Mark D. Ardema

Near-optimal propulsion-system operation for an air-breathing launch vehicle

A methodology for determining the near-optimal operation of the propulsion system of hybrid air-breathing launch vehicles is derived. The method is based on selecting propulsion-system modes and parameters that maximize a certain performance function. This function is derived from consideration of the energy-state model of the aircraft equations of motion. The vehicle model reflects the many interactions and complexities of the multimode air-breathing and rocket engine systems proposed for launch-vehicle use. The method is used to investigate the optimal throttle switching of air-breathing and rocket engine modes, and to investigate the desirability of using liquid-oxygen augmentation in air-breathing engine cycles, the oxygen either carried from takeoff or collected in flight. It is found that the air-breathing engine is always at full throttle, and that the rocket is on full at takeoff and at very high Mach numbers, but off otherwise. Augmentation of the air-breathing engine with stored liquid oxygen is beneficial, but only marginally so.

Optimal trajectories for hypersonic launch vehicles

In this paper, we derive a near-optimal guidance law for the ascent trajectory from earth surface to earth orbit of a hypersonic, dual-mode propulsion, lifting vehicle. Of interest are both the optimal flight path and the optimal operation of the propulsion system. The guidance law is developed from the energy-state approximation of the equations of motion. Because liquid hydrogen fueled hypersonic aircraft are volume sensitive, as well as weight sensitive, the cost functional is a weighted sum of fuel mass and volume; the weighting factor is chosen to minimize gross take-off weight for a given payload mass and volume in orbit.

An introduction to singular perturbations in nonlinear optimal control

Before turning to the general n-dimensional nonlinear optimal control problem, which is of primary interest in this paper, we will first consider a much simpler problem, namely the singularly perturbed, uncontrolled, autonomous, initial-value problem (1.1) where x(e,t) and y(e,t) are scalars, e < 0, and xo and yo are constants.

Brief paper: Nonlinear singularly perturbed optimal control problems with singular arcs

A third-order, nonlinear, singularly perturbed optimal control problem is considered under assumptions which assure that the full problem is singular and the reduced problem is nonsingular. The separation between the singular arc of the full problem and the optimal control law of the reduced one, both of which are hypersurfaces in state space, is of the same order as the small parameter of the problem. Boundary-layer solutions are constructed which are stable and reach the outer solution in a finite time. A uniformly valid composite solution is then formed from the reduced and boundary-layer solutions. The value of the approximate solution is that it is relatively easy to obtain and does not involve singular arcs. To illustrate the utility of the results, the technique is used to obtain an approximate solution of a simplified version of the aircraft minimum time-to-climb problem. A numerical example is included.

Minimum Heating Entry Trajectories for Reusable Launch Vehicles

A e nite control volume heat transfer analysis is coupled to a e ight-path optimization and integration algorithm forthepurposeofcalculatingconductiveheatratesand transienttemperatureeffectswithin thethermalprotection system of a reusable launch vehicle. Results are obtained for three different thermal protection system concepts: tile, blanket, and metallic. The optimization algorithm is based on the energy state approximation and is used to generate optimal entry trajectories minimizing the following three criteria: 1 ) the thermal energy absorbed at the vehicle surface, 2 ) the heat load applied to the vehicle, and 3 ) the thermal energy absorbed by the internal structure. Results indicate that allthreetrajectoriesproduce comparablepeak internal structure temperatures for a given thermal protection system, with the trajectory minimizing the heat load applied to the vehicle producing thelowestpeaktemperature.However,ifthemaximum stagnation temperatureconstraintatthenoseof thevehicle is increased from 3000 to 4000 ±F, the trajectory minimizing the thermal energy absorbed by the internal structure becomes superior. Further, the trajectory with the 4000 ±F limit gives a peak internal structure temperature 25 ±F less than the one with the 3000 ± F limit.

OPTIMIZATION OF SUPERSONIC TRANSPORT TRAJECTORIES

This paper develops a near-optimal guidance law for generating minimum fuel, time, or cost fixed-range trajectories for supersonic transport aircraft. The approach uses a choice of new state variables along with singular perturbation techniques to time-scale decouple the dynamic equations into multiple equations of single order (second order for the fast dynamics). Application of the maximum principle to each of the decoupled equations, as opposed to application to the original coupled equations, avoids the two point boundary value problem and transforms the problem from one of a functional optimization to one of multiple function optimizations. It is shown that such an approach produces well known aircraft performance results such as minimizing the Brequet factor for minimum fuel consumption and the energy climb path. Furthermore, the new state variables produce a consistent calculation of flight path angle along the trajectory, eliminating one of the deficiencies in the traditional energy state approximation. In addition, jumps in the energy climb path are smoothed out by integration of the original dynamic equations at constant load factor. Numerical results performed for a supersonic transport design show that a pushover dive followed by a pullout at nominal load factors are sufficient maneuvers to smooth the jump.

Singular perturbations and the sounding rocket problem

In this paper, Goddards problem of maximizing the final altitude of a sounding rocket (a singular problem of optimal control) is analyzed using singular perturbation methods. The problem is first cast in singular perturbation form and then solved to zero order by adding boundary-layer corrections to the reduced solution. For a quadratic drag law, a closed-form solution is obtained, although consideration of a numerical example indicates that this solution is not useful for practical sounding rockets. However, use of state variable transformations allows a very accurate numerical approximation to be constructed. It is concluded that application of singular perturbation methods to the well-known sounding rocket problem indicates that these methods may have utility in dealing with singular problems of optimal control.

Nonlinear Singularly Perturbed Optimal Control Problems with Singular Arcs

Abstract A third order, nonlinear, singularly perturbed optimal control problem is considered under assumptions which assure that the full problem is singular and the reduced problem is nonsingular. The separation between the singular arc of the full problem and the optimal control law of the reduced one, both of which are hypersurfaces in state space, is of the same order as the small parameter of the problem. Boundary layer solutions are constructed which are stable and reach the outer solution in a finite time. A uniformly valid composite solution is then formed from the reduced and boundary layer solutions. The value of the approximate solution is that it is relatively easy to obtain and does not involve singular arcs. To illustrate the utility of the results, the technique is used to obtain an approximate solution of a simplified version of the aircraft minimum time—to-climb problem. A numerical example is included.