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.

[1]  G. Fairweather,et al.  Numerical Methods for Two – Point Boundary Value Problems , 2008 .

[2]  P. Brunovský.  On optimal stabilization of nonlinear systems , 1967 .

[3]  A. P. Willemstein,et al.  Optimal regulation of nonlinear dynamical systems on a finite interval , 1977 .

[4]  Emiliano Cristiani,et al.  Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach , 2009, 0910.0521.

[5]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[6]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[7]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[8]  Kenji Fujimoto,et al.  Optimal trajectory generation for linear systems based on double generating functions , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[9]  Daniel J. Scheeres,et al.  Determination of optimal feedback terminal controllers for general boundary conditions using generating functions , 2006, Autom..

[10]  Hiroshi Yamakawa,et al.  A new optimal orbit control for two-point boundary-value problem using generating functions , 2009 .

[11]  Basílio E. A. Milani,et al.  Design of L-Q regulators for state constrained continuous-time systems , 1995, IEEE Trans. Autom. Control..

[12]  Arjan van der Schaft,et al.  Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation , 2008, IEEE Transactions on Automatic Control.

[13]  Vera Zeidan,et al.  Hamilton-Jacobi theory over time scales and applications to linear-quadratic problems , 2012 .

[14]  Hans P. Geering,et al.  Optimal control with engineering applications , 2007 .

[15]  D. Scheeres,et al.  Solving Relative Two-Point Boundary Value Problems: Spacecraft Formation Flight Transfers Application , 2004 .

[16]  D. Lukes.  Optimal Regulation of Nonlinear Dynamical Systems , 1969 .

[17]  S. Sieniutycz,et al.  Optimal Control: An Introduction , 2001 .

[18]  Kenji Fujimoto,et al.  Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions with a Quadratic Cost Function with Respect to the Input , 2012 .

[19]  Joseph J. Winkin,et al.  Convergence of the Time-Invariant Riccati Differential Equation towards Its Strong Solution for Stabilizable Systems , 1995 .

[20]  Lorenzo Ntogramatzidis,et al.  A parametrization of the solutions of the finite-horizon LQ problem with general cost and boundary conditions , 2005, Autom..