This problem describes the lifting of an airbreathing lower stage of a two-stage-to-orbit Sänger-type launch vehicle. We focus on the Ramjet-powered second part of the trajectory. The four state variables are the velocity v, the flight path angle , the altitude h, and the mass m. The three control variables are the lift coefficient , the thrust angle and the throttle setting , . The equations of motion are
The considered time interval is and is free. The following formulae are used for the thrust, the lift and the drag forces
The lift and drag model has a quadratic polar
The quantities , and are constants. For more details of the problem and for a three dimensional formulation cf. Chudej . The boundary conditions are
The objective is to maximize the final mass, i. e.,
Here, the direct collocation method was applied on a rather bad initial estimate
of the optimal trajectory. For the states, the boundary values have been
interpolated linearly and the controls have been set to zero.
The direct collocation method DIRCOL
converges in two macro
iteration steps to a solution with 21 grid points.
From this solution, the optimal states and the
adjoint variables have been estimated.
Based on this estimate, the multiple shooting method was applied
to solve the boundary value problem arising from
the optimality conditions (see ).
The final solutions are kg and s.
For these values, the solution of the direct collocation method
was accurate to four digits.
In Figs. 1 to 6, the solution of the direct collocation method is shown by a dashed line and the highly accurate solution of the multiple shooting method is shown by a solid line. In the figures, there is no visible difference between the suboptimal and the optimal state variables. Also, the estimated adjoint variables and the suboptimal controls of the direct collocation method show a pretty good conformity with the highly accurate ones. The approximation quality can furthermore be improved by increasing the number of grid points to more than 21. The optimal throttle setting equals one within the whole time interval as it is found by both methods.
=5.2cm =7.3cm ZZZG67_altitude.epsf
=5.2cm =7.3cm ZZZG67_fl_p_angle.epsf
Fig. 1: The altitude . Fig. 2: The flight path angle .
*[-3.0cm] =5.2cm =7.3cm ZZZG67_lambda_altitude.epsf =5.2cm =7.3cm ZZZG67_lambda_fl_p_angle.epsf
*[0.3cm] Fig. 3: The adjoint variable . Fig. 4: The adjoint variable .
=5.2cm =7.3cm ZZZG67_lift_coef.epsf
=5.2cm =7.3cm ZZZG67_thrust_angle.epsf
*[0.3cm] Fig. 5: The lift coefficient . Fig. 6: The thrust angle .