Optimal control of two-time-scale systems with state-variable inequality constraints
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Calise, A. J.↗
Engineering topics
Corban, J. E.
Publications and source records attributed to Corban, J. E..
Rapid near-optimal aerospace plane trajectory generation and guidance
Problems associated with onboard trajectory optimization, propulsion system cycle selection, and the synthesis of guidance laws are addressed for ascent to low earth orbit of an airbreathing, single-stage-to-orbit vehicle. A multicycle propulsion system is assumed that incorporates turbojet, ramjet, scramjet, and rocket engines. An energy state approximation is applied to a singularly perturbed, four-state dynamic model for flight of a point mass over a spherical nonrotating earth. An algorithm is then derived for generating both the fuel-optimal climb profile and the guidance commands required to follow that profile. In particular, analytic switching conditions are derived that, under appropriate assumptions, efficiently govern optimal transition from one propulsion cycle to another. The algorithm proves to be computationally efficient and suitable for real-time implementation. The paper concludes with the presentation of representative numerical results that illustrate the nature of the fuel-optimal climb paths and the tracking performance of the guidance algorithm.
Corban, J. E.↗
Rapid near-optimal aerospace plane trajectory generation and guidance
Effort was directed toward the problems of the real time trajectory optimization and guidance law development for the National Aerospace Plane (NASP) applications. In particular, singular perturbation methods were used to develop guidance algorithms suitable for onboard, real time implementation. The progress made in this research effort is reported.
Calise, A. J.↗
Nondimensional forms for singular perturbation analyses of aircraft energy climbs
This paper proposes a systematic approach for identifying the perturbation parameter in singular perturbation analysis of aircraft optimal guidance, and in particular considers a family of problems related to aircraft energy climbs. The approach, which is based on a nondimensionalization of the equations of motion, is used to evaluatae the appropriateness of forced singular perturbation formulations used in the past for transport and fighter aircraft, and to assess the applicability of energy state approximations and singular perturbation analysis for airbreathing transatmospheric vehicles with hypersonic cruise and orbital capabilities.
Calise, A. J.↗
Rapid near-optimal trajectory generation and guidance law development for single-stage-to-orbit airbreathing vehicles
General problems associated with on-board trajectory optimization, propulsion system cycle selection, and with the synthesis of guidance laws were addressed for an ascent to low-earth-orbit of an air-breathing single-stage-to-orbit vehicle. The NASA Generic Hypersonic Aerodynamic Model Example and the Langley Accelerator aerodynamic sets were acquired and implemented. Work related to the development of purely analytic aerodynamic models was also performed at a low level. A generic model of a multi-mode propulsion system was developed that includes turbojet, ramjet, scramjet, and rocket engine cycles. Provisions were made in the dynamic model for a component of thrust normal to the flight path. Computational results, which characterize the nonlinear sensitivity of scramjet performance to changes in vehicle angle of attack, were obtained and incorporated into the engine model. Additional trajectory constraints were introduced: maximum dynamic pressure; maximum aerodynamic heating rate per unit area; angle of attack and lift limits; and limits on acceleration both along and normal to the flight path. The remainder of the effort focused on required modifications to a previously derived algorithm when the model complexity cited above was added. In particular, analytic switching conditions were derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another for two cases: the case in which engine cycle operations can overlap, and the case in which engine cycle operations are mutually exclusive. The resulting guidance algorithm was implemented in software and exercised extensively. It was found that the approximations associated with the assumed time scale separation employed in this work are reasonable except over the Mach range from roughly 5 to 8. This phenomenon is due to the very large thrust capability of scramjets in this Mach regime when sized to meet the requirement for ascent to orbit. By accounting for flight path angle and flight path angle rate in construction of the flight path over this Mach range, the resulting algorithm provides the means for rapid near-optimal trajectory generation and propulsion cycle selection over the entire Mach range from take-off to orbit.
Calise, A. J.↗
Optimal control of singularly perturbed nonlinear systems with state-variable inequality constraints
The established necessary conditions for optimality in nonlinear control problems that involve state-variable inequality constraints are applied to a class of singularly perturbed systems. The distinguishing feature of this class of two-time-scale systems is a transformation of the state-variable inequality constraint, present in the full order problem, to a constraint involving states and controls in the reduced problem. It is shown that, when a state constraint is active in the reduced problem, the boundary layer problem can be of finite time in the stretched time variable. Thus, the usual requirement for asymptotic stability of the boundary layer system is not applicable, and cannot be used to construct approximate boundary layer solutions. Several alternative solution methods are explored and illustrated with simple examples.
Calise, A. J.↗
Optimal guidance and propulsion control for transatmospheric vehicles
Problems associated with on-board trajectory optimization and with the synthesis of guidance laws are addressed for ascent to LEO of an air-breathing, single-stage-to-orbit vehicle. A multimode propulsion system is assumed which incorporates turbojet, ramjet, scramjet, and rocket engines. An energy-state approximation is applied to a four-state dynamic model for flight of a point mass over a spherical nonrotating earth. An algorithm for generating fuel-optimal climb profiles is derived via singular perturbation theory. This algorithm results from application of the minimum principle to a low-order dynamic model that includes general functional dependence on angle of attack and a normal component of thrust. Switching conditions are derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another. The use of bank angle to modulate the magnitude of the vertical component of lift is shown to improve the index performance. Numerical results illustrate the nature of the resulting fuel-optimal climb paths.
Corban, J. E.↗
A real-time guidance algorithm for aerospace plane optimal ascent to low earth orbit
Problems of onboard trajectory optimization and synthesis of suitable guidance laws for ascent to low Earth orbit of an air-breathing, single-stage-to-orbit vehicle are addressed. A multimode propulsion system is assumed which incorporates turbojet, ramjet, Scramjet, and rocket engines. An algorithm for generating fuel-optimal climb profiles is presented. This algorithm results from the application of the minimum principle to a low-order dynamic model that includes angle-of-attack effects and the normal component of thrust. Maximum dynamic pressure and maximum aerodynamic heating rate constraints are considered. Switching conditions are derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another. A nonlinear transformation technique is employed to derived a feedback controller for tracking the computed trajectory. Numerical results illustrate the nature of the resulting fuel-optimal climb paths.
Calise, A. J.↗
Trajectory optimization and guidance law development for national aerospace plane applications
The work completed to date is comprised of the following: a simple vehicle model representative of the aerospace plane concept in the hypersonic flight regime, fuel-optimal climb profiles for the unconstrained and dynamic pressure constrained cases generated using a reduced order dynamic model, an analytic switching condition for transition to rocket powered flight as orbital velocity is approached, simple feedback guidance laws for both the unconstrained and dynamic pressure constrained cases derived via singular perturbation theory and a nonlinear transformation technique, and numerical simulation results for ascent to orbit in the dynamic pressure constrained case.
Calise, A. J.↗
Trajectory optimization and guidance law development for national aerospace plane applications
The problem of onboard trajectory optimization for an airbreathing, single-stage-to-orbit vehicle is examined. A simple model representative of the aerospace plane concept, including a dual-mode propulsion system composed of scramjet and rocket engines, is presented. Consideration is restricted to hypersonic flight within the atmosphere. An energy state approximation is used in a four-state model for flight of a point mass in a vertical plane. Trajectory constraints, including those of dynamic pressure and aerodynamic heating, are initially ignored. Singular perturbation methods are applied in solving the optimal control problem of minimum fuel climb. The resulting reduced solution for the energy state dynamics provides an optimal altitude profile dependent on energy level and control for rocket thrust. A boundary-layer analysis produces an approximate lift control solution in feedback form and accounts for altitude and flight path angle dynamics. The reduced solution optimal climb path is presented for the unconstrained case and the case for which a maximum dynamic pressure constraint is enforced.
Calise, A. J.↗