Engineering topics
Publications and source records attributed to Cliff, E. M..
Finite dimensional approximation schemes that work well for distributed parameter systems are often not suitable for the analysis and implementation of feedback control systems. The relationship between approximation schemes for distributed parameter systems and their application to optimal control problems is discussed. A numerical example is given.
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F. Kaiser's germinal 1944 report on his 'resultant-height' concept, now known as energy modelling, is reviewed. The data base for the Me. 262 jet fighter is recreated via spline-lattice representation of specific excess power. Minimum-time and 'distance'-climb trajectories are generated in an attempt to check Kaiser's results. Agreement is good for the minimum-time calculations but only qualitative agreement is obtained for the mysterious 'distance climbs' whose documentation is fragmentary. The character of optimal climb-dash trajectories in energy approximation is examined and illustrated.
Several topics in optimal symmetric flight of airbreathing vehicles are examined. In one study, an approximation scheme designed for onboard real-time energy management of climb-dash is developed and calculations for a high-performance aircraft presented. In another, a vehicle model intermediate in complexity between energy and point-mass models is explored and some quirks in optimal flight characteristics peculiar to the model uncovered. In yet another study, energy-modelling procedures are re-examined with a view to stretching the range of validity of zeroth-order approximation by special choice of state variables. In a final study, time-fuel tradeoffs in cruise-dash are examined for the consequences of nonconvexities appearing in the classical steady cruise-dash model. Two appendices provide retrospective looks at two early publications on energy modelling and related optimal control theory.
A real-time computer algorithm to control and optimize aircraft flight profiles is described and applied to a three-dimensional minimum-time intercept mission. The proposed scheme has roots in two well known techniques: singular perturbations and neighboring-optimal guidance. Use of singular-perturbation ideas is made in terms of the assumed trajectory-family structure. A heading/energy family of prestored point-mass-model state-Euler solutions is used as the baseline in this scheme. The next step is to generate a near-optimal guidance law that will transfer the aircraft to the vicinity of this reference family. The control commands fed to the autopilot (bank angle and load factor) consist of the reference controls plus correction terms which are linear combinations of the altitude and path-angle deviations from reference values, weighted by a set of precalculated gains. In this respect the proposed scheme resembles neighboring-optimal guidance. However, in contrast to the neighboring-optimal guidance scheme, the reference control and state variables as well as the feedback gains are stored as functions of energy and heading in the present approach. Some numerical results comparing open-loop optimal and approximate feedback solutions are presented.
A small but interesting class of optimal control problems featuring a scalar control appearing linearly is equivalent to the class of identically nonregular problems in the Calculus of Variations. It is shown that a condition due to Mancill (1950) is equivalent to the generalized Legendre-Clebsch condition for this narrow class of problems.
It is shown that care must be taken to ensure that finite dimensional approximations of distributed parameter systems preserve important system properties (i.e., controllability, observability, stabilizability, detectability, etc.). It is noted that, if the particular scheme used to construct the finite dimensional model does not take into account these system properties, the model may not be suitable for control design and analysis. These ideas are illustrated by a simple example, i.e., a cable-spring-mass system.
Cruise-dash flight performance is analyzed in the context of singular perturbations. Attention is given to the problem of determining an atmospheric flight path between given end points which minimizes a linear combination of time and fuel. It is shown that nonconvexity in the fuel-flow vs. airspeed graph has important consequences in optimum-cruise problems with time restrictions. Certain velocity regions are nonoptimal for cruise-dash and optimal cruise-dash sometimes requires time-shared operation between two altitude-airspeed points. Calculations are presented illustrating the occurrence of time-shared operation between two altitude-airspeed combinations for optimal cruise-dash.
Optimal flight in the vertical plane with a vehicle model intermediate in complexity between the point-mass and energy models is studied. Flight-path angle takes on the role of a control variable. Range-open problems feature subarcs of vertical flight and singular subarcs. The class of altitude-speed-range-time optimization problems with fuel expenditure unspecified is investigated and some interesting phenomena uncovered. The maximum-lift-to-drag glide appears as part of the family, final-time-open, with appropriate initial and terminal transient exceeding level-flight drag, some members exhibiting oscillations. Oscillatory paths generally fail the Jacobi test for durations exceeding a period and furnish a minimum only for short-duration problems.
Kaiser (1944) has introduced the concept of 'resultant height' in connection with aircraft minimum-time climbs. Its use as a state variable in trajectory work is attractive because it is a 'slower' variable than either altitude or velocity. Kelley (1972, 1973) has made an attempt to synthesize 'slow' state variables in connection with singular-perturbation procedures. In the present investigation, attempts are made to synthesize both 'fast' and 'slow' variables for the minimum-time-to-climb problem along lines explored by Kelley. Attention is given to climb equations, energy-modeling simplifications, 'slow'-variable choice, 'fast'-variable-choice considerations, a singular-perturbation analysis, the choice of a 'fast' variable, and the climb-dash problem.
On-board rear-optimal climb-dash energy management, optimal symmetric flight with an intermediate vehicle model, and energy states are presented.
On-board real time flight control is studied in order to develop algorithms which are simple enough to be used in practice, for a variety of missions involving three dimensional flight. The intercept mission in symmetric flight is emphasized. Extensive computation is required on the ground prior to the mission but the ensuing on-board exploitation is extremely simple. The scheme takes advantage of the boundary layer structure common in singular perturbations, arising with the multiple time scales appropriate to aircraft dynamics. Energy modelling of aircraft is used as the starting point for the analysis. In the symmetric case, a nominal path is generated which fairs into the dash or cruise state.
The present investigation is concerned with an examination of optimal symmetry flight on the basis of the intermediate vehicle model. The analysis is partly based upon an exploration of Euler solutions for the path-angle-as-control model carried out by Kelley (1958). The current analysis takes into account higher-order optimality conditions and "chattering-control' phenomena. Attention is given to details regarding the intermediate vehicle model, the Legendre-Clebsch necessary condition, the conjugate-point test, and the numerical solution of the time-range problem. It is found that the flight path angle takes on the role of control variable in the model. From physical considerations, it can be seen that when a positive margin of thrust over drag exists, the maximum-range climb trajectory without time or fuel constraints has no proper maximum nor an upper bound.
Kaiser (1944) has introduced the concept of "resultant height' in connection with aircraft minimum-time climbs. Its use as a state variable in trajectory work is attractive because it is a "slower' variable than either altitude or velocity. Kelley (1972, 1973) has made an attempt to synthesize "slow' state variables in connection with singular-perturbation procedures. In the present investigation, attempts are made to synthesize both "fast' and "slow' variables for the minimum-time-climb problem along lines explored by Kelley. Attention is given to climb equations, energy-modeling simplifications, "slow'-variable choice, "fast'-variable-choice considerations, a singular-perturbation analysis, the choise of a "fast' variable, and the climb-dash problem.
A real-time computer algorithm to control and optimize aircraft flight profiles is described and applied to a three-dimensional minimum-time intercept mission.
The present investigation is concerned with an examination of optimal symmetry flight on the basis of the intermediate vehicle model. The analysis is partly based upon an exploration of Euler solutions for the path-angle-as-control model carried out by Kelley (1958). The current analysis takes into account higher-order optimality conditions and 'chattering-control' phenomena. Attention is given to details regarding the intermediate vehicle model, the Legendre-Clebsch necessary condition, the conjugate-point test, and the numerical solution of the time-range problem. It is found that the flight path angle takes on the role of control variable in the model. From physical considerations, it can be seen that when a positive margin of thrust over drag exists, the maximum-range climb trajectory without time or fuel constraints has no proper maximum nor an upper bound.
On-board real time flight control is studied in order to develop algorithms which are simple enough to be used in practice, for a variety of missions involving three-dimensional flight. The intercept mission in symmetric flight is emphasized. Extensive computation is required on the ground prior to the mission but the ensuing on-board exploitation is extremely simple. The scheme takes advantage of the boundary layer structure common in singular perturbations, arising with the multiple time scales appropriate to aircraft dynamics. Energy modelling of aircraft is used as the starting point for the analysis. In the symmetric case, a nominal path is generated which fairs into the dash or cruise state. Previously announced in STAR as N84-16116