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NASA NTRS2007

Quaternion Averaging

Many applications require an algorithm that averages quaternions in an optimal manner. For example, when combining the quaternion outputs of multiple star trackers having this output capability, it is desirable to properly average the quaternions without recomputing the attitude from the the raw star tracker data. Other applications requiring some sort of optimal quaternion averaging include particle filtering and multiple-model adaptive estimation, where weighted quaternions are used to determine the quaternion estimate. For spacecraft attitude estimation applications, derives an optimal averaging scheme to compute the average of a set of weighted attitude matrices using the singular value decomposition method. Focusing on a 4-dimensional quaternion Gaussian distribution on the unit hypersphere, provides an approach to computing the average quaternion by minimizing a quaternion cost function that is equivalent to the attitude matrix cost function Motivated by and extending its results, this Note derives an algorithm that deterniines an optimal average quaternion from a set of scalar- or matrix-weighted quaternions. Rirthermore, a sufficient condition for the uniqueness of the average quaternion, and the equivalence of the mininiization problem, stated herein, to maximum likelihood estimation, are shown.

Markley, F. Landis↗

NASA NTRS1989

Algorithm For Optimal Control Of Large Structures

Cost of computation appears competitive with other methods. Problem to compute optimal control of forced response of structure with n degrees of freedom identified in terms of smaller number, r, of vibrational modes. Article begins with Hamilton-Jacobi formulation of mechanics and use of quadratic cost functional. Complexity reduced by alternative approach in which quadratic cost functional expressed in terms of control variables only. Leads to iterative solution of second-order time-integral matrix Volterra equation of second kind containing optimal control vector. Cost of algorithm, measured in terms of number of computations required, is of order of, or less than, cost of prior algoritms applied to similar problems.

Salama, Moktar A.↗

NASA NTRS1987

Information prioritization for control and automation of space operations

The applicability of a real-time information prioritization technique to the development of a decision support system for control and automation of Space Station operations is considered. The steps involved in the technique are described, including the definition of abnormal scenarios and of attributes, measures of individual attributes, formulation and optimization of a cost function, simulation of test cases on the basis of the cost function, and examination of the simulation scenerios. A list is given comparing the intrinsic importances of various Space Station information data.

Ray, Asock↗

NASA NTRS2012

On Optimal Control of Non-Autonomous Switched Systems with a Fixed Mode Sequence

We consider differentiability with respect to the switch times of the value function of an optimal control problem for a non-autonomous switched system. The control variables are the switch times between the modes and the input in each mode. We provide a method to compute the derivative of the cost function given a nominal input. Then, we view the optimal control problem as a parametrized optimization problem in which the switch times are the parameters and the optimization is over the set of feasible inputs of each mode. From this point of view, we provide conditions under which the continuity and differentiability of the optimal value function, that is the cost function optimized over the inputs, can be guaranteed.

Kamgarpour, Maryam↗

NASA NTRS1992

Traffic routing for multicomputer networks with virtual cut-through capability

Consideration is given to the problem of selecting routes for interprocess communication in a network with virtual cut-through capability, while balancing the network load and minimizing the number of times that a message gets buffered. An approach is proposed that formulates the route selection problem as a minimization problem with a link cost function that depends upon the traffic through the link. The form of this cost function is derived using the probability of establishing a virtual cut-through route. The route selection problem is shown to be NP-hard, and an algorithm is developed to incrementally reduce the cost by rerouting the traffic. The performance of this algorithm is exemplified by two network topologies: the hypercube and the C-wrapped hexagonal mesh.

Kandlur, Dilip D.↗

NASA NTRS1976

Complex symmetric root square locus with an application to a spinning drag-free satellite

The parameters and relations associated with optimal systems are examined, taking into account a quadratic performance index and a root square locus plot, including the characteristic roots of the optimal system and its adjoint system as a function of the cost function weights. The calculation of the locus is described and the employment of the considered relations in studies of a drag-free satellite is discussed. Attention is given to weights regarding the initial states, questions of rotating integral control, approaches for experimental verification, and the performance of various methods for the reduction of fuel consumption due to center of spin offsets.

Tashker, M. G.↗

NASA NTRS1980

Optimal regulation in systems with stochastic time sampling

An optimal control theory that accounts for stochastic variable time sampling in a distributed microprocessor based flight control system is presented. The theory is developed by using a linear process model for the airplane dynamics and the information distribution process is modeled as a variable time increment process where, at the time that information is supplied to the control effectors, the control effectors know the time of the next information update only in a stochastic sense. An optimal control problem is formulated and solved for the control law that minimizes the expected value of a quadratic cost function. The optimal cost obtained with a variable time increment Markov information update process where the control effectors know only the past information update intervals and the Markov transition mechanism is almost identical to that obtained with a known and uniform information update interval.

Montgomery, R. C.↗

NASA NTRS1992

Unified Optimization Of A Structure And Its Control System

Method for unified optimization of structure and its vibration-suppressing control system involves extension of established techniques of optimization via approach based on homotopy, in which cost function used as criterion for optimization of control system coupled to corresponding cost function of structure via new parameter. Applicable to such terrestrial problems as designing buildings with control actuators to prevent damage by earthquakes.

Milman, Mark H.↗

NASA NTRS1992

Minimum entropy deconvolution and blind equalisation

Relationships between minimum entropy deconvolution, developed primarily for geophysics applications, and blind equalization are pointed out. It is seen that a large class of existing blind equalization algorithms are directly related to the scale-invariant cost functions used in minimum entropy deconvolution. Thus the extensive analyses of these cost functions can be directly applied to blind equalization, including the important asymptotic results of Donoho.

Satorius, E. H.↗

NASA NTRS1985

Characteristic Elastic Systems of Time-Limited Optimal Maneuvers

Optimizing an elastic system and its active control is discussed. Maneuvers from an initial state to a final state in a finite time interval are considered. An active generalized control force that accomplishes the desired maneuver of a prespecified system is optimal if it minimizes a given quadratic cost function. By also varying a set of design parameters, the elastic system can be determined so as to further minimize the cost function. Here, the elastic system that minimizes the actual control cost is compared with the system that minimizes the ratio of actual cost to the cost of optimally maneuvering a rigid system of the same inertial properties. It is shown that an elastic system corresponding to an extremum of the ratio is actually a characteristic of the time-limited maneuver. Because both the spatial domain and the time interval are fixed, a characteristic elastic system is tuned to the specified temporal boundary conditions. The implication for rest-to-rest, spinup, and spin reversal maneuvers of spacecraft is that the optimal control for a characteristic elastic spacecraft is identical to the optimal control for the same spacecraft as if it were rigid.

Hale, A. L.↗

NASA NTRS1978

A new formulation for the epsilon method applied to the minimum-time-to-climb problem

Balakrishnan's epsilon technique is used to compute minimum-time profiles for the F-104 airplane. This technique differs from the classical gradient method in that a quadratic penalty on the error in satisfying the equation of motion is included in the cost function to be minimized as a means of eliminating the requirement of satisfying the equations of motion. Although the number of unknown independent functions is increased to include the state variables, the evaluation of the gradient of the cost function is simplified, resulting in considerable computational savings, thereby making it appear feasible to use the epsilon method for real-time application.

Taylor, L. W., Jr.↗

NASA NTRS2005

Quadratic Programming for Allocating Control Effort

A computer program calculates an optimal allocation of control effort in a system that includes redundant control actuators. The program implements an iterative (but otherwise single-stage) algorithm of the quadratic-programming type. In general, in the quadratic-programming problem, one seeks the values of a set of variables that minimize a quadratic cost function, subject to a set of linear equality and inequality constraints. In this program, the cost function combines control effort (typically quantified in terms of energy or fuel consumed) and control residuals (differences between commanded and sensed values of variables to be controlled). In comparison with prior control-allocation software, this program offers approximately equal accuracy but much greater computational efficiency. In addition, this program offers flexibility, robustness to actuation failures, and a capability for selective enforcement of control requirements. The computational efficiency of this program makes it suitable for such complex, real-time applications as controlling redundant aircraft actuators or redundant spacecraft thrusters. The program is written in the C language for execution in a UNIX operating system.

Singh, Gurkirpal↗

NASA NTRS2022

Earth System Model Parameter Adjustment Using a Green's Functions Approach

We demonstrate the practicality and effectiveness of using a Green's functions estimation approach for adjusting uncertain parameters in an Earth system model (ESM). This estimation approach has previously been applied to an intermediate-complexity climate model and to individual ESM components, e.g., ocean, sea ice, or carbon cycle components. Here, the Green's functions approach is applied to a state-of-the-art ESM that comprises a global atmosphere/land configuration of the Goddard Earth Observing System (GEOS) coupled to an ocean and sea ice configuration of the Massachusetts Institute of Technology general circulation model (MITgcm). Horizontal grid spacing is approximately 110 km for GEOS and 37–110 km for MITgcm. In addition to the reference GEOS-MITgcm simulation, we carried out a series of model sensitivity experiments, in which 20 uncertain parameters are perturbed. These “control” parameters can be used to adjust sea ice, microphysics, turbulence, radiation, and surface schemes in the coupled simulation. We defined eight observational targets: sea ice fraction, net surface shortwave radiation, downward longwave radiation, near-surface temperature, sea surface temperature, sea surface salinity, and ocean temperature and salinity at 300 m. We applied the Green's functions approach to optimize the values of the 20 control parameters so as to minimize a weighted least-squares distance between the model and the eight observational targets. The new experiment with the optimized parameters resulted in a total cost reduction of 9 % relative to a simulation that had already been adjusted using other methods. The optimized experiment attained a balanced cost reduction over most of the observational targets. We also report on results from a set of sensitivity experiments that are not used in the final optimized simulation but helped explore options and guided the optimization process. These experiments include an assessment of sensitivity to the number of control parameters and to the selection of observational targets and weights in the cost function. Based on these sensitivity experiments, we selected a specific definition for the cost function. The sensitivity experiments also revealed a decreasing overall cost as the number of control variables was increased. In summary, we recommend using the Green's functions estimation approach as an additional fine-tuning step in the model development process. The method is not a replacement for modelers' experience in choosing and adjusting sensitive model parameters. Instead, it is an additional practical and effective tool for carrying out final adjustments of uncertain ESM parameters.

Green's Function↗

NASA NTRS1985

More than you want to know about maximum likelihood estimation

The maximum likelihood estimator has been used to extract stability and control derivatives from flight data for many years. Most of the literature on aircraft estimation concentrates on new developments and applications, assuming familiarity with basic estimation concepts. Some of these basic concepts are presented. The maximum likelihood estimator is briefly discussed and the aircraft equations of motion that the estimator uses. The basic concepts of minimization and estimation are examined for a simple computed aircraft example. The cost functions that are to be minimized during estimation are defined and discussed. Graphic representations of the cost functions are given to help illustrate the minimization process. Finally, the basic concepts are generalized, and estimation from flight data is discussed. Some of the major conclusions for the computed example are also developed for the analysis of flight data.

Iliff, K. W.↗

NASA NTRS1985

Extraction of aerodynamic parameters for aircraft at extreme flight conditions

The maximum likelihood estimator has been used to extract stability and control derivatives from flight data for many years. Most of the literature on aircraft estimation concentrates on new developments and applications, assuming familiarity with basic concepts. This paper briefly discusses the maximum likelihood estimator and the aircraft equations of motion that the estimator uses. The current strength and limitations associated with obtaining flight-determined aerodynamic coefficients in extreme flight conditions is assessed. The importance of the careful combining of wind tunnel results (or calculations) and flight results and the thorough evaluation of the mathematical model is emphasized. The basic concepts of minimization and estimation are examined for a simple computed aircraft example, and the cost functions that are to be minimized during estimation are defined and discussed. Graphic representations of the cost functions are given to help illustrate the minimization process. Finally, the basic concepts are generalized, and estimation of stability and control derivatives from flight data is discussed.

Iliff, K. W.↗

NASA NTRS1985

Maximum Likelihood Estimation with Emphasis on Aircraft Flight Data

Accurate modeling of flexible space structures is an important field that is currently under investigation. Parameter estimation, using methods such as maximum likelihood, is one of the ways that the model can be improved. The maximum likelihood estimator has been used to extract stability and control derivatives from flight data for many years. Most of the literature on aircraft estimation concentrates on new developments and applications, assuming familiarity with basic estimation concepts. Some of these basic concepts are presented. The maximum likelihood estimator and the aircraft equations of motion that the estimator uses are briefly discussed. The basic concepts of minimization and estimation are examined for a simple computed aircraft example. The cost functions that are to be minimized during estimation are defined and discussed. Graphic representations of the cost functions are given to help illustrate the minimization process. Finally, the basic concepts are generalized, and estimation from flight data is discussed. Specific examples of estimation of structural dynamics are included. Some of the major conclusions for the computed example are also developed for the analysis of flight data.

Iliff, K. W.↗

NASA NTRS1985

Extraction of aerodynamic parameters for aircraft at extreme flight conditions

The maximum likelihood estimator was used to extract stability and control derivatives from flight data for many years. Most of the literature on aircraft estimation concentrates on new development and applications, assuming familiarity with basic concepts. The maximum likelihood estimator and the aircraft equations of motion that the estimator uses are discussed. The current strength and limitations associated with obtaining flight-determined aerodynamic coefficients in extreme flight conditions are assessed. The importance of the careful combining of wind tunnel results (or calculations) and flight results and the thorough evaluation of the mathematical model is emphasized. The basic concepts of minimization and estimation are examined for a simple computed aircraft example, and the cost functions that are to be minimized during estimation are defined and discussed. Graphic representations of the cost functions are given to help illustrate the minimization process. Finally, the basic concepts are generalized, and estimation of stability and control derivatives from flight data is discussed.

Iliff, K. W.↗

NASA NTRS1987

Aircraft parameter estimation

The aircraft parameter estimation problem is used to illustrate the utility of parameter estimation, which applies to many engineering and scientific fields. Maximum likelihood estimation has been used to extract stability and control derivatives from flight data for many years. This paper presents some of the basic concepts of aircraft parameter estimation and briefly surveys the literature in the field. The maximum likelihood estimator is discussed, and the basic concepts of minimization and estimation are examined for a simple simulated aircraft example. The cost functions that are to be minimized during estimation are defined and discussed. Graphic representations of the cost functions are given to illustrate the minimization process. Finally, the basic concepts are generalized, and estimation from flight data is discussed. Some of the major conclusions for the simulated example are also developed for the analysis of flight data from the F-14, highly maneuverable aircraft technology (HiMAT), and space shuttle vehicles.

Kenneth W Iliff↗