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Fractional-Order Digital Filters
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 number of fractional calculus applications has seen a rapid growth over the last decade. Fractional calculus can be easily defined as a generalization of integer-order calculus with the order of the differintegral operators as fractional. Its versatility in modeling and control theory has received a lot of attention recently, although it is still a concept insufficiently understood. This limits the wide acceptance of fractional calculus in industrial use. Fractional calculus has been regarded as a much better way to cover the dynamics of certain type of phenomena, such as anomalous diffusive characteristics , viscoelasticity , epidemic spreading , etc. At the same time, fractional calculus in controller design has increased their flexibility and robustness . Review papers dealing with the use of fractional calculus in control engineering have been published recently, such as .
However, apart from fractional-order models and controllers, the theoretical aspects of fractional calculus have been extended to cover adjacent areas of research, namely sensing and filtering. This has somewhat evolved as a logical step, since actual processes are better modeled using fractional-order systems . At the same time, state estimation is crucial in designing fractional-order controllers . Thus, for a robust state estimation and an efficient noise elimination in fractional-order systems, extensions to a fractional-order of the popular integer-order estimators have been proposed.
It has been widely proven that complex systems can be accurately described by power-law series . For the case of electronic devices, the behavior is given by the sum of various independent actions of the charge carriers, exhibiting the normal distribution. Unknown interactions in the electronic device leads to the second moment of distribution that fails to converge. For the case of real-time sampling, the mean converges rapidly towards infinity, while the standard deviation fluctuates. These systems are best described by the Generalized Law of Large Numbers, resulting in power-law series behavior, with an added α-stable component, proving the presence of fractional-order dynamics in any complex system .
Filters are one of the key elements in the signal processing field. Many filtering techniques have been developed throughout the years for noise reduction, signal modulation, demodulation, amplification, etc. Filters can be analog, consisting of electronic circuits that process the analog signal, or digital, consisting of mathematical filters that process the analog signal after its discretization. The popular field of fractional calculus has also infiltrated into filter design, for both analog and digital cases. For the case of analog filters, the creation of the fractance device, integrating fractional-order dynamics into electronic components such as the fractional-order capacitor has been the starting point of fractional-order analog filters. Fractional-order electronic components are used to create filters that have a larger frequency range and a better response than integer-order filters . However, due to the limitations present in fractional-order physical hardware, there are only a few studies covering the physical realization of fractional-order analog filters, which will be described later in the manuscript.
The applicability of digital fractional-order filters spans on a manifold of domains from data transmission and networking applications , electrical vehicle manufacturing (through the determination of state of charge in lithium-ion batteries , aerial vehicle orientation using fractional-order filtering of yaw, pitch and roll signals , air-quality assessments through pollution and humidity factors , civil engineering targeting the measurement and data processing of various characteristics of buildings such as stiffness and damping , different biomedical processes, image processing and many more. Most of the existing implementations of fractional-order filters are in the fields of data transmission and battery estimation, as will be shown in a dedicated section that highlights the benefits of fractional-order filters in real-life applications.
2. Applications of Fractional-Order Filters
2.1. Data Transmission and Networking
2.2. Applications Using Lithium-Ion Batteries
2.3. Other Applications
|Extended and Unscented Filtering Algorithms in Nonlinear Fractional-Order Systems with Uncertain Observations||2012|||
|Dual Estimation of Fractional Variable Order Based on the Unscented Fractional-Order Kalman Filter for Direct and Networked Measurements||2016|||
|State-of-Charge Estimation for Lithium-Ion Batteries Based on a Nonlinear Fractional Model||2017|||
|A Modified Fractional-Order Unscented Kalman Filter for Nonlinear Fractional-Order Systems||2018|||
|A novel cubature statistically linearized Kalman filter for fractional-order nonlinear discrete-time stochastic systems||2018|||
|Nonlinear Fractional-Order Estimator With Guaranteed Robustness and Stability for Lithium-Ion Batteries||2018|||
|Robust extended fractional Kalman filter for nonlinear fractional system with missing measurements||2018|||
|Fractional-order chaotic cryptography in colored noise environment using fractional-order interpolatory cubature Kalman filter||2019|||
|Fractional-order Kalman filters for continuous-time linear and nonlinear fractional-order systems using Tustin generating function||2019|||
|An adaptive unscented Kalman filter for a nonlinear fractional-order system with unknown order||2020|||
|Design of a Robust State Estimator for a Discrete-Time Nonlinear Fractional-Order System With Incomplete Measurements and Stochastic Nonlinearities||2020|||
|Extended Kalman Filters for Continuous-time Nonlinear Fractional-order Systems Involving Correlated and Uncorrelated Process and Measurement Noises||2020|||
|Extended Kalman filters for nonlinear fractional-order systems perturbed by colored noises||2020|||
|Hybrid extended-cubature Kalman filters for nonlinear continuous-time fractional-order systems involving uncorrelated and correlated noises using fractional-order average derivative||2020|||
|Hybrid extended-unscented Kalman filters for continuous-time nonlinear fractional-order systems involving process and measurement noises||2020|||
|Novel hybrid robust fractional interpolatory cubature Kalman filters||2020|||
|Adaptive fractional-order Kalman filters for continuous- time nonlinear fractional-order systems with unknown parameters and fractional orders||2021|||
This entry is adapted from 10.3390/s21175920
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