Fractional Extended Kalman Filter Research Articles

Kalman filter based on a fractional discrete-time stochastic augmented CoVid-19 model

In this paper, we study the dynamics of the CoVid-19 outbreak in Semarang, Indonesia, using a fractional CoVid-19 model. We first determine the effects of the isolation rate ∊ and infection rate β on the reproduction number R0 and infected number V. We find that R0 is directly proportional to β and inversely proportional to ∊. For V, the effect of physical distancing is not as significant as changing ∊. As ∊ increases, V decreases, the number of susceptible individuals increases, the number of quarantined individuals decreases sharply, and the number of recovered individuals decreases. Moreover, the effect of vaccination is also considered. The combination of physical distancing, isolation, and vaccination has a significant impact on reducing the number of infected individuals. Analysis of dynamical systems allows us to understand the characteristics of our model, such as its boundedness and non-negativity, the existence of equilibrium points, the existence and uniqueness of solutions, and the local and global stability. To validate our fractional CoVid-19 model, we introduce the fractional extended Kalman filter (FracEKF) as a prediction method and compare the results against reported CoVid-19 data. FracEKF is a modified version of the basic extended Kalman filter with a time-fractional memory effect. The prediction results illustrate the accuracy of this model in terms of the root mean square error (RMSE), normalized root mean square error (NRMSE), and mean absolute percentage error (MAPE) for each fractional-order. Varying ∊ reproduces the trends observed in the reported data for the number of infected individuals, i.e., when ∊ increases, the infected number decreases. Moreover, a higher fractional-order results in higher model accuracy. Furthermore, higher values of the process noise Qf give smaller errors, whereas higher values of the observation noise Rf produce higher errors. Qf and the fractional-order α are inversely proportional to RMSE,NRMSE, and MAPE, whereas Rf is directly proportional to RMSE,NRMSE, and MAPE.

Intersection collision risk evaluation and active collision avoidance strategies for autonomous vehicles

In order to ensure that the autonomous vehicle can predict and taking actions to avoid the collision in time when facing the obstacles with intersection collision risk, an intersection collision risk prediction system is proposed in this paper, and two kinds of active obstacle avoidance strategies are designed according to the system: braking strategy and steering strategy. The position information of the obstacle is predicted by Fractional extended Kalman filter, the collision risk rate is determined by the time difference between the vehicle and the obstacle through the intersection point, and a neural network is trained to quickly give the collision risk of the vehicle and the obstacle. Braking strategy and steering strategy are formulated according to collision risk, the braking deceleration and Sigmoid path parameters are given. Finally, the simulation results of PreScan and MATLAB show that the collision risk prediction system can accurately predict the collision between vehicles and obstacles, the braking and steering strategies can effectively avoid the collision.

A Framework for States Co-Estimation of Hybrid Energy Storage Systems Based on Fractional-Order Theory

Accurate state estimation is crucial to the control and management of the hybrid energy storage system (HESS). Although there have been many studies on state estimation of lithium-ion batteries (LIBs) and ultracapacitors (UCs), state estimation for HESS is scarce and poorly defined. To fill this gap, this article aims to propose a co-estimation framework of the state-of-charge (SoC) and remaining discharging time (RDT) for the HESS based on fractional-order theory. First, considering the impedance characteristics of the LIB and UC, the electrochemical impedance spectroscopy (EIS) experiment is conducted to establish the fractional-order models (FOMs). The grey wolf optimizer (GWO) is used to identify the model parameters. Then the fractional extended Kalman filter (FEKF) is used to obtain the SoC for the LIB and UC. A statistical method with the forgetting factor is used for load current forecasting. Finally, the co-estimation framework of the SoC and RDT for the HESS is presented and verified with experimental data. The root-mean-square error (RMSE) of the SoC estimation for the HESS is less than 0.92% in two cases. The experimental results indicate that the proposed co-estimation method of the HESS has achieved high precision and a fast convergence rate under the typical current working condition. This work can be used for a better energy management strategy development.

Estimation for Battery State of Charge Based on Temperature Effect and Fractional Extended Kalman Filter

The electric vehicle has become an important development direction of the automobile industry, and the lithium-ion power battery is the main energy source of electric vehicles. The accuracy of state of charge (SOC) estimation directly affects the performance of the vehicle. In this paper, the first order fractional equivalent circuit model of a lithium iron phosphate battery was established. Battery capacity tests with different charging and discharging rates and open circuit voltage tests were carried out under different ambient temperatures. The conversion coefficient of charging and discharging capacity and the simplified open circuit voltage model considering the hysteresis characteristics of the battery were proposed. The parameters of the first order fractional equivalent circuit model were identified by using a particle swarm optimization algorithm with dynamic inertia weight. Finally, the recursive formula of a fractional extended Kalman filter was derived, and the battery SOC was estimated under continuous Dynamic Stress Test (DST) conditions. The results show that the estimation method has high accuracy and strong robustness.

Battery state-of-charge estimation using fractional extended Kalman filter with Mittag-Leffler memory

In this paper, a Fractional-Order Extended Kalman Filter (FOEKF) to estimate the State-of-Charge (SOC) of a LiFePO4 battery is proposed. Today there is no sensor to measure the state of charge, due to this, it is necessary to use various methods to calculate an estimated value of this parameter. For designing the FOEKF a battery model based on an equivalent electric circuit and the fractional derivative of Atangana-Baleanu type were used. Different numerical simulations using Matlab® were carried out in order to compare it with experimental results from a laboratory prototype and show the performance and accuracy of the FOEKF for estimating the SOC.

Robust extended fractional Kalman filter for nonlinear fractional system with missing measurements

Accurate and effective state estimation is essential for nonlinear fractional system, since it can provide some vital operation information about the system. However, inevitably missing measurements and additive uncertainty in the gain will affect the performance of estimation result. Thus, in this paper, in order to deal with these problems, a novel robust extended fractional Kalman filter (REFKF) is developed for states estimation of nonlinear fractional system, by which the states can be estimated accurately even with missing measurements. Finally, simulation results are provided to demonstrate that the proposed method can achieve much better estimation performance than the conventional extended fractional Kalman filter (EFKF).

A novel cubature statistically linearized Kalman filter for fractional-order nonlinear discrete-time stochastic systems

In this paper, the state estimation problem for fractional-order nonlinear discrete-time stochastic systems is considered. A new method for the state estimation of fractional nonlinear systems using the statistically linearized method and cubature transform is presented. The fractional extended Kalman filter suffers from two problems. Firstly, the dynamic and measurement models must be differentiable and, secondly, nonlinearity is approximated by neglecting the higher order terms in the Taylor series expansion; by the proposed method in this paper, these problems can be solved using a statistically linearized algorithm for the linearization of fractional nonlinear dynamics and cubature transform for calculating the expected values of the nonlinear functions. The effectiveness of this proposed method is demonstrated through simulation results and its superiority is shown by comparing our method with some other present methods, such as the fractional extended Kalman filter.

Fractional extended Kalman filtering for non‐linear fractional system with Lévy noises

In this study, the extended Kalman filtering problem is discussed for fractional-order non-linear discrete system with Levy noises. Two different approximating methods are proposed, respectively, one is to obtain the approximated system measurement noise firstly, then calculates the iterative state vector and measurement vector, the other one is to approximate state vector and measurement vector directly, which seems to be more efficient. Based on the proposed approximating methods, two novel fractional extended Kalman filter designing strategies are developed for fractional-order non-linear discrete system with Levy noises, some detailed comparisons and discussions are presented to show that the obtained results are more general. Finally, some simulation results with necessary comparisons and discussions are given to illustrate the validity and correctness of the proposed methods.

Inverse Air-Pollutant Emission and Prediction Using Extended Fractional Kalman Filtering

It is essential to maintain air-quality standards and to take necessary measures when air-pollutant concentrations exceed permissible limits. Pollutants such as ground-level ozone ( $\text{O}_3$ ), nitrogen oxides ( $\text{NO}_X$ ), and volatile organic compounds (VOCs) emitted from various sources can be estimated at a particular location through integration of observation data obtained from measurement sites and effective air-quality models, using emission inventory data as input. However, there are always uncertainties associated with the emission inventory data as well as uncertainties generated by a meteorological model. This paper addresses the problem of improving the inverse air pollution emission and prediction over the urban and suburban areas using the air-pollution model with chemical transport model (TAPM-CTM) coupled with the extended fractional Kalman filter (EFKF) based on a Matern covariance function. Here, nitrogen oxide (NO), nitrogen dioxide ( $\text{NO}_2$ ), and $\text{O}_3$ concentrations are predicted by TAPM-CTM in the airshed of Sydney and surrounding areas. For improvement of the emission inventory, and hence the air-quality prediction, the fractional order of the EFKF is tuned using a genetic algorithm (GA). The proposed methodology is verified with measurements at monitoring stations and is then applied to obtain a better spatial distribution of $\text{O}_3$ over the region.

A new method of modeling and state of charge estimation of the battery

Accurately estimating the State of Charge (SOC) of the battery is the basis of Battery Management System (BMS). This paper has introduced a new modeling and state estimation method for the lithium battery system, which utilizes the fractional order theories. Firstly, a fractional order model based on the PNGV (Partnership for a New Generation of Vehicle) model is proposed after analyzing the impedance characteristics of the lithium battery and compared with the integer order model. With the observability of the discrete non-linear model of the battery confirmed, the method of the state observer based on the extended fractional Kalman filter (EFKF) and the least square identification method of battery parameters are studied. Then, it has been applied successfully to estimate the battery SOC using the measured battery current and voltage. Finally, a standard HPPC (Hybrid Pulse Power Characteristic) test is used for parameter identification and several experimental validations are investigated on a ternary manganese-nickel-cobalt lithium battery pack with a nominal capacity of 24 Ah which consists of ten Sony commercial cells (US18650GR G7) in parallels. The results demonstrate the effectiveness of the fractional order model and the estimation method.

An extended fractional Kalman filter for inferring gene regulatory networks using time-series data

In recent years, inference of gene regulatory networks has received ever increasing attention in the systems biology field. In this paper, for the first time, a fractional gene regulatory algorithm by extended fractional Kalman filter (EFKF) is proposed to estimate the hidden states as well as the unknown static parameters of the model, which can provide insight into the underlying regulatory relations among genes in the biological system. In the proposed method, gene regulatory networks are inferred via evolutionary modeling based on time-series microarray measurements. The gene regulatory network is considered as a fractional order discrete stochastic dynamic model that consists of the gene measurement equation and the gene regulation equation. After specifying the model structure, we apply the EFKF algorithm for identifying both the model parameters and the actual value of gene expression levels. In this paper, the main advantages of using fractional order systems, increasing the flexibility and improving the accuracy of the system state equation in EFKF are highlighted. The performance of the EFKF algorithm is compared with EKF and other nonlinear algorithms in predicting the parameters of gene regulatory networks from synthetic data and real biological data. Extensive computer simulations illustrate that the proposed algorithm outperforms EKF and other methods, and therefore, it can serve as a natural framework for inference gene regulatory networks with a nonlinear structure.

A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter

In recent years chaotic secure communication and chaos synchronization have received ever increasing attention. In this paper, for the first time, a fractional chaotic communication method using an extended fractional Kalman filter is presented. The chaotic synchronization is implemented by the EFKF design in the presence of channel additive noise and processing noise. Encoding chaotic communication achieves a satisfactory, typical secure communication scheme. In the proposed system, security is enhanced based on spreading the signal in frequency and encrypting it in time domain. In this paper, the main advantages of using fractional order systems, increasing nonlinearity and spreading the power spectrum are highlighted. To illustrate the effectiveness of the proposed scheme, a numerical example based on the fractional Lorenz dynamical system is presented and the results are compared to the integer Lorenz system.