Self-adaptive anti-interference control method and system for fuel cell system
By acquiring external interference parameters in real time and dynamically switching control strategies, the problems of signal instability and inaccurate control in fuel cell systems in complex electromagnetic environments are solved, realizing system-level electromagnetic interference control and improving the stability and reliability of the system.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-10
- Publication Date
- 2026-03-27
AI Technical Summary
Existing fuel cell systems are susceptible to electromagnetic interference in complex or highly interfering electromagnetic environments, leading to sensor signal fluctuations, controller program malfunctions, unstable communication links, false triggering of drive units, and reduced system reliability.
An adaptive anti-interference control method is adopted, which collects external interference parameters in real time, judges the interference level and dynamically switches the control strategy, including adjusting the communication protocol frequency, initiating multiple redundancy verification of data, delay compensation for important control loops and module hibernation, to achieve system-level electromagnetic anti-interference control.
It improves the signal stability and control accuracy of fuel cell systems in complex electromagnetic environments, reduces false alarms, enhances operational reliability, and reduces reliance on expensive shielding hardware.
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Figure CN121748439A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of fuel cell technology for new energy vehicles, and more specifically, to an adaptive anti-interference control method and system for a fuel cell system. Background Technology
[0002] In existing fuel cell control systems, the system primarily relies on real-time parameter feedback from multiple sensors (such as temperature, humidity, and pressure) for precise control. However, in complex or highly interfering electromagnetic environments (such as near high-voltage drive motors, DC / DC modules, inverters, and high-frequency charging devices), the system is susceptible to electromagnetic interference (EMI), leading to the following problems: 1. Fluctuations or distortions in sensor signals can cause the system to misjudge its state. 2. The controller program malfunctions or parameters are interrupted, causing the control strategy to fail; 3. Unstable communication links (such as CAN / LIN buses) can cause data loss or delays; 4. Mistriking of the drive unit, such as valves opening or closing accidentally, or compressor not responding; 5. The system frequently triggers error alarms or protection modes, reducing its reliability.
[0003] Therefore, in order to improve the signal stability, control accuracy and operational reliability of fuel cell systems in complex electromagnetic environments, it is necessary to develop an adaptive anti-interference control method for fuel cell systems in environments with high electromagnetic interference. Summary of the Invention
[0004] Based on this, in order to improve the signal stability, control accuracy, and operational reliability of fuel cell systems in complex electromagnetic environments, this invention provides an adaptive anti-interference control method and system for fuel cell systems, the specific technical solution of which is as follows: An adaptive anti-interference control method for a fuel cell system includes the following steps: Start the fuel cell system and collect at least one external disturbance parameter in real time; Determine the interference level based on external interference parameters; The control strategy is dynamically switched according to the interference level; Determine if the switched control strategy is effective; if not, enter safe operation mode.
[0005] The adaptive anti-interference control method of the fuel cell system collects at least one external interference parameter in real time, judges the interference level, and dynamically switches the control strategy according to the interference level. It can dynamically identify the interference source and respond quickly, adapt to complex operating environments, realize system-level electromagnetic anti-interference control capabilities, and ensure the stable operation of the fuel cell.
[0006] Preferably, the adaptive anti-interference control method further includes the following steps: Identify interference patterns based on external interference parameters; Determine the impact of interference modes on the core components of the fuel cell system.
[0007] Preferably, the dynamic switching control strategy based on the interference level specifically includes: adjusting the communication protocol frequency, initiating a data multi-redundancy verification mechanism, compensating for or re-initializing the delay of important control loops, and temporarily putting susceptible modules into hibernation, filtering, or bypassing.
[0008] Preferably, the specific method for determining the interference level based on external interference parameters includes: determining whether the external interference parameters exceed a preset parameter threshold; if so, determining the interference level based on the external interference parameters; if not, maintaining the normal operating logic.
[0009] Preferably, the external interference parameter is the electromagnetic interference intensity.
[0010] An adaptive anti-interference control system for a fuel cell system, used to implement the aforementioned adaptive anti-interference control method, includes: The interference monitoring module is used to collect at least one external interference parameter in real time and determine the interference level based on the external interference parameter. The controller is used to dynamically switch control strategies according to the interference level and determine whether the switched control strategy is effective. If it is ineffective, it enters the safe operation mode.
[0011] Preferably, the adaptive anti-interference control system further includes: The interference identification module is used to identify interference patterns based on external interference parameters and determine the impact of the interference patterns on the core components of the fuel cell system.
[0012] Preferably, the controller includes: The dynamic strategy switching unit is used to dynamically switch control strategies according to the interference level. The status feedback closed-loop unit is used to determine whether the control strategy after the switch is effective. If it is ineffective, it enters the safe operation mode.
[0013] Preferably, the interference identification module includes: The feature vector acquisition unit is used to acquire the feature vectors of external interference parameters; The fractional derivative acquisition unit is used to obtain the fractional derivative for quantifying the abrupt change features of the signal based on the feature vector; Interference identification unit, used to determine interference patterns based on fractional derivatives.
[0014] Preferably, the fractional derivative acquisition unit is based on the function Obtain the fractional derivative; in, These represent the eigenvector and the fractional order, respectively. These represent the current time and the integral time variable, respectively. Attached Figure Description
[0015] The invention will be further understood from the following description taken in conjunction with the accompanying drawings. The components in the drawings are not necessarily drawn to scale, but rather the emphasis is on illustrating the principles of the embodiments. In different views, the same reference numerals designate corresponding parts.
[0016] Figure 1 This is a schematic flowchart of an adaptive anti-interference control method for a fuel cell system according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the overall structure of an adaptive anti-interference control system for a fuel cell system according to an embodiment of the present invention; Figure 3 This is a schematic diagram of the anti-interference operation mode switching control process in one embodiment of the present invention; Figure 4 This is a schematic diagram comparing signal stability before and after interference in one embodiment of the present invention. Detailed Implementation
[0017] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to its embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of protection of the invention.
[0018] It should be noted that when an element is referred to as being "fixed to" another element, it can be directly attached to the other element or there may be an intervening element. When an element is referred to as being "connected to" another element, it can be directly connected to the other element or there may be an intervening element. The terms "vertical," "horizontal," "left," "right," and similar expressions used herein are for illustrative purposes only and do not represent the only possible implementation.
[0019] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.
[0020] In this invention, "first" and "second" do not represent a specific quantity or order, but are merely used to distinguish names.
[0021] like Figure 1 As shown, an embodiment of the present invention provides an adaptive anti-interference control method for a fuel cell system, which includes the following steps: S1, start the fuel cell system and collect at least one external disturbance parameter in real time.
[0022] The specific method for determining the interference level based on external interference parameters includes: determining whether the external interference parameters exceed the preset parameter threshold; if so, determining the interference level based on the external interference parameters; otherwise, maintaining the normal operating logic.
[0023] Specifically, the external interference parameters include, but are not limited to, electromagnetic interference intensity.
[0024] S2, determine the interference level based on external interference parameters.
[0025] S3 dynamically switches control strategies based on the interference level.
[0026] Preferably, the dynamic switching control strategy based on the interference level specifically includes: adjusting the communication protocol frequency, initiating a data multi-redundancy verification mechanism, compensating for or re-initializing the delay of important control loops, and temporarily putting susceptible modules into hibernation, filtering, or bypassing.
[0027] Specifically, different control strategies can be predefined for different interference levels, and then the specific control strategy can be dynamically switched according to the real-time interference level.
[0028] S4 determines whether the switched control strategy is effective. If it is not effective, it enters the safe operation mode.
[0029] The adaptive anti-interference control method of the fuel cell system collects at least one external interference parameter in real time, judges the interference level, and dynamically switches the control strategy according to the interference level. It can dynamically identify the interference source and respond quickly, adapt to complex operating environments, realize system-level electromagnetic anti-interference control capabilities, and ensure the stable operation of the fuel cell.
[0030] In one embodiment, the adaptive anti-interference control method further includes the following steps: identifying interference modes based on external interference parameters; and determining the impact of the interference modes on the core components of the fuel cell system.
[0031] like Figure 2 As shown, an embodiment of the present invention also provides an adaptive anti-interference control system for a fuel cell system, used to implement the aforementioned adaptive anti-interference control method, including an interference monitoring module and a controller. The interference monitoring module is used to collect at least one external interference parameter in real time and determine the interference level based on the external interference parameter; the controller is used to dynamically switch the control strategy according to the interference level and determine whether the switched control strategy is effective. If it is ineffective, it enters the safe operation mode.
[0032] Specifically, the interference monitoring module collects interference parameters such as external electromagnetic field strength, current fluctuations, and communication stability in real time to determine the interference level (e.g., low, medium, high, and extremely high).
[0033] Preferably, the adaptive anti-interference control system further includes an interference identification module. The interference identification module is used to identify interference patterns based on external interference parameters and determine the impact of the interference patterns on the core components of the fuel cell system.
[0034] Specifically, the interference identification module can identify interference patterns (continuous interference, sudden interference, etc.) based on empirical models, interference spectrum characteristics, or neural network algorithms, and determine their impact on the core components of the system.
[0035] The controller includes a dynamic strategy switching unit and a state feedback closed-loop unit.
[0036] The dynamic strategy switching unit is used to dynamically switch the control strategy according to the interference level; the status feedback closed-loop unit is used to determine whether the switched control strategy is effective. If it is ineffective, it enters the safe operation mode.
[0037] Dynamic switching control strategies include, but are not limited to, communication protocol frequency adjustment (such as switching to a more interference-resistant mode), data multiple redundancy verification mechanism, delay compensation or reinitialization of important control loops, and temporary hibernation, filtering or bypass processing for vulnerable modules.
[0038] The system monitors the effectiveness of the control strategy through a dual-channel feedback mechanism (such as primary and backup channels, dual MCU architecture). If the adjustment is found to be ineffective, the fault tolerance submodule is activated and the system enters a safe operation mode.
[0039] The specific control process of the control system includes the following steps: 1. The system starts up and performs an initialization self-test.
[0040] 2. The interference monitoring module is activated to continuously assess environmental interference indicators.
[0041] 3. If the interference exceeds the threshold, it enters the anti-interference working state.
[0042] 4. The control strategy adapts and switches according to the level of interference.
[0043] 5. If the interference is resolved, gradually restore the normal control logic.
[0044] like Figure 3As shown, after the system starts up, it first performs a real-time assessment of the current environmental electromagnetic interference intensity. If the interference exceeds a preset parameter threshold, the interference level is identified and the corresponding anti-interference control strategy is invoked, and the system enters an anti-interference operation state. If the interference does not exceed the preset parameter threshold, the system maintains its normal operating logic. This process ensures that the fuel cell system can achieve adaptive switching and dynamic response of strategies under different interference levels.
[0045] Figure 4 This graph compares the signal stability before and after interference, illustrating the signal changes in the fuel cell system before and after electromagnetic interference. The horizontal axis represents time, and the vertical axis represents the system signal amplitude. During normal operation, the signal remains stable; however, after interference occurs, the signal fluctuates violently, indicating a disturbed state. Once the system enters the anti-interference control phase, an adaptive strategy gradually suppresses the interference, reducing the signal fluctuation amplitude and eventually stabilizing. This graph clearly demonstrates that the anti-interference control method proposed in this invention can effectively reduce signal deviations caused by interference and improve system operational stability.
[0046] The table below shows the impact of electromagnetic interference (EMI) intensity on the key performance characteristics of a fuel cell system. The table illustrates the trends in output voltage fluctuation, control response delay, and communication bit error rate of the fuel cell system under different EMI intensities. It can be seen that as the EMI intensity increases, system performance deteriorates significantly, especially after the EMI intensity exceeds a certain threshold (e.g., 30V / m), where signal stability and control accuracy deteriorate sharply.
[0047]
[0048] (Table 1) In summary, the present invention has the following technical advantages: 1. Achieve system-level electromagnetic interference immunity control capabilities to ensure stable operation of fuel cells.
[0049] 2. Dynamically identify interference sources and respond quickly to adapt to complex operating environments.
[0050] 3. Reduce false alarms and misadjustments in the system, and improve operational accuracy.
[0051] 4. It can be extended to other physical interference scenarios such as vibration interference and thermal interference.
[0052] 5. Reduce costs and improve efficiency through software strategies, and decrease reliance on expensive shielded hardware.
[0053] To address the issues of delayed identification of sudden interference and missed detection of resonance interference, in one embodiment, the interference identification module includes a feature vector acquisition unit, a fractional derivative acquisition unit, and an interference identification unit.
[0054] The feature vector acquisition unit is used to acquire the feature vector of the external interference parameters; the fractional derivative acquisition unit is used to acquire the fractional derivative for quantifying the signal mutation characteristics based on the feature vector; and the interference identification unit is used to determine the interference mode based on the fractional derivative.
[0055] For example, the fractional derivative acquisition unit is based on the function Obtain the fractional derivative; where, These represent the eigenvector and the fractional order, respectively. These represent the current time and the integration time variable, respectively. The fractional order determines the sensitivity of the derivative to abrupt signals, which can be calibrated experimentally and is generally set to 0.8. Use the Gamma function to ensure the mathematical validity of fractional operations. It represents the first derivative of the eigenvector at time τ.
[0056] Feature vector .in, These represent the real-time electromagnetic field strength at time t, the average electromagnetic field strength within the sliding time window, and the standard deviation of the electromagnetic field strength, respectively. The real-time electromagnetic field strength is used to quantify the intensity of environmental electromagnetic interference, such as the instantaneous interference generated by high-voltage motors or charging devices. The standard deviation of the electromagnetic field strength characterizes the fluctuation of the electromagnetic environment; the larger the value, the more unstable the environment. In this calculation, the numerator is used to capture the instantaneous deviation of the current field strength from the baseline, while the denominator is mainly used to eliminate the influence of environmental baseline fluctuations.
[0057] These represent the first derivative of the current and the reference current value, respectively. In this formula, the numerator ignores direction and focuses on the amplitude of change, making it sensitive to high-frequency switching noise. The denominator is mainly used to eliminate the influence of load size.
[0058] These represent the two-dimensional Laplacian operator, the communication stability index, and the maximum allowable gradient threshold (usually taken as 0.5), respectively. The communication stability index S(t)∈[0,1], where 1 indicates perfect communication stability and 0 indicates complete failure. It can generally be calculated based on the bit error rate, i.e., S(t) = 1 - bit error rate.
[0059] The spatial distribution rate of change of the communication stability index S(t) is quantified using a two-dimensional Laplace operator. The gradient value is scaled to a uniform range in the denominator to avoid deviations caused by differences in spatial node density. Since instability in communication links (such as CAN / LIN buses) can lead to data loss or delays, The purpose is to identify the direction and extent of interference propagation, such as whether it is local or global interference.
[0060] when This indicates that local stability is greater than that of the surrounding area (the interference source is in a neighboring node). Local stability is lower than that of surrounding areas (local is the source of interference).
[0061] The aforementioned feature vectors extract features of multi-dimensional interference, including electromagnetic field strength deviation, current change rate, and communication spatial gradient. This enables precise quantification of interference characteristics across electromagnetic, current, and communication dimensions, facilitating the capture of sudden electromagnetic events, transient responses to high-frequency switching, and the location and propagation direction identification of interference sources. Calculating the interference level based on feature vector components can drive smooth strategy switching, avoiding hard switching oscillations.
[0062] Generally, high-frequency charging devices are prone to inducing resonance at the system's inherent frequency, causing the signal to exhibit chaotic oscillations (not simple periodicity). Therefore, under continuous interference, the fractional derivative is stable (with a small amplitude); under sudden interference, the fractional derivative exhibits a single-peak pulse; and under resonant interference, the fractional derivative oscillates continuously with a high amplitude.
[0063] If the fractional derivative is less than the lower limit of the sudden interference threshold, it is judged as continuous interference; if the lower limit of the sudden interference threshold is less than or equal to the fractional derivative and less than or equal to the lower limit of the resonance interference threshold, it is judged as sudden interference; if the fractional derivative is greater than or equal to the lower limit of the resonance interference threshold, it is judged as resonance interference.
[0064] Lower limit of sudden interference threshold Lower limit of resonance interference threshold .in, These represent the maximum electromagnetic field strength and the reference field strength within the current time window, respectively. The reference field strength can be taken as 50V / m. These are calibration coefficients, which can be calibrated by fitting historical data.
[0065] When the maximum electromagnetic field strength is >50V / m Automatic elevation to avoid misjudgment under strong interference; when the maximum electromagnetic field strength is >80V / m. It grows in a quadratic manner, ensuring strict determination of resonance interference.
[0066] This embodiment uses fractional-order calculus to quantize the singular characteristics of interference signals, which is beneficial for identifying resonance modes in complex electromagnetic environments.
[0067] In one embodiment, the interference identification module identifies interference patterns based on empirical models, interference spectrum features, or neural network algorithms.
[0068] A convolutional neural network (CNN) can be used to process the interference spectrum data and extract interference features. The formula for extracting the interference feature vector is as follows: .in, These represent the electromagnetic field strength, current fluctuation rate, and communication stability index at time t, respectively.
[0069] Define reward function This optimizes recognition accuracy. α and β are weighting coefficients that can be adjusted through online learning.
[0070] In one embodiment, a spatiotemporal convolutional field strength prediction model is introduced to address the delay in detecting sudden interference. For example, .
[0071] in, This represents the predicted electromagnetic field strength at the future time Δt. Let represent the measured electromagnetic field strength at time τ, the current fluctuation rate at time τ, the communication stability index at time τ, and the spatial Laplace operator for communication stability, respectively. Δt is the prediction time step. Weighting coefficients Quantum annealing weight function Phase compensation Used to align interference waveforms Dynamic fitting of interference spectrum characteristics allows the model to focus on the dominant frequency band. These represent the exponential decay term and the spectral modulation term, respectively. The exponential decay term assigns higher weights to recent data, and the parameter λ controls the decay rate. Traditional models use fixed weights, while the quantum annealing weight function adaptively adjusts according to the characteristics of the interference spectrum, which can improve the sensitivity to frequency conversion interference (such as DC / DC switching noise).
[0072] In environments with high electromagnetic interference (EMI), traditional sensors are prone to control failure due to signal delay. The proposed spatiotemporal convolutional field strength prediction model uses spatiotemporal convolution to fuse historical data, integrating electromagnetic field strength, current rate of change, and communication stability spatial gradient to predict the field strength at a future time Δt, rather than relying on the current instantaneous value. In the time dimension, the integral window [t...] [τ,t] covers the historical trend of interference evolution and can capture the early characteristics of sudden interference; in the spatial dimension, the spatial Laplace operator introduces the spatial gradient of communication nodes, such as the distribution of multiple CAN bus nodes in a vehicle system, which can detect the direction of interference spread in advance.
[0073] Since discrete policy switching may cause signal jumps and hard switching can easily lead to system oscillations, this embodiment provides a multi-objective policy weight allocation model as a preferred technical solution. .in, This represents the activation weight of strategy j, such as j=1 indicating communication protocol switching, and j=2 indicating redundancy check. These represent the current interference level quantization value (e.g., low / medium / high / extremely high are mapped to 1 / 2 / 3 / 4 respectively), the ideal interference level center of strategy j (preset optimal activation interference level), and the ideal interference level center of strategy k. represents the real-time performance score and weight allocation bandwidth coefficient of strategy j, respectively. The real-time performance score of strategy j is used to quantify the effectiveness of the strategy, and the weight allocation bandwidth coefficient controls the smoothness of the weight distribution. M represents the total number of strategies in the strategy library. The distance is Euclidean, and the current interference level is never quantified. With Strategy Center The differences.
[0074] The real-time performance score of strategy j can be calculated by weighting the corresponding volatility suppression rate, response time and energy efficiency, or by weighting the strategy response delay time and error rate. It is directly proportional to the volatility suppression rate and energy efficiency, and inversely proportional to the strategy response delay time and error rate.
[0075] Assuming the interference level abruptly changes from 2 (medium) to 3.5 (high → extremely high transition), j=1 represents communication protocol switching, j=2 represents data redundancy verification, j=3 represents delay compensation, and j=4 represents module hibernation. If, after calculation, the activation weights of strategies 3 and 4 are 65% and 35% respectively, then the system will execute delay compensation with 65% intensity and module hibernation with 35% intensity, avoiding voltage drops caused by directly switching to hibernation mode.
[0076] The multi-objective strategy weight allocation model uses Gaussian radial basis functions. The policy activation strength is expressed as a continuous function of the interference level, when near hour Maximum bandwidth, decays with distance. Weighted bandwidth coefficient. If the disturbance changes rapidly (such as sudden disturbance), increasing σ makes the weight distribution smoother and avoids drastic adjustments; if the disturbance is stable, decreasing σ improves the strategy's targeting. In summary, this multi-objective strategy weight allocation model has a smooth transition mechanism to avoid hard switching oscillations.
[0077] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.
[0078] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.
Claims
1. An adaptive anti-interference control method for a fuel cell system, characterized in that, The adaptive anti-interference control method includes the following steps: Start the fuel cell system and collect at least one external disturbance parameter in real time; Determine the interference level based on external interference parameters; The control strategy is dynamically switched according to the interference level; Determine if the switched control strategy is effective; if not, enter safe operation mode.
2. The adaptive anti-interference control method as described in claim 1, characterized in that, The adaptive anti-interference control method further includes the following steps: Identify interference patterns based on external interference parameters; Determine the impact of interference modes on the core components of the fuel cell system.
3. The adaptive anti-interference control method as described in claim 2, characterized in that, The dynamic switching control strategy based on the interference level includes: adjusting the communication protocol frequency, activating a data multi-redundancy verification mechanism, compensating for or re-initializing the delay of important control loops, and temporarily putting vulnerable modules into hibernation, filtering, or bypassing.
4. The adaptive anti-interference control method as described in claim 3, characterized in that, The specific method for determining the interference level based on external interference parameters includes: determining whether the external interference parameters exceed the preset parameter threshold; if so, determining the interference level based on the external interference parameters; otherwise, maintaining the normal operating logic.
5. The adaptive anti-interference control method as described in claim 4, characterized in that, The external interference parameter is the electromagnetic interference intensity.
6. An adaptive anti-interference control system for a fuel cell system, used to implement the adaptive anti-interference control method as described in any one of claims 1-5, characterized in that, The adaptive anti-interference control system includes: The interference monitoring module is used to collect at least one external interference parameter in real time and determine the interference level based on the external interference parameter. The controller is used to dynamically switch control strategies according to the interference level and determine whether the switched control strategy is effective. If it is ineffective, it enters the safe operation mode.
7. The adaptive anti-interference control system as described in claim 6, characterized in that, The adaptive anti-interference control system further includes: The interference identification module is used to identify interference patterns based on external interference parameters and determine the impact of the interference patterns on the core components of the fuel cell system.
8. The adaptive anti-interference control system as described in claim 7, characterized in that, The controller includes: The dynamic strategy switching unit is used to dynamically switch control strategies according to the interference level. The status feedback closed-loop unit is used to determine whether the control strategy after the switch is effective. If it is ineffective, it enters the safe operation mode.
9. The adaptive anti-interference control system as described in claim 8, characterized in that, The interference identification module includes: The feature vector acquisition unit is used to acquire the feature vectors of external interference parameters; The fractional derivative acquisition unit is used to obtain the fractional derivative for quantifying the abrupt change features of the signal based on the feature vector; Interference identification unit, used to determine interference patterns based on fractional derivatives.
10. The adaptive anti-interference control system as described in claim 9, characterized in that, The fractional derivative acquisition unit is based on the function Obtain the fractional derivative; in, These represent the eigenvector and the fractional order, respectively. These represent the current time and the integral time variable, respectively.