Singular trajectories in airplane cruise-dash optimizaiton

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Virginia Polytechnic Institute and State University


The problem of determining cruise-dash trajectories is examined for the case of time-fuel optimization using a linear combination of time and fuel as the performance index. These trajectories consist of a transient arc followed by a steady-state arc. For cases where the steady-state arc is flown with full throttle the associated skeletal transient trajectories are also flown with full throttle, and approach the cruise-dash points monotonically in an asymptotic fashion.

When the steady-state arc is flown at an intermediate throttle setting, the transient trajectories follow a singular control law and exhibit a complex structure that is different from the full-throttle transients. Singular transients in the vicinity of singular cruise-dash points are confined to a bounded singular surface. In state-space these trajectories trace out asymptotic spirals on the singular surface as they approach the steady-state arc. If the initial operating point lies outside the singular surface, then the transient trajectories are composites consisting of a full-throttle or zero-throttle segment flown until the singular surface is met, followed by a singular segment that fairs into the cruise-dash point.

Addressing the question of optimality of the steady-state arc, it was found that although steady-state cruise fails a Jacobi-type condition, steady-state cruise-dash can satisfy this condition if the emphasis on time is sufficiently large. The outcome of the Jacobi-type test appears to be connected with the eigenstructure of the linearized state-adjoint system.