A method for parameter identification of fractional time-delay systems based on Gegenbauer wavelet operation matrices

Abstract

A parameter identification method for fractional-order time-delay systems based on the Gegenbauer wavelet operation matrix is ​​presented, addressing the problem that existing parameter identification methods cannot identify fractional-order systems with time delays. This method constructs a Gegenbauer wavelet operation matrix for both fractional-order integral operators and time-delay operators. Then, the original fractional-order time-delay differential equation is transformed into a system of algebraic equations, thus avoiding direct numerical approximation of fractional-order differential terms. Furthermore, a nonlinear least squares method is used to jointly estimate the fractional-order order, time-delay parameters, and system parameters. Finally, the optimal fractional-order order and model parameters are determined based on minimizing the error between the model output and the actual output. This method effectively reduces solution complexity and improves the numerical stability and accuracy of parameter identification. It provides an effective tool for high-precision modeling and analysis of fractional-order time-delay systems.

Description

Technical Field

[0001] This invention relates to the field of parameter identification technology for fractional time-delay systems, specifically to a method for parameter identification of fractional time-delay systems based on the Gegenbauer wavelet operation matrix, applicable to parameter identification of fractional time-delay systems. Background Technology

[0002] As the complexity of engineering objects continues to increase, memory effects, time delay characteristics, and complex nonlocal dynamic behaviors in systems are becoming increasingly prominent, making traditional integer-order models insufficient for high-precision modeling requirements. Fractional-order systems, as an important extension of classical integer-order systems, can more accurately characterize the dynamic characteristics of complex engineering objects. Therefore, they have been widely applied in fields such as control engineering, signal processing, biomedicine, viscoelasticity, and electromagnetism. However, the accurate characterization of the dynamic characteristics of complex engineering objects by fractional-order systems largely depends on the high-precision identification of model parameters. Because these system models are typically structurally complex, the number of parameters to be identified is not only large, but there is also strong coupling between fractional-order parameters, time delay parameters, and system parameters. Changes in order are particularly sensitive to the dynamic characteristics of the system, making the parameter identification process more complex and prone to decreasing identification accuracy. The accumulation of parameter identification errors further weakens the model's ability to accurately characterize the dynamic behavior of the controlled object. Therefore, research on parameter identification methods for fractional-order systems to improve parameter identification accuracy and model representation capabilities has significant theoretical and engineering value.

[0003] Currently, scholars both domestically and internationally have proposed various parameter identification methods from different research perspectives to address the parameter identification problem of fractional-order time-delay systems. These related studies can be mainly categorized into two types: one is parameter identification methods based on optimization algorithms, and the other is algebraic identification methods based on orthogonal functions.

[0004] One type of research focuses on transforming the parameter identification problem of fractional-order time-delay systems into a nonlinear optimization problem, and then solving it using intelligent optimization algorithms. This type of method constructs an error objective function to jointly estimate the fractional-order differential order, system model parameters, and time-delay parameters. It has advantages such as flexible implementation and wide applicability, but it typically suffers from high computational cost, sensitivity to initial conditions, susceptibility to local optima, and sensitivity to noise. Especially in cases of complex system structures or a large number of parameters, its numerical stability and identification accuracy still need further improvement.

[0005] Another type of research takes a function approximation approach, using orthogonal functions to approximate fractional operators, transforming the original calculus equations into algebraic equations, thereby achieving parameter identification. By representing fractional differential or integral operators in the wavelet coefficient domain, the instability caused by direct numerical approximation of fractional differentials is avoided to some extent, improving the numerical stability and computational efficiency of parameter identification. However, such methods based on orthogonal polynomials or wavelet bases typically employ fixed-scale expansions, lacking flexible adjustment capabilities for basis function shapes and having limited adaptability to complex dynamic processes. Furthermore, the discrete or piecewise characteristics of some wavelet basis functions easily introduce approximation errors when approximating continuous fractional operators, and they are quite sensitive to initial conditions during algebraic solutions. Especially for fractional systems containing time-delay terms, nonlinear terms, and highly coupled parameters, existing methods still have certain limitations in terms of approximation accuracy, solution stability, and applicability. Unlike wavelets with fixed basis functions, Gegenbauer wavelets, by introducing adjustable parameters to form a family of functions, can flexibly adjust between smoothness and locality, providing a new approach for constructing high-precision parameter identification methods for fractional-order time-delay systems.

[0006] In summary, existing parameter identification methods still have shortcomings in characterizing the beam control characteristics of fractional-order time-delay systems, especially in cases with complex system structures or a large number of parameters. Their numerical stability and identification accuracy, as well as the identification accuracy in complex systems with highly coupled parameters and containing time-delay fractional-order systems, urgently need improvement. Summary of the Invention

[0007] This invention provides a parameter identification method for fractional time-delay systems based on the Gegenbauer wavelet operation matrix, in order to solve the problem that existing parameter identification methods cannot identify fractional time-delay systems.

[0008] A method for parameter identification of fractional time-delay systems based on the Gegenbauer wavelet operation matrix is ​​implemented by the following steps:

[0009] Step 1: Construct the Gegenbauer wavelet fractional integral operation matrix and the time delay integral operation matrix;

[0010] Step 2: Based on the Gegenbauer wavelet fractional integral operation matrix and time delay integral operation matrix constructed in Step 1, perform wavelet expansion and matrix transformation on the fractional time delay system to transform the fractional time delay system into the corresponding Gegenbauer wavelet operation matrix algebraic equation.

[0011] Step 3: Construct an identification model based on the algebraic equations obtained in Step 2, and use the error function between the output of the identification model and the output of the actual system as the optimization objective function; use the nonlinear least squares method to optimize and solve the problem, and select the fractional time delay system parameters and order corresponding to the minimum error value as the optimal solution; thus realizing the parameter identification of the fractional time delay system.

[0012] The beneficial effects of this invention are:

[0013] The parameter identification method described in this invention first utilizes fractional integral and time-delay integral operation matrices to transform fractional linear and nonlinear systems with time delays into algebraic equations, thereby constructing an identification model suitable for numerical computation. Finally, it combines the nonlinear least squares method as the objective function to complete the model parameter identification. Specifically, it possesses the following advantages:

[0014] 1. The parameter identification method described in this invention has better identification accuracy than parameter identification methods based on optimization algorithms and parameter identification methods based on function approximation. It can effectively identify parameters of fractional time-delay systems and maintains good robustness under different noise levels, providing an effective method for parameter identification of fractional time-delay systems.

[0015] 2. The parameter identification method described in this invention, combined with the Gegenbauer wavelet, introduces adjustable parameters to form a family of functions, which can flexibly adjust between smoothness and locality, thus achieving high-precision and robust parameter identification of fractional-order time-delay systems.

[0016] 3. The parameter identification method described in this invention provides theoretical support for the accurate identification of fractional-order time-delay model parameters of liquid crystal optical phased arrays. It can be used for rapid deflection of lidar and beam alignment of space optical communication, laying the foundation for the design of high-performance non-mechanical beam control systems. Attached Figure Description

[0017] Figure 1 This is a flowchart of a method for identifying parameters of a fractional time-delay system based on a Gegenbauer wavelet operation matrix, as described in this invention.

[0018] Figure 2 A schematic diagram of the X-channel input response curve of a liquid crystal optical phased array beam control system under X-channel input conditions;

[0019] Figure 3 A schematic diagram of the Y-channel input response curve of a liquid crystal optical phased array beam control system under Y-channel input conditions;

[0020] Figure 4 A schematic diagram of the Y-channel input response curve under the X-channel input of a liquid crystal optical phased array beam control system;

[0021] Figure 5This is a schematic diagram of the X-channel input response curve for a liquid crystal optical phased array beam control system with Y-channel input. Detailed Implementation

[0022] Specific Implementation Method 1: Combination Figure 1 This paper describes a method for parameter identification of fractional-order time-delay systems based on Gegenbauer wavelets. The method first constructs the Gegenbauer wavelet operation matrix of the fractional-order integral operator and the time-delay operator. Then, it transforms the fractional-order time-delay differential equation into an algebraic equation, thus avoiding direct numerical approximation of the fractional-order differential terms. Furthermore, it combines the nonlinear least squares method to jointly estimate the fractional-order order, time-delay parameters, and system parameters. Finally, the optimal fractional-order order and model parameters are determined based on minimizing the error between the model output and the actual output. This method effectively reduces the solution complexity and improves the numerical stability and accuracy of parameter identification. It provides an effective tool for high-precision modeling and analysis of fractional-order time-delay systems.

[0023] The parameter identification method described in this embodiment is implemented by the following steps:

[0024] Step 1. Construct the Gegenbauer wavelet fractional operation matrix and the time delay operation matrix to realize the matrix representation of the fractional differential operator and the time delay operator, providing a foundation for the algebraic solution of fractional time delay systems; the specific implementation process is as follows:

[0025] First, starting with fractional calculus, we analyze the definitions of fractional calculus by Grünwald–Letnikov, Riemann–Liouville, and Caputo, concluding that the Caputo definition is more consistent with the physical context and is the most commonly used form in scientific and engineering applications. The mathematical expression of the Caputo fractional differential definition is as follows:

[0026] (1)

[0027] in, express Caputo fractional differential operator of order 1. and These are the lower and upper bounds of the differential operator, respectively. For fractional order, To meet positive integers, Let be the function to be determined. for of Integer derivative of order 1 This represents the gamma function.

[0028] The mathematical expression for the definition of Caputo's fractional integral is:

[0029] (2)

[0030] in It is a fractional integral operator.

[0031] Based on the properties of convolution, the definition of Caputo fractional integral (2) can be simplified to:

[0032] (3)

[0033] In this embodiment, the Gegenbauer wavelet is a class of wavelet functions constructed by scaling and translating orthogonal Gegenbauer polynomials. With five parameters in the interval [0,1], the Gegenbauer wavelet is defined as follows:

[0034] (4)

[0035] in, This represents the order of the Gegenbauer polynomial at different scales. These are the Gegenbauer polynomial weight parameters. Let be any positive integer, representing the resolution level, or scale. Represents the normalization constant. Representing translation parameters at different scales, Represents the standardized time of the independent variable;

[0036] The representation is defined in the interval. The Gegenbauer polynomial within the range is represented as follows:

[0037] (5)

[0038] The normalization constant is represented as follows:

[0039] (6)

[0040] The Gegenbauer polynomial satisfies the following recurrence formula:

[0041] (7)

[0042] Arbitrary function to be solved Both can be expanded using Gegenbauer wavelets:

[0043] (8)

[0044] in, These are the Gegenbauer wavelet expansion coefficients. , To represent the inner product, we can write formula (8) in vector form. Define the Gegenbauer wavelet coefficient vector and wavelet basis function vector as follows:

[0045] (9)

[0046] (10)

[0047] To obtain the Gegenbauer wavelet function matrix, discretization and numerical calculation were performed, and uniform midpoint sampling nodes were set.

[0048] (11)

[0049] make Then the Gegenbauer wavelet function matrix can be expressed as:

[0050] (12)

[0051] To obtain the fractional-order integral operation matrix of the Gegenbauer wavelet, a block impulse function is introduced. The block impulse function is defined on [0,T] as follows:

[0052] (13)

[0053] in, Let this be the system termination time. Let the block pulse function vector be... Then its fractional integral can be expressed as:

[0054] (14)

[0055] In the formula, For block pulse basis vectors, The integral operation matrix of the block impulse function is represented by the following formula:

[0056] (15)

[0057] Among them, the weighting coefficient ;

[0058] An integrable function can be expanded in the form of a block impulse function over a given interval [0, T]. Then the Gegenbauer wavelet vector... Representing the form of a block impulse function:

[0059] (16)

[0060] Combining formulas (14) and (16) By performing fractional integral operations, we can obtain:

[0061] (17)

[0062] According to the definition of Caputo fractional integral (2), combined with formula (16) and the fractional integral property of block impulse functions (14), the fractional integral of the Gegenbauer wavelet basis function vector can be expressed as:

[0063] (18)

[0064] in, It is the fractional integral operation matrix of the Gegenbauer wavelet.

[0065] Combining formulas (17) and (18), we can obtain the fractional integral operation matrix of the Gegenbauer wavelet. The expression is:

[0066] (19)

[0067] In this embodiment, the time delay operation matrix is ​​used to transform the basis function vectors Translate along the time axis The expression for each sampling point is:

[0068] (20)

[0069] in, This is the time delay operation matrix for the block impulse function, typically a shift matrix. Time delay It can be represented as:

[0070] (twenty one)

[0071] in, This is the sampling step size.

[0072] (twenty two)

[0073] Therefore, the time delay operation matrix of the block impulse function It can be represented as:

[0074] (twenty three)

[0075] in, Indicates row index, Indicates column index, Indicates the number of time delay steps (number of sampling points). Indicates the number of data points. Describes the... The first line Set the column to 1 and the rest to 0.

[0076] In this embodiment, in order to better understand the block pulse delay matrix The following example is provided. If , ,at this time ,and Then you can get Right now Represented as:

[0077] (twenty four)

[0078] To obtain the Gegenbauer wavelet time delay matrix, one can... It can be represented in the following form:

[0079] (25)

[0080] in, This is the time delay operation matrix for the Gegenbauer wavelet.

[0081] According to equations (16) and (20), It can be rewritten as:

[0082] (26)

[0083] Combining (25) and (26), we can obtain:

[0084] (27)

[0085] Therefore, the Gegenbauer wavelet time delay operation matrix It can be represented as:

[0086] (28)

[0087] Based on the definition of Caputo fractional calculus and formulas (18) and (25), the Gegenbauer wavelet time-delay integral operation matrix is ​​obtained:

[0088] (29)

[0089] Through the above derivation, the fractional-order time-delay integral operation matrix of the Gegenbauer wavelet is established. Under the Gegenbauer wavelet basis, any function... It can be represented in fractional integral matrix form:

[0090] (30)

[0091] Step 2: Based on the constructed Gegenbauer wavelet fractional operation matrix and time delay operation matrix, perform wavelet expansion and matrix transformation on the fractional time delay system to transform it into the corresponding Gegenbauer wavelet operation matrix algebraic equation.

[0092] In this embodiment, the Gegenbauer wavelet fractional integral operation matrix discretizes fractional calculus and differential operators into standard matrix form, transforming the original fractional time-delay system into directly solvable algebraic equations, thereby significantly improving the accuracy, numerical stability, and efficiency of parameter identification. This embodiment presents parameter identification strategies for both linear and nonlinear fractional time-delay systems.

[0093] Fractional-order linear time-delay systems are an extension of traditional integer-order linear time-invariant systems in two dimensions. First, the dynamics of the system are described by several fractional-order calculus operators of different orders, which can more precisely characterize the complex dynamic characteristics of the system. Second, there is often a time delay between the system input and output, which is used to reflect the non-negligible lag processes in actuators, transmission links and feedback control.

[0094] Step A1: The linear time-delay differential equation of the fractional-order time-delay system can be expressed as:

[0095] (31)

[0096] Transforming it into a transfer function, it can be expressed as:

[0097] (32)

[0098] in, and Let them represent the fractional derivative operators of the system, are any positive real numbers, representing the order of the fractional order system; This represents the system's parameters to be identified; Indicates the system's input and output. Indicates a time delay.

[0099] Step A2: Determine the order of the fractional order. Parameters to be identified and time delay As parameters to be identified in a fractional-order time-delay system, this embodiment uses the time-delay operation matrix of fractional-order calculus to achieve the purpose of identifying these parameters. To facilitate representation using fractional integrals and avoid singular values ​​during the identification process, both the numerator and denominator are divided by the highest order of the numerator. This yields the following form:

[0100] (33)

[0101] Then the fractional integral equation can be expressed as:

[0102] (34)

[0103] In the formula, and The parameters to be identified by the system; and The fractional order of the system; For time delay parameters; using Indicates the superscripts of I, express Fractional integral operator.

[0104] In step A3, in the fractional linear time-delay differential equation (31), the system input... and output Expanding using the Gegenbauer wavelet function yields:

[0105] (35)

[0106] (36)

[0107] Step A4: Since the system's input and output are in the form of integral functions, according to the definition of Caputo's fractional integral, we obtain... and The fractional integral expressions are as follows:

[0108] (37)

[0109] (38)

[0110] in, and These are known vectors, representing the input signals respectively. and output signal The expansion coefficients under the Gegenbauer wavelet basis. By combining formulas (18), (29), (37) and (38), the basis function expansion of the fractional integral equation of formula (34) is obtained, i.e., the algebraic equation;

[0111] (39)

[0112] As can be seen from formula (39), complex fractional differential equations are transformed into algebraic equations through the Gegenbauer wavelet integral operation matrix and the time delay operation matrix.

[0113] Step 3: In order to obtain the optimal estimates of the parameters, an identification model is constructed based on the algebraic equations obtained in Step 2. The error function between the output of the identification model and the output of the actual system is calculated, and this error function is used as the optimization objective function. The fractional order, time delay, and system parameters are jointly estimated by combining the nonlinear least squares method. Finally, the optimal fractional order and model parameters are determined based on the criterion of minimizing the error function between the output of the identification model and the output of the actual system.

[0114] In this embodiment, all terms of the fractional-order linear time-delay system (algebraic equation) are reduced to matrix form and expressed by algebraic matrix relations:

[0115] (40)

[0116] In the formula, all those related to Related terms are merged into a matrix :

[0117] (41)

[0118] All of them with Related terms are merged into a matrix :

[0119] (42)

[0120] By transforming formula (40), the Gegenbauer wavelet coefficient vector is obtained and output. It can be represented as:

[0121] (43)

[0122] Substituting equation (43) into the Gegenbauer wavelet expansion (36) of the system output signal, we can obtain the output signal of the fractional-order linear time-delay system. The final matrix expression of the matrix is ​​the output of the actual system:

[0123] (44)

[0124] Equation (44) gives the system output. The operation matrix form is used to represent it. It transforms the complex fractional differential equation, which originally contained fractional derivatives and integral operators, into an algebraic expression containing only matrix multiplication. Therefore, equation (44) corresponds to a more concise system expression method than differential equations.

[0125] In this embodiment, it should be noted that the matrix and The system model contains the parameters to be identified. Fractional order and time delay Therefore, equation (44) not only describes the operator relationship between the system input and output, but also provides the possibility for system parameter estimation, enabling the simultaneous estimation of these unknown parameters through the observation of the input and output signals.

[0126] make , , , The coefficients of the system are respectively , and the order of the system , The estimated value, and the system output obtained according to formula (44) can be expressed as:

[0127] (45)

[0128] in, , It is obtained from estimation , , , The constructed operation matrix, The time delay operation matrix is ​​constructed from the estimated time delay. These three matrices contain the parameters of the fractional linear time delay system to be identified. Formula (45) gives the operation matrix expression of the system output signal after the parameter estimation is completed, that is: the output of this formula is the output of the identification model.

[0129] In this embodiment, the nonlinear least squares error between the actual system output and the identification model output is selected as the criterion function, and the objective function is minimized. The parameter identification estimates are obtained.

[0130] (46)

[0131] in, , The search range for system parameters. To identify the number of data points for the system.

[0132] The fractional-order time-delay system coefficients, time-delay parameters, and system order output by the identification system described in this embodiment are applied to the dynamic modeling and parameter characterization of the beam control system of the liquid crystal spatial light modulator.

[0133] In step 2 of the parameter identification method for fractional-order time-delay systems described in this embodiment, in practical applications, besides fractional-order linear time-delay systems, many systems also contain nonlinear structures, thus forming fractional-order nonlinear time-delay systems. These systems further superimpose nonlinear terms on top of fractional-order memory and time-delay effects, causing the relationship between input and state to no longer satisfy the linear superposition principle. Therefore, their parameter identification is more complex than that of linear models. The construction process of the nonlinear time-delay system is as follows:

[0134] Step B1: To describe this type of system, the input-output relationship is represented by the differential equations of a fractional-order nonlinear time-delay system. Its general form can be written as:

[0135] (47)

[0136] in, and These represent the system's input and output, respectively. Represents the nonlinear part of the system. and Let represent the orders of the nonlinear terms corresponding to the system input and output, respectively. Using Gegenbauer wavelet expansion, the system input delay term can be expressed as... Similarly, the system output can be represented as This allows the input and output of the original nonlinear fractional time-delay system to be represented in a form similar to that of a fractional linear time-delay system. Furthermore, by utilizing the fractional time-delay integral operation matrix of the Gegenbauer wavelet, the nonlinear fractional time-delay differential equation of the system can be written in the following matrix form, which facilitates subsequent parameter identification.

[0137] (48)

[0138] Step B2: According to formula (48), the output of the nonlinear terms can be obtained as follows:

[0139] (49)

[0140] Substituting formula (16) into formula (48) yields:

[0141] (50)

[0142] According to formula (13), we can obtain:

[0143] (51)

[0144] By substituting formula (51) into (50) and simplifying, we can obtain:

[0145] (52)

[0146] Step B3: Based on formulas (16) and (18), formula (52) can be redefined as:

[0147] (53)

[0148] Similarly, the nonlinear term in the output part can be expressed as:

[0149] (54)

[0150] Step B4: According to formula (54), substitute the system equation into the Gegenbauer wavelet expansion (48) to transform all fractional differential terms in the system into Gegenbauer wavelet integral operation matrices. Thus, all fractional terms in the system input and output can be written in the following form: Nonlinear algebraic equation:

[0151] (55)

[0152] In this embodiment, all terms of the fractional-order nonlinear time-delay system are reduced to matrix form, thereby obtaining the overall matrix expression of the system:

[0153] (56)

[0154] By performing a matrix transformation on formula (56), we can obtain the Gegenbauer wavelet coefficient vector. The expression is:

[0155] (57)

[0156] Substituting equation (57) into the Gegenbauer wavelet expansion (36) of the system output signal, we can obtain the system output signal. The final matrix expression, i.e., the identification model, is obtained as follows:

[0157] (58)

[0158] This model fully describes the output behavior of a fractional-order nonlinear time-delay system.

[0159] Specific Implementation Method Two: Combination Figures 2 to 5This embodiment describes a verification example of the fractional-order time-delay system parameter identification method based on the Gegenbauer wavelet operation matrix described in Specific Embodiment 1. To verify the feasibility of the fractional-order time-delay system parameter identification method described in Specific Embodiment 1 in practical engineering, this embodiment studies a liquid crystal optical phased array beam control system. A beam control dynamics model is constructed, a beam control modeling data experimental platform is built, and model parameter identification is performed based on this platform.

[0160] Liquid crystal spatial light modulators can effectively control the orientation state of liquid crystal molecules within different pixel array elements by applying voltages of varying amplitudes, thereby altering the effective refractive index distribution of the liquid crystal layer. This change in refractive index distribution further induces phase difference modulation between adjacent array elements, ultimately achieving beam deflection. The beam manipulation behavior of the liquid crystal spatial light modulator can be equivalently represented as a dual-input dual-output dynamic system, the model of which is as follows:

[0161] (59)

[0162] in, , These represent the beam deflection angles output by the liquid crystal spatial light modulator beam control system along the X and Y axes, respectively. , The phase difference is input along the X and Y axes to the beam control system of the liquid crystal spatial light modulator, respectively. , Let X and Y represent the fractional transfer functions in the X and Y directions, respectively. , These represent the two-axis cross-coupled transfer functions.

[0163] The main channel transfer function can be expressed as:

[0164] (60)

[0165] The transfer functions of the coupled channels can be expressed as follows:

[0166] (61)

[0167] (62)

[0168] in, This represents the coupling coefficient between the X-axis phase difference and the Y-axis deflection angle. This represents the coupling coefficient between the Y-axis phase difference and the X-axis deflection angle. and These represent the viscosity coefficients for unfolding and bending, respectively. Indicates the intermolecular coupling coefficient. These represent the viscous memory strength of the unfolded state, the viscous memory strength of the bent state, and the historical dependence of the molecular coupling process, respectively. The time delay is represented by the Laplace domain transfer function. The above model uses a Laplace domain transfer function to describe the beam modulation process of the liquid crystal spatial light modulator, facilitating parameter identification and system analysis. Its corresponding fractional-order time-domain dynamic expression can be obtained through the inverse Laplace transform.

[0169] A parameter estimation method based on the Gegenbauer wavelet fractional-order time-delay system is used to estimate the parameters of the beam modulation dynamics model of a liquid crystal spatial light modulator. The system output response and identification model are shown below.

[0170] First, the dynamic responses of the system's X-axis and Y-axis outputs are identified, and its fractional-order time-delay model is obtained, as shown in equation (63). The system step response obtained based on the identification results is compared with the actual system output response as follows: Figure 2 and Figure 3 As shown in the figure, the output of the identification model has a high degree of consistency with the response curve of the real system. It can track the dynamic changes of the real system well in the rising, transition and steady-state phases of the system, indicating that the established fractional time delay model can effectively describe the dynamic characteristics of the liquid crystal beam control system.

[0171] (63)

[0172] Furthermore, the coupled dynamic characteristics of the system from X-axis input to Y-axis output are modeled, and its fractional-order time-delay transfer function is shown in equation (64). The corresponding system output response is compared with the identification model as follows: Figure 4 As shown in the figure. The results show that the identification model can fit the dynamic coupling relationship of the system in different directions well, verifying the effectiveness of the proposed method in modeling dual-input dual-output coupled systems.

[0173] (64)

[0174] Furthermore, the dynamic response of the system from Y-axis input to X-axis output is identified, and its model expression is shown in equation (64). The corresponding system output response is compared with the identified model as follows: Figure 5 As shown.

[0175] (65)

[0176] comprehensive Figures 2 to 5It can be seen that the output of the identification model and the output of the real system (real model) maintain good consistency in both overall trend and dynamic change process, indicating that the parameter identification method of the fractional time delay system based on the Gegenbauer wavelet operation matrix of the present invention can accurately characterize the dynamic characteristics of the liquid crystal spatial light modulator beam control system and has high parameter identification accuracy.

[0177] 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.

[0178] 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. A method for parameter identification of fractional order time-delay systems based on Gegenbauer wavelet operation matrix, characterized in that: This method is implemented by the following steps: Step 1: Construct the Gegenbauer wavelet fractional integral operation matrix and the time delay integral operation matrix; Step 2: Based on the Gegenbauer wavelet fractional integral operation matrix and time delay integral operation matrix constructed in Step 1, perform wavelet expansion and matrix transformation on the fractional time delay system to transform the fractional time delay system into the corresponding Gegenbauer wavelet operation matrix algebraic equation. Step 3: Construct an identification model based on the algebraic equations obtained in Step 2, and use the error function between the output of the identification model and the output of the actual system as the optimization objective function; use the nonlinear least squares method to optimize and solve the problem, and select the fractional time-delay system parameters and order corresponding to the minimum value of the error function as the optimal solution; Implement parameter identification for fractional time-delay systems.

2. The recognition method of claim 1, wherein: The specific process of step two is as follows: Step A1: Construct the linear time-delay differential equation for the fractional-order time-delay system, expressed as follows: ； wherein, and are fractional derivative operators of the system, respectively, are the orders of the fractional derivative of the system, respectively; are the parameters to be identified of the system, respectively; are the input and output of the system, respectively, is the time delay; The order of the fractional order Parameters to be identified and time delay As parameters to be identified in a fractional-order linear time-delay system; Step A2, Input to the system and output Represented using the Gegenbauer wavelet function expansion: ； ； In the formula, This is the Gegenbauer wavelet time delay operation matrix. ; The Gegenbauer wavelet function matrix is... The time delay operation matrix for the block pulse function; The Gegenbauer wavelet basis function vector; and Both are known vectors, representing the input signals respectively. and output signal Expansion coefficients under the Gegenbauer wavelet basis; Step A3: Obtain the system input according to the definition of Caputo fractional integral. and output The fractional integral expression is: ； ； In the formula, This is the fractional integral operation matrix for the Gegenbauer wavelet. , This is the matrix for the integration operation of the block pulse function; Step A4: By combining the fractional integral operation matrix of the Gegenbauer wavelet and the time-delay integral operation matrix of the Gegenbauer wavelet, and The fractional integral expression expanded using the basis functions of the fractional integral equation is as follows: ； In the formula, and The parameters to be identified by the system; and The fractional order of the system; For time delay; utilize The superscripts of I are represented. express Fractional integral operator; .

3. The identification method according to claim 2, characterized in that: In step three, all terms of the algebraic equation are reduced to matrix form and expressed using algebraic matrix relations: ； In the formula, will be with Related terms are merged into a matrix ; as shown in the following formula: ； will with Related terms are merged into a matrix ; as shown in the following formula: ； By transforming the algebraic matrix of the fractional-order time-delay system, the Gegenbauer wavelet coefficient vector is obtained and output. Represented as: ； The wavelet coefficient vector Substituting the Gegenbauer wavelet expansion of the system output signal, we obtain the output of the fractional-order linear time-delay system. The final matrix expression of the matrix is ​​the output of the actual system: ； In the formula, the matrix and Including the system's parameters to be identified Fractional order and time delay .

4. The number identification method according to claim 3, characterized in that: Construct a recognition model; , , , These are respectively used as the parameters to be identified in the system. , and the order of the system , The estimated value, based on the system output obtained from the actual system identification, is expressed as: ； In the formula, , To estimate the obtained , , , The constructed operation matrix, The time delay operation matrix constructed from the estimated time delay; , and The three matrices each contain the parameters of the fractional-order time-delay system to be identified.

5. The identification method according to claim 4, characterized in that: The nonlinear least squares error between the actual system output and the identification model output is selected as the criterion function, and the objective function is minimized. The parameter identification estimate is obtained and expressed by the following formula: ； In the formula, , For time delay The estimated value, The search range for system parameters. The number of data points identified by the system.

6. The identification method according to any one of claims 1-5, characterized in that: In step two, parameter identification is achieved using a fractional-order nonlinear time-delay system based on Gegenbauer wavelets.

7. The identification method according to claim 6, characterized in that: The process of converting a fractional-order nonlinear time-delay system to obtain the corresponding algebraic equation is as follows: Step B1, the nonlinear time-delay differential equation of the fractional-order time-delay system is in the form of: ； In the formula, For the nonlinear part of the system, and Let be the orders of the nonlinear terms corresponding to the system input and output, respectively; using Gegenbauer wavelet expansion, the system input delay term is expressed as... The system output items are represented as Then, using the fractional time-delay integral operation matrix of the Gegenbauer wavelet, the nonlinear time-delay differential equation of the system can be written in the following matrix form: ； Step B2: Based on the above matrix form, output the nonlinear terms to obtain: ； Gegenbauer wavelet vector Substituting into the nonlinear time-delay differential equation of the system, we get: ； In the formula, Let be the block impulse function vector. Furthermore, based on the definition of the block impulse function, the above equation can be simplified to: ； Step B3: Based on the Gegenbauer wavelet vector Using the operational matrix of the block impulse function and the fractional integral of the Gegenbauer wavelet, the above equation is redefined as: ； Similarly, the nonlinear term in the output part is expressed as: ； Step B4: Substitute the system equations into the Gegenbauer wavelet expansion to transform all fractional differential terms in the system into Gegenbauer wavelet integral matrices. All fractional terms in the system input and output can be written in the following form: 。 8. The identification method according to claim 7, characterized in that: By reducing all terms of the nonlinear time-delay differential equation of the fractional-order time-delay system to matrix form, the overall matrix expression of the system is obtained. Then, a matrix transformation is performed to obtain the Gegenbauer wavelet coefficient vector. The expression is: ； Substituting the above equation into the Gegenbauer wavelet expansion of the system output signal, we obtain the system output signal. The final matrix expression: 。

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