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6Article available at http://www.matec-conferences.org or http://dx.doi.org/10.1051/matecconf/20152203016To compare the merits, flaws and adaptation of different optimization algorithm, this article uses respectively direct shooting method, Gauss pseudo spectral method in the direct method and sequential gradient-restoration algorithm in the indirect method to conduct an optimization analysis on minimum time problem in the diving segment of hypersonic air-vehicle, and comprehensively considers multiple constraints such as dynamic pressure, overload, angle of attack and route angle of terminal flight.
GENERAL DESCRIPTION OF OPTIMAL CONTROL PROBLEMThe general optimal control problem can be described as: determine the allowable control ( ) The direct shooting method is the most common one in the direct method that simply makes the control variables discrete in the time domain; obtains the status value by explicit numerical integration and further obtains the performance index and constraint value. However, due to its sensitiveness to initial value and great computational burden, each time of iteration requires integration for the status variable, and it is just suitable for optimal control problem with less requirement of accuracy. The direct shooting method can be used to approximately convert the optimal control problem in finite dimension to non-linear planning problem, and use secondary planning of sequence to solve the non-linear planning problem above. The detailed calculation procedure is as shown in the figure 1.
f x u p t t t t c x u p t d x u p t x t x x t p
Gauss pseudo spectral methodGauss pseudo spectral method is a direct point collocation method based on global interpolation multinomial and, compared to regular direct point collocation method, has less nodes and higher accuracy [9] . Gauss pseudo spectral method makes the status and control variable discrete in a series of Gauss points, and constructs Lagrange interpolation multinomial to approximate to the status and control variable by those discrete points.( , , which will be taken as the primary function to construct approximate expression of status and control variable, namely:(2)Dynamics differential equation (confirmed Gauss differential matrix)The approximation of status variable can be realized by global interpolation multinomial, and the approximation of its derivative can be realized by derivation of Lagrange interpolation multinomial in order to convert the dynamics differential equation constraint to algebra constraint, which is:The optimal control problem generally includes terminal status constraint. Make the terminal constraint discrete and use Gauss integration for approximation, and then we get:(4) Approximation of performance index function Approximate the integration item in Bolza-type performance ind...