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NASA NTRS2009

Analytic Confusion Matrix Bounds for Fault Detection and Isolation Using a Sum-of-Squared- Residuals Approach

Given a system which can fail in 1 or n different ways, a fault detection and isolation (FDI) algorithm uses sensor data in order to determine which fault is the most likely to have occurred. The effectiveness of an FDI algorithm can be quantified by a confusion matrix, which i ndicates the probability that each fault is isolated given that each fault has occurred. Confusion matrices are often generated with simulation data, particularly for complex systems. In this paper we perform FDI using sums of squares of sensor residuals (SSRs). We assume that the sensor residuals are Gaussian, which gives the SSRs a chi-squared distribution. We then generate analytic lower and upper bounds on the confusion matrix elements. This allows for the generation of optimal sensor sets without numerical simulations. The confusion matrix bound s are verified with simulated aircraft engine data.

Simon, Dan↗

NASA NTRS2011

Automated Methods to Maintain Aircraft Separation

The air traffic control system in the United States has a great track-record for safety. As more aircraft enter the system at a given time, the situation becomes more complex though. Researchers at NASA are attempting to leverage advances in many fields including optimization, data mining, and numerical modeling of systems to improve the air-transportation system maintaining safety while increasing throughput and reducing delays. This talk will give a brief overview of the research at NASA towards modernizing the air-transportation system. It will then focus on the specific area of automation tools for maintaining physical separation between aircraft known as Separation Assurance.

Lauderdale, Todd↗

NASA NTRS2019

Solid Fuel Ignition and Extinction (SoFIE) Project on ISS

The Solid Fuel Ignition and Extinction (SoFIE) project studies ignition and flammability of solid spacecraft materials (fuels) in practical geometries and realistic atmospheric conditions. It is an experiment insert designed for use within the existing Combustion Integrated Rack (CIR). The CIR chamber provides a level of containment and permits testing at variable oxygen concentrations and pressures representative of current and planned NASA Space Exploration Atmospheres. The applications of SoFIE include: (1) Determining safer selection of cabin materials and validating NASA materials flammability selection using 1-g test protocols for low-gravity fires, (2) Improving understanding of early fire growth behavior, (3) Validating material flammability numerical models, (4) Determining optimal suppression techniques for burning materials by diluents, flow reduction, and venting, (5) Obtaining burning behavior of actual engineering materials planned for spacecraft, (6) Developing corresponding models of microgravity flame spread, flammability, and extinction, and use the results to improve normal gravity combustion models for terrestrial applications. The hardware permits a wide range of solid-material combustion and fire suppression studies. It supports multiple investigations using common infrastructure including sample holders, flow control, test sections, external radiant heaters, igniters, and diagnostics. SoFIE has been developed to meet the requirements of five unique investigations. It is currently being built and slated to begin operations on the ISS in July 2021. Given the general capabilities of the hardware insert, it is intended to be used as a facility for future researchers who can propose to NASA for related solid combustion studies.

Ferkul, Paul↗

NASA NTRS2021

A Functional Interpolation Approach to Compute Period Orbits in the Circular Restricted Three-body Problem

In this paper, we develop a method to solve for periodic orbits, i.e., Lyapunov and Halo orbits, using a functional interpolation scheme called the Theory of Func- tional Connections (TFC). Using this technique, a periodic constraint is analyti- cally embedded into the TFC constrained expression. By doing this, the system of differential equations governing the three-body problem is transformed into an unconstrained optimization problem where simple numerical schemes can be used to find a solution, e.g., nonlinear least-squares is used. This allows for a simpler numerical implementation with comparable accuracy and speed to the traditional differential corrector method.

Mortari, Daniele↗

NASA NTRS1989

Parallel and vector computation for stochastic optimal control applications

A general method for parallel and vector numerical solutions of stochastic dynamic programming problems is described for optimal control of general nonlinear, continuous time, multibody dynamical systems, perturbed by Poisson as well as Gaussian random white noise. Possible applications include lumped flight dynamics models for uncertain environments, such as large scale and background random atmospheric fluctuations. The numerical formulation is highly suitable for a vector multiprocessor or vectorizing supercomputer, and results exhibit high processor efficiency and numerical stability. Advanced computing techniques, data structures, and hardware help alleviate Bellman's curse of dimensionality in dynamic programming computations.

Hanson, F. B.↗

DOE OSTI2025

A simple introduction to the SiMPL method for density-based topology optimization

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as “the simple method”) optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi–Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. In conclusion, to outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

Calculus of Variations and Optimization↗

NASA NTRS2002

Multidimensional indexing structure for use with linear optimization queries

Linear optimization queries, which usually arise in various decision support and resource planning applications, are queries that retrieve top N data records (where N is an integer greater than zero) which satisfy a specific optimization criterion. The optimization criterion is to either maximize or minimize a linear equation. The coefficients of the linear equation are given at query time. Methods and apparatus are disclosed for constructing, maintaining and utilizing a multidimensional indexing structure of database records to improve the execution speed of linear optimization queries. Database records with numerical attributes are organized into a number of layers and each layer represents a geometric structure called convex hull. Such linear optimization queries are processed by searching from the outer-most layer of this multi-layer indexing structure inwards. At least one record per layer will satisfy the query criterion and the number of layers needed to be searched depends on the spatial distribution of records, the query-issued linear coefficients, and N, the number of records to be returned. When N is small compared to the total size of the database, answering the query typically requires searching only a small fraction of all relevant records, resulting in a tremendous speedup as compared to linearly scanning the entire dataset.

Bergman, Lawrence David↗

NASA NTRS1986

Numerical difficulties associated with using equality constraints to achieve multi-level decomposition in structural optimization

There has been recent interest in multidisciplinary multilevel optimization applied to large engineering systems. The usual approach is to divide the system into a hierarchy of subsystems with ever increasing detail in the analysis focus. Equality constraints are usually placed on various design quantities at every successive level to ensure consistency between levels. In many previous applications these equality constraints were eliminated by reducing the number of design variables. In complex systems this may not be possible and these equality constraints may have to be retained in the optimization process. In this paper the impact of such a retention is examined for a simple portal frame problem. It is shown that the equality constraints introduce numerical difficulties, and that the numerical solution becomes very sensitive to optimization parameters for a wide range of optimization algorithms.

Thareja, R.↗

NASA NTRS1993

An analytical optimization of electric propulsion orbit transfer vehicles

Due to the electric propulsion's inherent propellant mass savings over chemical propulsion, electric propulsion orbit transfer vehicles (EPOTV's) are highly efficient mode of orbit transfer. When selecting an electric propulsion device (ion, MPD, or arcjet) and propellant for a particular mission, it is preferable to use quick, analytical system optimization methods instead of time intensive numerical integration methods. It is also of interest to determine each thruster's optimal operating characteristics for a specific mission. Analytical expressions are derived which determine the optimal specific impulse (Isp) for each type of electric thruster to maximize payload fraction for a desired thrusting time. These expressions take into account the variation of thruster efficiency with specific impulse. Verification of the method is made with representative electric propulsion values on a LEO-to-GEO mission. Application of the method to specific missions is discussed.

Oleson, Steven R.↗

NASA NTRS2009

Optimization of Layer Densities for Spacecraft Multilayered Insulation Systems

Numerous tests of various multilayer insulation systems have indicated that there are optimal densities for these systems. However, the only method of calculating this optimal density was by a complex physics based algorithm developed by McIntosh. In the 1970's much data were collected on the performance of these insulation systems with many different variables analyzed. All formulas generated included number of layers and layer density as geometric variables in solving for the heat flux, none of them was in a differentiable form for a single geometric variable. It was recently discovered that by converting the equations from heat flux to thermal conductivity using Fourier's Law, the equations became functions of layer density, temperatures, and material properties only. The thickness and number of layers of the blanket were merged into a layer density. These equations were then differentiated with respect to layer density. By setting the first derivative equal to zero, and solving for the layer density, the critical layer density was determined. Taking a second derivative showed that the critical layer density is a minimum in the function and thus the optimum density for minimal heat leak, this is confirmed by plotting the original function. This method was checked and validated using test data from the Multipurpose Hydrogen Testbed which was designed using McIntosh's algorithm.

Johnson, W. L.↗

NASA NTRS1994

Theory and computation of optimal low- and medium-thrust transfers

This report presents two numerical methods considered for the computation of fuel-optimal, low-thrust orbit transfers in large numbers of burns. The origins of these methods are observations made with the extremal solutions of transfers in small numbers of burns; there seems to exist a trend such that the longer the time allowed to perform an optimal transfer the less fuel that is used. These longer transfers are obviously of interest since they require a motor of low thrust; however, we also find a trend that the longer the time allowed to perform the optimal transfer the more burns are required to satisfy optimality. Unfortunately, this usually increases the difficulty of computation. Both of the methods described use small-numbered burn solutions to determine solutions in large numbers of burns. One method is a homotopy method that corrects for problems that arise when a solution requires a new burn or coast arc for optimality. The other method is to simply patch together long transfers from smaller ones. An orbit correction problem is solved to develop this method. This method may also lead to a good guidance law for transfer orbits with long transfer times.

Chuang, C.-H.↗

NASA NTRS2014

Optimal Design of Calibration Signals in Space Borne Gravitational Wave Detectors

Future space borne gravitational wave detectors will require a precise definition of calibration signals to ensure the achievement of their design sensitivity. The careful design of the test signals plays a key role in the correct understanding and characterization of these instruments. In that sense, methods achieving optimal experiment designs must be considered as complementary to the parameter estimation methods being used to determine the parameters describing the system. The relevance of experiment design is particularly significant for the LISA Pathfinder mission, which will spend most of its operation time performing experiments to characterize key technologies for future space borne gravitational wave observatories. Here we propose a framework to derive the optimal signals in terms of minimum parameter uncertainty to be injected to these instruments during its calibration phase. We compare our results with an alternative numerical algorithm which achieves an optimal input signal by iteratively improving an initial guess. We show agreement of both approaches when applied to the LISA Pathfinder case.

LISA Path_nder mission↗

DOE OSTI2024

Identifiability and characterization of transmon qutrits through Bayesian experimental design

Robust control of a quantum system is essential to utilize the current noisy quantum hardware to its full potential, such as quantum algorithms. To achieve such a goal, a systematic search for an optimal control for any given experiment is essential. The design of optimal control pulses requires accurate numerical models and, therefore, accurate characterization of the system parameters. We present an online Bayesian approach for quantum characterization of qutrit systems, which automatically and systematically identifies optimal experiments that provide maximum information on the system parameters, thereby greatly reducing the number of experiments that need to be performed on the quantum testbed. Unlike most characterization protocols that provide point-estimates of the parameters, the proposed approach is able to estimate their probability distribution. The applicability of the Bayesian experimental design technique was demonstrated on test problems, where each experiment was defined by a parameterized control pulse. In addition to this, we also present an approach for iterative pulse extension, which is robust under uncertainties in transition frequencies and coherence times, and shot noise, despite being initialized with wide uninformative priors. Furthermore, we provide a mathematical proof of the theoretical identifiability of the model parameters and present conditions on the quantum state under which the parameters are identifiable. The proof and conditions for identifiability are presented for both closed and open quantum systems using the Schrödinger equation and the Lindblad master equation, respectively.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1972

Optimal trajectories between Earth and Mars in their true planetary orbits.

The optimal transfers from Earth to Mars and from Mars to Earth, considering the actual planetary orbits, are presented as functions of the corresponding idealized Hohmann transfers. The numerically exact two-impulse optimal trajectories are given in graphical form for all possible Hohmann windows. The two-impulse transfers which are absolute optimals and those for which a third impulse provides the absolute optimal are delineated. These data are designed to provide all the information necessary for quick orbit calculations for preliminary Martian mission analysis. In this form, they are as easy to use as the standard Hohmann transfer approximations and provide much greater accuracy.

Gravier, J. P.↗

NASA NTRS1989

Theory of the optimally coupled Q-switched laser

The general equations describing Q-switched laser operation are transcendental in nature and require numerical solutions, which greatly complicates the optimization of real devices. Here, it is shown that, using the mathematical technique of Lagrange multipliers, one can derive simple analytic expressions for all of the key parameters of the optimally coupled laser, i.e., one which uses an optimum reflector to obtain maximum laser efficiency for a given pump level. These parameters can all be expressed as functions of a single dimensionless variable z, defined as the ratio of the unsaturated small-signal gain to the dissipative (nonuseful) optical loss, multiplied by a few simple constants. Laser design tradeoff studies and performance projections can be accomplished quickly with the help of several graphs and a simple hand calculator. Sample calculations for a high-gain Nd:YAG and a low-gain alexandrite laser are presented as illustrations of the technique.

Degnan, John J.↗

NASA NTRS1999

Data Structure and Parallel Decomposition Considerations on a Fibonacci Grid

The Fibonacci grid, proposed by Swinbank and Purser (see companion abstract), provides attractive properties for global numerical atmospheric prediction by offering an optimally homogeneous, geometrically regular, and approximately isotropic discretization, with only the polar regions requiring special numerical treatment. It is a mathematical idealization, applied to the sphere, of the multi-spiral patterns often found in botanical structures, such as in pine cones and sunflower heads. Computationally, it is natural to organize the domain, into zones, in each of which the same pair, or triple, of "Fibonacci spirals" dominate. But the further subdivision of such zones into "tiles" of a shape and size suitable for distribution to the processors of a massively parallel computer requires very careful consideration if the subsequent spatial computations along the respective spirals, especially those computations (such as compact differencing schemes) that involve recursion, can be implemented in an efficient "load-balanced "manner without requiring excessive amounts of inter-processor communications. In this paper we show how certain "number theoretic" properties of the Fibonacci sequence (whose numbers prescribe the multiplicity of successive spirals) may be exploited in the decomposition of grid zones into tidy arrangements of triangular grid tiles, each tile possessing one side approximately parallel to the constant-latitude zone boundary. We also describe how the spatially recursive processes may be decomposed across such a tiling, and the directionality of the recursions reversed on alternate grid lines, to ensure a very high degree of load balancing throughout the execution of the computations required for one time step of a global model.

Michalakes, John↗

NASA NTRSDate not supplied

Performance Analysis of Magnetohydrodynamic Drag Modulation for Actively Controlled Aerocapture at Neptune

Missions to the Ice Giants are a top priority for flagship missions this coming decade. However, a fully propulsive orbital insertion into these planets requires an immense amount of fuel, taking a significant portion of the spacecraft mass and restricting the scientific payload. To mitigate this, aerocapture has been heavily investigated. Although simulations have shown that aerodynamically controlled aerocapture can successfully insert into an orbit around both Ice Giants, the deep atmospheric pass required necessitates a complex, mass expensive, and sometimes prohibitive thermal protection system. Magnetohydrodynamic drag modulation serves as a potential alternative control method for aerocapture which could not only save propellant mass compared to fully propulsive orbital insertion, but also save thermal protection system mass compared to conventional aerocapture methods. Both aerodynamically controlled and magnetohydrodynamically controlled aerocapture methodologies were simulated in NASA Langley’s high-fidelity six degree-of-freedom flight dynamics code, the Program to Optimize Simulated Trajectories II. Each method was simulated to identical missions to Neptune using a numerical predictor-corrector algorithm to optimize the control towards the target orbit. The results concluded that magnetohydrodynamic drag modulation can successfully capture and performs on par with aerodynamic drag modulation while significantly reducing the heatflux and aeroshell complexity.

Danny N Nguyen↗

NASA NTRS1976

Application of optimal input synthesis to aircraft parameter identification

The Frequency Domain Input Synthesis procedure is used in identifying the stability and control derivatives of an aircraft. By using a frequency-domain approach, one can handle criteria that are not easily handled by the time-domain approaches. Numerical results are presented for optimal elevator deflections to estimate the longitudinal stability and control derivatives subject to root-mean square constraints on the input. The applicability of the steady state optimal inputs to finite duration flight testing is investigated. The steady state approximation of frequency-domain synthesis is good for data lengths greater than two time cycles for the short period mode of the aircraft longitudinal motions. Phase relationships between different frequency components become important for shorter data lengths. The frequency domain inputs are shown to be much better than the conventional doublet inputs.

Gupta, N. K.↗