(61ab) A Comparison of Nonlinear Optimal Control Trajectory Sensitivity Formulations

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 of a Non-Linear 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 OCP gradient under a uniform time grid. A uniform time grid is the most commonly used time grid which requires switching times of all multiple inputs to occur synchronously as opposed to an emerging non-uniform time grid in which the input changes occur asynchronously. The first two approaches consider the control policy switching time (Approach 1) or the length of the time period (Approach 2) of the original OCP as the decision variables respectively. The time period OCP formulation may be transformed by a time scaling transformation named Control Parameterization Enhancing Transform (CPET) technique to evaluate the gradient of transformed OCP (Approach 3). Studies addressing Approach 1 and Approach 3 have been reported in the literature and this work involves slight improvements and supplements to Approach 1 and 3: The sensitivity equation of Approach 1 is re-derived in a slightly more rigorous and general form. With regard to Approach 3, this work presents an equivalently time scaling transformation and provides sensitivity and gradient formulae for the converted problem. The sensitivity equation of Approach 2 under the uniform time grid regarding time stage length as decision variable is derived from that under the non-uniform time grid and Approach 2 will be compared to Approach 1 and 3. Results show that employing the CPET technique with trajectory sensitivity method (Approach 3) is able to circumvent the numerical difficulties shown in Approach 1 and 2.