Fractional derivative estimation method based on sliding mode technology

Abstract

The present invention discloses a fractional-order derivative estimation method based on sliding mode technology. By using the properties of fractional-order calculus, a strategy for estimating the fractional-order derivatives of any bounded continuous signal or contaminated signal is designed. Then, an estimation system structure diagram is constructed to form a fractional-order derivative estimator, that is, a typical negative feedback closed-loop system + filter model. Then, with the help of stability theory and statistical linearization methods, relevant parameters of the above-mentioned fractional-order derivative estimator are determined, and an adaptive component is further added in the case of an upper bound of the fractional-order derivative that is unknown in advance, so that the adaptive gain can be automatically adjusted online as the error system changes. At the same time, computer software is used to perform simulation tests, and the gain coefficients at various locations in the optimization structure diagram are adjusted to improve the estimation effect. Then, the designed method is applied to practice to estimate the fractional-order derivatives of contaminated signals, and the design of a fractional-order controller for two inverted pendulums is realized.

Description

Technical Field

[0001] The present invention relates to the field of fractional derivative estimation, and specifically to a fractional derivative estimation method based on a sliding mode technique. Background Art

[0002] Fractional calculus is a generalization of integer-order calculus. In recent years, fractional calculus has been successfully applied to various aspects of natural science and engineering applications to process various collected signal data. A signal is a carrier of information, and the transmission and exchange of information are the specific contents of signal processing. The description method of a signal is generally a mathematical expression, and most of these expressions are functions of time. For example, in the aerospace field, the speed and acceleration of an aircraft's attitude and azimuth adjustment from the current position to the target position need to be estimated and measured in advance by taking the time derivative of the angular position signal (Design and Experimental Results of an Adaptive Fractional-Order Controller for a Quadrotor, 2022, 6, 204); in the design of a fractional-order PI λ D μ controller for ship control, the input signal of the controller needs to be synthesized based on the fractional derivative of the error (Fractional-order PI λ D μ -controller using adaptive neural fuzzy model for course control of underactuated ships, 2022, 12(11), 5604), so as to control the ship's speed and heading. Therefore, in many cases in the fields of information science and control engineering, it is necessary to determine or estimate the integer-order or fractional-order time derivative of a given signal.

[0003] However, due to the weak singularity in the definition of fractional calculus, the fractional derivatives of most given signal functions cannot give specific analytical expressions like integer-order derivatives. Especially when the input signal of the controller is contaminated, it is still extremely difficult to determine or estimate the fractional derivative of a given signal to achieve the control of specific target objects (such as aircraft, ships, etc.) (Applications of Fractional Operators in Robotics: A Review, 2022, 104, 63). Although some scholars have conducted in-depth research in this area and made some progress, for example, Professor A. Oustaloup approximated the amplitude-frequency characteristics of fractional operators with a set of broken lines in the frequency domain, making the fractional response curve of the signal close to the true value; Professor D. Liu used methods such as the modulation function method, algebraic parameter method, and kernel function method in the time domain to identify the fractional derivatives of unknown signals, making their approximate values close to the true value. However, in practical engineering applications, the methods in the frequency domain are difficult to implement, and the algorithms in the time domain are relatively complex, resulting in poor control effects on target objects, such as problems like delays in target actions (such as attitude, azimuth, and speed and heading, etc.) or failure to achieve the control purpose, which is not conducive to engineering implementation.

[0004] Therefore, there is an urgent need for a convenient, simple, fast, effective, stable, and reliable fractional derivative estimation strategy and method, especially one that can still work properly and maintain good robustness when the signal is contaminated. Summary of the Invention

[0005] The technical problem to be solved by the present invention is: how to solve the problem of determining or estimating the time fractional derivative of a given signal function in the existing fields of information science and control engineering, and provide a fractional derivative estimation method based on sliding mode technology, which is also applicable to the fractional derivative estimation of contaminated signals.

[0006] The estimation method of the present invention is different from the existing fractional derivative estimation methods and ideas. Its main components include three major modules: a nonlinear module, a relay module, and a filter module. For any given bounded continuous signal or contaminated signal, it can quickly, effectively, and accurately obtain the estimated signal of its fractional derivative, and achieve good estimation effects by selecting and adjusting parameters;

[0007] The present invention first utilizes the properties of fractional calculus to design a strategy (or scheme) for estimating fractional derivatives, then constructs the structural diagram of the estimation system to form a fractional derivative estimator (i.e., a typical negative feedback closed-loop system + filter model), and then determines the relevant parameters of the above-mentioned fractional derivative estimator with the assistance of stability theory and statistical linearization methods. Since the upper bound of the fractional derivative needs to be known during implementation, this will impose many limitations on the design; therefore, the design is further optimized, that is, an adaptive component is added considering the case where the upper bound of the fractional derivative is not known in advance, and the adaptive gain can be automatically adjusted online as the error system changes; finally, computer software (MATLAB) is combined for simulation testing, and the gain coefficients at various places in the structural diagram are adjusted and optimized to improve the estimation effect, and then the designed method is applied to the actual situation to estimate the fractional derivatives of contaminated signals, realizing the design of fractional controllers for two inverted pendulums.

[0008] The present invention solves the above technical problems through the following technical solutions. The present invention includes:

[0009] (1) Before entering the design of the fractional derivative estimator, first use the properties of fractional calculus to perform an equivalent transformation on the object to be estimated to obtain the equation model of the fractional error system, providing a clear idea for giving the strategy or scheme for estimating fractional derivatives. The equation expression of the fractional error system is:

[0010] ,

[0011] where represents the Riemann-Liouville (R-L) fractional derivative, , and respectively represent the input of the signal to be estimated, the error signal, and the controller to be designed.

[0012] (2) Design a strategy or scheme for estimating the fractional derivatives of any bounded continuous signal or contaminated signal, and give the various components of the structural diagram of the estimation system, specifically including: the signal to be estimated, the controller, the fractional integrator, and the filter. Among them, the main components: the controller and the filter are designed as follows:

[0013] The first part of the controller: Construct a nonlinear module , where the parameter is the nonlinear gain coefficient, satisfies: (i) , ( can be ); (ii) . For example: ( and is an odd number), or , or ( is a saturation function).

[0014] Controller second part: Construct the relay module , where the parameter is the relay gain coefficient, satisfies: (i) ; (ii) ; (iii) .

[0015] Select appropriate matrices , and , and construct the filter module: . For example: when , and , then is a first-order low-pass filter module; when , and , then is a second-order low-pass filter module , (where and are filter coefficients), and so on, a third-order low-pass filter can also be given. Additionally, when all filter coefficients are zero, this module is a constant 1.

[0016] (3) Based on step (2), first connect the nonlinear module and the relay module in parallel and then connect them in series with the fractional-order filter module, and lead out a negative feedback signal before the fractional-order filter module to after the input function to form a closed-loop circuit, and then obtain the estimated system structure diagram.

[0017] (4) Based on and in step (2), and then combined with the estimated system structure diagram, determine the parameters and of the controller module through the statistical linearization method. The specific principle is as follows: First, obtain the closed-loop transfer function from the structure diagram, where is the equivalent transfer function coefficient; then, according to the statistical linearization method, the expression for the equivalent transfer function coefficient is given as:

[0018] ,

[0019] where is the standard deviation of the error signal , ( and is odd). Obviously is closely related to the standard deviation of the error signal and parameters , , , , and there are and . If the parameters , , , are properly selected such that the overall is very small, then in a low-frequency noise environment or under perturbation, the closed-loop transfer function is equivalent to a fractional-order differentiator; on the contrary, in a high-frequency noise environment or under perturbation, the closed-loop transfer function , that is, the signal contaminated by noise does not undergo fractional-order differentiation operation when passing through here. Therefore, the selection of the parameter is very crucial. Considering that is the switching gain, the larger it is, the more severe the chattering is, that is cannot be taken too large, and only can be taken larger, so as to ensure that the designed fractional-order derivative estimation device can still work normally when the signal is contaminated by random noise (or uncertain perturbation).

[0020] (5) Based on in step (2), combined with the estimation system structure diagram, determine the filter coefficients and through stability theory. The specific principle is as follows: According to the obtained fractional-order derivative estimation of any bounded continuous signal, it may contain "harmful" noise, and a low-pass filter module needs to be set to filter out the "harmful" signal, and there may still be an error between the obtained estimated value and the exact value. The error expression is:

[0021]

[0022] where , is the minimum negative real part eigenvalue of is a higher-order infinitesimal.

[0023] When is a first-order low-pass filter, its solution is:

[0024]

[0025] The error expression is: , where is a higher-order infinitesimal.

[0026] When is a second-order low-pass filter, its solution is:

[0027]

[0028] The error expression is: , where is a higher-order infinitesimal. Therefore, based on the above solution expression and error expression, the selection of parameters and should be as small as possible, but not too small to prevent the distortion of the linear filter and have an adverse impact on the optimization estimation performance. In addition, by equivalently converting the fractional-order error system into a continuous frequency distribution state weight model, then constructing an energy function, and using the Lyapunov stability theory to give whether the error signal can reach the predetermined index range within the finite time , the finite time satisfies the relation:

[0029]

[0030] According to the above formula, the parameters and can be determined, thus providing support for improving the estimation efficiency.

[0031] (6) For further optimized design, considering the case of adding an adaptive component (equation) without knowing the upper bound of the fractional derivative in advance: , where is a positive constant, is the adaptive gain, and the adaptive rate of is adjusted by to make the adaptive gain automatically adjustable online as the error system changes.

[0032] (7) Based on the above steps, use software for simulation testing, and finely adjust the gain coefficients at various places in the optimized structure diagram to improve the estimation effect. At the same time, the parameters can be further finely adjusted according to the actual signal and the external noise (or interference) signal, so that any bounded continuous signal can well obtain its fractional derivative signal after passing through this device, and it is applied to the design of the fractional-order controller for two inverted pendulums.

[0033] The present invention has the following advantages compared with the prior art: The fractional derivative estimation method based on the sliding mode technology avoids the weak singularity in the definition of fractional calculus, reduces the difficulty of theoretical analysis of fractional calculus by using the continuous frequency distribution state weight model, and solves the problem of estimating (or solving) the fractional derivative of a signal function by proposing a fractional derivative estimation method of a signal function under the framework of control theory. In particular, it can still work properly and maintain good robustness when the signal is contaminated. That is, in practical applications, through the fractional derivative estimation method of the present application, the control effect of the controller on the target control object can be better. The method and results proposed by the present invention can be widely applied to many fields such as the identification, control, and signal processing of signal systems, expanding the scope of fractional calculus in the field of engineering technology and making the system more worthy of popularization and use. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 is the working principle diagram of the method of the present invention.

[0035] Figure 2 is the closed-loop control structure diagram used in the present invention.

[0036] Figure 3 is the working flow chart of the present invention.

[0037] Figure 4 is the evolution diagram of the estimated value and the exact value of the R-L fractional derivative when the bounded continuous signal is not contaminated by noise in the embodiment of the present invention.

[0038] Figure 5 is in the embodiment of the present invention, when the bounded continuous signal is not contaminated by noise and the parameters and are modified, the evolution diagram of the estimated value and the exact value of the R-L fractional derivative.

[0039] Figure 6 is the evolution diagram of the estimated value and the exact value of the R-L fractional derivative, and the error between the estimated value and the exact value when the bounded continuous signal is contaminated by noise in the embodiment of the present invention.

[0040] Figure 7 is in the embodiment of the present invention, when the bounded continuous signal is contaminated by noise and the parameters , , and are modified, the evolution diagram of the estimated value and the exact value of the R-L fractional derivative, and the error between the estimated value and the exact value.

[0041] Figure 8 is in the embodiment of the present invention, for the bounded continuous signal function when the parameter is modified to the adaptive equation The estimated and exact values of the R-L fractional derivative, and the evolution diagram of the adaptive parameter .

[0042] Figure 9 is the bounded continuous signal function in the embodiment of the present invention after being contaminated by Gaussian white noise and the adaptive equation is The estimated and exact values of the R-L fractional derivative, and the evolution diagram of the adaptive parameter .

[0043] Figure 10 is the mathematical model of two inverted pendulums on a cart in the embodiment of the present invention

[0044] Figure 11 is the controller design structure diagram of the inverted double pendulum on a cart in the embodiment of the present invention

[0045] Figure 12 is the angle of the pendulum under the action of the controller . Detailed implementation method

[0046] The following details the embodiments of the present invention. These embodiments are implemented on the premise of the technical solution of the present invention, and detailed implementation methods and specific operation processes are given. However, the protection scope of the present invention is not limited to the following embodiments

[0047] This embodiment provides a technical solution: a fractional derivative estimation method based on sliding mode technology, as Figure 1 and Figure 2 shown, and Figure 2 is Figure 1 the concretization process of, where and respectively represent the input signal and the error signal to be estimated, is the output signal, is the R-L fractional integrator, the controller is , and the filter is . If the signal is regarded as the input and the output is regarded as the R-L fractional derivative signal to be estimated, then the fractional error system equation is:

[0048] or

[0049] where , and respectively represent the R-L fractional integral and derivative, 、 and respectively represent the input of the signal to be estimated, the error signal, and the controller to be designed. It should be noted that the above relational expressions have established because is equivalent to the initial moment when the controller has not yet taken effect. Then at this time and should be equal.

[0050] The current goal is to design a fractional derivative estimation strategy and construct a controller such that the error system reaches the sliding mode surface within a finite time , and then the error signal is controlled within the predetermined index range . Therefore, the designed controller is:

[0051]

[0052] where is the nonlinear gain coefficient, is the relay gain coefficient, and is the switching coefficient of the controller; satisfies: (i) , , ( can be taken as ); (ii) , ; satisfies: (i) , ; (ii) , ; (iii) , . When the error system reaches the sliding mode surface, at this moment , then

[0053]

[0054] By constructing the Lyapunov function , the finite time to reach the sliding mode surface can be calculated as:

[0055]

[0056] where . It can be seen from the above formula that

[0057] ,

[0058] then Therefore, when designing the controller, the error system is placed on the sliding surface, that is, there is no switching control at this time. The error system starts from the sliding surface and passes through the controller to control the error signal to the predetermined index range .

[0059] Next, the performance of the controller will be analyzed, especially the reason why this estimation method can still work properly when the signal contains random noise and uncertain disturbances. Specifically as follows:

[0060] If the input signal is contaminated by noise (here assumed to be Gaussian white noise), since Gaussian white noise is a stationary random process with a mean of 0 and a non-zero spectral density function or an idealized random process composed of a series of uncorrelated random variables, thus in the present invention, perhaps the input signal can be written as the sum of the useful signal and the noise signal , that is .

[0061] Assume that there exists a certain such that the spectral density satisfies the relationship and , where and are the spectral densities of the useful signal and the noise signal respectively. This is a common assumption used to distinguish the useful signal and the noise signal at .

[0062] Let be the variance of the error signal , be the expectation of the error signal ; then where and are the expectations of the useful signal part and the noise signal part respectively. Similar to the previous case, perhaps the error signal can also be divided into the useful signal part and the noise signal part, that is , then from it can be obtained that

[0063]

[0064] In the formula is the transfer function input to the error (i.e., the AB segment in Figure 1 ). The following will give this transfer function. First, the double-dashed line part in Figure 2 (i.e.,Figure 1 in ), perform statistical linearization. Since the signal consists of a useful signal and a noise signal, where the noise signal is a Gaussian random signal with a mean of zero, then the probability density function of

[0065] .

[0066] Now assume is 's linear approximation. According to the minimum mean square error criterion, then its minimum mean square error is

[0067]

[0068] To make minimum, according to and we get:

[0069] .

[0070] Subsequently, we have

[0071]

[0072] In the formula, ( and is odd).

[0073] Therefore, according to the working principle Figure 1 (or the control structure Figure 2 ), the transfer functions of the AB segment and the AC segment can be obtained as

[0074] and

[0075] In the formula . Then we can get and , and there is

[0076]

[0077] Let Substitute it into We get

[0078] ,

[0079] Then its logarithmic amplitude-frequency characteristic is:

[0080]

[0081] Therefore, select appropriate parameters such that when there is and when there is . From the assumptions in the previous text, we have

[0082]

[0083] It can be seen from the above approximate analysis that , indicating that the parameter is related to , that is, related to . If the parameters , , , are properly selected such that the overall is very small, then in a low-frequency noise environment or disturbance situation, the closed-loop transfer function is equivalent to a fractional-order differentiator; on the contrary, in a high-frequency noise environment or disturbance situation, the closed-loop transfer function , that is, the signal contaminated by noise does not undergo fractional-order differentiation operation when passing through here. Therefore, the selection of the parameter is very crucial. Considering that is the switching gain, the larger is, the more severe the chattering is, that is cannot be taken too large, and only can be taken larger, so as to ensure that the designed fractional-order derivative estimation device can still work properly when the signal is contaminated by random noise (or uncertain disturbance).

[0084] Next, analyze whether the error signal can reach the predetermined index range under the action of the controller . First, transform the fractional-order error system equation into a continuous frequency distribution state weight model

[0085]

[0086] where , , . Let and . If the Lyapunov function (or energy function)

[0087] is taken, when and , then there is

[0088] .

[0089] If ( and is odd), then and further

[0090]

[0091] where is the time when the error signal enters from into When the error signal arrives it will not leave, so the designed controller can well estimate the fractional derivative of the signal However, the output signal may still contain "harmful" noise at this time. Therefore, a low-pass filter module is set in the CD section and transformed into an equation form: or

[0092] or

[0093] Its solution is:

[0094]

[0095] The error expression is:

[0096]

[0097] where , is the minimum negative real part eigenvalue of and

[0098] When is a first-order low-pass filter, its solution is:

[0099] .

[0100] The error expression is: , where is a higher-order infinitesimal.

[0101] When is a second-order low-pass filter, its solution is:

[0102] .

[0103] The error expression is: , where is a higher-order infinitesimal. Therefore, based on the above error expression, the selection of parameters and should be as small as possible, but not too small to prevent distortion of the linear filter and have an adverse effect on the optimized estimation performance.

[0104] From the above analysis, it can be seen that the implementation of the present invention requires prior knowledge of the parameter , and The upper bound of is generally not easy to know. Therefore, the parameter is changed to an adaptive equation: , where is the adaptive gain, and the adaptive rate of is adjusted by . At this moment, the parameter will be automatically adjusted online as the error system changes, and the Lyapunov function (or energy function) is taken as:

[0105] ,

[0106] where the unknown parameter is The upper bound of. When , there is

[0107] .

[0108] If ( and is odd), then , and then there is

[0109] .

[0110] When the error signal arrives will not leave, so the designed controller can still estimate the fractional derivative of the signal well . In addition, the adaptive equation can also be modified to: , (where is a positive constant) to prevent the Windup effect and improve the adaptive online adjustment ability of the gain coefficient as the error system changes.

[0111] The following verifies the method of the present invention through an embodiment: If the fractional order is taken, and then the bounded continuous signal functions and are taken respectively. At this time, the R-L fractional derivative of can be accurately given as , while The R-L fractional derivative was similarly given by Professors Oldham and Spanier in their monograph (The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Dover Publications, 2006). When the initial time is

[0112] and ,

[0113] and as time progresses, there are and . Additionally, from and , it can be deduced that

[0114] .

[0115] Therefore, combining the process Figure 3 and setting the initial values of the signal error to and during MATLAB simulation; when the input signal is not contaminated by noise, taking the parameters , , , , , , and , the simulation results are as shown in Figure 4 . It can be seen from Figure 4 that the estimated values of the R-L fractional derivatives of the bounded continuous signal functions and at the fractional order can track the exact values, but there is a chattering phenomenon.

[0116] Now set the parameters , , and the other parameters are the same as Figure 4 ; then when the input signal is not contaminated by noise, the estimated values of the R-L fractional derivatives of the bounded continuous signal functions and at the fractional order can still track the exact values and the chattering disappears, as shown in Figure 5 .

[0117] When the input signal is contaminated by noise, for example, the bounded continuous signal function After being contaminated by Gaussian white noise with a standard deviation of , taking parameters , , , , , , and the initial value , then at the fractional order , the estimated value of the R-L fractional derivative tracks the exact value, and the error between the estimated value and the exact value is shown in Figure 6 . Obviously, it can be seen from the figure that there are obvious high-frequency vibrations, which reduces the estimation performance. If the parameters , , and are changed and other parameters remain unchanged, it can be found that the high-frequency vibrations disappear and the estimation performance is improved, as shown in Figure 7 .

[0118] Now, considering the case where the upper bound of the fractional derivative is unknown in advance, an adaptive component (equation) is added to the estimator: , where is a positive constant, is the adaptive gain, and the adaptive rate of is adjusted by to enable the gain coefficient to be automatically adjusted online as the error system changes.

[0119] When taking parameters , , , , , , , the initial value and . The estimated value of the R-L fractional derivative of the bounded continuous signal function at the fractional order tracks the exact value, and the evolution of the adaptive parameter over time is shown in Figure 8 .

[0120] When taking parameters , , , , , , , the initial value and . The bounded continuous signal function after being contaminated by Gaussian white noise with a standard deviation of at the fractional order The estimated value of the R-L fractional derivative tracks the exact value at this time, as well as the adaptive parameters and their evolution over time, as Figure 9 shown.

[0121] From the above Figures 4 - 9 it can be seen that for different continuously bounded signals to be estimated, the estimation method provided by the present invention can simply and effectively give their fractional derivative signals, and the parameters can be finely adjusted according to the actual needs of the error precision and the external noise (or interference) signals, thereby optimizing the gain coefficients at various places in the structure diagram and improving the estimation effect. Now, it is further applied to the design of the fractional-order sliding mode controller for two inverted pendulums, where the two inverted pendulums are connected by a moving spring installed on two carts, as Figure 10 shown.

[0122] The dynamic equations of the double inverted pendulum on the cart can be described as:

[0123] ,

[0124] where , , , , .

[0125] Denote , , , , , , , then the above equation can be simplified to:

[0126]

[0127] where , , , , . Now design the controller to make the angle of each pendulum with mass converge to zero in a finite time. First, define the variable , and set , where (i) , ( ); (ii) , ( ), then , is a positive constant. If the controller is taken, and (i) , ( ); (ii) , ( ), where , ; , while , 、 and are positive constants. From the properties of fractional calculus, we can obtain:

[0128] .

[0129] Now design the nominal signal to satisfy: , then

[0130]

[0131] When and are measurable and differentiable, let be the output of the fractional derivative estimator. Thus, we can obtain , where is the error of the fractional derivative estimator, and its value is very small according to step (5) in the invention content. If we denote , then the implementation block diagram is as shown in Figure 11 . Furthermore, define: . If satisfies the inequality:

[0132] ,

[0133] where is an unknown positive constant, then the angle of each pendulum with mass converges to zero in finite time. The specific process is as follows: First, construct the candidate Lyapunov function: , where , and we get , that is, reaches zero in finite time . Then construct the candidate Lyapunov function: , then we can obtain , that is, reaches zero in finite time . That is to say, the angle of each pendulum with mass converges to zero in finite time, as shown in Figure 12 , where the parameters of the inverted double pendulum are taken as , , , , , , , ; Take the design parameters of the controlled object , , , , , , , , , , , , ; The initial value is taken as . From Figure 12 It can be seen that the estimator of the fractional-order derivative of the present invention can fully realize the control of two inverted pendulums. In addition, the present invention can also be used in many fields such as fractional-order signal system identification, control, and signal processing.

[0134] In addition, the terms "first" and "second" are only used for descriptive purposes and cannot be construed as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include at least one of the features. In the description of the present invention, "a plurality" means at least two, such as two, three, etc., unless otherwise specifically defined.

[0135] In the description of this specification, the description with reference to terms such as "one embodiment", "some embodiments", "example", "specific example", or "some examples" means that the specific features, structures, materials, or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described can be combined in a suitable manner in any one or more embodiments or examples. In addition, without contradiction, those skilled in the art can combine and combine the different embodiments or examples described in this specification and the features of different embodiments or examples.

[0136] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

Claims

1. A fractional derivative estimation method based on sliding mode technology, characterized in that: The following steps are involved: (1) Before entering the design of the fractional-order derivative estimator, the properties of fractional-order calculus are used to perform an equivalent transformation on the estimated object to obtain the fractional-order error system model, whose mathematical expression is: ， in represents the Riemann-Liouville fractional derivative, , and They represent the input signal to be estimated, the error signal and the controller to be designed respectively; (2) Design a fractional derivative estimation strategy for any bounded continuous signal or contaminated signal, and give the components of the estimation system structure diagram, including: the input signal to be estimated, the controller, the fractional-order integrator, and the filter; (3) Based on step (2), the nonlinear module and the relay module are first connected in parallel and then in series with the fractional-order filter module, and a negative feedback signal is led out before the fractional-order filter module to the input function to form a closed loop, thereby obtaining an estimation system structure diagram; (4) Based on step (2), the parameters of the controller module are determined by statistical linearization method. and ; (5) Based on step (2), determine the filter coefficients using stability theory and ; (6) To further optimize the design, an adaptive component is added when the upper bound of the fractional derivative is unknown in advance: ,in is a normal number, is the adaptive gain, The adaptive rate is given by Adjustment, so that the gain coefficient can be automatically adjusted online as the error system changes; (7) Computer software is used to carry out simulation tests, and the gain coefficients of each part in the fractional-order derivative estimator are fine-tuned and optimized. The parameters are further fine-tuned according to the actual signal and the external noise signal, so that any bounded continuous signal can obtain its fractional-order derivative signal after passing through this device, which is applied to the design of fractional-order controllers for two inverted pendulums.

2. The fractional-order derivative estimation method based on sliding mode technology according to claim 1, characterized in that: The components of step (2): controller and filter, are designed as follows: Controller Part I: Constructing Nonlinear Modules ,in is the nonlinear gain coefficient, Satisfies: (i) , (ii) ; Controller Part II: Constructing the Relay Module , where the parameters is the relay gain coefficient, Satisfies: (i) (ii) (iii) ; Select the appropriate matrix , and , build the filter module: ;when , and When It is a first-order low-pass filter module; when , and When It is a second-order low-pass filter module ,in and is the filter coefficient. By analogy, a third-order low-pass filter can also be given. In addition, when the filter coefficients are all zero, this module is a constant 1.

3. The fractional-order derivative estimation method based on sliding mode technology according to claim 2, characterized in that: The step (4) is combined with the step (2) and , and combined with the estimated system structure diagram, the parameters of the controller module are determined by statistical linearization method and .

4. The fractional-order derivative estimation method based on sliding mode technology according to claim 2, characterized in that: The parameters of the controller module are determined and The specific principle is as follows: First, the closed-loop transfer function is obtained according to the structure diagram ,in is the equivalent transfer function coefficient; then according to the statistical linearization method, the equivalent transfer function coefficient expression is given as: ， is the error signal The standard deviation of , and is an odd number, The standard deviation of the error signal ,parameter , , , The choice is related, and there is and ; Choose appropriate parameters , , , Make the whole Small, then in a low-frequency noise environment or disturbance, the closed-loop transfer function is equivalent to a fractional-order differentiator; on the contrary, in a high-frequency noise environment or disturbance, the closed-loop transfer function , that is, the signal contaminated by noise does not undergo fractional differential operation when passing through this point. and The selection of must ensure that the designed fractional-order derivative estimation device can still work normally when the signal is contaminated by random noise or uncertain disturbances.

5. The fractional-order derivative estimation method based on sliding mode technology according to claim 2, characterized in that: The step (5) is combined with the step (2) , combined with the estimated system structure diagram, the filter coefficients are determined by stability theory. The filter coefficients include and .

6. The fractional-order derivative estimation method based on sliding mode technology according to claim 1, characterized in that: The filter coefficients are determined and The specific principle is as follows: According to the obtained fractional derivative estimate of any bounded continuous signal, it may contain "harmful" noise, and a low-pass filter module needs to be set to filter out the "harmful" signal. In addition, there may be errors between the estimated value and the exact value. The error expression is: in , yes The smallest negative real eigenvalue of It is a higher-order infinitesimal; when When it is a first-order low-pass filter, the solution is: ; The error expression is: ,in It is a higher-order infinitesimal; when When it is a second-order low-pass filter, the solution is: ; The error expression is: ,in It is a higher-order infinitesimal; By converting the fractional-order error system into a continuous frequency distribution state weight model, and then constructing the energy function, the error signal is given by using the Lyapunov stability theory. In limited time Whether the predetermined target range can be achieved within the , limited time Satisfies the relationship: According to the above formula, the parameters can be determined and .

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