Fixed-Range Optimal Trajectories of Supersonic Aircraft by First-Order Expansions

Abstract

A near-optimal guidance law that generates minimum-fuel, minimum-time, or direct operating cost e xed-range trajectories for supersonic transport aircraft is developed. The approach uses singular perturbation techniques to timescaledecoupletheequationsofmotion into threesetsof dynamics, twoofwhicharestudied here:weight /range and energy. The two-point boundary-value problems obtained by application of the maximum principle to the dynamic systems are solved using the method of matched asymptotic expansions. Both the weight /range and the energy dynamic solutions are carried out to e rst order. The two solutions are combined using the matching principletoformauniformlyvalidapproximationofthefulle xed-rangetrajectory.Resultsshowthattheminimumfuel trajectory has three segments: a minimum-fuel energy climb, a cruise climb, and a minimum-drag glide. The minimum-timetrajectory also has three segments: a maximum dynamicpressureclimb, a constant altitudecruise, andamaximumdynamicpressuredescent.Theminimumdirectoperatingcosttrajectoryisanoptimalcombination of the preceding two trajectories. It is shown that for representative costs of fuel and e ight time, the minimum direct operating cost trajectory is very similar to the minimum-fuel trajectory.