Time Scaling Transformation Avoiding Sensitivity Discontinuity for Nonlinear Optimal Control

Abstract

When using gradient-based numerical optimization methods, the cost function, the constraint functions, and their respective gradient information must be calculated at each iteration. The gradient information on a nonlinear programming (NLP) optimal control problem (OCP) can be evaluated by the trajectory sensitivity method. This work is a study of the trajectory sensitivity method comparing three approaches for calculating the gradient under a single time grid. A single time grid requires switching times for multiple inputs to occur synchronously, as opposed to the multiple time grids under which the input changes occur asynchronously. The first two approaches presented consider the control policy switching times (Approach 1) or the lengths of the time periods (Approach 2) of the original OCP as the decision variables. The piecewise constant OCP formulation may be transformed by time scaling using the control parameterization enhancing transform (CPET) to evaluate the gradient of the transformed OCP (Approach 3). Studies addressing Approach 1 and Approach 3 have been reported in the literature, and this work combines and complements these two approaches. The sensitivity equation of Approach 1 is rederived in a more rigorous form. The sensitivity equation of Approach 2 under the single time grid regarding time stage lengths as decision variables is derived from a previous work under multiple time grids. Regarding Approach 3, this work presents an equivalent time scaling transformation and provides a gradient formula for the converted problem. Results are illustrated with several benchmarks, where both the derived formulas and results show that the sensitivity equation employing the CPET technique (Approach 3) is able to avoid the sensitivity discontinuity and dividing by zero issues encountered in Approaches 1 and 2, respectively. However, the division by zero issues encountered in the proposed Approach 2 can be easily addressed, making Approach 2 closely resemble the CPET technique in terms of practicality and applicability.