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Publications and source records attributed to Cliff, Eugene M..
A new proof is given for Jacobi's necessary condition. It is shown that the existence of a conjugate point in the interior of an extremal implies the existence of control perturbations which lead to a reduction in cost. The present analysis is restricted to linear-quadratic optimal control problems.
The Goddard problem is that of maximizing the final altitude for a vertically ascending, rocket-powered vehicle under the influence of an inverse square gravitational field and atmospheric drag. The present paper deals with the effects of two additional constraints, namely, a dynamic pressure limit and specified final time. Nine different switching structures involving zero-thrust arcs, full-thrust arcs, singular-thrust arcs, and state-constrained arcs are obtained when the value of the dynamic pressure limit is varied between zero and infinity and the final time is specified between the minimum possible time within which all of the fuel can be burned and the natural final time that emerges for the problem with final time unspecified. For all points in the aforementioned domain of dynamic pressure limit and prescribed final time, the associated optimal switching structure is clearly identified. Finally, a simple intuitive feedback law is presented for the free time problem. For all values of prescribed dynamic pressure limit, this strategy yields a loss in final altitude of less than 3 percent with respect to the associated optimal solution.
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This report is a summary of work done under Grant NAG1-946 for the period 19 December 1988 through 15 August 1992. Most of the technical work is described in the paper produced, including the NASA Contractor Report 4393 - 'Optimal Control Problems with Switching Points'. In the following pages we present a list of the papers along with a brief abstract. The technical work during the last months of the grant has not yet been submitted for publication so we present a summary of that work in an Appendix.
A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.
An account is given of key geometrical concepts involved in the use of a hodograph as an optimal control theory resource which furnishes a framework for geometrical interpretation of the minimum principle. Attention is given to the effects of different convexity properties on the hodograph, which bear on the existence of solutions and such types of controls as chattering controls, 'bang-bang' control, and/or singular control. Illustrative aircraft trajectory optimization problems are examined in view of this use of the hodograph.
The appearance of touch points in state constrained optimal control problems with general vector-valued control is studied. Under the assumption that the Hamiltonian is regular, touch points for first-order state inequalities are shown to exist only under very special conditions. In many cases of practical importance these conditions can be used to exclude touch points a priori without solving an optimal control problem. The results are demonstrated on a simple example.
An extension of the Generalized Legendre-Clebsch Condition is obtained for problems with singular control along arcs with active state or control constraints. This is achieved by first transforming the Accessory Minimum Problem associated with constrained singular arcs into an unconstrained singular, linear quadratic problem. In a second step whose theorems and proofs are largely based on Goh's work (1966), necessary conditions are derived for such singular linear quadratic problems to yield nonnegative cost.
A guidance solution for aerospace planes is given based on a family of energy-heading extremal solutions that can be used for feedback implementations. Test conditions are developed as steady-cruise solutions by means of energy-heading extremal solutions to a time/fuel-range flight-optimization problem. Feedback charts are derived that give control as a function of energy and heading-to-go, and the reduced-order model is compared to numerically optimized point-mass solutions for flight optimization. The energy-heading model is shown to provide feedback-control charts that yield fuel-efficient turn/acceleration maneuvers with significant weight-change as compared to the point-mass solutions. Such variables as Mach number, dynamic pressure, and lift are found to influence the solutions significantly.
A new dynamic model for aircraft motions is presented. This model can be viewed as intermediate between a point-mass model, in which the body attitude angles are control-like, and a rigid-body model, in which the body-attitude angles evolve according to Newton's Laws. Specifically, consideration is given to the case of symmetric flight, and a model is constructed in which the body roll-rate and the body pitch-rate are the controls. In terms of this body-rate model a minimum-time heading change maneuver is formulated. When the bounds on the body-rates are large the results are similar to the point-mass model in that the model can very quickly change the applied forces and produce an acceleration to turn the vehicle. With finite bounds on these rates, the forces change in a smooth way. This leads to a measurable effect of agility.
In the quest for understanding problems of supermaneuverability for combat aircraft, a study is presented about the role the dihedral-effect can have in fuselage-reorientation maneuvers that involve high angles of attack. A mathematical model for attitude maneuvers is developed, which accurately represents the High Angle-of-Attack Research Vehicle, including thrust-vectoring generated propulsive moments. The fuselage-reorientation problem is posed as an unconstrained time-optimal control problem. Results for a few families of extremal trajectories are obtained, and the global role of the dihedral effect in the course of the corresponding maneuvers is discussed. In addition, a detailed case-study is presented for an extremal trajectory which utilizes the dihedral effect with considerable benefit.
An analysis is presented for a family of regular extremal attitude-maneuvers for the High Angle-of-Attack Research Vehicle that has thrust-vectoring capability. Different levels of dynamic coupling are identified in the combat aircraft attitude model, and the characteristic extremal-family motion is explained. It is shown why the extremal-family trajectories develop small sideslip-angles, a highly desirable feature from a practical viewpoint.
A time-range-fuel optimization problem is formulated for an airbreathing, hypersonic vehicle. Singular perturbation theory is used to decompose the problem into simpler subproblems. Analysis of the cruise-dash problem shows the importance of a Mach-limit. Energy-heading transients are studied and a family of trajectories, fairing asymptotically to a cruise condition, are generated.
A robust feedback algorithm is presented for a near-minimum-fuel ascent of a two-stage launch vehicle operating in the equatorial plane. The development of the algorithm is based on the ideas of neighboring optimal control and can be derived into three phases. In phase 1, the formalism of optimal control is employed to calculate fuel-optimal ascent trajectories for a simple point-mass model. In phase 2, these trajectories are used to numerically calculate gain functions of time for the control(s), the total flight time, and possibly, for other variables of interest. In phase 3, these gains are used to determine feedback expressions for the controls associated with a more realistic model of a launch vehicle. With the Advanced Launch System in mind, all calculations are performed on a two-stage vehicle with fixed thrust history, but this restriction is by no means important for the approach taken. Performance and robustness of the algorithm is found to be excellent.
Range optimal trajectories for an aircraft flying in the vertical plane are obtained from Pontryagin's Minimum Principle. Control variables are load factor n which appears nonlinearly in the equations of motion and throttle setting eta, which appears only linearly. Both controls are subject to fixed bounds, namely eta between values of 0 and 1 and absolute value of n not greater than n(max). Additionally, a dynamic pressure limit is imposed, which represents a first-order state-inequality constraint. For fixed flight time, fixed initial coordinates, and partially fixed final coordinates, the effect of the load factor limit absolute value of n not greater than n(max) is studied. Upon varying n(max), six different switching structures are obtained. All trajectories involve singular control along arcs with active dynamic pressure limit.
A new proof is given for Jacobi's no-conjugate-point necessary condition. For a certain class of linear-quadratic optimal control problems it is shown that the existence of a conjugate point in the interior of the extremal implies the existence of control perturbations that lead to a reduction in cost. In a well-known way, through the concept of the accessory minimum problem, this results in a no-conjugate-point condition for general optimal control problems. Important ideas used in this work are adopted from Breakwell and Ho (1965). In contrast to earlier results, the new proof also applies if the coefficient functions of time associated with the accessory minimum problem have any finite number of discontinuities.
It is shown that it is possible to construct a controller for the inner loop which is consistent with the performance index used in shaping the trajectory. Although the complete analysis has been carried out only for a model without an RCS, the resulting pitch-loop control is of interest. The gains depend on the system only through the parameter Omega and do not depend on the outer performance index in any way.
A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.