Optimal periodic cruise for a hypersonic vehicle with constraints

It is known that the periodic cruise control can save fuel for many aircraft and engine models. This paper applies the periodic cruise control to a hypersonic SCRAMJet transport since fuel saving is critical to its performance in maximizing range. The model was constructed by using numerical data and figures from available literature in the area of space planes. In particular, a heating-rate and a load-factor constraint are considered to make this model more realistic than other previous models. The control on the boundary needs to be determined by a nonlinear programming, and the Lagrange multipliers exhibit jump phenomenon at the entry points of the on-boundary arcs. These constraints increase difficulty in obtaining numerical solution and also increase the sensitivity of the initial guess for a convergence. By assuming the shape of altitude as a sinusoidal function of range and by using a bang-bang thrust control, a sub-optimal solution is obtained for the vehicle without the heating rate and load factor constraints. This sub-optimal solution serves as a very good initial guess for the optimal solution generated by the minimizing-boundary-condition method. The optimal solution shows a fuel saving of 8.12% over the steady-state cruise, a maximum heating rate of 1202.4 watts per square centimeters, and a maximum load factor of 8.27. The heating rate and load factor constraints are then added to the problem. With a maximum heating rate of 400 watts per square centimeters, the fuel saving reduces to 2.45%. With a load factor of seven, the fuel saving does not change much from the non-constrained solution. An optimal periodic-cruise solution with the maximum heating rate of 1158.0 watts per square centimeters and simultaneously with the maximum load factor of seven is also determined with a fuel saving of Nomenclature AS = Inlet area (m^) Aw = Wing area (m^) a = Speed of sound at sea level or at the standard temperature (m/s) bj = Lapse rate between j-th and (j+l)-th junction points (j=0,2,4, and 6) Copyright © 1996 by Chuang and Morimoto. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. C = Load factor constraint function Cj_> = Drag coefficient CDO = Zero-lift drag coefficient CL = Lift coefficient CLO = Zero angle-of-attack lift coefficient CLCC = Lift-curve dope Cxmax = Maximum thrust coefficient D = Drag(N) f = System dynamics vector g = Gravity acceleration at sea level (m/sec) h = Altitude (km) ha = Amplitude corresponding to frequency co of a specified sinusoidal altitude curve hb = Amplitude corresponding to frequency 2co of a specified sinusoidal altitude curve r^ = Offset of a specified sinusoidal altitude curve !sp = Specific impulse (s) J = Cost function K = Induced drag parameter L = Lift(N) M = Mach number (defined as a normalized velocity by a constant speed of sound at sea level) m = Mass of the vehicle (kg) n = Load factor Umax = Specified maximum load factor Q = Heating rate (w/cm) Qmax = Specified maximum heating rate (w/cm) q = Dynamic pressure (N/m) R = Specific gas constant for air (J/kgK) RQ = Radius of the earth (km) r = Range (km) ri = Coordinate of range of an entry point to a boundary rd = Coordinate of range where a specified throttle falls down from 1 to 0 ru = Coordinate of range where a specified throttle rises up from 0 to 1 S = Heating-rate constraint function s = Throttle T = Thrust (N) Ti = Temperature at i-th junction point (i=0,1, 2, 3,4,5, and 6) (K) American Institute of Aeronautics and Astronautics u = Control vector V = Velocity (m/s) x = State vector cc = Angle of attack (deg) y = Flight-path angle (deg) 1 = Lagrange multiplier vector (i = Lagrange multiplier function associated with a load-factor constraint v = Lagrange multiplier constant associated with periodic boundary condition on the state rc = Lagrange multiplier constant associated with heating rate constraint at an entry point p = Density of air (kg/m) (o = Frequency with respect to range of a specified sinusoidal altitude curve ^ = Lagrange multiplier function associated with a heating-rate constraint Introduction Ever since the fuel efficiency of oscillatory cruise paths over the steady-state cruise paths was first recognized, research on periodic optimal processes has been an active subject, especially for fuel saving considerations for aircraft and hypersonic vehicles" . The steady-state cruise solution was shown not to be fuel-optimal although it satisfies the first-order necessary conditions along the steady-state path. Fueloptimal periodic trajectories were numerically determined for aircraft and hypersonic vehicles". An optimal periodic control problem for a hypersonic vehicle with a maximum load-factor constraint was recently solved-'. Those studies, however, did not consider aerodynamic heating that vehicles experience along the optimal trajectories. Thus, in order to protect the crew and materials of the vehicle from the heat, it is necessary to impose a restriction on the heating rate. This restriction means that the optimal periodic control problem must be solved including an additional state inequality constraint related to the heating rate constraint. In the following sections, the vehicle's dynamics is stated, the atmospheric density model, the aerodynamicforce models, and the engine thrust model are described. A more realistic aircraft model and a more precise atmospheric model than models used before are adopted here. An approximation to the optimal control solution was obtained by parametrizing the control time histories and using nonlinear programming techniques to obtain the optimum values of the parameters. This solution in turn, which we call it a sub-optimal solution, was used as an initial guess in solving the optimal control problem. The unconstrained optimal periodic solution shows that the vehicle has a peak aerodynamic heating rate of over 1200 watts/cm and a peak load factor of over 8g's along the optimal trajectory. Thus, we next present the optimal periodic control solution constrained by a maximum load factor of 7.0g's and the optimal periodic control solution constrained by a maximum heating rate of 400 watts/cm. The load factor constraint constitutes a state-control inequality constraint, and the maximum heating rate constitutes a state inequality constraint. Finally optimal periodic control solutions are presented with both the maximum load factor and maximum heating rate constraint. Vehicle's Dynamic Model The equations of motion for flight in a vertical plane over the non-rotating spherical Earth with range as the independent variable are dh . .„ h . — = (tany)(l + —) dr R0 dM (Tcosa-D-mgsiny) . h . dr Mamcosy R0 (1)