Engineering topics
Publications and source records attributed to Ardema, M. D..
The formulation and basic theorems of singularly perturbed nonlinear optimal control problems are discussed and the solution of such problems using matched asymptotic expansions is described.
The problem of minimum-time interception of a target moving in a horizontal plane is studied. The target may fly a trajectory that is known at the start of the interception or try to evade the intercepting aircraft. The interceptor's optimal motion is described independently of the target in terms of an extremal trajectory map. The latter is used to develop guidelines for suboptimal approximations to the extremals. A method of constructing the feedback solution by drawing isochrones (constant minimum-time loci) is developed. Sections of the feedback solution for the interception of a straight-flying target are presented, and the construction of an isochrone for the interception of an actively evading target is demonstrated.
Matched asymptotic expansions (MAE) are used to obtain a first order approximation to the solution of a singularly perturbed second order system. A special case is considered in which the uniform asymptotic solution obtained by MAE is shown to converge to the exact solution. Ways in which the method can be used to sole higher-order linear systems, including those which are not singularly perturbed, are also discussed.
Minimum time interception of a target moving in a horizontal plane is analyzed as a one-player differential game. Dispersal points and points on the barrier are located for a class of pursuit evasion and interception problems. These points are determined by constructing cross sections of the isochrones and hence obtaining the barrier, dispersal, and control level surfaces. The game solution maps the controls as a function of the state within the capture region.
Several buoyant-vehicle (airship) concepts proposed for short hauls of heavy payloads are described. Numerous studies identified operating cost and payload capacity advantages relative to existing or proposed heavy-lift helicopters for such vehicles. Applications involving payloads of from 15 tons up to 800 tons were identified. The buoyant quad-rotor concept is discussed in detail, including the history of its development, current estimates of performance and economics, currently perceived technology requirements, and recent research and technology development. It is concluded that the buoyant quad-rotor, and possibly other buoyant vehicle concepts, has the potential of satisfying the market for very heavy vertical lift but that additional research and technology development are necessary. Because of uncertainties in analytical prediction methods and small-scale experimental measurements, there is a strong need for large or full-scale experiments in ground test facilities and, ultimately, with a flight research vehicle.
Worldwide civil markets for heavy lift airships were investigated. Substantial potential market demand was identified for payloads of from 13 to 800 tons. The largest markets appear to be in applications to relieve port congestion, construction of power generating plants, and, most notably, logging. Because of significant uncertainties both in vehicle and market characteristics, further analysis will be necessary to verify the identified market potential of heavy lift airship concepts.
The problem of evaluating the combat performance of fighter/attack aircraft is discussed, and the mathematical nature of the problem is examined. The following approaches to air combat analysis are reviewed: (1) differential-turning differential game and (2) coplanar differential game. Selected numerical examples of these approaches are presented. The relative advantages and disadvantages of each are analyzed, and it is concluded that air combat analysis is an extremely difficult mathematical problem and that no one method of approach is best for all purposes. The paper concludes with a discussion of how the two approaches might be used in a complementary manner.
A method for estimating the specific productivity of both hybrid and fully buoyant airships is developed. Various methods of estimating structural weight of deltoid hybrids are discussed and a derived weight estimating relationship is presented. Specific productivity is used as a figure of merit in a parametric study of fully buoyant ellipsoidal and deltoid hybrid semi-buoyant vehicles. The sensitivity of results as a function of assumptions is also determined. No airship configurations were found to have superior specific productivity to transport airplanes.
The lighter than air (LTA) systems technology conference is reviewed. Highlights of the conference were: (1) the interest shown in patrol and surveillance airships, particularly for coastal patrol missions; (2) the session devoted to overviews of foreign activity; and (3) heavy lift and long range transport aircraft design considerations.
Ardema (1974) has formally linearized the two-point boundary value problem arising from a general optimal control problem, and has reviewed the known stability properties of such a linear system. In the present paper, Ardema's results are applied to the minimum time-to-climb problem. The linearized zeroth-order boundary layer equations of the problem are derived and solved.
Singular perturbation techniques are studied for dealing with singular arc problems by analyzing a relatively low-order but otherwise general system. This system encompasses many flight mechanic problems including Goddard's problem and a version of the minimum time-to-climb 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.
In this paper, Goddard's 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.
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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.
Attention is given to the NASA program, Feasibility Study of Modern Airships, initiated to investigate potential research and technology programs associated with airship development. A historical survey of the program is presented, including the development of past airship concepts, aerodynamical and design improvements, structure and material concepts, and research in controls, avionics, instrumentation, flight operations, and ground handling. A mission analysis was carried out which considered passenger and cargo transportation, heavy-lift, short-haul applications, surveillance missions, and the transportation of natural gas. A vehicle parametric analysis examined the entire range of airship concepts, discussing both conventional airships and hybrids. Various design options were evaluated, such as choice of structural materials, use of boundary-layer control, and choice of lifting gas.
Three promising modern airship system concepts and their associated missions were studied: (1) a heavy-lift airship, employing a nonrigid hull and a significant amount of rotor lift, used for short-range transporting and positioning of heavy military and civil payloads; (2) a VTOL (vertical take-off and landing), metal-clad, partially buoyant airship used as a short-haul commercial transport; (3) a fully-buoyant airship used for long-endurance Navy missions. The heavy-lift airship concept offers a dramatic increase in vertical lift capability over existing systems at significantly lower costs per ton-mile. The VTOL airship transport concept appears to be economically competititve with other VTOL aircraft concepts but can attain significantly lower noise levels. The fully-buoyant airship concept can provide an airborne platform with long endurance that satisfies many Navy mission requirements.
The economic characteristics and historical trends of long-haul air transportation are discussed. Various published estimates of modern airship direct operating costs are presented and comparatively analyzed. The large discrepancies in these estimates are in large measure explained, allowing airship direct operating costs to be estimated with some confidence. Indirect costs are also briefly discussed. Projected airship operating costs are compared with actual costs of competing modes such as airplanes, trucks, rail, and pipelines, and it is concluded that airships cannot economically compete with other long-haul modes. Thus attention is drawn to short-haul applications of the airship; and because short-haul missions lead to vehicle requirements that are considerably different from those for long-haul missions, it is concluded that the civil airship of the future is likely to bear little resemblance to those of the past.
The paper reports the results of a conceptual design study of new near-term fuel-conservative aircraft. A parametric study was made to determine the effects of cruise Mach number and fuel cost on the optimum configuration characteristics and relative economic performance. Supercritical wing technology and advanced engine cycles were assumed. For each design, the wing geometry was selected to maximize an economic figure of merit which reflects the potential rate of return on investment. Based on the results of the parametric study, a reduced energy configuration was selected. Compared with existing transport design, the reduced energy design has a higher aspect ratio wing with lower sweep, and cruises at a slightly lower Mach number. It yields about 30% more seat-miles/gal than current wide-body aircraft. At the higher fuel costs anticipated in the future, the reduced energy design has about the same economic performance as existing designs with the same technology level. As an example of a far-term technology application, a design with a composite material wing was also investigated.