Fractional-order Kalman Filter Research Articles

Intelligent Predictive Maintenance of Dry-Type Transformer Based on Vibration

The normal operation of the transformer is an important guarantee for the safe operation of the whole power network. Usually, the periodic maintenance work increases the workload of the operation and maintenance personnel. In this paper, an intelligent predictive and maintenance system for dry-type transformer based on vibration is proposed to monitor the vibration state of the transformer. The fractional-order Kalman filter and normalization method are used to preprocess the vibration data to reduce noise interference, and then the fault is classified by one-dimensional convolution neural network. On this basis, an improved grey Verhulst model is established to predict the fault time of the transformer. The experimental results show that the classification accuracy of the one-dimensional convolution neural network can reach more than 95%, and the improved grey Verhulst model can predict the fault of the dry-type transformer one week in advance.

Fractional variable-order calculus based state of charge estimation of Li-ion battery using dual fractional order Kalman filter

Although fractional order based models and approaches have been proposed to determine battery key state characteristics, the inflexible order has a detrimental impact on battery modeling and SOC calculation. In this study, a novel SOC estimation strategy based on the combination of simplified fractional variable-order equivalent circuit model (FVOECM) and FVO adaptive dual Kalman filter (FVOADKF) algorithm is proposed. The simplified FVOECM is designed to represent the battery's dynamic external electrical characteristics. The FVO-based adaptive extended Kalman filter algorithm is utilized for the battery SOC estimation, while the FVO-based adaptive unscented Kalman filter algorithm is used to determine the order of the FVOECM. Consequently, the updated order effectively tracks the transaction of the battery FVOECM thus improving the SOC estimation accuracy. Furthermore, the influences of SOC initialization error and FO backtracking length on the algorithm convergence are analyzed. The effectiveness of the FVOECM is confirmed under two different conditions, and four experiment results indicate that the effectiveness of the FVOADKF is superior to other estimation methods.

State-of-Charge Estimation for Lithium-Ion Batteries Based on Temperature-Based Fractional-Order Model and Dual Fractional-Order Kalman Filter

Accurate state-of-charge (SOC) estimation of batteries is of great significance for electric vehicles. Ambient temperature influences the relationship between open-circuit voltage (OCV) and SOC as well as model parameter and, accordingly, influences the accuracy of the battery SOC estimation for model-based methods. To address the temperature dependence of battery modeling and SOC estimation, an SOC estimation method for lithium-ion batteries based on a temperature-based fractional first-order RC circuit model and dual fractional-order Kalman filter (DFOKF) was proposed. The OCV-SOC look-up table corresponding to ambient temperature and an offset function regarding to ambient temperature were applied to the developed model for improving modeling accuracy. One of dual filters was used to estimate the SOC, and the other was employed to update the model parameters online for addressing the temperature dependence of model parameter. Comparisons of the SOC estimation results between the developed model and the original model that ignores the influence of ambient temperature under the US06 Highway Driving Schedule and the Federal Urban Driving Schedule (FUDS) tests at eight specified temperatures were performed. The results show that the developed model combined with the DFOKF algorithm improves the accuracy of SOC estimation. For the application of SOC estimation at untested temperature, we proposed to construct the corresponding OCV-SOC look-up table by linear interpolation of the measured OCV-SOC look-up table at the tested temperature. Comparisons of results between using the two kinds of look-up tables to estimate the SOC demonstrate the effectiveness of the proposed approach.

A Review of Recent Advances in Fractional-Order Sensing and Filtering Techniques.

The present manuscript aims at raising awareness of the endless possibilities of fractional calculus applied not only to system identification and control engineering, but also into sensing and filtering domains. The creation of the fractance device has enabled the physical realization of a new array of sensors capable of gathering more information. The same fractional-order electronic component has led to the possibility of exploring analog filtering techniques from a practical perspective, enlarging the horizon to a wider frequency range, with increased robustness to component variation, stability and noise reduction. Furthermore, fractional-order digital filters have developed to provide an alternative solution to higher-order integer-order filters, with increased design flexibility and better performance. The present study is a comprehensive review of the latest advances in fractional-order sensors and filters, with a focus on design methodologies and their real-life applicability reported in the last decade. The potential enhancements brought by the use of fractional calculus have been exploited as well in sensing and filtering techniques. Several extensions of the classical sensing and filtering methods have been proposed to date. The basics of fractional-order filters are reviewed, with a focus on the popular fractional-order Kalman filter, as well as those related to sensing. A detailed presentation of fractional-order filters is included in applications such as data transmission and networking, electrical and chemical engineering, biomedicine and various industrial fields.

Adaptive fractional-order Kalman filters for continuous-time nonlinear fractional-order systems with unknown parameters and fractional-orders

In this paper, two types of adaptive Kalman filters are proposed by using the Grünwald-Letnikov (G-L) difference method to estimate the state information of continuous-time nonlinear fractional-order systems with unknown parameters and fractional-orders. An adaptive extended Kalman filter is designed by using the first-order Taylor expansion to deal with the nonlinear function in a nonlinear fractional-order system with unknown parameters and fractional-order. Based on the third-degree spherical-radial rule, an adaptive cubature Kalman filter as another adaptive fractional-order Kalman filter discussed in this paper is provided by the cubature points to deal with the nonlinear function. The augmented vector consisting of the unknown state vectors, parameters and fractional-order is constructed, and the corresponding augmented equation is established to solve the estimation problem with unknown parameters and fractional-order. The state estimations of nonlinear fractional-order systems with unknown parameters and fractional-orders are carried out by the augmented vector method. Finally, four examples are given to verify the effectiveness of the proposed adaptive Kalman filters with unknown parameters and fractional-orders in this paper.

Innovation-based fractional order adaptive Kalman filter

Abstract Kalman Filter (KF) is the most widely used estimator to estimate and track the states of target. It works well when noise parameters and system models are well defined in advance. Its performance degrades and starts diverging when noise parameters (mainly measurement noise) changes. In the open literature available researchers has used the concept of Fractional Order Kalman Filter (FOKF) to stabilize the KF. However in the practical application there is a variation in the measurement noise, which will leads to divergence and degradation in the FOKF approach. An Innovation Adaptive Estimation (IAE) based FOKF algorithm is presented in this paper. In order to check the stability of the proposed method, Lyapunov theory is used. Position tracking simulation has been performed for performance evaluation, which shows the better result and robustness.

Sensing Data Fusion for Enhanced Indoor Air Quality Monitoring

Multisensor fusion of air pollutant data in smart buildings remains an important input to address the well-being and comfort perceived by their inhabitants. An integrated sensing system is part of a smart building where real-time indoor air quality data are monitored round the clock using sensors and operating in the Internet-of-Things (IoT) environment. In this work, we propose an air quality management system merging indoor air quality index (IAQI) and humidex into an enhanced indoor air quality index (EIAQI) by using sensor data on a real-time basis. Here, indoor air pollutant levels are measured by a network of waspmote sensors while IAQI and humidex data are fused together using an extended fractional-order Kalman filter (EFKF). According to the obtained EIAQI, overall air quality alerts are provided in a timely fashion for accurate prediction with enhanced performance against measurement noise and nonlinearity. The estimation scheme is implemented by using the fractional-order modeling and control (FOMCON) toolbox. A case study is analysed to prove the effectiveness and validity of the proposed approach.

A New State of Charge Estimation Method for Lithium-Ion Battery Based on the Fractional Order Model

This paper presents that the fractional order Kalman filter (FOKF) method is used to estimate the state of charge (SOC) for lithium-ion battery based on the fractional order model. First, a fractional order battery model was established which can better reflect the dynamic characteristics of the battery. The fractional orders were identified by genetic algorithm. Then, compared with three other modeling methods in four aspects: maximum absolute error, maximum relative error, computational complexity and number of model parameters, it is shown that the fractional order model proposed in this paper is more accurate and reliable. The results shows that the maximum absolute error of the terminal voltage is 0.014 V under constant current discharge test. The accuracy improves 0.058 V comparing to the integer order model. Finally, the SOC was estimated through two methods. The results shows that the maximum absolute estimation error of SOC is under 0.02 by FOKF, which has higher accuracy and faster convergence speed compared with extend Kalman filter (EKF) method.

Fitting of experimental data using a fractional Kalman-like observer

In this paper, a fractional order Kalman filter (FOKF) is presented, this is based on a system expressed by fractional differential equations according to the Riemann–Liouville definition. In order to get the best fitting of the FOKF, the cuckoo search optimization algorithm (CS) was used. The purpose of using the CS algorithm is to optimize the order of the observer, the fractional Riccati equation and the FOKF tuning parameters. The Grünwald–Letnikov approximation was used to compute the numerical solution of the FOKF. To show the effectiveness of the proposed FOKF, four examples are presented, the brain activity, the cutaneous potential recordings of a pregnant woman, the earthquake acceleration, and the Chua’s circuit response.

Kalman Filters for Continuous-time Fractional-order Systems Involving Fractional-order Colored Noises Using Tustin Generating Function

This study presents fractional-order Kalman filers for linear fractional-order systems with colored noises using Tustin generating function. A continuous-time fractional-order system with the fractional-order colored process noise is discretized by Tustin generating function. The augmented vector consists of the state and the colored noise is offered to construct an augmented system based on the discretized state equation of a fractional-order system and the colored process noise. The Tustin fractional-order Kalman filter is designed based on the augmented system to obtain the state estimation, effectively. Besides, the colored noise involved in the measurement of a continuous-time fractional-order system is also discussed, and the corresponding Tustin fractional-order Kalman filter is provided in this study. Two illustrative examples are given to verify the effectiveness of Tustin fractional-order Kalman filters for the colored process and measurement noises.

Kalman filters for linear continuous‐time fractional‐order systems involving coloured noises using fractional‐order average derivative

In this study, Kalman filters for continuous-time linear fractional-order systems are studied with fractional-order coloured process and measurement noise, respectively. By fractional-order average derivative, linear fractional-order systems with coloured fractional-order process or measurement noise are discretised. To deal with coloured noises, the authors construct an augmented system with respect to the state, the process noise and the measurement noise. Furthermore, fractional-order Kalman filter using the fractional-order average derivative is proposed. This filter improves the accuracy of the state estimation and the filtering effect for coloured process and measurement noises. Finally, they give two examples to verify the correctness and validity of the proposed algorithm.

Fractional-order Kalman filters for continuous-time fractional-order systems involving colored process and measurement noises

This study proposes fractional-order Kalman filers using Tustin generating function and the average value of fractional-order derivative to estimate the state of fractional-order systems involving colored process and measurement noises. By Tustin generating function, a fractional-order differential equation is provided to approximate the dynamics of a continuous-time fractional-order system and colored process and measurement noises. By constructing an augmented system with respect to state, the process noise and the measurement noise to deal with colored noises, the fractional-order Kalman filter using Tustin generating function is proposed to improve the estimation accuracy. Besides, the average value of fractional-order derivative is proposed, and the corresponding fractional-order Kalman filter by the augmented system method is presented to reduce estimation error. Finally, three illustrative examples are given to illustrate that the proposed two kinds of Kalman filters are more effective than fractional-order Kalman filter based on Gru¨nwald–Letnikov definition.

Fractional-order Kalman filters for continuous-time linear and nonlinear fractional-order systems using Tustin generating function

ABSTRACTThis paper presents the fractional-order Kalman filters using Tustin generating function for linear and nonlinear fractional-order systems involving process noise and measurement noise. By using the Tustin generating function, the differential equation model is obtained by discretising the investigated continuous-time fractional-order system. The two kinds of fractional-order Kalman filters are given for the correlated and uncorrelated cases in terms of the process noise and measurement noise for linear fractional-order system, respectively. In addition, based on the first-order Taylor expansion formula, the extended fractional-order Kalman filter using Tustin generating function is proposed to improve the accuracy of state estimation. Finally, three examples are illustrated to verify the effectiveness of the Tustion fractional-order Kalman filters for linear and nonlinear fractional-order systems.

A Practical Method for Designing Linear Quadratic Regulator for Commensurate Fractional-Order Systems

The methods currently available for designing a linear quadratic regulator for fractional-order systems are either based on sufficient-type conditions for the optimality of functionals or generate very complicated analytical solutions even for simple systems. It follows that the use of such methods is limited to very simple problems. The present paper proposes a practical method for designing a linear quadratic regulator (assuming linear state feedback), Kalman filter, and linear quadratic Gaussian regulator/controller for commensurate fractional-order systems (in Caputo sense). For this purpose, considering the fact that in dealing with fractional-order systems the cost function of linear quadratic regulator has only one extremum, the optimal state feedback gains of linear quadratic regulator and the gains of the Kalman filter are calculated using a gradient-based numerical optimization algorithm. Various fractional-order linear quadratic regulator and Kalman filter design problems are solved using the proposed approach. Specifically, a linear quadratic Gaussian controller capable of tracking step command is designed for a commensurate fractional-order system which is non-minimum phase and unstable and has seven (pseudo) states.

Fractional modeling and SOC estimation of lithium-ion battery

This paper proposes a state of charge (SOC) estimator of Lithium-ion battery based on a fractional order impedance spectra model. Firstly, a battery fractional order impedance model is derived on the grounds of the characteristics of Warburg element and constant phase element (CPE) over a wide range of frequency domain. Secondly, a frequency fitting method and parameter identification algorithm based on output error are presented to identify parameters of the fractional order model of Lithium-ion battery. Finally, the fractional order Kalman filter approach is introduced to estimate the SOC of the lithium-ion battery based on the fractional order model. The simulation results show that the fractional-order model can ensure an acceptable accuracy of the SOC estimation, and the error of estimation reaches maximally up to 0.5% SOC.

Simultaneous state estimation and parameter identification in linear fractional order systems using coloured measurement noise

ABSTRACTIn the present paper, the identification and estimation problem of a single-input–single-output (SISO) fractional order state-space system will be addressed. A SISO state-space model is considered in which parameters and also state variables should be estimated. The canonical fractional order state-space system will be transformed into a regression equation by using a linear transformation and a shift operator that are appropriate for identification. The identification method provided in this paper is based on a recursive identification algorithm that has the capability of identifying the parameters of fractional order state-space system recursively. Another subject that will be addressed in this paper is a novel fractional order Kalman filter suitable for the systems with coloured measurement noise. The promising performance of the proposed methods is verified using two stable fractional order systems.

Dual Estimation of Fractional Variable Order Based on the Unscented Fractional Order Kalman Filter for Direct and Networked Measurements

The paper is devoted to variable order estimation process when measurements are obtained in two different ways: directly and by lossy network. Since the problem of fractional order estimation is highly nonlinear, dual estimation algorithm based on Unscented Fractional Order Kalman filter has been used. In dual estimation process, state variable and order estimation have been divided into two sub-processes. For estimation state variables and variable fractional order, the Fractional Kalman filter and the Unscented Fractional Kalman filter have been used, respectively. The order estimation algorithms were applied to numerical examples and to real fractional variable order inertial system realized as an analog circuit.

Vehicle Tracking in Video using Fractional Feedback Kalman Filter

Object tracking is a challenging and important area of research. The object tracking system must be capable of tracking abrupt variations in object state. Kalman filter is fundamental and widely used as an optimal state estimator in object tracking. With known noise and system parameters, Kalman filter tends to stabilize the gain. However, during sudden transitions of the object tracked, constant gain Kalman filter may diverge. This paper proposes a modified steady-state gain of the Kalman filter, which is achieved by introducing a fractional feedback loop across the Kalman gain. The modified Kalman gain is estimated by minimizing the cost function of the proposed Kalman filter. Results show that the accuracy and robustness of the Kalman filter are significantly improved. The performance of the proposed method is compared with that of the standard Kalman filter, fractional-order Kalman filter, and unscented Kalman filter. In this work, the root mean square error is used as the performance metric. The proposed method has been tested for different datasets, including traffic videos. Experiments show that RMSE is improved upto $17\%$ by using the proposed modified Kalman filter.

On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment

In this paper via a novel method of discretized continuous-time Kalman filter, the problem of synchronization and cryptography in fractional-order systems has been investigated in presence of noisy environment for process and output signals. The fractional-order Kalman filter equation, applicable for linear systems, and its extension called the extended Kalman filter, which can be used for nonlinear systems, are derived. The result is utilized for chaos synchronization with the aim of cryptography while the transmitter system is fractional-order, and both the transmitter and transmission channel are noisy. The fractional-order stochastic chaotic Chen system is then presented to apply the proposed method for chaotic signal cryptography. The results show the effectiveness of the proposed method.

Simulation and experimental tools for fractional order control education

The paper presents the simulation toolkit in MATLAB/Simulink® for the fractional order discrete, state-space system education. The toolkit has been written as a set of C-MEX S-functions which simulate several fractional order blocks - e.g. fractional order difference, fractional order discrete state-space representation, fractional order Kalman filter. This way the C code generation capability (using Real Time Workshop) has been provided. This allowed to easily implement the control algorithms in the experimental setup. For analysis of fractional order systems the ultracapacitor has been chosen. It represents the real-life system which is inherently of fractional order. Altogether the simulation toolkit and the experimental testbed form the basis for the advanced fractional order control lab at Institute of Control and Industrial Electronics, Warsaw University of Technology.