Optimal Trajectories for Multidimensional Nonlinear Processes by Iterated Dynamic Programming

A trajectory optimization technique for multidimensional nonlinear processes is presented. Problems which are cast in a discrete-time mold are considered. The method is based on dynamic programming and employs a combination of the technique of functional approximation and the method of region-limiting strategies. The cost function at each stage is approximated by a quadratic polynomial in a region which is restricted to be of a size judged appropriate to reduce the error in the approximation. Minimal costs are evaluated at a set of points, called base points. A new control trajectory and an improved state trajectory are then generated within an extrapolation region. The iterative application of this procedure yields an optimal trajectory. Contained in the algorithm is a simple procedure which eliminates matrix inversion to determine the coefficients of the approximating polynomial. The present algorithm is applicable to problems with one bounded control action. It accounts for inequality constraints on state variables in a straightforward manner. The algorithm is applied to solve a number of trajectory optimization problems.