遺伝的アルゴリズムによる終端拘束付き最適制御問題の数値解法

Abstract

In this paper, a new numerical computation method using the genetic algorithm for an optimal control problem with terminal constraints and singular arcs is proposed. The input functions are parameterized using spline interpolation, which has devices that can represent discontinuous input functions. In order to treat the terminal constraints properly, Lagrange multipliers that are contained in genetic information of chromosomes are introduced. On the singular arcs, coefficients of inputs in the Hamiltonian vanish, so the coefficients on the arcs are included in the extended performance index. The weighting coefficents of the extended performance index are changed adaptively at every generation of the genetic algorithm. A simple example is solved using this method, which verifies the efficiency of the genetic algorithm in the computation of optimal control. In this paper, by using the genetic algorithm, GA, we develop a new numerical method to solve the optimal control problem with terminal constraints, and especially including singular arcs. When the equations describing the plant and the integrand of the performance index are affine with respect to inputs, the Hamiltonian is also affine with respect to inputs, and thus the optimal inputs during a certain interval of time when the coefficients of inputs in the Hamiltonian are zero cannot be determined using the minimum principle. Such a case is called singular case, and the optimal control problems with the possibility of the appearance of the optimal singular control have not been solved theoretically except in several particular cases. From the point of view of the numerical solution, since the Hessian matrix Huu(·) of the Hamiltonian with respect to inputs becomes singular, all the numerical methods using the regularity of the Hessian matrix cannot apply to the singular optimal control problems. For examples of numerical methods that are applicable to singular cases, we can refer to studies 1)∼3) that require some a-priori information. Moreover, the � -algorithm of Jacobson et al., and the � -α(·)-algorithm 4) that is an improved version of the � -algorithm can be applied to the singular control. In the � -α(·)-algorithm, a small term of quadratic form of