Simultaneous solution of non-linear algebraic equations Technical memorandum no. 163
Digital computer program for nonlinear equation sets characteristic of filter circuit analysis
NONLINEAR EQUATION↗
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Results for “Nonlinear filtering”
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Constructing an Efficient Self-Tuning Aircraft Engine Model for Control and Health Management Applications
Self-tuning aircraft engine models can be applied for control and health management applications. The self-tuning feature of these models minimizes the mismatch between any given engine and the underlying engineering model describing an engine family. This paper provides details of the construction of a self-tuning engine model centered on a piecewise linear Kalman filter design. Starting from a nonlinear transient aerothermal model, a piecewise linear representation is first extracted. The linearization procedure creates a database of trim vectors and state-space matrices that are subsequently scheduled for interpolation based on engine operating point. A series of steady-state Kalman gains can next be constructed from a reduced-order form of the piecewise linear model. Reduction of the piecewise linear model to an observable dimension with respect to available sensed engine measurements can be achieved using either a subset or an optimal linear combination of "health" parameters, which describe engine performance. The resulting piecewise linear Kalman filter is then implemented for faster-than-real-time processing of sensed engine measurements, generating outputs appropriate for trending engine performance, estimating both measured and unmeasured parameters for control purposes, and performing on-board gas-path fault diagnostics. Computational efficiency is achieved by designing multidimensional interpolation algorithms that exploit the shared scheduling of multiple trim vectors and system matrices. An example application illustrates the accuracy of a self-tuning piecewise linear Kalman filter model when applied to a nonlinear turbofan engine simulation. Additional discussions focus on the issue of transient response accuracy and the advantages of a piecewise linear Kalman filter in the context of validation and verification. The techniques described provide a framework for constructing efficient self-tuning aircraft engine models from complex nonlinear simulations.Self-tuning aircraft engine models can be applied for control and health management applications. The self-tuning feature of these models minimizes the mismatch between any given engine and the underlying engineering model describing an engine family. This paper provides details of the construction of a self-tuning engine model centered on a piecewise linear Kalman filter design. Starting from a nonlinear transient aerothermal model, a piecewise linear representation is first extracted. The linearization procedure creates a database of trim vectors and state-space matrices that are subsequently scheduled for interpolation based on engine operating point. A series of steady-state Kalman gains can next be constructed from a reduced-order form of the piecewise linear model. Reduction of the piecewise linear model to an observable dimension with respect to available sensed engine measurements can be achieved using either a subset or an optimal linear combination of "health" parameters, which describe engine performance. The resulting piecewise linear Kalman filter is then implemented for faster-than-real-time processing of sensed engine measurements, generating outputs appropriate for trending engine performance, estimating both measured and unmeasured parameters for control purposes, and performing on-board gas-path fault diagnostics. Computational efficiency is achieved by designing multidimensional interpolation algorithms that exploit the shared scheduling of multiple trim vectors and system matrices. An example application illustrates the accuracy of a self-tuning piecewise linear Kalman filter model when applied to a nonlinear turbofan engine simulation. Additional discussions focus on the issue of transient response accuracy and the advantages of a piecewise linear Kalman filter in the context of validation and verification. The techniques described provide a framework for constructing efficient self-tuning aircraft engine models from complex nonlinear simulatns.
Armstrong, Jeffrey B.↗
Optimal washout for control of a moving base simulator
A general form of an optimal washout filter is derived using state-space linear optimal control theory, and this is applied to the design of washout filters of various types of moving base motion simulators, including the NASA's vertical motion simulator. Attention is given to the linear elements of a washout filter. One of the nonlinearities considered is braking which may be required near the end of the simulator excursion to prevent a crash. Although the general form of the optimal washout filter is applicable to time-variant system, the applications analyzed in the study are restricted to time-invariant cases.
Kurosaki, M.↗
Signal-Preserving Erratic Noise Attenuation via Iterative Robust Sparsity-Promoting Filter
Sparse domain thresholding filters operating in a sparse domain are highly effective in removing Gaussian random noise under Gaussian distribution assumption. Erratic noise, which designates non-Gaussian noise that consists of large isolated events with known or unknown distribution, also needs to be explicitly taken into account. However, conventional sparse domain thresholding filters based on the least-squares (LS) criterion are severely sensitive to data with high-amplitude and non-Gaussian noise, i.e., the erratic noise, which makes the suppression of this type of noise extremely challenging. Here, in this paper, we present a robust sparsity-promoting denoising model, in which the LS criterion is replaced by the Huber criterion to weaken the effects of erratic noise. The random and erratic noise is distinguished by using a data-adaptive parameter in the presented method, where random noise is described by mean square, while the erratic noise is downweighted through a damped weight. Different from conventional sparse domain thresholding filters, definition of the misfit between noisy data and recovered signal via the Huber criterion results in a nonlinear optimization problem. With the help of theoretical pseudoseismic data, an iterative robust sparsity-promoting filter is proposed to transform the nonlinear optimization problem into a linear LS problem through an iterative procedure. The main advantage of this transformation is that the nonlinear denoising filter can be solved by conventional LS solvers. Lastly, tests with several data sets demonstrate that the proposed denoising filter can successfully attenuate the erratic noise without damaging useful signal when compared with conventional denoising approaches based on the LS criterion.
97 MATHEMATICS AND COMPUTING↗
Comparing Consider-Covariance Analysis with Sigma-Point Consider Filter and Linear-Theory Consider Filter Formulations
Recent literature in applied estimation theory reflects growing interest in the sigma-point (also called unscented ) formulation for optimal sequential state estimation, often describing performance comparisons with extended Kalman filters as applied to specific dynamical problems [c.f. 1, 2, 3]. Favorable attributes of sigma-point filters are described as including a lower expected error for nonlinear even non-differentiable dynamical systems, and a straightforward formulation not requiring derivation or implementation of any partial derivative Jacobian matrices. These attributes are particularly attractive, e.g. in terms of enabling simplified code architecture and streamlined testing, in the formulation of estimators for nonlinear spaceflight mechanics systems, such as filter software onboard deep-space robotic spacecraft. As presented in [4], the Sigma-Point Consider Filter (SPCF) algorithm extends the sigma-point filter algorithm to the problem of consider covariance analysis. Considering parameters in a dynamical system, while estimating its state, provides an upper bound on the estimated state covariance, which is viewed as a conservative approach to designing estimators for problems of general guidance, navigation and control. This is because, whether a parameter in the system model is observable or not, error in the knowledge of the value of a non-estimated parameter will increase the actual uncertainty of the estimated state of the system beyond the level formally indicated by the covariance of an estimator that neglects errors or uncertainty in that parameter. The equations for SPCF covariance evolution are obtained in a fashion similar to the derivation approach taken with standard (i.e. linearized or extended) consider parameterized Kalman filters (c.f. [5]). While in [4] the SPCF and linear-theory consider filter (LTCF) were applied to an illustrative linear dynamics/linear measurement problem, in the present work examines the SPCF as applied to nonlinear sequential consider covariance analysis, i.e. in the presence of nonlinear dynamics and nonlinear measurements. A simple SPCF for orbit determination, exemplifying an algorithm hosted in the guidance, navigation and control (GN&C) computer processor of a hypothetical robotic spacecraft, was implemented, and compared with an identically-parameterized (standard) extended, consider-parameterized Kalman filter. The onboard filtering scenario examined is a hypothetical spacecraft orbit about a small natural body with imperfectly-known mass. The formulations, relative complexities, and performances of the filters are compared and discussed.
Lisano, Michael E.↗
A spatial disorientation predictor device to enhance pilot situational awareness regarding aircraft attitude
An effort was initiated at the Armstrong Aerospace Medical Research Laboratory (AAMRL) to investigate the improvement of the situational awareness of a pilot with respect to his aircraft's spatial orientation. The end product of this study is a device to alert a pilot to potentially disorienting situations. Much like a ground collision avoidance system (GCAS) is used in fighter aircraft to alert the pilot to 'pull up' when dangerous flight paths are predicted, this device warns the pilot to put a higher priority on attention to the orientation instrument. A Kalman filter was developed which estimates the pilot's perceived position and orientation. The input to the Kalman filter consists of two classes of data. The first class of data consists of noise parameters (indicating parameter uncertainty), conflict signals (e.g. vestibular and kinesthetic signal disagreement), and some nonlinear effects. The Kalman filter's perceived estimates are now the sum of both Class 1 data (good information) and Class 2 data (distorted information). When the estimated perceived position or orientation is significantly different from the actual position or orientation, the pilot is alerted.
Chelette, T. L.↗
Nonlinear hierarchical unit cell for passive, amplitude-dependent filtering of acoustic waves
Not Available
Mork, Nehemiah↗
Hierarchical unit cell employing a nonlinear energy sink for passive, low-pass amplitude filtering of acoustic waves
Not Available
Mork, Nehemiah (ORCID:0000000163158508)↗
Model-Based Control of a Nonlinear Aircraft Engine Simulation using an Optimal Tuner Kalman Filter Approach
This paper covers the development of a model-based engine control (MBEC) methodology featuring a self tuning on-board model applied to an aircraft turbofan engine simulation. Here, the Commercial Modular Aero-Propulsion System Simulation 40,000 (CMAPSS40k) serves as the MBEC application engine. CMAPSS40k is capable of modeling realistic engine performance, allowing for a verification of the MBEC over a wide range of operating points. The on-board model is a piece-wise linear model derived from CMAPSS40k and updated using an optimal tuner Kalman Filter (OTKF) estimation routine, which enables the on-board model to self-tune to account for engine performance variations. The focus here is on developing a methodology for MBEC with direct control of estimated parameters of interest such as thrust and stall margins. Investigations using the MBEC to provide a stall margin limit for the controller protection logic are presented that could provide benefits over a simple acceleration schedule that is currently used in traditional engine control architectures.
Control Theory↗
Accuracy of telemetry signal power loss in a filter as an estimate for telemetry degradation
When telemetry data is transmitted through a communication link, some degradation in telemetry performance occurs as a result of the imperfect frequency response of the channel. The term telemetry degradation as used here is the increase in received signal power required to offset this filtering. The usual approach to assessing this degradation is to assume that it is equal to the signal power loss in the filtering, which is easily calculated. However, this approach neglects the effects of the nonlinear phase response of the filter, the effect of any reduction of the receiving system noise due to the filter, and intersymbol interference. Here, an exact calculation of the telemetry degradation, which includes all of the above effects, is compared with the signal power loss calculation for RF filtering of NRZ data on a carrier. The signal power loss calculation is found to be a reasonable approximation when the filter follows the point at which the receiving system noise is introduced, especially if the signal power loss is less than 0.5 dB. The signal power loss approximation is less valid when the receiving system noise is not filtered.
Koerner, M. A.↗
Flight-test results using nonlinear control with the F-8C digital fly-by-wire aircraft
The design and operation of the cooperative advanced digital research experiment (CADRE) to develop nonlinear pitch flight control algorithms is described, and the results of an in-flight evaluation using the F-8C digital fly-by-wire (DFBW) research aircraft are presented. The CADRE controller is described, including the initial filter, linear command prefilter, nonlinear command prefilter, and gain scheduling. The variable-integral control-to-optimize response of the controller is considered, and CADRE parameter combinations are addressed. The remotely-augmented-vehicle interface used in the DFBW aircraft experiment is discussed. The distanct-tracking and close-formation tracking evaluation tasks for the aircraft are described along with evaluation configurations, and the test results are presented and discussed. The latter indicate that a nonlinear adaptive controller is a feasible control system technique for the fighter tracking task.
Larson, R. R.↗
Satellite Angular Rate Estimation From Vector Measurements
This paper presents an algorithm for estimating the angular rate vector of a satellite which is based on the time derivatives of vector measurements expressed in a reference and body coordinate. The computed derivatives are fed into a spacial Kalman filter which yields an estimate of the spacecraft angular velocity. The filter, named Extended Interlaced Kalman Filter (EIKF), is an extension of the Kalman filter which, although being linear, estimates the state of a nonlinear dynamic system. It consists of two or three parallel Kalman filters whose individual estimates are fed to one another and are considered as known inputs by the other parallel filter(s). The nonlinear dynamics stem from the nonlinear differential equation that describes the rotation of a three dimensional body. Initial results, using simulated data, and real Rossi X ray Timing Explorer (RXTE) data indicate that the algorithm is efficient and robust.
Azor, Ruth↗
Metrics for Intercomparison of Remapping Algorithms (MIRA) protocol applied to Earth system models
Abstract. Strongly coupled nonlinear phenomena such as those described by Earth system models (ESMs) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESMs is to remap or interpolate component solution fields defined on their computational mesh to another mesh with a different combinatorial structure and decomposition, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESMs. However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations is conducted based on a progression of scalar fields from idealized and smooth to more general climate data with strong discontinuities and strict bounds. We examine four remapping algorithms with distinct design approaches, namely ESMF Regrid (Hill et al., 2004), TempestRemap (Ullrich and Taylor, 2015), generalized moving least squares (GMLS) (Trask and Kuberry, 2020) with post-processing filters, and WLS-ENOR (Li et al., 2020). By repeated iterative application of the high-order remapping methods to the test fields, we verify the accuracy of each scheme in terms of their observed convergence order for smooth data and determine the bounded error propagation using challenging, realistic field data on both uniform and regionally refined mesh cases. In addition to retaining high-order accuracy under idealized conditions, the methods also demonstrate robust remapping performance when dealing with non-smooth data. There is a failure to maintain monotonicity in the traditional L2-minimization approaches used in ESMF and TempestRemap, in contrast to stable recovery through nonlinear filters used in both meshless GMLS and hybrid mesh-based WLS-ENOR schemes. Local feature preservation analysis indicates that high-order methods perform better than low-order dissipative schemes for all test cases. The behavior of these remappers remains consistent when applied on regionally refined meshes, indicating mesh-invariant implementations. The MIRA intercomparison protocol proposed in this paper and the detailed comparison of the four algorithms demonstrate that the new schemes, namely GMLS and WLS-ENOR, are competitive compared to standard conservative minimization methods requiring computation of mesh intersections. The work presented in this paper provides a foundation that can be extended to include complex field definitions, realistic mesh topologies, and spectral element discretizations, thereby allowing for a more complete analysis of production-ready remapping packages.
58 GEOSCIENCES↗
Experimental observation of acoustic sub-harmonic diffraction by a grating
A diffraction grating is a spatial filter causing sound waves or optical waves to reflect in directions determined by the frequency of the waves and the period of the grating. The classical grating equation is the governing principle that has successfully described the diffraction phenomena caused by gratings. However, in this work, we show experimental observation of the so-called sub-harmonic diffraction in acoustics that cannot be explained by the classical grating equation. Experiments indicate two physical phenomena causing the effect: internal scattering effects within the corrugation causing a phase shift and nonlinear acoustic effects generating new frequencies. This discovery expands our current understanding of the diffraction phenomenon, and it also makes it possible to better design spatial diffraction spectra, such as a rainbow effect in optics with a more complicated color spectrum than a traditional rainbow. The discovery reveals also a possibly new technique to study nonlinear acoustics by exploitation of the natural spatial filtering effect inherent to an acoustic diffraction grating.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Model-Based Engine Control Architecture with an Extended Kalman Filter
This paper discusses the design and implementation of an extended Kalman filter (EKF) for model-based engine control (MBEC). Previously proposed MBEC architectures feature an optimal tuner Kalman Filter (OTKF) to produce estimates of both unmeasured engine parameters and estimates for the health of the engine. The success of this approach relies on the accuracy of the linear model and the ability of the optimal tuner to update its tuner estimates based on only a few sensors. Advances in computer processing are making it possible to replace the piece-wise linear model, developed off-line, with an on-board nonlinear model running in real-time. This will reduce the estimation errors associated with the linearization process, and is typically referred to as an extended Kalman filter. The nonlinear extended Kalman filter approach is applied to the Commercial Modular Aero-Propulsion System Simulation 40,000 (C-MAPSS40k) and compared to the previously proposed MBEC architecture. The results show that the EKF reduces the estimation error, especially during transient operation.
engine control↗
Studies in astronomical time series analysis. IV - Modeling chaotic and random processes with linear filters
While chaos arises only in nonlinear systems, standard linear time series models are nevertheless useful for analyzing data from chaotic processes. This paper introduces such a model, the chaotic moving average. This time-domain model is based on the theorem that any chaotic process can be represented as the convolution of a linear filter with an uncorrelated process called the chaotic innovation. A technique, minimum phase-volume deconvolution, is introduced to estimate the filter and innovation. The algorithm measures the quality of a model using the volume covered by the phase-portrait of the innovation process. Experiments on synthetic data demonstrate that the algorithm accurately recovers the parameters of simple chaotic processes. Though tailored for chaos, the algorithm can detect both chaos and randomness, distinguish them from each other, and separate them if both are present. It can also recover nonminimum-delay pulse shapes in non-Gaussian processes, both random and chaotic.
Scargle, Jeffrey D.↗
Real-time Identification and Control of Satellite Signal Impairments Solution and Application of the Stratonovich Equation Part 1. Theoretical Development
As satellite communications systems become both more complex and reliant with respect to their operating environment, it has become imperative to be able to identify, during real-time operation, the onset of one or more impairments to the quality of overall communications system integrity. One of the most important aspects to monitor of a satellite link operating within the Earth's atmosphere is the signal fading due to the occurrence of rain and/or phase scintillations. This, of course, must be done in the presence of the associated measurement uncertainty or potentially faulty measurement equipment such as in the Advanced Communication Technology Satellite (ACTS) experiment. In the present work, an approach originally suggested in 1991, and apparently still considered iconoclastic, will be significantly developed and applied to the satellite communications link on which the deleterious composite signal fade is the result of one or many component fade mechanisms. Through the measurement (with the attendant uncertainty or 'error' in the measurement) of such a composite fading satellite signal, it is desired to extract the level of each of the individual fading mechanisms so they can be appropriately mitigated before they impact the overall performance of the communications network. Rather than employing simple-minded deterministic filtering to the real-time fading, the present approach is built around all the models and/or descriptions used to describe the individual fade components, including their dynamic evolution. The latter is usually given by a first-order Langevin equation. This circumstance allows the description of the associated temporal transition probability densities of each of the component processes. By using this description, along with the real-time measurements of the composite fade (along with the measurement errors), one can obtain statistical estimates of the levels of each of the component fading mechanisms as well as their predicted values into the future. This is all accomplished by the use of the well-known Stratonovich integro-differential equation that results from the model of the measured signal fade that is also tailored to adaptively adjust the values of the parameters used in the statistical models of the individual fade mechanisms. Three examples of increasing complexity are addressed and solved for the iterative determination of fade component levels from the measured composite signal fade in the presence of measurement error and, in the last case, with uncertainty in the model parameters.
Countermeasure↗
Degradation of the contrast of a short light pulse in a CPA system
We have calculated the effect of four-wave mixing of the main radiation pulse with the background radiation components on the value of the output radiation contrast in a chirped pulse amplification system of petawatt laser facilities. Formulae are presented that make it possible to quantify the contributions of two mechanisms, i.e. Kerr nonlinearity of the refractive index of the amplifying medium and gain saturation. For typical values of the nonlinear refractive index, both contributions are comparable. The possibility of mutual quenching of nonlinear contributions is noted. The influence of spectral filtering of radiation on the output pulse contrast is considered. (paper)
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗