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NASA NTRS2012

Wavefront Sensing for WFIRST with a Linear Optical Model

In this paper we develop methods to use a linear optical model to capture the field dependence of wavefront aberrations in a nonlinear optimization-based phase retrieval algorithm for image-based wavefront sensing. The linear optical model is generated from a ray trace model of the system and allows the system state to be described in terms of mechanical alignment parameters rather than wavefront coefficients. This approach allows joint optimization over images taken at different field points and does not require separate convergence of phase retrieval at individual field points. Because the algorithm exploits field diversity, multiple defocused images per field point are not required for robustness. Furthermore, because it is possible to simultaneously fit images of many stars over the field, it is not necessary to use a fixed defocus to achieve adequate signal-to-noise ratio despite having images with high dynamic range. This allows high performance wavefront sensing using in-focus science data. We applied this technique in a simulation model based on the Wide Field Infrared Survey Telescope (WFIRST) Intermediate Design Reference Mission (IDRM) imager using a linear optical model with 25 field points. We demonstrate sub-thousandth-wave wavefront sensing accuracy in the presence of noise and moderate undersampling for both monochromatic and polychromatic images using 25 high-SNR target stars. Using these high-quality wavefront sensing results, we are able to generate upsampled point-spread functions (PSFs) and use them to determine PSF ellipticity to high accuracy in order to reduce the systematic impact of aberrations on the accuracy of galactic ellipticity determination for weak-lensing science.

Jurling, Alden S.↗

NASA NTRS2000

Parameterization of Model Validating Sets for Uncertainty Bound Optimizations

Given measurement data, a nominal model and a linear fractional transformation uncertainty structure with an allowance on unknown but bounded exogenous disturbances, easily computable tests for the existence of a model validating uncertainty set are given. Under mild conditions, these tests are necessary and sufficient for the case of complex, nonrepeated, block-diagonal structure. For the more general case which includes repeated and/or real scalar uncertainties, the tests are only necessary but become sufficient if a collinearity condition is also satisfied. With the satisfaction of these tests, it is shown that a parameterization of all model validating sets of plant models is possible. The new parameterization is used as a basis for a systematic way to construct or perform uncertainty tradeoff with model validating uncertainty sets which have specific linear fractional transformation structure for use in robust control design and analysis. An illustrative example which includes a comparison of candidate model validating sets is given.

Lim, K. B.↗

NASA NTRS2022

Low-Earth Orbit Trajectory Optimization in the Presence of Atmospheric Uncertainty

The previous 20 to 25 years have seen a tremendous increase in space exploration, and with that an increase in the level of logistics planning needed to ensure mission success. For spacecraft that are designed to be periodically re-supplied, a key logistics consumable is propellant, as it constitutes the greatest up-mass on re-supply vehicles. A trajectory design strategy is therefore desired that minimizes propellant usage in order to ease the demand for propellant re-supply missions. This thesis develops such a strategy in three stages, and uses the International Space Station (ISS) as its testbed, as no other LEO spacecraft is more challenging from a space logistics standpoint. First, the ISS trajectory planning problem is formulated as a constrained burn optimization problem assuming a deterministic atmosphere. The cost function is total ∆v, with constraints imposed on longitude of ascending viii node (LAN) and semi-major axis (SMA) altitude. Analytic derivatives are constructed for both the cost and constraints, which are necessary given the 6-week to 2-year time frames being considered. A gradient-based optimizer is then utilized to find locally-optimal solutions to real-world ISS trajectory planning problems. Second, atmospheric uncertainty is addressed by constructing a probabilistic model of space weather data using Gaussian Processes (GPs). Bayesian inference is performed using the GP model to generate mean and covariance estimates for space weather predictions, whose pedigree is assessed against test data. The predictions are then mapped into atmospheric density via the analytic Jacchia-Roberts density model, and the effect of space weather uncertainty on orbital lifetime is examined. Third, an ISS burn execution uncertainty model is developed. This model, along with the space weather uncertainty model, are deployed in a linear covariance analysis to ascertain their combined effect on LAN and SMA altitude dispersions. The deterministic constraints from the original problem are re-formulated as stochastic constraints, where now the constraint uncertainty interval is required to fall within specified bounds. An updated optimization framework is constructed using the original ∆v cost function along with the stochastic constraints to solve the trajectory optimization problem under atmospheric uncertainty. Finally, the complete architecture is summarized for deployment in an operational setting.

Trajectory Optimization↗

NASA NTRSDate not supplied

Gradient-Based Optimization of the Common Research Model Wing Subject to CFD-Based Gust and Flutter Constraints

The linearized frequency-domain method was recently implemented in the stabilized finite element solver in NASA’s FUN3D code. Previous work by the authors used this method for enforcing flutter constraints during gradient-based optimizations. More recently, the solver was expanded to account for continuous (also known as stochastic) gust responses. This paper expands on recent Common Research Model wing optimization work, which demonstrated gradient-based optimization with flutter and stochastic gust constraints, among others. While that work utilized FUN3D for static aeroelastic solutions but relied on doublet lattice aerodynamics for gust and flutter responses, the present work replaces these unsteady aerodynamic analyses with those of FUN3D’s linearized frequency-domain solver. With analytic derivatives available, gradient-based optimization is performed through the use of the OpenMDAO/MPhys libraries with over 700 shape, structural, and aerodynamic design variables and over 10 nonlinear constraints. Comparisons of analysis results and optimized designs are made between doublet lattice and linearized frequency-domain solutions.

aeroelasticity↗

NASA NTRS1984

Investigation and appreciation of optimal output feedback. Volume 1: A convergent algorithm for the stochastic infinite-time discrete optimal output feedback problem

The stochastic, infinite time, discrete output feedback problem for time invariant linear systems is examined. Two sets of sufficient conditions for the existence of a stable, globally optimal solution are presented. An expression for the total change in the cost function due to a change in the feedback gain is obtained. This expression is used to show that a sequence of gains can be obtained by an algorithm, so that the corresponding cost sequence is monotonically decreasing and the corresponding sequence of the cost gradient converges to zero. The algorithm is guaranteed to obtain a critical point of the cost function. The computational steps necessary to implement the algorithm on a computer are presented. The results are applied to a digital outer loop flight control problem. The numerical results for this 13th order problem indicate a rate of convergence considerably faster than two other algorithms used for comparison.

Halyo, N.↗

NASA NTRS1990

Parallel/distributed direct method for solving linear systems

A new family of parallel schemes for directly solving linear systems is presented and analyzed. It is shown that these schemes exhibit a near optimal performance and enjoy several important features: (1) For large enough linear systems, the design of the appropriate paralleled algorithm is insensitive to the number of processors as its performance grows monotonically with them; (2) It is especially good for large matrices, with dimensions large relative to the number of processors in the system; (3) It can be used in both distributed parallel computing environments and tightly coupled parallel computing systems; and (4) This set of algorithms can be mapped onto any parallel architecture without any major programming difficulties or algorithmical changes.

Lin, Avi↗

NASA NTRS1996

A Higher Harmonic Optimal Controller to Optimise Rotorcraft Aeromechanical Behaviour

Three methods to optimize rotorcraft aeromechanical behavior for those cases where the rotorcraft plant can be adequately represented by a linear model system matrix were identified and implemented in a stand-alone code. These methods determine the optimal control vector which minimizes the vibration metric subject to constraints at discrete time points, and differ from the commonly used non-optimal constraint penalty methods such as those employed by conventional controllers in that the constraints are handled as actual constraints to an optimization problem rather than as just additional terms in the performance index. The first method is to use a Non-linear Programming algorithm to solve the problem directly. The second method is to solve the full set of non-linear equations which define the necessary conditions for optimality. The third method is to solve each of the possible reduced sets of equations defining the necessary conditions for optimality when the constraints are pre-selected to be either active or inactive, and then to simply select the best solution. The effects of maneuvers and aeroelasticity on the systems matrix are modelled by using a pseudo-random pseudo-row-dependency scheme to define the systems matrix. Cases run to date indicate that the first method of solution is reliable, robust, and easiest to use, and that it was superior to the conventional controllers which were considered.

Leyland, Jane Anne↗

NASA NTRS1990

Factorization and the synthesis of optimal feedback gains for distributed parameter systems

An approach based on Volterra factorization leads to a new methodology for the analysis and synthesis of the optimal feedback gain in the finite-time linear quadratic control problem for distributed parameter systems. The approach circumvents the need for solving and analyzing Riccati equations and provides a more transparent connection between the system dynamics and the optimal gain. The general results are further extended and specialized for the case where the underlying state is characterized by autonomous differential-delay dynamics. Numerical examples are given to illustrate the second-order convergence rate that is derived for an approximation scheme for the optimal feedback gain in the differential-delay problem.

Milman, Mark H.↗

NASA NTRS1988

Sensitivity of control-augmented structure obtained by a system decomposition method

The verification of a method for computing sensitivity derivatives of a coupled system is presented. The method deals with a system whose analysis can be partitioned into subsets that correspond to disciplines and/or physical subsystems that exchange input-output data with each other. The method uses the partial sensitivity derivatives of the output with respect to input obtained for each subset separately to assemble a set of linear, simultaneous, algebraic equations that are solved for the derivatives of the coupled system response. This sensitivity analysis is verified using an example of a cantilever beam augmented with an active control system to limit the beam's dynamic displacements under an excitation force. The verification shows good agreement of the method with reference data obtained by a finite difference technique involving entire system analysis. The usefulness of a system sensitivity method in optimization applications by employing a piecewise-linear approach to the same numerical example is demonstrated. The method's principal merits are its intrinsically superior accuracy in comparison with the finite difference technique, and its compatibility with the traditional division of work in complex engineering tasks among specialty groups.

Sobieszczanskisobieski, Jaroslaw↗

NASA NTRS1991

Sensitivity of control-augmented structure obtained by a system decomposition method

Sobieszczanski-Sobieski, Jaroslaw↗

NASA NTRS1986

Synthesis of control laws for optimally designed large space structures

In this study, the vibration control of large space structures using the linear quadratic regulator technique is investigated. Emphasis is made on the control of both optimally designed structures and also the original (uniform) structures using the cantilever beam as an example. The open loop and closed loop eigenvalues are compared and the transient responses are obtained to determine the effectiveness of the control system design.

Satyanarayana, K.↗

NASA NTRS1988

A feedback linearization approach to spacecraft control using momentum exchange devices

Recent developments in the area of nonlinear control theory have shown how coordiante changes in the state and input spaces can be used with nonlinear feedback to transform certain nonlinear ordinary differential equations into equivalent linear equations. These feedback linearization techniques are applied to resolve two problems arising in the control of spacecraft equipped with control moment gyroscopes (CMGs). The first application involves the computation of rate commands for the gimbals that rotate the individual gyroscopes to produce commanded torques on the spacecraft. The second application is to the long-term management of stored momentum in the system of control moment gyroscopes using environmental torques acting on the vehicle. An approach to distributing control effort among a group of redundant actuators is described that uses feedback linearization techniques to parameterize sets of controls which influence a specified subsystem in a desired way. The approach is adapted for use in spacecraft control with double-gimballed gyroscopes to produce an algorithm that avoids problematic gimbal configurations by approximating sets of gimbal rates that drive CMG rotors into desirable configurations. The momentum management problem is stated as a trajectory optimization problem with a nonlinear dynamical constraint. Feedback linearization and collocation are used to transform this problem into an unconstrainted nonlinear program. The approach to trajectory optimization is fast and robust. A number of examples are presented showing applications to the proposed NASA space station.

Dzielski, John Edward↗

NASA NTRS1989

Optimal moving grids for time-dependent partial differential equations

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of partial differential equation solutions in the least squares norm.

Wathen, A. J.↗

NASA NTRS1992

Optimal moving grids for time-dependent partial differential equations

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm are reported.

Wathen, A. J.↗

NASA NTRS1963

Mathematical Optimization Techniques

The papers collected in this volume were presented at the Symposium on Mathematical Optimization Techniques held in the Santa Monica Civic Auditorium, Santa Monica, California, on October 18-20, 1960. The objective of the symposium was to bring together, for the purpose of mutual education, mathematicians, scientists, and engineers interested in modern optimization techniques. Some 250 persons attended. The techniques discussed included recent developments in linear, integer, convex, and dynamic programming as well as the variational processes surrounding optimal guidance, flight trajectories, statistical decisions, structural configurations, and adaptive control systems. The symposium was sponsored jointly by the University of California, with assistance from the National Science Foundation, the Office of Naval Research, the National Aeronautics and Space Administration, and The RAND Corporation, through Air Force Project RAND.

Bellman, R.↗

NASA NTRS2013

Regression Analysis of Top of Descent Location for Idle-thrust Descents

In this paper, multiple regression analysis is used to model the top of descent (TOD) location of user-preferred descent trajectories computed by the flight management system (FMS) on over 1000 commercial flights into Melbourne, Australia. The independent variables cruise altitude, final altitude, cruise Mach, descent speed, wind, and engine type were also recorded or computed post-operations. Both first-order and second-order models are considered, where cross-validation, hypothesis testing, and additional analysis are used to compare models. This identifies the models that should give the smallest errors if used to predict TOD location for new data in the future. A model that is linear in TOD altitude, final altitude, descent speed, and wind gives an estimated standard deviation of 3.9 nmi for TOD location given the trajec- tory parameters, which means about 80% of predictions would have error less than 5 nmi in absolute value. This accuracy is better than demonstrated by other ground automation predictions using kinetic models. Furthermore, this approach would enable online learning of the model. Additional data or further knowl- edge of algorithms is necessary to conclude definitively that no second-order terms are appropriate. Possible applications of the linear model are described, including enabling arriving aircraft to fly optimized descents computed by the FMS even in congested airspace. In particular, a model for TOD location that is linear in the independent variables would enable decision support tool human-machine interfaces for which a kinetic approach would be computationally too slow.

trajectory prediction↗

NASA NTRS1983

Structural optimization with dynamic behavior constraints

The minimum weight optimum design of damped linearly elastic structural systems subjected to periodic loading with behavior constraints on maximum deflections and side constraints on design variables is addressed. Attention is focused on the two major impediments to an optimal solution: (1) the time parametric nature of the behavior constraints; and (2) the severe nonconvexity of the design space. A solution method based on upper bound approximations for the behavior constraints and an innovative mathematical programming scheme for seeking the optimal frequency subspace is set forth. Numerical results for several test problems illustrate the effectiveness of the method reported.

Mills-Curran, W. C.↗

NASA NTRS1998

Design Process for High Speed Civil Transport Aircraft Improved by Neural Network and Regression Methods

A key challenge in designing the new High Speed Civil Transport (HSCT) aircraft is determining a good match between the airframe and engine. Multidisciplinary design optimization can be used to solve the problem by adjusting parameters of both the engine and the airframe. Earlier, an example problem was presented of an HSCT aircraft with four mixed-flow turbofan engines and a baseline mission to carry 305 passengers 5000 nautical miles at a cruise speed of Mach 2.4. The problem was solved by coupling NASA Lewis Research Center's design optimization testbed (COMETBOARDS) with NASA Langley Research Center's Flight Optimization System (FLOPS). The computing time expended in solving the problem was substantial, and the instability of the FLOPS analyzer at certain design points caused difficulties. In an attempt to alleviate both of these limitations, we explored the use of two approximation concepts in the design optimization process. The two concepts, which are based on neural network and linear regression approximation, provide the reanalysis capability and design sensitivity analysis information required for the optimization process. The HSCT aircraft optimization problem was solved by using three alternate approaches; that is, the original FLOPS analyzer and two approximate (derived) analyzers. The approximate analyzers were calibrated and used in three different ranges of the design variables; narrow (interpolated), standard, and wide (extrapolated).

Hopkins, Dale A.↗