Genetic Algorithm Approach for Optimal Control Problems with Linearly Appearing Controls

Abstract

For optimal control problems in Mayer form with all controls appearing only linearly in the equations of motion, this paper presents a method for calculating the optimal solution without user-specified initial guesses and without a priori knowledge of the optimal switching structure. The solution is generated in a sequence of steps involving a genetic algorithm (GA), nonlinear programming, and (multiple) shooting. The centerpiece of this method is a variant of the GA that provides reliable initial guesses for the nonlinear programming method, even for large numbers of parameters. As a numerical example, minimum-time spacecraft reorientation trajectories are generated. The described procedure never failed to correctly determine the optimal solution. INDING the solution to an optimal control problem is a dif- ficult and time-consuming task. By employing Pontryagin's minimum principle in conjunction with simple or multiple shoot- ing to solve the resulting boundary-value problem (BVP), this task becomes equivalent to finding the numerical values of the costates (Lagrange multipliers) associated with the physical states of the underlying dynamic system at discrete times. Thus, the problem of solving an optimal control problem can be reduced to solving a non- linear system of equations. Usually, Newton-Raphson methods are well suited for this type of problem. However, due to the sensitiv- ity of the state-costate dynamical system, the task of finding initial guesses that lie within the domain of convergence can become arbi- trarily difficult. In addition, if a control appears only linearly in the equations of motion, the optimal solution is known to consist of a sequence of bang-bang and, possibly, singular subarcs. The switch- ing structure, however, is not known in advance and has to be found by trial and error. The present paper introduces a method for generating the opti- mal control solution for problems in which all controls appear only linearly in the equations of motion. In this method, the user need not provide initial guesses for the state history, the control history, the costate history, or the switching structure. Initial guesses that lie within the domain of convergence of a gradient search method are generated with a genetic algorithm (GA) using substring length 1 for each individual control parameter. A theoretical justification of the approach is given through hodograph analysis and convexity arguments. General convergence arguments pertaining to the GA are mainly heuristic and based on practical experience. Because of the probabilistic nature of GAs, this seems to be unavoidable.