• Research paper •

### Time-varying optimal control via canonical transformation of hamiltonian system

Zhigang Wu Shujun Tan

1. State Key Lab of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116023, China State Key Lab of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116023, China
• Received:2007-05-15 Revised:1900-01-01 Online:2008-01-25 Published:2007-12-25

Abstract: This paper presents a unified canonical transformation and generating function approach, including associated numerical algorithms, for linear time-varying optimal control problems with various terminal constraints. Generating functions are employed to find the optimal control law by solving Hamiltonian two-point-boundary-value problems. The time-varying optimal control laws constructed by the second type generating function do not have infinite feedback gain at terminal time, which is different from other existing solutions. Motivated by practical design of time-varying optimal control systems, a structure-preserving matrix recursive algorithm is proposed to solve coupled time-varying matrix differential equations of the generating function; derivation of the recursive algorithm is based on symplectic formulation of canonical transformation. To preserve symplectic structure of matrices in the recursive computation, state transition matrices of the Hamiltonian system are calculated by Magnus series. In fact, the canonical transformation and generating function method leads to a geometric perspective to the design and computation of optimal control systems. %control systems synthesis and computation.