Fading Memory Filter Research Articles

Reducing echo state network size with controllability matrices.

Echo state networks are a fast training variant of recurrent neural networks excelling at approximating nonlinear dynamical systems and time series prediction. These machine learning models act as nonlinear fading memory filters. While these models benefit from quick training and low complexity, computation demands from a large reservoir matrix are a bottleneck. Using control theory, a reduced size replacement reservoir matrix is found. Starting from a large, task-effective reservoir matrix, we form a controllability matrix whose rank indicates the active sub-manifold and candidate replacement reservoir size. Resulting time speed-ups and reduced memory usage come with minimal error increase to chaotic climate reconstruction or short term prediction. Experiments are performed on simple time series signals and the Lorenz-1963 and Mackey-Glass complex chaotic signals. Observing low error models shows variation of active rank and memory along a sequence of predictions.

Discrete-Time Signatures and Randomness in Reservoir Computing.

A new explanation of the geometric nature of the reservoir computing (RC) phenomenon is presented. RC is understood in the literature as the possibility of approximating input-output systems with randomly chosen recurrent neural systems and a trained linear readout layer. Light is shed on this phenomenon by constructing what is called strongly universal reservoir systems as random projections of a family of state-space systems that generate Volterra series expansions. This procedure yields a state-affine reservoir system with randomly generated coefficients in a dimension that is logarithmically reduced with respect to the original system. This reservoir system is able to approximate any element in the fading memory filters class just by training a different linear readout for each different filter. Explicit expressions for the probability distributions needed in the generation of the projected reservoir system are stated, and bounds for the committed approximation error are provided.

Echo state networks are universal

This paper shows that echo state networks are universal uniform approximants in the context of discrete-time fading memory filters with uniformly bounded inputs defined on negative infinite times. This result guarantees that any fading memory input/output system in discrete time can be realized as a simple finite-dimensional neural network-type state-space model with a static linear readout map. This approximation is valid for infinite time intervals. The proof of this statement is based on fundamental results, also presented in this work, about the topological nature of the fading memory property and about reservoir computing systems generated by continuous reservoir maps.

Kalman Filter Based Optimal Controllers in Free Space Optics Communication

There is no doubt that adaptive optics (AO) is the most promising method to compensate wavefront disturbance in free space optics communication (FSO). In order to improve the performance of the AO system described by discrete-time linear system model with time-delay and implicit phase turbulent model, new controllers based on a Kalman filter and its extensions are proposed. Based on the standard Kalman filter, we propose a fading memory filter to deal with the ruleless strong interference; sequential and U-D filters are applied to reduce implementation complexity for the embedded controllers. Theoretical analysis and the numerical simulations show that the proposed fading memory filter can upgrade the performance for AO systems in consideration of the unforeseen strong pulse interference, and the sequential and U-D filters perform well compared with a Kalman filter.

Erratum to “Adaptive Fading Memory Filter Design for Compensation of Delayed Components in Self Powered Flux Detectors” [Aug 15 1857-1864

In <citerefgrp xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><citeref refid="ref1"/></citerefgrp> , we presented an adaptive fading memory <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${{\rm H}_\infty }$</tex></formula> filter for delay compensation of Self Powered Flux Detectors (SPFDs). Unfortunately, we committed some errors in the text as well as in equations which were pointed by Mr. A. K. Mishra, BARC, India. In this erratum, we correct these errors. We also confirm that, all the corrections are editorial in nature and has no effect on the overall studies presented in the manuscript.

Research on the Effectiveness of Different Estimation Algorithm on the Autonomous Orbit Determination of Lagrangian Navigation Constellation

The accuracy of autonomous orbit determination of Lagrangian navigation constellation will affect the navigation accuracy for the deep space probes. Because of the special dynamical characteristics of Lagrangian navigation satellite, the error caused by different estimation algorithm will cause totally different autonomous orbit determination accuracy. We apply the extended Kalman filter and the fading–memory filter to determinate the orbits of Lagrangian navigation satellites. The autonomous orbit determination errors are compared. The accuracy of autonomous orbit determination using fading-memory filter can improve 50% compared to the autonomous orbit determination accuracy using extended Kalman filter. We proposed an integrated Kalman fading filter to smooth the process of autonomous orbit determination and improve the accuracy of autonomous orbit determination. The square root extended Kalman filter is introduced to deal with the case of inaccurate initial error variance matrix. The simulations proved that the estimation method can affect the accuracy of autonomous orbit determination greatly.

Fixed Target 3D Localization Based on Range Data Only: a Recursive Least Squares Approach

This problem of identifying the position of a fixed target by a vehicle moving in 3D space may occur in several applications as underwater or aerospace monitoring. Assuming that only range measurements to the target are available and that the vehicle is equipped with a navigation system providing self-localization, the problem of localizing the target is analyzed. A recursive least squares fading memory filter solution is proposed. The dependence of the covariance of the estimated target position on the velocity profile of the vehicle is discussed.

Impacting small Near Earth Objects

The design of a low-cost spacecraft to impact on a small, faint Near Earth Object (NEO), poses major challenges. This paper focuses on the terminal phase of such impact mission, analyzing the capability of autonomous Guidance, Navigation and Control (GNC) systems to compensate the deviations in the impact point to achieve a successful collision. The autonomous GNC system employs the information of the optical sensors to estimate the parameters allowing the computation of divert maneuvers to achieve the impact. GMV has developed a simulator, with different levels of sophistication, and a set of different GNC algorithms to help in the design process. This tool is used for different purposes such as: dimensioning the sensors and actuators, verifying mission requirements, computing figures of merit of different SC configurations and evaluating GNC performances. Four GNC algorithms are compared: low-thrust proportional navigation using a fading memory filter, high-thrust predictive guidance using either a Kalman filter or a batch-sequential least-squares filter, and a mid-thrust hybrid predictive-proportional guidance using a fading memory filter. Monte Carlo analysis using global-performances models of the optical sensors for each of these GNC algorithms are presented for two different asteroids (1989 ML and 2002 AT4), showing the mission parameters driving the mission performances. In addition, single-runs with high-fidelity optical sensors models are presented to validate the Monte Carlo simulations.

Kalman filter with hypothesis testing: A tool for estimating uncertain parameters

In this paper we present a simple and practical algorithm for the estimation of uncertain parameters of linear systems. The uncertainty is twofold, involving random observation noise, and possible jumps in the parameter values. The jumps may occur at unknown points in time, and are of unknown magnitudes and directions. The algorithm is based on the Kalman filter, with a single-sample hypothesis test, which is used to employ a three-state decision rule (yes, no, maybe). The "maybe" choice invokes a fading memory Kalman filter. The overall algorithm contains the constant parameter filter, fading memory filter, and the set of tests and rules that enable it to switch back and forth between the two filters. Application examples are presented.

Simplified Solution for a Class of Fading Memory Filters

A fading memory filter is a least squares estimator (LSE) thatapplies an exponentially decaying weight to past measurements.When compared with a standard Kalman filter, its key advantagesare asymptotic stability and reduced sensitivity to modeling errors.This paper derives a simple solution for a class of fading memoryfilters, resulting in a reduction in computational complexity. Steadystate filter solutions are obtained for second-and third-order filtersused in a global positioning system (GPS) receiver for high dynamicvehicles.

Additions and Corrections to "Simplified Solution for a Class of Fading Memory Filters"

Additions and Corrections to "Simplified Solution for a Class of Fading Memory Filters"

On the steady-state error of the fading memory filter

The steady-state error of the fading memory filter is studied for stable, two-dimensional models with an uncertainty in the form of a constant, unknown control input. For a set of these models, it is observed that any age-weighting of the data can cause an error greater than that of the Kalman filter. This contradicts previous assumptions about the fading memory filter.

Recursive fading memory smoothing

The recursive fading memory filter equations for time-continuous and time-discrete systems presented in Ref. 1 is extended here to the fixed-interval smoothing case. The three standard forms of the smoothing equations are derived, and practical aspects of their implementation are discussed.

Filter-and-Hold Technique for Television Densitometry

A simplified television densitometer was developed utilizing a filter-and-hold technique. The system circumvents the use of complex digital circuitry by using a fading memory filter which yields an approximate integral of the optical density in a window in the area of interest. The response is linear for features whose optical density changes in the parallel direction to the raster lines, and linear for 40% changes of optical density in the vertical direction.

Nonlinear extensions of the fading memory filter

The equations for a recursive extended Kalman filter with exponential age-weighting of data and dynamics are derived. A similar result is given for a second-order filter. It is seen that the filter equations are essentially of the same form as their unfaded counterparts. This technique offers promise in controlling the divergence problem that recursive filtering often encounters.

Recursive fading memory filtering

A recursive, fading memory filter for time-continuous and time-discrete systems is presented as a means for overcoming the destructive influence of model errors in Kalman filter applications that lead to the occurrence of divergence. This fading memory filter is shown to be uniformly asymptotically stable under basically the same conditions as the Kalman filter and bounds on the error covariance matrix of the filter are given. An adaptive procedure for implementing the procedure is discussed in terms of a scalar example. The ease of implementation of this filter and the highly satisfactory nature of numerical results indicate the efficacy of the procedure as a desirable recursive data processing method.