Nonlinear Kalman Filter Research Articles

Minimum Error Entropy Kalman Filter

To date, most linear and nonlinear Kalman filters (KFs) have been developed under the Gaussian assumption and the well-known minimum mean square error (MMSE) criterion. In order to improve the robustness with respect to impulsive (or heavy-tailed) non-Gaussian noises, the maximum correntropy criterion (MCC) has recently been used to replace the MMSE criterion in developing several robust Kalman-type filters. To deal with more complicated non-Gaussian noises such as noises from multimodal distributions, in this article, we develop a new Kalman-type filter, called minimum error entropy KF (MEE-KF), by using the minimum error entropy (MEE) criterion instead of the MMSE or MCC. Similar to the MCC-based KFs, the proposed filter is also an online algorithm with the recursive process, in which the propagation equations are used to give prior estimates of the state and covariance matrix, and a fixed-point algorithm is used to update the posterior estimates. In addition, the MEE extended KF (MEE-EKF) is also developed for performance improvement in the nonlinear situations. The high accuracy and strong robustness of MEE-KF and MEE-EKF are confirmed by experimental results.

The correction method of the aircraftastro-inertial system

The accuracy increasing problem of the unmanned aerial vehicle navigation complex in an algorithmic way is investigated. The algorithmic correction schemes of navigation systems for modern high-precision unmanned aerial vehicles are considered. The error compensating method of the inertial navigation system corrected by astro-system signals is proposed. Correction is carried out in the structure of the inertial navigation system using a non-linear Kalman filter and a control algorithm. A nonlinear control algorithm based on the SDC representation method is used. Keywords unmanned aerial vehicle; inertial navigation system; astro-system; navigation complex; nonlinear Kalman filter; error model; regulator; SDC method

Robust Gain Scheduling Control for Wave Energy Conversion

Enhancing the power performance of wave energy converters is undoubtedly a step required to reduce the cost of energy from this source of renewable energy, thereby making it competitive to other sources of renewable energy. Increasing the power absorption can be achieved by utilizing smart and advanced control algorithms. There are theories for a variety of advanced control algorithms, but few have proved stable and reliable for real applications. An often used, and robust method for practical applications, is to apply a simple gain scheduling controller where the control gains are parameterized in function of the sea state, and not at wave-by-wave level. This paper presents a wave-by-wave adaptive controller, which has proven a robust method that can increase the power absorption performance. The use of the wave by wave adaptive controller is achieved by the identification of the instantaneous fundamental frequency in real time. One numerical procedure to achieve this frequency is using non-linear Kalman filters. But the pitfall of these non-linear filters is their sensitivity to the parameter tuning, which decreases practical usability, reliability and robustness. This paper focuses on two complementary topics. The first topic will tackle the implementation of a reliable filter for the identification of the instantaneous fundamental frequency using a particle filter. The second topic will demonstrate the implementation of the wave-by-wave gain scheduling controller. The case study is a scaled absorber of the Floating Power Plant wave and wind energy converter.

Stochastic evaluation of orientation parameters for an antenna complex using strapdown inertial system measurements

When solving the problem of high-precision spatial orientation of antennas of radio engineering complexes placed on high-rise masts of various designs, one of the main tasks is to increase the accuracy of their angular orientation in the conditions of inevitable disturbances of the mast base (due to wind disturbances, operation of various units, etc.) and interference of measurement of antenna orientation parameters by various measuring systems. Currently used methods based on measurements of satellite navigation systems and autonomous measurements of attitude and heading reference system do not provide the required accuracy for solving the problem of antenna orientation, located on a highly dynamic mobile base. This circumstance is due both to the weak noise immunity of algorithms of information processing and the inability of accounting for the dynamic properties of orientation parameters measured also in the presence of noise of high intensity, whose properties also usually are not taken into account. In this regard, the article considers a dynamic algorithm for estimating the stochastic parameters of the antenna orientation of the radio engineering complexes, invariant to the nature of the movement of the mast base and providing stability and the required accuracy of the evaluation under the most general assumptions about the nature of interference of the sensing elements attitude and heading reference system. As the observed state vector – the vector of antenna orientation parameters, the vector of Rodriguez–Hamilton parameters is used, and as the vector of observation of it – the vector of measurements of angular velocity sensors of a strapdown inertial system, which consists of three accelerometers and three angular velocity sensors located orthogonally in the center of mass of the antenna. Based on the nonlinear stochastic equations of dynamics of change of the vector of parameters of the current orientation attitude and heading reference system on the moving base and stochastic models of the measuring signals sensing elements attitude and heading reference system, a non-linear (generalized) Kalman filter, providing the desired solution of the problem of estimation of the parameters of the current orientation of the antenna on the perturbed basis, was built. The results of the numerical experiment allow us to conclude that it can be used to solve the problem of operational orientation of radio engineering complexes antennas placed on masts, using medium-and high-precision attitude and heading reference system without correction for a long time.

Development of the navigation complex structure for a high-precision unmanned aerial vehicle for flight in the atmosphere

The navigation systems as part of the navigation complex of a high-precision unmanned aerial vehicle in conditions of different altitude flight are investigated. The working contours of the navigation complex with correction algorithms for an unmanned aerial vehicle during high-altitude and low-altitude flights are formed. Mathematical models of inertial navigation system errors used in non-linear and linear Kalman filters are presented. The results of mathematical modeling demonstrate the effectiveness of the working contours effectiveness of the navigation complex with correction algorithms. Keywords high-precision unmanned aerial vehicle; navigation complex; multi-altitude flight; work circuit; passive noises; Kalman filter; correction

Two-Layer Nonlinear FIR Filter and Unscented Kalman Filter Fusion With Application to Mobile Robot Localization

In this paper, we propose a new state estimator called the two-layer nonlinear finite impulse response (TLNF) filter and adopt this new filter and unscented Kalman filter (UKF) as subfilters to create the fusion TLNF/UK filter. The TLNF filter is constructed with measurements that are redefined by weighting the estimated states acquired through minimizing the cost function based on the Frobenius norm. The efficient iterative form of the TLNF filter is also developed in this paper. Using the fact that the UKF and the TLNF filter each takes a different type of memory structure, the fusion TLNF/UK filter is designed as a robust nonlinear state estimator taking both advantages of each filter. To obtain the best fusion estimates, probabilistic weights are computed based on Bayes’ rule and the likelihood of each filter. Both simulation and experimental results for mobile robot indoor localization have shown that the fusion TLNF/UK filter achieves a higher level of accuracy and robustness under practical situations.

Physical Layer Authentication Based on Nonlinear Kalman Filter for V2X Communication

Authentication is an important guarantee for vehicle to everything (V2X) commercial deployment. Currently, V2X security often use identity authentication schemes based on public key infrastructure (PKI). These schemes need to transmit certificates and signatures when sending safety-related information such as basic safety messages (BSMs), which need to occupy extra bandwidth and reduce the available channel capacity. So, V2X communication efficiency will be seriously affected in traffic congestion. In this paper, we propose a V2X authentication model based on physical layer characteristics. Then we use the Kalman filter to refine the iterative model and threshold model. The iterative model mainly realizes the priori and posteriori estimation of the current time based on the physical layer characteristics of the previous time, which provides the basis for the entire authentication process. The threshold model analyzes the mathematical characteristics of the priori estimation, and gives the calculation method of the authentication threshold. Since the conventional Kalman filter can only be used for linear discrete system, we use extended Kalman filter and unscented Kalman filter to extend the characteristics used for authentication to non-linearity. At the same time, iterative model and threshold model are improved according to these two algorithms. In terms of security and performance, we compare the proposed schemes with the conventional V2X authentication scheme and physical layer authentication scheme, and the effects of these schemes are analyzed by experiment. We select three characteristics for simulation: received signal strength indication (RSSI), the distance between the two vehicles, and the relative speed between the two vehicles. Then we analyze the process and effect of these two filters, and the factors that affect the threshold. Through experiments, the proposed authentication schemes can effectively take the responsibility of identity authentication in the V2X environment, and have high security level and low overhead, which can reduce the consumption of communication resources by security.

Improving the nonlinear models characteristics of dynamical systems

The qualitative characteristics of the models are studied: the degree of observability, controllability and parametric identity. The numerical criteria of linear models qualitative characteristics of one class in the SDC representation are used to determine such characteristics. Usage methods of the models with improved characteristics in the aircraft navigation system of the aircraft are described. Keywords aircraft; navigation complex; adaptive non-linear Kalman filter; method of arguments group accounting; nonlinear control algorithm; SDC representation; observability degree criterion; controllability degree criterion; parametric identity degree criterion

Distributed nonlinear Kalman filter with communication protocol

This paper proposes an optimal design of the general distributed nonlinear Kalman-based filtering algorithm to tackle the discrete-time estimation problem with noisy communication networks. The algorithm extends the Kalman filter by enabling it to predict the noisy communication data and fuse it with the received neighboring information to produce a posterior estimate value. In the prediction step, the unscented transformations of the estimate values and covariances originated in the Unscented Kalman Filter (UKF) are exploited. In the update step, a communication protocol is appended to the posterior estimator, which consequently leads to a modified posterior error covariance containing the covariance of the communication term with its communication gain. Both Kalman and communication gains are then optimised to collectively minimise the mean-squared estimation error. Afterwards, stochastic stability analysis is performed to guarantee its exponential boundedness. To exemplify the performance, this algorithm is applied to a group of robots in a sensor network assigned to estimate an unknown information distribution over an area in the optimal coverage control problem. Comparative numerical experiments finally verify the effectiveness of our design.

Jet transport-based nonlinear state and parameter estimation for geostationary spacecraft

Based on the jet transport technique, this paper proposes a novel nonlinear Kalman filter for simultaneously estimating the spacecraft state vector and uncertain parameters, either physically related with the spacecraft or with the measurement procedure. Two different coordinate representations, including Cartesian and hybrid geostationary orbital elements, are exploited in the new nonlinear estimators. The performance and sensitivity analyses of the proposed jet transport-based nonlinear estimators are assessed by numerical simulations and compared with the classical extended Kalman filter.

Non‐linear Kalman filters comparison for generalised autoregressive conditional heteroscedastic clutter parameter estimation

In this work, the authors analyse the estimation of the generalised autoregressive conditional heteroscedastic (GARCH) process conditional variance based on three non-linear filtering approaches: extended Kalman filter (EKF), unscented Kalman filter and cubature Kalman filter. The authors present a state model for a GARCH process and derive an EKF including second-order non-linear terms for simultaneous estimation of state and parameters. Using synthetic data, the authors evaluate the consistency and the correlation of the innovations for the three filters, by means of numerical simulations. The authors also study the performance of smoothed versions of the non-linear Kalman filters using real clutter data in comparison with a conventional quasi-maximum likelihood estimation method for the GARCH process coefficients. The authors show that with all methods the process coefficients estimates are of the same order and the resulting conditional variances are commensurable. However, the non-linear Kalman filters greatly reduce the computational load. These kind of filters could be used for the radar detector based on a GARCH clutter model that uses an adaptive threshold that demands the conditional variance at each decision instant.

Robust estimation of voltage harmonics in a single‐phase system

A frequency adaptive technique relying on a linear Kalman filter (KF) is presented here for robust estimation of voltage harmonics under variable frequency conditions in a single-phase system. A relatively simple frequency-locked loop (FLL) is combined with the linear KF (LKF-FLL) to achieve frequency adaptive ability and avoid the use of a non-linear KF. In contrast to the non-linear extended KF (EKF), the LKF-FLL technique has several advantages such as robustness, linearity, simple tuning, having fewer states, requiring no derivative actions, while offering low complexity, excellent convergence, and computational efficiency. When compared to the non-linear extended real KF, it can generate a faster dynamic response and more accurate steady-state estimation of the harmonics under frequency variations. It can also provide an improved estimation for off-nominal frequency conditions when compared to the discrete Fourier transform (DFT) method. The effectiveness of the technique is verified by various simulated and real-time experimental case studies.

Improving bearings-only target state estimation tracking problem by using adaptive and nonlinear kalman algorithms

<p><span style="font-family: 'Times New Roman'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">Finding the best estimate of the process state from noisy data is the main problem in tracking systems, many efforts and researches have been done to remove this noise. More useful information about the target’s state can be extracted from observations by using a more appropriate model for the target’s motion or using additional sensors. In this paper, we will introduce two methods to improve the estimation of bearing-only target tracking problem in two dimensions (2D). The first method is by adding a third sensor and making a good alignment of those sensors, and at the same time an extended Kalman filter (EKF), unscented Kalman filter (UKF) and cubature Kalman filter (CKF) are implemented. The second method is by applying an adaptive nonlinear Kalman filter (ANKF) for two sensors to solve the problem of measurement variance uncertainty.</span></p>

Navigation complex with adaptive non-linear Kalman filter for unmanned flight vehicle

Navigation complex with adaptive non-linear Kalman filter for unmanned flight vehicle

Accuracy analysis of three deterministic sampling nonlinear filtering algorithms

To study the estimation accuracy of nonlinear Kalman filter in transfer alignment system, the Taylor expansion of multivariable function is utilized to analyze the estimation accuracy of the three typical deterministic sampling nonlinear Kalman filters, such as high-degree cubature Kalman filter (HCKF), cubature Kalman filter (CKF) and unscented Kalman filter (UKF). The Taylor expansion analyses demonstrate that CKF and UKF produce truncation error since the fourth degree term, while HCKF can capture fifth degree term of Taylor series expansion. The three nonlinear filtering algorithms are applied to carrier-aircraft transfer alignment system under large misalignment angle. The simulation results show that CKF and UKF have the same accuracy, and HCKF has higher estimation accuracy than the other two, which is consistent with the theoretical analysis results.

Overlay Cognitive Radio Based on OFDM with Channel Estimation Issues

Cognitive radio (CR) has been proposed as a technology to improve the spectrum efficiency by giving an opportunistic access of the licensed-user spectra to unlicensed users. We consider an overlay CR consisting of a primary macro-cell and cognitive small cells of cooperative secondary base stations (SBS). We suggest studying a CR where an orthogonal frequency division multiplexing is used for both the primary users (PU) and the secondary users (SU). In order to cancel the interferences, a precoding is required at the SBS. Therefore, we first derive the interferences expression due to SU at the PU receiver. Then, zero forcing beamforming (ZFBF) is considered to cancel the interferences. However, applying ZFBF depends on the channels between the SBS and the PU. A channel estimation is hence necessary. For this purpose, we propose to approximate the channel by an autoregressive process (AR) and to consider the channel estimation issue by using a training sequence. The received signals, also called the observations, are considered to be disturbed by an additive white measurement noise. In that case, the AR parameters and the channel can be jointly estimated from the received noisy signal by using a recursive approach. Nevertheless, the corresponding state space representation of the system is non-linear. Then, we propose to carry out a complementary study by compare non-linear Kalman filter based approaches.

Effects of Model Horizontal Grid Resolution on Short- and Medium-Term Daily Temperature Forecasts for Energy Consumption Application in European Cities

A short-term forecast of energy consumption is affected by different factors related to the demand in residential, commercial, thermoelectric, and industrial sectors. This demand can be strongly constrained by weather variables, especially temperatures, whose forecast may be very useful to predict the balances between supply and demand, minimizing the risk of price volatility. Energy companies use the relationship between meteorological forecast output and energy request to provide an effective scheduling of national gas and power grids and reduce operational costs in critical periods. This work reports a comparison analysis for short- and medium-term daily temperature forecasts during the period 2013-2014 by using the weather model e-kmf™ (eni-kassandra meteo forecast), currently adopted in gas and power applications where meteorological output has a key role. This weather forecast system uses different models and initial data to develop probabilistic predictions from a perspective of eleven days ahead. In particular, a set of model runs with horizontal grid spacing of 5.5, 8, 13, and 18 km with the same domain size are undertaken to assess the sensitivity of temperature to horizontal resolutions. A nonlinear Kalman filter has been also applied to postprocess forecasted data in eight European cities (Milano, Roma, Torino, Napoli, Munich, Paris, Brussels, and London). Filtered forecasts over these cities have been compared to local observations taken from SYNOP (surface synoptic observations) and METAR (meteorological Aerodrome Report) stations. Skill scores of performance have been used to generally assess the forecast reliability up to day +11. In order to understand the sensitivity to the horizontal resolution, investigations have been carried out even during four specific periods of two weeks with stable and unstable weather conditions.

Application of adaptive square root cubature Kalman filter in turbofan engine gas path performance monitoring

Kalman filters are very popular in turbofan engine community for health monitoring purposes. In this study, in order to get better performance in terms of filtering accuracy and noise adaptability, an Adaptive Square Root Cubature Kalman Filter (ASRCKF) was proposed to estimate health parameters of the turbofan engine gas path components. In the ASRCKF algorithm, the mean value and covariance of the engine nonlinear function were calculated by cubature rule-based numerical integration method and used as a substitute for nonlinear model in nonlinear Kalman filter. The latest information of measurement parameters in the recursion and filtering process was used to estimate and self-adjust the noises cross-covariance by removable window method to get higher filtering accuracy. Compared with the Extended Kalman Filter (EKF) and Square Root Cubature Kalman Filter (SRCKF), the simulation results in the gradual and rapid deterioration process of turbofan engine gas path components indicate that the higher accuracy and faster convergence can be obtained through ASRUKF, which can be used in health parameters estimation and condition monitoring of turbofan engine gas path.1

An implementation of the cubature Kalman filter for estimating trajectory parameters and air data of a hypersonic vehicle

Trajectory parameters (including the position, velocity, and attitude angles of a vehicle) and air data (consisting of the flow angles, the Mach number, and the freestream static pressure) are vital data for the analysis and evaluation process in the hypersonic flight tests. This paper describes a data fusion estimation algorithm for a flush air data sensing system/inertial navigation system/global positioning system integrated system, which is used to estimate the trajectory parameters and air data for an unpowered hypersonic vehicle. In the approach, the raw outputs of flush air data sensing system (i.e. the surface pressure measurements) are integrated with global positioning system results (the vehicle’s position and velocity) and inertial navigation system measurements (including the acceleration and the angular velocity measurements) by using a nonlinear Kalman filter algorithm. Firstly, the system state vector is defined with the trajectory parameters, the biases of the inertial sensors and the winds. Then, the system dynamic models are built based on the motion equations of an unpowered hypersonic vehicle, the inertial sensor error models and the wind model. Besides, the system measurement vector is designed with the global positioning system results and the flush air data sensing system raw outputs. Based on these works, the system state is directly estimated by using the cubature Kalman filter algorithm. After that, the air data is calculated based on the estimated values and a high-fidelity model of atmosphere. Simulation cases are implemented to assess the performance of the proposed algorithm. The results show that the proposed method could estimate the trajectory parameters and air data for hypersonic vehicle with a high precision.

Fault detection and diagnosis algorithms for transient state of an open-cycle liquid rocket engine using nonlinear Kalman filter methods

This paper deals with an application of the fault detection and diagnosis algorithm based on nonlinear Kalman filter methods for transient state of an open-cycle liquid propellant rocket engine. In order to develop the algorithm, we designed two types of the nonlinear Kalman filter which are the extended Kalman filter and unscented Kalman filter with non-linear model of the liquid propellant rocket engine. Then using the measurement data of some important parameters of the engine, the residuals from the nonlinear Kalman filters are obtained. Using the residuals, we can detect and diagnose faults in the components of the engine by using the multiple model method. To confirm the fault detection and diagnosis algorithm, we developed mathematical model of an open-cycle liquid propellant rocket engine and artificially injected various faults such as decreasing turbopump efficiency. And then, perform the fault detection and diagnosis algorithm and check the performance of the algorithm. This process is numerically demonstrated for the open-cycle liquid propellant rocket engine under start-up process by using the simulated measurement data from the mathematical model of the engine.