Optimal control methods with application to flight planning


Optimal control

Optimal control is a powerful mathematical optimization technique that can be used to find the optimal control input for a dynamic system subject to constraints. Indirect methods, such as Pontryagin’s maximum principle and calculus of variations, solve the necessary conditions for optimality, while direct methods discretize the problem and solve a nonlinear programming problem. In aircraft trajectory optimization, the goal is to find the optimal path that a plane should take to minimize fuel usage, flight time, or emissions, while meeting safety and operational constraints. Optimal control techniques can be applied to aircraft trajectory optimization to find the optimal control input that minimizes a cost function while satisfying these constraints. These techniques have the potential to significantly reduce the environmental impact of air travel and improve the efficiency of air traffic management systems.

Perturbed-analytic direct transcription for optimal control (PADOC)

The herein coined PADOC (perturbed-analytic direct transcription for optimal control) stands for a new transcription method for direct trajectory optimiza- tion. We construct PADOC on a novel segmented decomposition method that provides series solutions for nonlinear problems. To transcribe the infinite dimensional problem into a finite one, PADOC comes along with a new solution approach, namely, the herein coined average nonlinear programming (aNLP). The aNLP theorem suggests generating a staircase optimal solution (low reso- lution) and turning it into a distributed optimal solution (high resolution) by exploiting the Hamiltonian of the problem. The analytic architecture provides PADOC with an analytic connection of a truncated order between the discrete nodes of the solution.

This renders stability, accuracy within an analytic res- olution, robustness, and a cut-down in the number of the decision variables in the frame of NLP. We also prove that the multipliers associated with the Karush–Kuhn–Tucker optimality conditions in the frame of NLP (as transcribed by PADOC) correspond to a backward analytic solution for the costate equations. We finally show distinct features of PADOC through some examples.

Minimum-Time Aircraft Trajectory in Climbing Phase

Highlighted publications

  • Amin Jafarimoghaddam and Manuel Soler. Time-Fuel-Optimal Navigation of a Commercial Aircraft in Cruise with Heading and Throttle Controls using Pontryagin’s Maximum Principle«. IEEE Control Systems Letters (L-CSS). DOI: 10.1109/LCSYS.2023.3288471
  • Fast 4D Flight Planning under Uncertainty through Parallel Stochastic Path Simulation. by González-Arribas, Daniel; Andrés, Eduardo; Soler, Manuel; Jardines, Aniel; García-Heras, Javier. Transportation Research Part C. https://doi.org/10.1016/j.trc.2023.104018
  • Jafarimoghaddam A, Soler M. Perturbed-analytic direct transcription for optimalcontrol (PADOC). Optimal Control Applications and Methods. 2023;1-33. doi: 10.1002/oca.2965