A Reachable Set Analysis Method for Generating Near-Optimal Trajectories of Constrained Multiphase Systems

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Abstract

Few sophisticated problems in the optimal control of a dynamical system can be solved analytically. There are many numerical solution methods, but most, especially those with the most potential accuracy, work iteratively and must be initialized with a guess of the solution. Satisfactory guesses may be very difficult to generate. In this work, a “Reachable Set Analysis” (RSA) method is developed to find near-optimal trajectories for multiphase systems with no a priori knowledge. A multiphase system is a generalization of a dynamical system that includes possible changes on the governing equations throughout the trajectory; the traditional dynamical system where the governing equations do not change is included in the formulation as a special case. The RSA method is based on a combination of metaheuristic algorithms and nonlinear programming. A particularly beneficial aspect of the solution found using RSA is that it satisfies the system governing equations and comes arbitrarily close (to a degree chosen by the planner) to satisfying given terminal conditions. Three qualitatively different multiphase problems, such as a low-thrust transfer from Earth to Mars, a system with chattering arcs in the optimal control, and a motion planning problem with obstacles, are solved using the near-optimal trajectories found by RSA as initial guesses to show the effectiveness of the new method.

Original languageEnglish (US)
Pages (from-to)161-194
Number of pages34
JournalJournal of Optimization Theory and Applications
Volume167
Issue number1
DOIs
StatePublished - Oct 14 2015

Keywords

  • Evolutionary algorithms
  • Nonlinear programming
  • Obstacle avoidance
  • Reachable sets
  • Trajectory optimization

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

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