Implicit Filter Research Articles

An Implicit UKF for Satellite Stellar Refraction Navigation System

The stellar refraction navigation is a type of autonomous celestial navigation method for satellites that uses high-accuracy star sensors to obtain the stellar refraction. The apparent height and the stellar refraction angle are two kinds of measurements for the satellite stellar refraction navigation system. A better navigation performance can be achieved by the stellar refraction angle rather than the refraction apparent height. However, the measurement model of the stellar refraction angle is an implicit function, which needs an implicit filter method. A new implicit unscented Kalman filter (UKF) method is proposed and applied to the satellite stellar refraction navigation system in this paper. The main difference between the classical UKF and the implicit UKF lies in the measurement update. First, since the implicit measurement model can be rearranged into an equation whose one side is zero, the vector of zeros is regarded as the equivalent measurement, and the difference between the predicted measurement and its true value is utilized for updating the predicted state and its covariance matrix. Second, the measurement noise and its covariance matrix are transformed based on the unscented transformation principle. In addition, the implicit UKF is compared with the implicit extended Kalman filter (IEKF), the iterative IEKF, and the UKF. Simulations show that the proposed implicit UKF method can obtain about 19.0% and 17.5% improvements in three-dimensional root mean square position estimation compared to the IEKF and the iterative IEKF, respectively. Although the navigation performance of the UKF is close to the implicit UKFs, the computational time of the implicit UKF is much shorter than the UKFs. In addition, the sensitivities of the four different Kalman filters to the star sensor's faults and the atmosphere perturbation are compared, respectively.

A quantification method for numerical dissipation in quasi-DNS and under-resolved DNS, and effects of numerical dissipation in quasi-DNS and under-resolved DNS of turbulent channel flows

LES for industrial applications with complex geometries is mostly characterised by: a) a finite volume CFD method using a non-staggered arrangement of the flow variables and second order accurate spatial and temporal discretisation schemes, b) an implicit top-hat filter, where the filter length is equal to the local computational cell size, and c) eddy-viscosity type LES models. LES based on these three main characteristics is indicated as industrial LES in this paper.It becomes increasingly clear that the numerical dissipation in CFD codes typically used in industrial applications with complex geometries may inhibit the predictive capabilities of explicit LES. Therefore, there is a need to quantify the numerical dissipation rate in such CFD codes. In this paper, we quantify the numerical dissipation rate in physical space based on an analysis of the transport equation for the mean turbulent kinetic energy. Using this method, we quantify the numerical dissipation rate in a quasi-Direct Numerical Simulation (DNS) and in under-resolved DNS of, as a basic demonstration case, fully-developed turbulent channel flow. With quasi-DNS, we indicate a DNS performed using a second order accurate finite volume method typically used in industrial applications. Furthermore, we determine and explain the trends in the performance of industrial LES for fully-developed turbulent channel flow for four different Reynolds numbers for three different LES mesh resolutions. The presented explanation of the mechanisms behind the observed trends is based on an analysis of the turbulent kinetic energy budgets.The presented quantitative analyses demonstrate that the numerical errors in the industrial LES computations of the considered turbulent channel flows result in a net numerical dissipation rate which is larger than the subgrid-scale dissipation rate.No new computational methods are presented in this paper. Instead, the main new elements in this paper are our detailed quantification method for the numerical dissipation rate, the application of this method to a quasi-DNS and under-resolved DNS of fully-developed turbulent channel flow, and the explanation of the effects of the numerical dissipation on the observed trends in the performance of industrial LES for fully-developed turbulent channel flows.

Estimation of thermospheric zonal and meridional winds using a Kalman filter technique

AbstractKnowledge of the thermospheric neutral wind and its horizontal components is critical for an improved understanding of F region dynamics and morphology. However, to date their reliable estimation remains a challenge because of difficulties in both measurement and modeling. We present a new method to estimate the climatology of the zonal and meridional components of thermospheric neutral wind at low and middle latitudes using a Kalman filter technique. First, the climatology of the magnetic meridional wind is obtained by assimilating seasonal maps of F region ionosphere peak parameters (NmF2 and hmF2), obtained from Constellation Observing System for Meteorology, Ionosphere, and Climate radio occultation data, into the Global Assimilation of Ionospheric Measurements Full Physics (GAIM‐FP) model. GAIM‐FP provides the 3‐D electron density throughout the ionosphere, together with the magnetic meridional wind. Next, the global zonal and meridional wind components are estimated using a newly developed Thermospheric Wind Assimilation Model (TWAM). TWAM combines magnetic meridional wind data obtained from GAIM‐FP with a physics‐based 3‐D thermospheric neutral wind model using an implicit Kalman filter technique. Ionospheric drag and ion diffusion velocities, needed for the wind calculation, are also taken from GAIM‐FP. The obtained wind velocities are in close agreement with measurements made by interferometers and with wind values from the Horizontal Wind Model 93 (HWM93) over Millstone Hill, Arecibo, and Arequipa during December and June solstices, and March equinox. In addition, it is shown that compared to HWM93 the winds from TWAM significantly improve the accuracy of the Ionosphere/Plasmasphere Model in reproducing the observed electron density variation over the Weddell Sea Anomaly.

Formula omitted]-norm of fractional transfer functions of implicit type

This paper studies the H2-norm (or impulse response energy) of fractional transfer functions of implicit type. Stability conditions are first shown to be identical as in rational systems with all poles located in the open left half complex plane. Then, analytical expressions of the H2-norm are derived for elementary fractional transfer functions of the first and the second kind cascaded with a pure fractional integrator. Next, general boundedness conditions are established in terms of transfer function relative degree. Three illustrative examples are finally proposed. The first one evaluates the quality of a rational approximation of a fractional model of implicit type on the basis of the H2-norm of the error signal. The second one evaluates the Integral Squared Error of a CRONE control loop and compares it to a classical proportional-derivative controller in a vehicle suspension. Finally, the third one allows to set up an implicit fractional preshaping filter for closed-loop control.

Transient response of general one-dimensional distributed systems through eigenfunction expansion with an implicit filter scheme

The current work is concerned with eigenfunction expansion of the transient response of general one-dimensional non-self-adjoint distributed dynamic systems. The main goal of the current work is to propose an alternative eigenfunction expansion approach for transient analysis of general one-dimensional non-self-adjoint systems which avoids complex-valued eigenvalue problems. With the implicit filter scheme proposed here one splits the solution into two additive components, namely, a filtered solution for which all boundary conditions are homogeneous and a carefully selected filter. Moreover, the eigenvalue problem associated with the differential equation for the filtered solution is self-adjoint and, therefore, all eigenvalues are real-valued and the eigenfunctions satisfy the classical orthogonality relations. The unknown time-dependent coefficients of the filter are then solved simultaneously with the modal co-ordinates of the filtered solution in order to satisfy the original boundary conditions. The proposed approach is demonstrated on a distributed system composed of a cantilever beam with end mass, viscous damper and linear spring.

A scale-dependent blending scheme for WRFDA: impact on regional weather forecasting

Abstract. Due to limitation of the domain size and limited observations used in regional data assimilation and forecasting systems, regional forecasts suffer a general deficiency in effectively representing large-scale features such as those in global analyses and forecasts. In this paper, a scale-dependent blending scheme using a low-pass Raymond tangent implicit filter was implemented in the Data Assimilation system of the Weather Research and Forecasting model (WRFDA) to reintroduce large-scale weather features from global model analysis into the WRFDA analysis. The impact of the blending method on regional forecasts was assessed by conducting full cycle data assimilation and forecasting experiments for a 2-week-long period in September 2012. It is found that there are obvious large-scale forecast errors in the regional WRFDA system running in full cycle mode without the blending scheme. The scale-dependent blending scheme can efficiently reintroduce the large-scale information from National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) analyses, and keep small-scale information from WRF analyses. The blending scheme is shown to reduce analysis and forecasting error of wind, temperature and humidity up to 24 h compared to the full cycle experiments without blending. It is also shown to increase precipitation prediction skills in the first 6 h forecasts.

IMU and Multiple RGB-D Camera Fusion for Assisting Indoor Stop-and-Go 3D Terrestrial Laser Scanning

Autonomous Simultaneous Localization and Mapping (SLAM) is an important topic in many engineering fields. Since stop-and-go systems are typically slow and full-kinematic systems may lack accuracy and integrity, this paper presents a novel hybrid “continuous stop-and-go” mobile mapping system called Scannect. A 3D terrestrial LiDAR system is integrated with a MEMS IMU and two Microsoft Kinect sensors to map indoor urban environments. The Kinects’ depth maps were processed using a new point-to-plane ICP that minimizes the reprojection error of the infrared camera and projector pair in an implicit iterative extended Kalman filter (IEKF). A new formulation of the 5-point visual odometry method is tightly coupled in the implicit IEKF without increasing the dimensions of the state space. The Scannect can map and navigate in areas with textureless walls and provides an effective means for mapping large areas with lots of occlusions. Mapping long corridors (total travel distance of 120 m) took approximately 30 minutes and achieved a Mean Radial Spherical Error of 17 cm before smoothing or global optimization.

A Numerical Method Based on Leapfrog and a Fourth-Order Implicit Time Filter

Abstract A time-stepping scheme is proposed. It is based on the leapfrog method and a fourth-order time filter. The scheme requires only one evaluation per time step and uses an implicit filter, but the effort needed to implement it in an explicit manner is trivial. Comparative tests demonstrate that the proposed scheme produces numerical approximations that are more stable and highly accurate compared to the standard Robert–Asselin (RA) and the Robert–Asselin–Williams (RAW) filtered leapfrog scheme, even though both methods use filter coefficients that are tuned such that the 2Δt modes are damped at the same rate. Formal stability analysis demonstrates that the proposed method generates amplitude errors of O[(Δt)4], implying third-order accuracy. This contrasts with the O[(Δt)2] errors produced by the standard RA and RAW filtered leapfrog. A second scheme that produces amplitude errors of O[(Δt)6] is also presented. The proposed scheme is found to do well at controlling numerical instabilities arising in the diffusion equation and in nonlinear computations using Lorenz’s system and the global shallow-water spectral model. In addition to noticeably improving the resolution of the physical modes, the proposed method is simple to implement and has a wider region of stability compared to the existing time-filtered leapfrog schemes. This makes the proposed method a potential alternative for use in atmospheric, oceanic, and climate modeling.

A Study on the Filtering Approach and Turbulence Modeling for LES

the past decades, Large-Eddy Simulation (LES) has been demonstrated to be a useful research tool for understanding the physics of turbulence as well as an accurate and sophisticated predictive method for flows of engineering interest. The LES is numerical technique and is based on the separation between large and small scales in which the large- scale motion is exactly calculated and the effects of small sales or so called sub grid-scale motions are modelled. It is also important to note that the explicit or implicit filter representations like spectral cut-offs or numerical discretizations are commonly used in LES of turbulent flows. Strictly we can say that in LES we need to filter the Navier- Stokes equations in turbulence. Therefore, the study on the filtering approach in turbulence is the main objects of the present research, and in this study we have elaborately studied on this filtering approach and analyzed some general algebraic properties of the filtered representations. It is shown that the averaged equations are the same in terms of the generalized central moments, and then we have defined the resolved turbulence using these average properties. The algebraic consistency rules related with the resolved quantities to the turbulent stresses are derived and their possible use in sub grid-scale modelling is examined. In this study, we have also discussed about the standard Smagorinsky model for LES and then we derived an expression to determine the Smagorinsky constant dynamically, which suppose to be assured the consistency between the filter and the sub grid-scale model. Finally, we have derived the governing equations for LES by applying the filtering approach to the Navier-Stokes equations.

Application of the implicit particle filter to a model of nearshore circulation

The implicit particle filter is applied to a stochastically forced shallow water model of nearshore flow, and found to produce reliable state estimates with tens of particles. The state vector of this model consists of a height anomaly and two horizontal velocity components at each point on a 128 × 98 regular rectangular grid, making for a state dimension O(104). The particle filter was applied to the model with two parameter choices representing two distinct dynamical regimes, and performed well in both. Demands on computing resources were manageable. Simulations with as many as a hundred particles ran overnight on a modestly configured workstation. In this case of observations defined by a linear function of the state vector, taken every time step of the numerical model, the implicit particle filter is equivalent to the optimal importance filter, i.e., at each step any given particle is drawn from the density of the system conditioned jointly upon observations and the state of that particle at the previous time. Even in this ideal case, the sample occasionally collapses to a single particle, and resampling is necessary. In those cases, the sample rapidly reinflates, and the analysis never loses track. In both dynamical regimes, the ensembles of particles deviated significantly from normality.

State estimation in large-scale open channel networks using sequential Monte Carlo methods: Optimal sampling importance resampling and implicit particle filters

[1] This article investigates the performance of Monte Carlo-based estimation methods for estimation of flow state in large-scale open channel networks. After constructing a state space model of the flow based on the Saint-Venant equations, we implement the optimal sampling importance resampling filter to perform state estimation in a case in which measurements are available at every time step. Considering a case in which measurements become available intermittently, a random-map implementation of the implicit particle filter is applied to estimate the state trajectory in the interval between the measurements. Finally, some heuristics are proposed, which are shown to improve the estimation results and lower the computational cost. In the first heuristics, considering the case in which measurements are available at every time step, we apply the implicit particle filter over time intervals of a desired size while incorporating all the available measurements over the corresponding time interval. As a second heuristic method, we introduce a maximum a posteriori (MAP) method, which does not require sampling. It will be seen, through implementation, that the MAP method provides more accurate results in the case of our application while having a smaller computational cost. All estimation methods are tested on a network of 19 tidally forced subchannels and 1 reservoir, Clifton Court Forebay, in Sacramento-San Joaquin Delta in California, and numerical results are presented.

Simulation of the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms

The main purpose of the current work is to employ an integral transform approach based on eigenfunction expansion and on an implicit filter scheme in order to solve the governing equations for the transverse vibrations of a cantilever beam clamped at one end and with an eccentric tip mass in the axial direction at the other end. Numerical results are obtained for both the undamped and damped natural frequencies of the system, as well as for its transverse displacement due to arbitrarily time-varying load and imposed displacement at the clamped end. The numerical results reported in the current work are highly accurate and new in the literature. New exact results are also provided for the transient displacement and its higher-order spatial derivatives to allow computation of bending stresses and strains. The relative merits of the proposed approach are finally pointed out.

Holistic Biquadratic IIR Filter Design for Communication Systems Using Differential Evolution

Digital IIR filter implementations are important building blocks of most communication systems. The chosen number format (fixed-point, floating-point; precision) has a major impact on achievable performance and implementation cost. Typically, filter design for communication systems is based on filter specifications in the frequency domain. We consider IIR filter design as an integral part of communication system optimisation with implicit filter specification in thetime domain(via symbol/bit error rate). We present a holistic design flow with the system's bit error rate as the main objective. We consider a discrete search space spanned by the quantised filter coefficients.Differential Evolutionis used for efficient sampling of this huge finite design space. We present communication system performance (based on bit-true simulations) and both measured and estimated receiver IIR chip areas. The results show that very small number formats are acceptable for complex filters and that the choice between fixed-point and floating-point number formats is nontrivial if precision is a free parameter.

An exploration of the equivalent weights particle filter

AbstractParticle filters are fully nonlinear data assimilation techniques that aim to represent the probability distribution of the model state given the observations (the posterior) by a number of particles. In high‐dimensional geophysical applications, the number of particles required by the sequential importance resampling (SIR) particle filter (in order to capture the high‐probability region of the posterior) is too large to make them usable. However particle filters can be formulated using proposal densities, which give greater freedom in how particles are sampled and allow for a much smaller number of particles. Here a particle filter is presented which uses the proposal density to ensure that all particles end up in the high‐probability region of the posterior probability density function. This gives rise to the possibility of nonlinear data assimilation in large‐dimensional systems. The particle filter formulation is compared to the optimal proposal density particle filter and the implicit particle filter, both of which also utilise a proposal density. We show that, when observations are available every time step, both schemes will be degenerate when the number of independent observations is large, unlike the new scheme. The sensitivity of the new scheme to its parameter values is explored theoretically and demonstrated using the Lorenz (1963) model. Copyright © 2012 Royal Meteorological Society

Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation

Abstract. Implicit particle filtering is a sequential Monte Carlo method for data assimilation, designed to keep the number of particles manageable by focussing attention on regions of large probability. These regions are found by minimizing, for each particle, a scalar function F of the state variables. Some previous implementations of the implicit filter rely on finding the Hessians of these functions. The calculation of the Hessians can be cumbersome if the state dimension is large or if the underlying physics are such that derivatives of F are difficult to calculate, as happens in many geophysical applications, in particular in models with partial noise, i.e. with a singular state covariance matrix. Examples of models with partial noise include models where uncertain dynamic equations are supplemented by conservation laws with zero uncertainty, or with higher order (in time) stochastic partial differential equations (PDE) or with PDEs driven by spatially smooth noise processes. We make the implicit particle filter applicable to such situations by combining gradient descent minimization with random maps and show that the filter is efficient, accurate and reliable because it operates in a subspace of the state space. As an example, we consider a system of nonlinear stochastic PDEs that is of importance in geomagnetic data assimilation.

Fault-Tolerant Attitude Control for Spacecraft Under Loss of Actuator Effectiveness

ATPRESENT, a large number of engineers have assorted to faulttolerant control (FTC) to enhance the system reliability and to guarantee the control performance (see, e.g., [1–4] and references therein), and several investigations on the application of FTC to spacecraft attitude maneuvers have also been considered. For instance, based on the dynamics inversion and time-delay control, a passive fault-tolerant controller was developed in [5] to achieve attitude tracking of a rigid satellite. Cai et al. [6] proposed an indirect adaptive fault scheme for spacecraft attitude tracking problem under thruster faults. In [7], a control augmentation method, similar to adaptive fault-tolerant control, was adopted for the flexible spacecraft attitude tracking. However, all the preceding FTC results can only tolerate limited predetermined faults and have great conservativeness due to its implementing a fixed controller only. In this work, the attitude stabilization of rigid spacecraft in the presence of partial loss of actuator effectiveness is investigated. Based on adaptive backstepping technique, a nominal attitude controller is derived first for the normal spacecraft system in the face of external disturbances. Then the case of the partial loss of reactionwheel-effectiveness fault is considered, and the faulty attitude system with time-varying gain is ultimately decoupled into three auxiliary systems by appropriate transformation. Moreover, for each auxiliary system, an implicit estimation filter is proposed to estimate the actuator fault correspondingly, and in the meantime, a new adaptive fault-tolerant controller is synthesized according to the estimated fault to guarantee that outputs of the auxiliary system can follow the normal attitude control command signals. By viewing the tracking error as another disturbance entering the system dynamics, with the robustness of the nominal controller to external disturbances, the online fault tolerance can be achieved. In contrast to preceding FTC results, the main contributions of this study include the following: 1) The proposed strategy can react to the fault online and in real time, and thus the conservativeness of the controller can be reduced largely. 2) Although three implicit estimation filters are developed in the fault-tolerant-controller design, it does not require the precise reconstruction of the faults. Thus, large computation power can be saved and also the response time can be reduced effectively. This Note is organized as follows: Sec. II briefly presents the spacecraft attitude model and control problems. In Secs. III and IV, adaptive backstepping attitude controllers are derived, respectively, with and without partial loss of actuator effectiveness fault. Simulation results of a rigid spacecraft are given in Sec. V, followed by conclusions in Sec. VI.

Grid-independent large-eddy simulation using explicit filtering

The governing equations for large-eddy simulation are derived from the application of a low-pass filter to the Navier–Stokes equations. It is often assumed that discrete operations performed on a particular grid act as an implicit filter, causing results to be sensitive to the mesh resolution. Alternatively, explicit filtering separates the filtering operation, and hence the resolved turbulence, from the underlying mesh distribution alleviating some of the grid sensitivities. We investigate the use of explicit filtering in large-eddy simulation in order to obtain numerical solutions that are grid independent. The convergence of simulations using a fixed filter width with varying mesh resolutions to a true large-eddy simulation solution is analyzed for a turbulent channel flow at Reτ=180, 395, and 640. By using explicit filtering, turbulent statistics and energy spectra are shown to be independent of the mesh resolution used.

Implicit large-Eddy simulation with a moving least squares-based finite volume method

In this contribution we present the application of a numerical method based on a mesh free interpolation technique (Moving Least-Squares (MLS)) in a finite volume framework to the computation of turbulent flows. Our approach is based on the monotonically implicit Large Eddy Simulation (MILES). The main idea of MILES methodology is the absence of any explicit subgrid scale (SGS) model in the numerical algorithm to solve turbulent flows. As MLS approximations are used to compute the gradients and higher-order derivatives, the parameters of the kernel function plays a crucial role on the properties of the numerical scheme, particularly on the dispersion and dissipation of the method. This allows us to define an implicit filter that acts as a SGS model in turbulence computations. We present the results for the decay of compressible isotropic turbulence, which are very promising, and the our first results computing turbulence flows in presence of walls.

Analysis of errors occurring in large eddy simulation

We analyse the effect of second- and fourth-order accurate central finite-volume discretizations on the outcome of large eddy simulations of homogeneous, isotropic, decaying turbulence at an initial Taylor-Reynolds number Re(lambda)=100. We determine the implicit filter that is induced by the spatial discretization and show that a higher order discretization also induces a higher order filter, i.e. a low-pass filter that keeps a wider range of flow scales virtually unchanged. The effectiveness of the implicit filtering is correlated with the optimal refinement strategy as observed in an error-landscape analysis based on Smagorinsky's subfilter model. As a point of reference, a finite-volume method that is second-order accurate for both the convective and the viscous fluxes in the Navier-Stokes equations is used. We observe that changing to a fourth-order accurate convective discretization leads to a higher value of the Smagorinsky coefficient C(S) required to achieve minimal total error at given resolution. Conversely, changing only the viscous flux discretization to fourth-order accuracy implies that optimal simulation results are obtained at lower values of C(S). Finally, a fully fourth-order discretization yields an optimal C(S) that is slightly lower than the reference fully second-order method.

Forward-calculated analytical interferograms in pass-through photon-based biomedical transillumination

Recently, we have introduced a transillumination technique for biomedical diagnosis. The technique, pass-through photon-based transillumination, relies on interferometric measurements to recover the information of interest. In this work, we present the forward-calculated analytical interferograms that describe the behavior of the system. Stochastic modeling of radiation interacting with tissue enables determination of amplitude and phase parameters, indispensable for computation of the interferograms. Sample variability is assessed by studying tissue phantoms similar to those used in the experimental verification of the technique and that are representative of (abnormal) dental tissues. For tissue characterization, perfect recovery of the integrated attenuation ensues by employing spatially compact radiation sources. For tissue imaging, spatially extended sources with broad bandwidth are superior due to the implicit longitudinal coherence filter. For both applications, sample variability issues may be neutralized by permitting spatial divergence of scattered photons.