Optimal trajectory generation for nonlinear systems based on double generating functions

A method based on the generating function for the finite time optimal control problems was proposed recently. A single generating function can generate a family of optimal inputs which are functions of the state for different boundary conditions. Therefore, a family of optimal trajectories for different boundary conditions can be obtained by numerical integration along the system dynamic equation. This paper proposes a method to compute a family of optimal trajectories for a nonlinear optimal control problem on a finite time interval by using a pair of generating functions. The proposed method reduces the online computational effort in calculating the numerical integration required in the method using a single generating function. It is useful to on-line nonlinear trajectory generation problems such as model predictive control. A numerical example illustrates the effectiveness of the proposed method.