Some Recent Developments in Computational Optimal Control

Three recently introduced approaches to computational optimal control are discussed. All three methods fall into the category of direct approaches, i.e. they are based on some type of problem discretization and require nonlinear programming techniques to determine the optimal solution within certain finite-dimensional parameter spaces.The first method is called Trajectory Optimization via Differential Inclusion (TODI). This method can be viewed as a finite difference discretization of optimal control problems represented in differential inclusion format, and results in a problem formulation completely devoid of control parameters. The second method, dubbed Concatenated Approach to Trajectory Optimization (CATO) solves an iterative sequence of low-order, discretized subproblems on short, overlapping subarcs of the original trajectory. The technique, which can be paired with practically any direct optimization approach to perform the inner-loop iterations, is fast and requires very little computer memory. Furthermore, it is ideally suited for parallel processing. Finally, the third method, which requires even less computing power, is based on an intuitive dense-sparse discretization scheme, with nodes placed densely near the initial time to capture immediate dynamics, and nodes placed sparsely over the rest of the trajectory to capture general trends. This method, too, can be paired with a large variety of off-the-shelf direct optimization techniques.An F-15 minimum time-to-climb problem is used to test and compare all three methods.