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The efficiency of the hardware implementations of fractional Kalman filter (FKF) heavily relies on the efficiency of realising the fractional‐order derivative operator. In this paper, a generic software and hardware implementation of the FKF based on the Grunwald–Letnikov approximation is proposed and verified on a field‐programmable gate array. Th...
Citations
... However, for carrier moving state, which is closely related to previous moments, integer-order systems and fractional-order nonlinear systems differ significantly. The main distinction is that the integerorder systems can be regarded as the first-order Markov process, while the fractional-order systems cannot because fractional calculus is related to all of the previous state, which can be viewed as a "memory" property (Xu et al. 2022;Liu et al. 2019a, b). Fractional derivatives allow for a more flexible and accurate representation of dynamic systems, enabling better adaptability to complex systems with nonlinear or non-Gaussian characteristics. ...
The global navigation satellite system (GNSS) is widely employed in location-based services (LBS) as a pivotal technology for high-precision navigation and positioning. However, measurement errors cannot be fully eliminated in practical applications, potentially impacting positioning accuracy and reliability. Based on robust estimation and fractional calculus, we construct a robust fractional-order extended Kalman filter (RFEKF) model with a Huber function model. First, we introduce a fractional-order extended Kalman filter (FEKF) model. Second, the RFEKF is constructed by incorporating an equivalence weight matrix that introduces redundancy and the statistical properties of predicted residuals. The RFEKF model adapts the gain matrix through iterative adjustment, obtaining optimal solutions and enhancing the operational efficiency of the model. Finally, simulation experiment and practical implementation are carried out to verify the proposed RFEKF model in GNSS navigation and positioning. The results demonstrate that the RFEKF significantly improves the accuracy of navigation and positioning in the presence of gross errors, surpassing the performance of the REKF.
... Robust Kalman filter can effectively overcome the abnormal values in the observation model, however, the state of carrier is closely related to the previous moments in the movement process. In comparison with integer order system, fractional order nonlinear system has a "memory" characteristic, in which the state estimate is related not only to the state at the current moment, but also to state changes over the previous continuous time horizon (Xu et al. 2022;Liu et al. 2019b;Liu et al. 2019c). The main distinction between integer order system and fractional order nonlinear system is that integer order system can be regarded as the first-order Markov process, while fractional order system cannot, because fractional order calculus depends on the state of all previous moments, and there is lack of research on the influence of gross error in navigation and positioning in the fractional order field. ...
Based on the Huber function, this paper constructs a model of the robust fractional order extended system (RFEKF), which effectively reduces the influence of the gross error. Firstly, by analyzing the abnormal observation characteristics of gross error in fractional order system, the RFEKF is constructed by an equivalence weight matrix of introducing redundancy and the statistical characteristics of predicted residuals, and thus it adjusts the gain matrix and obtains the optimal solution through iteration. Secondly, combined with the RFEKF, the robust estimation of the observed values with gross error is carried out, which further improves the real-time operational efficiency of the model. Finally, simulation experiment and practical implementation are carried out to verify the proposed RFEKF model in GNSS positioning and navigation field, the results show that the RFEKF can still navigate correctly in the case of gross error, compared with EKF, the accuracy of navigation positioning can be significantly improved.
... Due to advances in digital computing, the Kalman filter has been a useful tool for a variety of various applications [8,9]. Although the Kalman filter was originally developed for the case of discrete observations that enter into the estimation of the state variables at discrete times, the observations could be continuous as with analog measuring devices [10][11][12][13][14]. They might on some occasions be considered nearly continuous if the data rate is very high. ...
... Further information on sensitivity analysis can be referred to Gelb [3] and Jwo [10]. ...
This paper provides a useful supplement note for implementing the Kalman filters. The material presented in this work points out several significant highlights with emphasis on performance evaluation and consistency validation between the discrete Kalman filter (DKF) and the continuous Kalman filter (CKF). Several important issues are delivered through comprehensive exposition accompanied by supporting examples, both qualitatively and quantitatively for implementing the Kalman filter algorithms. The lesson learned assists the readers to capture the basic principles of the topic and enables the readers to better interpret the theory, understand the algorithms, and correctly implement the computer codes for further study on the theory and applications of the topic. A wide spectrum of content is covered from theoretical to implementation aspects, where the DKF and CKF along with the theoretical error covariance check based on Riccati and Lyapunov equations are involved. Consistency check of performance between discrete and continuous Kalman filters enables readers to assure correctness on implementing and coding for the algorithm. The tutorial-based exposition presented in this article involves the materials from a practical usage perspective that can provide profound insights into the topic with an appropriate understanding of the stochastic process and system theory.
Extensive research has demonstrated that memristors or trigonometric functions can enhance the complexity of discrete chaotic maps. This article introduces a four-dimensional trigonometric-based memristor hyperchaotic map (4D-TBMHM). By combining discrete memristors, sine, and cosine, the 4D-TBMHM exhibits complex dynamical behaviors. Bifurcation and multistability phenomena are demonstrated using numerical methods. The 4D-TBMHM demonstrates symmetric or attractor self-growth based on iteration length for various system control parameters and initial states, and its complicated complex fractal structure and exemplary performance metrics are also highlighted. Furthermore, an attractor hybrid control, capable of arbitrary positioning and shaping, is proposed. An field-programmable gate array (FPGA)-based hardware prototype is developed, and the attractors are experimentally captured. Moreover, by combining 4D-TBMHM with an arrayed linear feedback shift register, an ultra-fast pseudorandom number generator (UFPRNG) with a throughput of 195.2 Gbps is achieved on an FPGA, surpassing contemporary techniques. Last, the generated UFPRNG is employed for 2.8 Gbps signal generation with noise, and experimental results illustrate its remarkable real-time capability and the PRNG's inherent randomness, facilitating signal generation with any signal-to-noise ratios.
Fractional order memory elements (FOMEs), which exhibit marked memory and nonlinear characteristics, have found broad utilization across an array of interdisciplinary fields. Unfortunately, a significant impediment to their integration into engineering applications is the scarcity of commercially available nanomaterial-based FOMEs. In this paper, we propose a hardware architecture with high FPGA resource utilization efficiency based on the Grünwald–Letnikov approximation and sampling theorem. This architecture can simultaneously realize three fractional calculus (FC) algorithms: the fixed window length (FWL), K-piecewise linear function (
$K$
PLF), and FWL&
$K$
PLF without modifying the FPGA binary file. The method proposed in this paper has lower resource consumption, fewer memory requirements, and higher computational performance, resulting in lower step length by
$10^6$
and higher operating frequencies compared with existing methods. Moreover, we design an FC experimental platform based on a digital oscilloscope that is capable of realizing FOMEs of any order by modifying parameters. In addition, a universal digital circuit of FOMEs is designed in FPGA, which helps to directly display signals at both ends of FOMEs in the digital domain, avoiding interference from circuit noises. Hardware experiments and digital simulations show that the proposed FPGA-based FOMEs hardware circuit method has high precision and reconfigurability, which further accelerates the engineering application of FOMEs.
This paper focuses on the fractional-order Kalman filter’s growth, development, and application. Numerous advancement and the need for various application is critically investigated and summarized. The review work is done on fractional-order Kalman filters as they are best suited for constantly changing systems and can be used for estimating hidden variables. They provide the optimal solution to the filtering problem because it minimizes the state estimation error variance. They are also used to predict and update the location and velocity of an object given a video stream and detections on each of the frames. Many applications are described and analysed in the paper, including battery management, weather forecasting, stochastic state space systems, navigation of the system, and many others.
