William F. Powers

Systems and Microcomputer Approach to Anticoagulant Therapy

A modem systems approach to the warfarin (an oral anticoagulant drug) dosage control problem is developed and analyzed on numerous patient data sets. Except for the initial statistics generation, all of the computer programs are developed for the relatively inexpensive Commodore PET 2001 microcomputer. A stabilized extended Kalman filter is employed to determine two patient dependent parameters from a series of Quick prothrombin time measurements in addition to estimating the prothrombin complex activity. The resultant parameters are then employed in an interactive graphics simulation program to determine a near-optimal dosage strategy.

Asymptotic Solution to the Problem of Optimal Low-Thrust Energy Increase

HE problem to be considered is that of optimal ascent from an initial circular planetary orbit to some specified final energy level by a spacecraft equipped with a low-thrust engine. Optimal will be defined as minimum time; and since it will be assumed that the engine produces continuous thrust with constant thrust-acceleration, fuel expenditure is minimized. Analytical solutions to this problem have previously been found by Lawden, 1 and by Breakwell and Rauch.2 Until now, Lawdens was the only analysis which used a small parameter perturbation approach; and his results failed to predict the oscillatory nature of the optimal control program. Breakwell and Rauchs work was directed primarily toward the analytical representation of a nominal trajectory and guidance coefficients for a neighboring optimal guidance scheme. Their solution is basically a series solution in the radial distance, but also contains some additional correction terms which were found by developing a set of defining differential equations, assuming a periodic solution with variable coefficients, and employing the method of averaging to determine those coefficients. The solution correctly, represents the characteristics of the control program and trajectory and, for at least eight revolutions, matches a numerically generated solution to within 1%. The use of the radial distance as independent variable and the series form of the solution, however, make an analysis of the motion and a comparison with other spiral trajectories difficult. The purpose of this Note is to present an accurate small parameter perturbation solution to the problem. In addition, the optimal trajectory is analyzed and compared with a tangential thrust trajectory. Analysis The problem is formulated using the equations of motion

Singular Optimal Control Computation

A quadratic functional version of the linear quadratic optimal control problem is employed to develop insights into the computation and theory of singular optimal controls. It is shown that sufficiency, existence, and uniqueness conditions and convergence conditions for gradient-type algorithms require the same basic assumption, namely a positivity assumption on the second-variation operator and an inclusion requirement with regard to the range space of the second variation operator. The theory is interpreted for both the nonsingular and singular problems to show the inherent differences between the two problems. It is shown that the gradient, conjugate gradient, and Davidon-Fletcher-Powell (DFP) algorithms converge if the existence conditions for the optimal control are satisfied, and that the rate of convergence is superlinear for the DFP method applied to nonsingular problems. Operational aspects for improving the rate of convergence on singular problems are discussed along with an informative comparison of the behavior of gradient and accelerated gradient methods on singular problems.

Computation of Optimal Earth-Mars and Earth-Venus Trajectories

Accurate solutions of minimal time Earth-Mars and Earth-Venus heliocentric trajectories are calculated with a shooting-Newton method. The flight times are less and the steering histories are diferent than those presented in [1], thus contradicting the optimality claims in [1].

Convergence of Gradient-Type Methods on Singular Parameter Optimization Problems

The convergence properties of the gradient, conjugate gradient, and Davidon-Fletcher-Powell methods for the singular, finite-dimensional quadratic minimization problem are developed. It is shown that for all of the methods, except the gradient method, that the minimum is obtained in at most m iterates, where m is the dimension of the range of the Hessian matrix, as opposed to n >m iterates for nonsingular problems. A class of associated nonsingular quadratic problems is defined to show that the gradient method has slower convergence on singular problems than on corresponding nonsingular approximations to the singular problems while the conjugate direction methods have more rapid convergence. This implies that slow convergence attributed to singular problems is actually a property of the gradient method as opposed to the singularity of the problem.

Some Applications of Optimization Techniques to Water Quality Modeling and Control

Dynamic water quality modeling methodology is presented along with a number of applications of optimization theory to deterministic water quality modeling and control. Two of the applications are basically parameter estimation problems. The first involves the generation of a dynamic model for the behavior of coliform bacteria in Grand Traverse Bay, while the second is concerned with the estimation of parameters in an existing model for the growth of bacteria in organic carbon limited media. The third application utilizes the proposition that the laws of nature are in some sense optimal to aid in the characterization of an algae inhibition function. The basic problem involves the development of a model of phytoplankton succession in lakes involving two species of algae and two nutrients. An unknown algal inhibition function is treated as an unknown control that is to be determined to minimize the loss of nutrients from the system, which is considered as an approximate optimality principle for the system. The resultant solution possesses an optimal singular subarc that has an interesting ecological interpretation. An active control application utilizing the same model is also presented.

Optimal low-thrust takeoff from an orbit about an oblate planet

Future space missions to the outer planets may depend upon the use of low-thrust propulsion systems. As these planets are decidedly oblate, the question of the effect of that oblateness on a low-thrust trajectory is of some interest. In this paper the problem of optimal energy increase is attacked under the assumption that the coefficients for the second zonal harmonic, i.e.,J2, and the nondimensional thrust acceleration are the same order of magnitude. Using a two variable asymptotic expansion technique, a near optimal control program is generated and the first order uniformly valid approximation for the corresponding trajectory is obtained. Tangential thrust is shown to be a good near-optimal thrust program even in the presence of oblateness effects. The optimal control program is found to be oscillatory and quite similar to the optimal control for energy increase in an inverse square gravitational field.

Iterative explicit guidance for low thrust spacecraft.

A retargeting procedure is developed for use as a nonlinear low thrust guidance scheme. The selection of a control program composed of a sequence of inertially fixed thrust-acceleration vectors permits all trajectory computations to be made with closed form expressions, and allows the controls to be represented by constant parameters, thrust-acceleration vectors and thrusting times. By requiring each trajectory to be time optimal, the guidance problem is transformed into a parameter optimization problem which is solved by the conjugate gradient method. The scheme is applied to a low thrust capture mission, and the results of computer simulations are presented.