, 工学部, 助教授 (00109480)

Project Period (FY) 
1991 – 1992

Project Status 
Completed (Fiscal Year 1992)

Budget Amount *help 
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1992: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1991: ¥1,000,000 (Direct Cost: ¥1,000,000)

Keywords  controlled path integral / finite term expression / functional series expansion / timevarying system / output invariance / output controllability / Hamiltonian control system / nondissipative control / 制御径路積分 / 一般化LegendreClebsch条件 / 余接バンドル上のシステム / ハミルトン制御系 / 等価性 / Nondissipative control / Fliess型表現 / Volterra型表現 / 入出力無干渉化 / 入出力線形化 / Dyndmic Extension Algorithm / 多目的制御 
Research Abstract 
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
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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 timevarying 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 LegendreClebsch 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 nondissipative 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

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