Eugene M. Cliff

Parameter Estimation and Identification for Systems with Delays

Parameter identification problems for delay systems motivated by examples from aerody- namics and biochemistry are considered. The problem of estimation of the delays is included. Using approximation results from semigroup theory, a class of theoretical approximation schemes is developed and two specific cases (“averaging” and “spline” methods) are shown to be included in this treatment. Convergence results, error estimates, and a sample of numerical findings are given.

Airfoil Design by an All-At-Once Method

Abstract The all-at-once approach is implemented to solve an optimum airfoil design problem. The airfoil design problem is formulated as a constrained optimization problem in which flow variables and design variables arc viewed as independent and the coupling steady state Euler equation is included as a constraint, along with geometry and other constraints. In this formulation, the optimizer computes a sequence of points which tend toward feasibility and optimalily at the same lime (all-at-once). This decoupling of variables typically makes the problem less non-linear and can lead to more efficient solutions. In this paper an existing optimization algorithm is combined with an existing flow code. The problem formulation, its discretization, and the underlying solvers arc described. Implementation issues arc presented and numerical results are given which indicate that the cost of solving the design problem is appropriately six times the cost of solving a single analysis problem.

Direct calculation of aerodynamic force derivatives - A sensitivity-equation approach

The objective of the present work is to investigate the sensitivity-equation approach to computing stability derivatives using a single non-linear solution to the Navier-Stokes equations. The sensitivity equations are presented in integral form with the necessary boundary conditions. Flow computations are presented for a flat plate boundary layer for validation. The lift-curve slope is computed at several angles of attack for a laminar airfoil. Stability characteristics are analyzed for the YB-49 flying wing.

Time-optimal lateral maneuvers of an aircraft

Results and analysis are presented from a study of time-optimal lateral maneuvers for an aircraft during the power-on-approach-to-landing portion of the flight, typically used for landing on an aircraft carrier. A full sixdegree-of-freedom model is used to model the motions of the aircraft. The optimal control problems of interest are formulated and a family of optimal solutions obtained for two classes of lateral maneuvers. These include an unconstrained maneuver and one with bank-angle and sideslip-angle constraints imposed on the approach trajectory. The control powers of elevator, rudder, and aileron are varied individually, and thus an estimate of the change in downrange distance to perform the lateral maneuvers due to the control power change is obtained.

An optimal policy for a fish harvest

A dynamical model for harvesting a fish population system is proposed by introducing control into the known Verhulst-Pearl model. An optimal control problem including some parameters is stated, and the usual necessary conditions are applied. For specific parameter values, the candidate control policy is deduced, and optimality is verified by applying a sufficiency theorem. The optimal trajectories may contain maximum and minimum control arcs as well as a singular subarc. The significance of the singular arc is interpreted in terms of the system dynamics.

A state-space model for an aeroelastic system

A complete dynamic model is formulated for a system in which the elastic motions of a structure are coupled with the motions of the surrounding fluid. While certain aspects of the problem are well-studied, the emphasis here is on development of a well-posed state-space formulation. Such models have proven conceptual and computational value in problems of optimal control and parameter identification.

Goddard problem in presence of a dynamic pressure limit

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.

Range optimal trajectories for an aircraft flying in the vertical plane

Range optimal trajectories for an aircraft flying in the vertical plane are obtained from P on try agins minimum principle. Control variables are the load factor, which appears nonlinearly in the equations of motion, and the throttle setting, which appears only linearly. Both controls are subject to fixed bounds. Additionally, a dynamic pressure limit is imposed, which represents a first-order state-inequality constraint. For fixed flight time, initial coordinates, and final coordinates of the trajectory, the effect of the load factor limit on the resulting optimal switching structure is studied. All trajectories involve singular control along arcs with active dynamic pressure limit. For large flight times the optimal switching structures have not yet been found.

Neighboring optimal control based feedback law for the advanced launch system

In this paper a robust feedback algorithm is presented for a near-minimum-fuel ascent of a generic 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 divided 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), for 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 in this paper 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. cg ycg *TB yxB JCTC

Energy management of three-dimensional minimum-time intercept

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.