Nonlinear Estimation Algorithm Research Articles

Retrieving vertical profile of aerosol extinction by multi-axis differential optical absorption spectroscopy

Using the oxygen dimer (O4 information measured by multi-axis differential optical absorption spectroscopy (MAX-DOAS)), an inversion method of vertical profile of aerosol extinction based on nonlinear optimal estimation algorithm is developed. At first we study how to calculate some parameters (weighting function, the covariance matrices of measurement and a priori) of the algorithm and design nonlinear iteration strategy suited to Chinese region where aerosol usually shows rapid variation and high load. Then this inversion method is verified by computer simulation combined with discussion about error source in three typical cases of low, high and elevated aerosol. After that a continuous observation is reported in the city of Hefei. The aerosol optical depth (AOD) derived from MAX-DOAS is compared with that of CE318 sun photometer and the correlation coefficient is 0.94. The total relative error of AOD is about 20%. In addition the aerosol extinction in the lowest altitude (0-0.3 km) is compared with that of visibility meter and the correlation coefficient is about 0.65. The total relative error of surface-near aerosol extinction is about 25%. Both of simulation verification and comparison experiment indicate that the inversion method can well rebuild the vertical profile of aerosol extinction in the troposphere.

Measuring tropospheric vertical distribution and vertical column density of NO2 by multi-axis differential optical absorption spectroscopy

The inversion method of the vertical profile and vertical column density (VCD) of tropospheric NO2 using multi-axis differential optical absorption spectroscopy (MAX-DOAS) is investigated in this paper. An inversion method of two-step procedure is operated. In this method firstly the aerosol vertical profile is retrieved. Then the vertical distribution of trace gases is retrieved based on the corresponding aerosol status. Nonlinear optimal estimation algorithm is extended to acquire NO2 profile to reduce the dependence of the inversion on priori information. It is more advantageous to automatically obtain trace gases profile. At first we investigate how to calculate some parameters (weighting function, the covariance matrices of measurement, and a priori information) of the algorithm and design nonlinear iteration strategy suited to the region where NO2 vertical distribution usually shows rapid variation. Then this inversion algorithm is verified by computer simulation in the cases of box profile and elevated profile of NO2. It is indicated that the distribution of NO2 below 2 km could be well rebuilt and the retrieval accuracy of surface-near NO2 volume mixing ratio is 0.6%. The study of how accurately this algorithm can rebuild the same true profile in three aerosol status of low aerosol, high aerosol and elevated aerosol indicates that similar retrieval results could be acquired. In addition, the effect of wrong aerosol status on the retrieving of NO2 profile and the error sources of this algorithm are analyzed. After that a continuous observation is reported in the city of Hefei. NO2 VCDs derived from MAX-DOAS are compared with those from satellite observations, and the correlation coefficient is 0.85. The surface-near NO2 concentrations measured by MAX-DOAS are compared with those from LP-DOAS, and the correlation coefficient is 0.76. In addition, the simplified MAX-DOAS inversion method of obtaining the trace gas profile usually uses invariable typical aerosol status as input. The comparison with the tropospheric NO2 VCD from simplified method indicates that the using of invariable typical aerosol status would cause large deviation of NO2 VCD, and its maximum relative deviation is about 112%. So exactly acquiring aerosol status, aerosol optical density especially, is necessary to exactly retrieve tropospheric NO2 vertical column density.

Spacecraft Attitude and System Identification Using Marginal Reduced UKF Utilizing the Sun and Calibrated TAM Sensors

This paper deals with attitude determination, parameter identification and reference sensor calibration simultaneously. A LEO satellite’s attitude, inertia tensor as well as calibration of Three-Axis-Magnetometer (TAM) are estimated during a maneuver designed to satisfy persistency of excitation condition. For this purpose, kinematic and kinetic state equations of spacecraft motion are augmented for the determination of inertia tensor and TAM calibration parameters including scale factors, misalignments and biases along three body axes. Attitude determination is a nonlinear estimation problem. Unscented Kalman Filter (UKF) as an advanced nonlinear estimation algorithm with good performance can be used to estimate satellite attitude but its computational cost is considerably larger than the widespread, low accuracy, Extended Kalman Filter (EKF). Reduced Sigma Points Filters provide good solutions and also decrease run time of UKF. However, in contrast to nonlinear problem of attitude determination, parameter identification and sensor calibration have linear dynamics. Therefore, a new Marginal UKF (MUKF) is proposed that combines the utility of Kalman Filter with Modified UKF (MMUKF). The proposed MMUKF utilizes only 14 sigma points to achieve the complete 25-dimensional state vector estimation. Additionally, a Monte Carlo simulation has demonstrated a good accuracy for concurrent estimation of attitude, inertia tensor as well as TAM calibration parameters in significantly less time with respect to sole utilization of the UKF.

The influence of different PAST-based subspace trackers on DaPT parameter estimation

Abstract. In the context of parameter estimation, subspace-based methods like ESPRIT have become common. They require a subspace separation e.g. based on eigenvalue/-vector decomposition. In time-varying environments, this can be done by subspace trackers. One class of these is based on the PAST algorithm. Our non-linear parameter estimation algorithm DaPT builds on-top of the ESPRIT algorithm. Evaluation of the different variants of the PAST algorithm shows which variant of the PAST algorithm is worthwhile in the context of frequency estimation.

Characterization of the PEM Fuel Cell Catalyst Layer Microstructure by Nonlinear Least-Squares Parameter Estimation

Models of polymer electrolyte membrane fuel cells (PEMFC) have become increasingly complex, using many parameters to define the behavior of species and the performance of the cell. A framework is presented here to couple an agglomerate electrode based, multi-dimensional, multi-physics mathematical model of membrane electrode assembly (MEA) model with an optimization-based nonlinear least-squares parameter estimation algorithm. The framework is used to estimate the micro-structural parameters of the catalyst layer such as agglomerate size. The results show that a set of data over a range of operating conditions can be accurately described by using a unique set of structural parameters that match experimental visualization of the catalyst layer. Extension of this methodology can be used to systematically estimate any model parameters in order to reduce uncertainty in model predictions.

Sensor Calibration using In-Situ Celestial Observations to Estimate Bias in Space-Based Missile Tracking

The work presented here demonstrates a new methodology of using star observations and advanced nonlinear estimation algorithms to improve the ability of a space-based electro-optical (EO) tracking system to track targets in space. Nominally, the tracking system consists of two satellites flying in a lead-follower formation tracking a ballistic or space target. Each satellite is equipped with a narrow-view EO sensor that provides azimuth and elevation measurements to the target. The tracking problem is made more difficult due to a constant, nonvarying or slowly varying bias error present in each sensor's line of sight (LOS) measurements. The conventional sensor calibration process occurs prior to the start of the tracking process and does not account for subsequent changes in the sensor bias. This study develops a technique to estimate the sensor bias from celestial observations while simultaneously tracking the target. As stars are detected during the target tracking process, the instantaneous sensor pointing error can be calculated as the difference between a measurement of the celestial observation and the known position of the star. The system then utilizes a bias filter to estimate the bias value based on these measurements and correct the target line of sight measurements. After bias correction, we compare the ability of three advanced nonlinear state estimators to update the target state vector: a linearized Kalman filter (LKF); an extended Kalman filter (EKF); and an unscented Kalman filter (UKF). The bias correction-state estimation algorithm is validated with a number of scenarios that were created using The Satellite Toolkit (STK). The target position error resulting from the nonlinear estimation filters are compared to the Cramer-Rao lower bound (CRLB) and a filter consistency check. The results of this research provide a potential solution to sensor calibration while simultaneously tracking a space target with a space-based sensor system.

Sparse-grid quadrature nonlinear filtering

In this paper, a novel nonlinear filter named Sparse-grid Quadrature Filter (SGQF) is proposed. The filter utilizes weighted sparse-grid quadrature points to approximate the multi-dimensional integrals in the nonlinear Bayesian estimation algorithm. The locations and weights of the univariate quadrature points with a range of accuracy levels are determined by the moment matching method. Then the univariate quadrature point sets are extended to form a multi-dimensional grid using the sparse-grid theory. Compared with the conventional point-based methods, the estimation accuracy level of the SGQF can be flexibly controlled and the number of sparse-grid quadrature points for the SGQF is a polynomial of the dimension of the system, which alleviates the curse of dimensionality for high dimensional problems. The Unscented Kalman Filter (UKF) is proven to be a subset of the SGQF at the level-2 accuracy. The performance of this filter is demonstrated by an orbit estimation problem. The simulation results show that the SGQF achieves higher accuracy than the Extended Kalman Filter (EKF), the UKF, and the Cubature Kalman Filter (CKF). In addition, the SGQF is computationally much more efficient than the multi-dimensional Gauss–Hermite Quadrature Filter (GHQF) with the same performance.

Maritime radar display unit based on PC for safe ship navigation

A prototype radar display unit was implemented using inexpensive off-the-shelf components, including a nonlinear estimation algorithm for the target tracking in a clutter environment. Two custom designed boards; an analog signal processing board and a DSP board, can be plugged into an expansion slot of a personal computer (PC) to form a maritime radar display unit. Our system provided all the functionality specified in the International Maritime Organization (IMO) resolution A422(XI). The analog signal processing board was used for A/D conversion as well as rain and sea clutter suppression. The main functions of the DSP board were scan conversion and video overlay operations. A host PC was used to run the tracking algorithm of targets in clutter, using the discrete-time Bayes optimal (nonlinear, and non-Gaussian) estimation method, and the graphic user interface (GUI) software for Automatic Radar Plotting Aid (ARPA). The proposed tracking method recursively found the entire probability density function of the target position and velocity by converting into linear convolution operations.

Fusion of an Extended H∞ Filter and Cubature Kalman filter

AbstractIn this paper, we present a new nonlinear state estimation algorithm based on fusion of an extended H∞ (EH∞) filter and cubature Kalman filter (CKF), and is called as cubature H∞ filter. The proposed algorithm is developed from a recently developed cubature Kalman filter and an extended H∞ filter. The CKF is a Gaussian approximation of Bayesian filter and its performance over non-Gaussian approximation may degrade. An EH∞ filter does not make any such assumption about the statistics of process or measurement noises. But it require Jacobians during the state estimation, which makes it to performs well only in presence of ‘mild’ nonlinearities. The proposed approach neither require Jacobians like EH∞ filter nor statistical properties of noises like CKF. The efficacy of the proposed is verified on state estimation of univariate nonlinear model and an induction motor example.

Nonlinear Estimation of Hypersonic State Trajectories in Bayesian Framework with Polynomial Chaos

This paper presents a nonlinear estimation algorithm to estimate state trajectories of a hypersonic vehicle with initial condition uncertainty. Polynomial chaos theory is used to predict the evolution of state uncertainty of the nonlinear system, and a Bayesian estimation algorithm is used to estimate the posterior probability density function of the nonlinear random process. The nonlinear estimation algorithm is then applied to the hypersonic reentry of a spacecraft in Martian atmosphere. Its performance is compared with estimators based on an extended Kalman filtering and unscented Kalman filtering framework. It is observed that for the particular application, the proposed estimator outperforms extended Kalman filtering and unscented Kalman filtering, highlighting its need in the current scenario.

A Direct Approach for the Frequency-Adaptive Feedforward Cancellation of Harmonic Disturbances

This paper is concerned with the robust rejection of harmonic disturbances with unknown frequency and amplitude affecting uncertain linear system. The developed control scheme combines the properties of adaptive feedforward cancellation (AFC) techniques with the phase and frequency detection capabilities provided by a nonlinear frequency estimation algorithm. Under mild assumptions on the nominal model of the system to be controlled, the proposed scheme is proven to achieve the complete rejection of harmonic disturbances at the input or at the output of the system. A detailed stability analysis based on averaging provides useful informations on the effect of the tuning parameters over the convergence of the estimator. The effectiveness of the proposed approach is demonstrated by a two-time scale averaging analysis.

Modified unscented recursive nonlinear dynamic data reconciliation for constrained state estimation

In state estimation problems, often, the true states satisfy certain constraints resulting from the physics of the problem, that need to be incorporated and satisfied during the estimation procedure. Amongst various constrained nonlinear state estimation algorithms proposed in the literature, the unscented recursive nonlinear dynamic data reconciliation (URNDDR) presented in [1] seems to be promising since it is able to incorporate constraints while maintaining the recursive nature of estimation. In this article, we propose a modified URNDDR algorithm that gives superior performance when compared with the basic URNDDR. The improvements are obtained via better constraint handling and are demonstrated on representative case studies [2,3]. In addition to this modification, an efficient strategy combining basic unscented Kalman filter (UKF), URNDDR and modified URNDDR is also proposed in this article for solving large scale state estimation problems at relatively low computational cost. The utility of the proposed strategy is demonstrated by applying it to the Tennessee Eastman challenge process [4].

Optimal minimum variance estimation for non-linear discrete-time multichannel systems

A non-linear operator approach to estimation in discrete-time multivariable systems is described. It involves inferential estimation of a signal which enters a communication channel that contains non-linearities and transport delays. The measurements are assumed to be corrupted by a coloured noise signal correlated with the signal to be estimated. The solution of the non-linear estimation problem is obtained using non-linear operators. The signal and noise channels may be grossly non-linear and are represented in a very general non-linear operator form. The resulting so-called Wiener non-linear minimum variance estimation algorithm is relatively simple to implement. The optimal non-linear estimator is derived in terms of the non-linear operators and can be implemented as a recursive algorithm using a discrete-time non-linear difference equation. In the limiting case of a linear system, the estimator has the form of a Wiener filter in discrete-time polynomial matrix system form. A non-linear channel equalisation problem is considered for the design example.

An effective modelling approach to estimate nonlinear bending behaviour of cantilever type conducting polymer actuators

This paper reports on the establishment of an effective yet comprehensive modelling approach that enables (i) to determine the large and nonlinear bending displacements (i.e., deflections) of conducting polymer actuators, widely known as artificial muscles, and (ii) estimate the actuator parameters such as the effective modulus of elasticity. These actuators are fundamentally one-end fixed and the other end free cantilever structures, undergoing large deflections under an electrical potential difference. The classical beam theory fails to predict their bending behaviour such as the tip deflections accurately. Based on the operation principle of the electroactive polymer actuators and the large deflection Euler–Bernoulli equation, the bending displacement models are formulated for a cantilever beam under a continuously distributed load. These nonlinear models have been used to estimate the moduli of elasticity of the actuators, utilizing a nonlinear least square estimation algorithm, and experimentally measured longitudinal and transverse tip deflections. Parametric models relating the voltage input to the cylindrical coordinates of the tip deflection are also identified and experimentally validated. These models were further validated for a new set of experimental data such that the modelling approach is effective enough to estimate nonlinear deflection of the actuators and estimate actuator parameters such as the moduli of elasticity of the materials constituting polymer actuators. The modelling approach can be extended to mimic the bending behaviour of other ionic-type conducting polymer actuators.

An Adaptive Nonlinear Filter for System Identification

The primary difficulty in the identification of Hammerstein nonlinear systems (a static memoryless nonlinear system in series with a dynamic linear system) is that the output of the nonlinear system (input to the linear system) is unknown. By employing the theory of affine projection, we propose a gradient-based adaptive Hammerstein algorithm with variable step-size which estimates the Hammerstein nonlinear system parameters. The adaptive Hammerstein nonlinear system parameter estimation algorithm proposed is accomplished without linearizing the systems nonlinearity. To reduce the effects of eigenvalue spread as a result of the Hammerstein system nonlinearity, a new criterion that provides a measure of how close the Hammerstein filter is to optimum performance was used to update the step-size. Experimental results are presented to validate our proposed variable step-size adaptive Hammerstein algorithm given a real life system and a hypothetical case.

A software framework and tool for nonlinear state estimation

The goal of the article is to describe a software framework designed for nonlinear state estimation of discrete time dynamic systems. The framework was designed with the aim to facilitate implementation, testing and use of various nonlinear state estimation methods in mind. The main strength of the framework is its versatility due to the possibility of either structural or probabilistic description of the problem. Besides the well-known basic nonlinear estimation methods such as the extended Kalman filter, the divided difference filters and the unscented Kalman filter, the framework implements particle filter with advanced features as well. As the framework is designed on the object oriented basis, further extension by user-specified nonlinear estimation algorithms is extremely easy. The paper provides a brief introduction into nonlinear state estimation problem and describes the individual components of the framework, their key features and use. The strengths of the framework are presented in two examples.

A modified H 2 algorithm for improved frequency response function and nonlinear parameter estimation

H 1, H 2 and H v are the most commonly used algorithms for frequency response function estimation. H 1 and H v can be defined for multiple input–multiple output systems, whereas the H 2 estimator requires that the number of inputs equals the number of outputs for a unique solution. Otherwise, a non-square matrix needs to be inverted leading to numerical problems. These numerical problems can affect the Nonlinear Identification through Feedback of the Outputs (NIFO) nonlinear parameter estimation algorithm in which nonlinearities are treated as internal feedback forces (or inputs), which can lead to more inputs than outputs. In this paper, a modified form of the H 2 algorithm is presented that enables accurate estimates using NIFO. The modification to the H 2 algorithm is the addition of a correlated output, based on the nature of the nonlinearity in the system being identified, in order to make the matrix to be inverted square. Analytical models with multiple nonlinearities are used to show that this modification to H 2 leads to accurate estimates that are robust to noise on both the input and the output and that the modified algorithm is more robust to simultaneous noise on the input and output measurements than the H 1 and the H v algorithms. Mathematical reasoning is used to explain the greater robustness of the modified H 2 algorithm over the traditional H 1 algorithm.

Extension of a two-step calibration methodology to include nonorthogonal sensor axes

We present an extension of the nonlinear two-step estimation algorithm originally developed for the calibration of solid-state strapdown magnetometers. We expand the algorithm to include nonorthogonality within a sensor set for both two- and three-axis sensors. Nonorthogonality can result from manufacturing issues, installation geometry, and in the case of magnetometers, from soft iron bias errors. Simulation studies for both two- and three-axis sensors show convergence of the improved algorithm to the true values, even in the presence of realistic measurement noise. Finally the algorithm is experimentally validated on a low-cost solid-state three-axis magnetometer set, which shows definite improvement postcalibration. We note that the algorithm is general and can be applied to any two- or three-axis sensor set (such as accelerometers) with an error model consisting of scale, offset, and nonorthogonality errors.

Position and Velocity Tracking in Mobile Networks Using Particle and Kalman Filtering With Comparison

This paper presents several methods based on signal strength and wave scattering models for tracking a user. The received-signal level method is first used in combination with maximum likelihood (ML) estimation and triangulation to obtain an estimate of the location of the mobile. Due to nonline-of-sight conditions and multipath propagation environments, this estimate lacks acceptable accuracy for demanding services, as the numerical results reveal. The 3-D wave scattering multipath channel model of Aulin is employed, together with the recursive nonlinear Bayesian estimation algorithms to obtain improved location estimates with high accuracy. Several Bayesian estimation algorithms are considered, such as the extended Kalman filter (EKF), the particle filter (PF), and the unscented PF (UPF). These algorithms cope with nonlinearities in order to estimate mobile location and velocity. Since the EKF is very sensitive to the initial state, we propose the use of the ML estimate as the initial state of the EKF. In contrast to the EKF tracking approach, the PF and UPF approaches do not rely on linearized motion models, measurement relations, and Gaussian assumptions. Numerical results are presented to evaluate the performance of the proposed algorithms when the measurement data do not correspond to the ones generated by the model. This shows the robustness of the algorithm based on modeling inaccuracies.

Spacecraft Parameter Estimation by Using Predictive Filter Algorithm

An approach to estimate spacecraft system parameters is proposed in this paper. Oftentimes, spacecraft is under uncertainties of the system as well as internal and external disturbances which have different aspects each other. The moment of inertia of spacecraft is unknown especially when it is under considerable fuel consumption or equipped with deployable structures. This study aims to estimate the moment of inertia of the spacecraft body as well as its attitude and angular rate by using predictive filtering algorithm. Crassidis and Markley developed a predictive filtering algorithm for nonlinear estimation under a large system model error. This approach focuses not on the specific sources of model error described in the equations of motion but the resultant model error vector driving errors on the system dynamics. Therefore, we are not able to update the system model since the estimated model error has no additional information about the system. This paper establishes a method to apply the predictive filtering algorithm for nonlinear spacecraft parameter estimation by defining a new model error vector for parameters. This study shows different sources of the model error can be separated for estimation, and reveals excellent estimation results. Proposed algorithm is verified by numerical simulation studies.