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NASA NTRS1994

Gain weighted eigenspace assignment

This report presents the development of the gain weighted eigenspace assignment methodology. This provides a designer with a systematic methodology for trading off eigenvector placement versus gain magnitudes, while still maintaining desired closed-loop eigenvalue locations. This is accomplished by forming a cost function composed of a scalar measure of error between desired and achievable eigenvectors and a scalar measure of gain magnitude, determining analytical expressions for the gradients, and solving for the optimal solution by numerical iteration. For this development the scalar measure of gain magnitude is chosen to be a weighted sum of the squares of all the individual elements of the feedback gain matrix. An example is presented to demonstrate the method. In this example, solutions yielding achievable eigenvectors close to the desired eigenvectors are obtained with significant reductions in gain magnitude compared to a solution obtained using a previously developed eigenspace (eigenstructure) assignment method.

Davidson, John B.↗

NASA NTRS1994

Optimal control theory determination of feasible return-to-launch-site aborts for the HL-20 Personnel Launch System vehicle

The personnel launch system (PLS) being studied by NASA is a system to complement the space shuttle and provide alternative access to space. The PLS consists of a manned spacecraft launched by an expendable launch vehicle (ELV). A candidate for the manned spacecraft is the HL-20 lifting body. In the event of an ELV malfunction during the initial portion of the ascent trajectory, the HL-20 will separate from the rocket and perform an unpowered return to launch site (RTLS) abort. This work details an investigation, using optimal control theory, of the RTLS abort scenario. The objective of the optimization was to maximize final altitude. With final altitude as the cost function, the feasibility of an RTLS abort at different times during the ascent was determined. The method of differential inclusions was used to determine the optimal state trajectories, and the optimal controls were then calculated from the optimal states and state rates.

Dutton, Kevin E.↗

NASA NTRS1993

FASTER - A tool for DSN forecasting and scheduling

FASTER (Forecasting And Scheduling Tool for Earth-based Resources) is a suite of tools designed for forecasting and scheduling JPL's Deep Space Network (DSN). The DSN is a set of antennas and other associated resources that must be scheduled for satellite communications, astronomy, maintenance, and testing. FASTER consists of MS-Windows based programs that replace two existing programs (RALPH and PC4CAST). FASTER was designed to be more flexible, maintainable, and user friendly. FASTER makes heavy use of commercial software to allow for customization by users. FASTER implements scheduling as a two pass process: the first pass calculates a predictive profile of resource utilization; the second pass uses this information to calculate a cost function used in a dynamic programming optimization step. This information allows the scheduler to 'look ahead' at activities that are not as yet scheduled. FASTER has succeeded in allowing wider access to data and tools, reducing the amount of effort expended and increasing the quality of analysis.

Werntz, David↗

NASA NTRS1994

Optimal trajectories for hypersonic launch vehicles

In this paper, we derive a near-optimal guidance law for the ascent trajectory from earth surface to earth orbit of a hypersonic, dual-mode propulsion, lifting vehicle. Of interest are both the optical flight path and the optimal operation of the propulsion system. The guidance law is developed from the energy-state approximation of the equations of motion. Because liquid hydrogen fueled hypersonic aircraft are volume sensitive, as well as weight sensitive, the cost functional is a weighted sum of fuel mass and volume; the weighting factor is chosen to minimize gross take-off weight for a given payload mass and volume in orbit.

Ardema, Mark D.↗

NASA NTRS1995

Parallel Monte Carlo Simulation for control system design

The research during the 1993/94 academic year addressed the design of parallel algorithms for stochastic robustness synthesis (SRS). SRS uses Monte Carlo simulation to compute probabilities of system instability and other design-metric violations. The probabilities form a cost function which is used by a genetic algorithm (GA). The GA searches for the stochastic optimal controller. The existing sequential algorithm was analyzed and modified to execute in a distributed environment. For this, parallel approaches to Monte Carlo simulation and genetic algorithms were investigated. Initial empirical results are available for the KSR1.

Schubert, Wolfgang M.↗

NASA NTRS1992

Enhanced orbital gyrocompassing by the optical flow sensed by an Earth-pointing camera

A new method for improving the orbital gyrocompassing process involving the attitude angle estimation of an earth-pointing satellite in low-Earth orbit uses an electro-optical sensor for direct measurement of the satellite azimuth angle. Simulations have shown that this additional measurement drastically reduces the estimator convergence time, especially when the sun sensor is rendered ineffective, e.g., by high solar elevations. The azimuth-sensing method is based on estimation of the image shift between successive picture frames of an on-board, Earth-pointing, charge-coupled device (CCD) full-matrix camera. The shift-estimation algorithm is based on minimizing a cost function which expresses mean-squared differences in brightness patterns of selected areas of the two frames. An extensive evaluation program with a computer-controlled 2-axis light table and actual satellite images has demonstrated high robustness for a wide range of variation of parameters including image texture content; camera focal length; sampling rate; and number of pixels processed. It was shown to be possible to estimate the azimuth angle within 0.1-0.2 degrees, for a suitably chosen parameter set.

Topaz, Leora↗

NASA NTRS1996

Experimental and Theoretical Results in Output-Trajectory Redesign for Flexible Structures

In this paper we study the optimal redesign of output trajectory for linear invertible systems. This is particularly important for tracking control of flexible structures because the input-state trajectories that achieve the required output may cause excessive vibrations in the structure. A trade-off is then required between tracking and vibrations reduction. We pose and solve this problem as the minimization of a quadratic cost function. The theory is developed and applied to the output tracking of a flexible structure and experimental results are presented.

Dewey, J. S.↗

NASA NTRS1996

Subspace Based Signal Analysis of Partially Polarized Light Reflected by Plant Canopies

The analysis of partially polarized light reflected by plant canopies is presented. Linear Least-Squares Method is developed, followed by the method based on a nonlinear cost function.

Fontaine, Fred L.↗

NASA NTRS1996

Analysis of the Hessian for Aerodynamic Optimization: Inviscid Flow

In this paper we analyze inviscid aerodynamic shape optimization problems governed by the full potential and the Euler equations in two and three dimensions. The analysis indicates that minimization of pressure dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and the Euler equations. However the optimization problems in two and three dimensions are inherently different. While the two dimensional optimization problems are well-posed the three dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. A natural choice of such a regularization is derived. The analysis also gives an estimate of the Hessian's condition number which implies that the problems at hand are ill-conditioned. Infinite dimensional approximations for the Hessians are constructed and preconditioners for gradient based methods are derived from these approximate Hessians.

Arian, Eyal↗

NASA NTRS1997

Application of the Two-Step Filter to Process Ranging Measurements for Relative Navigation in an Elliptical Orbit

This estimator breaks a nonlinear estimation problem into a set of over determined 'first step' states which are linear in the observations and 'second step' states which are ultimately the states of interest. Linear estimation methods are applied to filter the observations and produce the optimal first step state estimate. The 'second step' states are obtained through iterative nonlinear parameter estimation considering the first step states as observations. It has been shown that this process exactly minimizes the least squares cost function for static problems and provides a better solution than the iterated extended Kalman filter (EKF) for dynamic problems. The two step filter is applied in this paper to process range and range rate measurements between the two spacecraft. Details of the application of the two step estimator to this problem will be given, highlighting the use of a test for ill-conditioned covariance estimates that can result from the first order covariance propagation. A comparison will be made between the performance of the two step filter and the IEKF.

Garrison, James L.↗

NASA NTRS1997

Aerodynamic Design on Unstructured Grids for Turbulent Flows

An aerodynamic design algorithm for turbulent flows using unstructured grids is described. The current approach uses adjoint (costate) variables for obtaining derivatives of the cost function. The solution of the adjoint equations is obtained using an implicit formulation in which the turbulence model is fully coupled with the flow equations when solving for the costate variables. The accuracy of the derivatives is demonstrated by comparison with finite-difference gradients and a few example computations are shown. In addition, a user interface is described which significantly reduces the time required for setting up the design problems. Recommendations on directions of further research into the Navier Stokes design process are made.

Anderson, W. Kyle↗

NASA NTRS1997

Hypersonic Vehicle Trajectory Optimization and Control

Two classes of neural networks have been developed for the study of hypersonic vehicle trajectory optimization and control. The first one is called an 'adaptive critic'. The uniqueness and main features of this approach are that: (1) they need no external training; (2) they allow variability of initial conditions; and (3) they can serve as feedback control. This is used to solve a 'free final time' two-point boundary value problem that maximizes the mass at the rocket burn-out while satisfying the pre-specified burn-out conditions in velocity, flightpath angle, and altitude. The second neural network is a recurrent network. An interesting feature of this network formulation is that when its inputs are the coefficients of the dynamics and control matrices, the network outputs are the Kalman sequences (with a quadratic cost function); the same network is also used for identifying the coefficients of the dynamics and control matrices. Consequently, we can use it to control a system whose parameters are uncertain. Numerical results are presented which illustrate the potential of these methods.

Balakrishnan, S. N.↗

NASA NTRS1999

Feasibility of Decentralized Linear-Quadratic-Gaussian Control of Autonomous Distributed Spacecraft

A distributed satellite formation, modeled as an arbitrary number of fully connected nodes in a network, could be controlled using a decentralized controller framework that distributes operations in parallel over the network. For such problems, a solution that minimizes data transmission requirements, in the context of linear-quadratic-Gaussian (LQG) control theory, was given by Speyer. This approach is advantageous because it is non-hierarchical, detected failures gracefully degrade system performance, fewer local computations are required than for a centralized controller, and it is optimal with respect to the standard LQG cost function. Disadvantages of the approach are the need for a fully connected communications network, the total operations performed over all the nodes are greater than for a centralized controller, and the approach is formulated for linear time-invariant systems. To investigate the feasibility of the decentralized approach to satellite formation flying, a simple centralized LQG design for a spacecraft orbit control problem is adapted to the decentralized framework. The simple design uses a fixed reference trajectory (an equatorial, Keplerian, circular orbit), and by appropriate choice of coordinates and measurements is formulated as a linear time-invariant system.

Carpenter, J. Russell↗

NASA NTRS1999

Optimal Frequency-Domain System Realization with Weighting

Several approaches are presented to identify an experimental system model directly from frequency response data. The formulation uses a matrix-fraction description as the model structure. Frequency weighting such as exponential weighting is introduced to solve a weighted least-squares problem to obtain the coefficient matrices for the matrix-fraction description. A multi-variable state-space model can then be formed using the coefficient matrices of the matrix-fraction description. Three different approaches are introduced to fine-tune the model using nonlinear programming methods to minimize the desired cost function. The first method uses an eigenvalue assignment technique to reassign a subset of system poles to improve the identified model. The second method deals with the model in the real Schur or modal form, reassigns a subset of system poles, and adjusts the columns (rows) of the input (output) influence matrix using a nonlinear optimizer. The third method also optimizes a subset of poles, but the input and output influence matrices are refined at every optimization step through least-squares procedures.

Juang, Jer-Nan↗

NASA NTRS1999

Optimal System Realization in Frequency Domain

Several approaches are presented to identify an experimental system model directly from frequency response data. The formulation begins with a matrix-fraction description as the model structure. Frequency weighting such as exponential weighting is introduced to solve a weighted least-squares problem to obtain the coefficient matrices for the matrix-fraction description. A multi-variable state-space model can then be formed using the coefficient matrices of the matrix-fraction description. An approach is introduced to fine-tune the model using non-linear programming methods to minimize the desired cost function. The method deals with the model in the real Schur or modal form and reassigns a subset of system poles using a nonlinear optimizer. At every optimization step, the input and output influence matrices are refined through least-squares procedures. The proposed approaches are used to identify an analytical model for a NASA testbed from experimental data.

Juang, Jer-Nan↗

NASA NTRS1999

Feasibility of Decentralized Linear-Quadratic-Gaussian Control of Autonomous Distributed Spacecraft

Carpenter, J. Russell↗

NASA NTRS1998

Experimental and Theoretical Results in Output Trajectory Redesign for Flexible Structures

In this paper we study the optimal redesign of output trajectories for linear invertible systems. This is particularly important for tracking control of flexible structures because the input-state trajectores, that achieve tracking of the required output may cause excessive vibrations in the structure. We pose and solve this problem, in the context of linear systems, as the minimization of a quadratic cost function. The theory is developed and applied to the output tracking of a flexible structure and experimental results are presented.

Dewey, J. S.↗

NASA NTRS1999

Integration of a Decentralized Linear-Quadratic-Gaussian Control into GSFC's Universal 3-D Autonomous Formation Flying Algorithm

A decentralized control is investigated for applicability to the autonomous formation flying control algorithm developed by GSFC for the New Millenium Program Earth Observer-1 (EO-1) mission. This decentralized framework has the following characteristics: The approach is non-hierarchical, and coordination by a central supervisor is not required; Detected failures degrade the system performance gracefully; Each node in the decentralized network processes only its own measurement data, in parallel with the other nodes; Although the total computational burden over the entire network is greater than it would be for a single, centralized controller, fewer computations are required locally at each node; Requirements for data transmission between nodes are limited to only the dimension of the control vector, at the cost of maintaining a local additional data vector. The data vector compresses all past measurement history from all the nodes into a single vector of the dimension of the state; and The approach is optimal with respect to standard cost functions. The current approach is valid for linear time-invariant systems only. Similar to the GSFC formation flying algorithm, the extension to linear LQG time-varying systems requires that each node propagate its filter covariance forward (navigation) and controller Riccati matrix backward (guidance) at each time step. Extension of the GSFC algorithm to non-linear systems can also be accomplished via linearization about a reference trajectory in the standard fashion, or linearization about the current state estimate as with the extended Kalman filter. To investigate the feasibility of the decentralized integration with the GSFC algorithm, an existing centralized LQG design for a single spacecraft orbit control problem is adapted to the decentralized framework while using the GSFC algorithm's state transition matrices and framework. The existing GSFC design uses both reference trajectories of each spacecraft in formation and by appropriate choice of coordinates and simplified measurement modeling is formulated as a linear time-invariant system. Results for improvements to the GSFC algorithm and a multiple satellite formation will be addressed. The goal of this investigation is to progressively relax the assumptions that result in linear time-invariance, ultimately to the point of linearization of the non-linear dynamics about the current state estimate as in the extended Kalman filter. An assessment will then be made about the feasibility of the decentralized approach to the realistic formation flying application of the EO-1/Landsat 7 formation flying experiment.

Folta, David C.↗