A high-precision piezoelectric ceramic wide-frequency hysteresis calculation and adaptive parameter identification method

Abstract

The application belongs to the technical field of chattering detection of aircraft model support systems, and discloses a high-precision piezoelectric ceramic wide-frequency hysteresis calculation and adaptive parameter identification method. First, the output displacement is decomposed into a linear displacement component and a nonlinear hysteresis displacement component, and then a voltage-rate cross correction term is introduced to realize accurate characterization of the wide-frequency asymmetric hysteresis characteristic. Then, the core parameters are adaptively identified by using the recursive least square method with a forgetting factor, the parameter vector is initialized, the regression vector is constructed, and the parameter and covariance matrix are iteratively updated at each time instant, so that the parameters converge quickly. Compared with the traditional model, the fitting accuracy of the model is improved by 75%, and the number of parameters is reduced by more than 50%, so that high-precision, simplified parameter and full-frequency adaptive modeling of the piezoelectric ceramic wide-frequency hysteresis characteristic can be realized. The method provides a reliable tool for hysteresis error compensation of an aviation high-precision vibration measurement system, and has strong engineering application.

Description

Technical Field

[0001] This invention belongs to the field of vibration detection technology for aircraft model support systems, and relates to a high-precision method for constructing a broadband piezoelectric ceramic hysteresis model and an adaptive parameter identification method. Background Technology

[0002] As next-generation aircraft upgrade towards higher maneuverability, longer endurance, and precision payload carrying, their vibration stability under all operating conditions has become a core indicator determining flight safety and mission accuracy. In particular, the tolerance threshold for vibration interference for key equipment such as high-precision optical payloads and inertial navigation components has been reduced to the micrometer level. Against this backdrop, piezoelectric ceramics (PZT), which combine rapid dynamic response, sub-micrometer positioning accuracy, and miniaturized structure, are gradually becoming the core functional material for aviation micro-vibration measurement and suppression platforms. Through bidirectional conversion of mechanical energy and electrical energy, PZT can achieve real-time sensing and rapid compensation of vibration signals, demonstrating irreplaceable application potential in scenarios such as aircraft structural vibration monitoring and load micro-vibration suppression. However, the inherent hysteresis characteristics of piezoelectric ceramic materials result in a strong nonlinear coupling between the input driving voltage and the output mechanical displacement. This characteristic not only exhibits "path dependence" (different displacement outputs for the same voltage input) but also shows significant asymmetric distortion as the excitation frequency increases. Under the wide-frequency (0~2000Hz) conditions of aerospace vibration, the introduced nonlinear error can reach 10%~20% of the nominal displacement range, directly causing the amplitude and phase deviations of vibration measurements to exceed the system accuracy threshold, becoming a key technical bottleneck restricting its engineering application in high-precision aerospace scenarios. Existing research has proposed classic hysteresis models such as Duhem and Preisach to address the modeling and compensation of the hysteresis characteristics of piezoelectric ceramics. However, traditional models have two major drawbacks: first, they do not consider the hysteresis asymmetry under high-frequency excitation of aerospace vibrations, and the fitting error increases significantly with increasing frequency; second, the models need to approximate nonlinear terms with high-order polynomials, resulting in high parameter redundancy (usually requiring 9~12 parameters to be identified), making it difficult to adapt to the stringent real-time and computational resource requirements of aerospace systems. Against this backdrop, constructing an improved hysteresis model that combines high-frequency asymmetric hysteresis adaptability, parameter simplification, and high fitting accuracy has become a necessary technical path to overcome the limitations of piezoelectric ceramics in the field of aerospace vibration measurement.

[0003] In his patent "An Asymmetric Hysteresis and Creep Model of a Piezoelectric Actuator and Its Inverse Compensation Method" (CN113759716A), Su Liangcai proposed a composite model integrating asymmetric hysteresis and creep. This model describes the asymmetric hysteresis characteristics of the piezoelectric actuator by introducing a piecewise polynomial correction term, employs an immune differential algorithm for parameter identification, and finally achieves hysteresis compensation through an inverse model. However, the asymmetric correction term in this patent is only based on "voltage amplitude" and does not introduce a cross term between voltage change rate and voltage. Under the wide-frequency (0~2000Hz) operating conditions of aircraft vibration measurement, when the excitation frequency exceeds 100Hz, the asymmetric distortion of the hysteresis curve cannot be accurately fitted, failing to meet the high-precision requirements of aviation. Furthermore, the model contains 12 parameters to be identified, requiring tens of thousands of iterations using the immune differential algorithm to converge, resulting in long computation times and making it difficult to adapt to the stringent real-time and computational resource constraints of aviation systems.

[0004] In his patent "Dynamic Hysteresis Modeling and Feedforward Control Method for Piezoelectric Actuators Based on MGPI," Wang Wen proposed an "improved generalized MGPI model." This model introduces a rate-dependent envelope function to correct the hysteresis operator threshold, adapting to dynamic hysteresis characteristics, and then constructs a feedforward compensation controller based on this model. However, the hysteresis characteristics of piezoelectric ceramics in aerospace scenarios exhibit strong asymmetry. Symmetrical threshold correction results in excessive fitting errors, failing to meet the accuracy requirements of vibration measurements. Furthermore, the envelope function parameters of this model need to be individually identified for specific excitation frequencies. When the vibration frequency of an aircraft dynamically changes over a wide range, parameter recalibration is required, making full-band adaptive modeling impossible and limiting its engineering practicality.

[0005] Based on the aforementioned problems, a high-precision method for calculating and identifying broadband hysteresis of piezoelectric ceramics is proposed, which combines high-precision fitting of asymmetric hysteresis, full-band adaptive generalization capability, and simplified parameter structure. This method has significant engineering value and practical significance for overcoming the application bottleneck of piezoelectric ceramics in the field of high-precision vibration measurement in aerospace. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, this invention proposes a high-precision method for wideband hysteresis calculation and adaptive parameter identification of piezoelectric ceramics. This method utilizes a customized piezoelectric ceramic hysteresis characteristic testing platform for experimental data acquisition and model verification. Accurate voltage-displacement measurement data is obtained through the platform, and combined with model design and algorithm optimization, high-precision modeling of wideband hysteresis characteristics is achieved. By decomposing the output displacement into linear and nonlinear hysteresis displacements, a voltage-rate cross-correction term is introduced into the classic Duhem model. Parameter identification is performed using a recursive least squares method with a forgetting factor. Leveraging the model's accurate characterization of asymmetric hysteresis and efficient parameter simplification, and supported by high-precision experimental data from the testing platform, accurate modeling of hysteresis characteristics under wideband operating conditions of piezoelectric ceramics is achieved. This overcomes the limitations of insufficient high-frequency fitting accuracy and parameter redundancy in traditional models, thereby improving modeling accuracy, enhancing full-band generalization ability, and reducing the complexity of engineering applications.

[0007] The technical solution of the present invention: A high-precision method for calculating broadband hysteresis and identifying adaptive parameters of piezoelectric ceramics, comprising the following steps: The first step is to build a high-precision test platform for the broadband hysteresis characteristics of piezoelectric ceramics; This high-precision piezoelectric ceramic broadband hysteresis characteristic test platform includes a pillow-shaped piezoelectric ceramic actuator unit 1, a precision polypropylene insulating sheet 2, a pillow-shaped piezoelectric ceramic actuator unit semi-covered limit cap 3, a piezoelectric ceramic mounting reference platform end face cover plate 4, a piezoelectric ceramic mounting reference platform end face cover plate limit screw 5, a piezoelectric ceramic mounting reference platform 6, a trapezoidal piezoelectric ceramic limit insert plate 7, a controllable pre-tightening and dissipation helical spring 8, two low dielectric loss piezoelectric transmission lines 9, and a broadband piezoelectric sensing signal preamplifier unit 10. The piezoelectric ceramic mounting reference platform 6 is horizontally calibrated bidirectionally to ensure that its flatness meets the test accuracy requirements. Then, the end face cover plate 4 of the piezoelectric ceramic mounting reference platform 6 is symmetrically locked on both sides of the platform using the limiting screws 5. Two pillow-shaped piezoelectric ceramic actuator units 1 are placed on the left and right sides of the piezoelectric ceramic mounting reference platform 6, and half-covering limiting caps 3 are placed over them to ensure that the half-covering limiting caps 3 are in close contact with the end face cover plate 4. Finally, the platform is moved towards the pre-set limit in the middle of the piezoelectric ceramic mounting reference platform 6. A trapezoidal piezoelectric ceramic limiting plate 7 is inserted into the slot to achieve lateral constraint and positioning of the pillow-shaped piezoelectric ceramic actuator unit 1. A low-dielectric-loss piezoelectric transmission line 9 is used to connect the four components in series in the order of input signal, two pillow-shaped piezoelectric ceramic actuator units 1, and grounding terminal. Then, a precision polypropylene insulating sheet 2 is attached and laid on the semi-covering limiting caps 3 of the pillow-shaped piezoelectric ceramic actuator units on both sides. A controllable pre-tightening release helical spring 8 is installed between the precision polypropylene insulating sheet 2 and the piezoelectric ceramic mounting reference platform 6. After connecting another low-dielectric-loss piezoelectric transmission line 9 to the electrode of the longitudinal displacement sensor on the precision polypropylene insulating sheet 2, it is then connected to the broadband piezoelectric sensing signal preamplifier unit 10. The second step is to read the output displacement signal from the broadband piezoelectric sensing signal preamplifier unit 10 and decompose it into linear displacement components and nonlinear hysteresis displacement components. in, The output displacement of the pillow-type piezoelectric ceramic actuator unit 1 at time t is a function of time t; The linear displacement component of the pillow-type piezoelectric ceramic actuator unit 1 at time t is a function of time t; The nonlinear hysteresis displacement component of the pillow-type piezoelectric ceramic actuator unit 1 at time t is a function of time t; The third step is to introduce a voltage-rate cross-correction term. Define the parameter vector to be identified: in, for The derivative with respect to time t; Let be the input drive voltage applied to the pillow-type piezoelectric ceramic actuator unit 1 at time t, which is a function of time t; for The derivative with respect to time t; Linear static gain, unit ; Linear dynamic gain, unit ; The hysteresis strength coefficient is expressed in units of 1 / V. The unit is the lag memory coefficient. ; Voltage-rate cross-correction factor, unit ; The fourth step is to identify parameters based on the recursive least squares method with a forgetting factor. First, set the sampling period to... , at t= Discretize at any time. ; Discretization of voltage change rate: Discretization of hysteresis displacement rate: Combining the above equations, we get: in, It is the regression vector at time t=k0. It is the parameter vector to be identified The parameter extension vector of the hysteresis displacement at the previous moment; Secondly, define the forgetting factor. Initial expansion vector Initial covariance matrix , It is a 6th order identity matrix; Then, the fitting residual of the recursive least squares method with forgetting factor at time t=k0 is calculated. and parameter gain vector : Using the fitting residual at time t=k0 and parameter gain vector Update the covariance matrix of the recursive least squares method with forgetting factor at time k0. and parameter extension vector : The final parameter extension vector is obtained when the iterations converge. and the convergence time kfinal : in, Expand the vector for the final parameters. These are linear static gains. Linear dynamic gain Hysteresis strength coefficient Delayed memory coefficient Voltage-rate cross correction factor The final convergence value; Step 5: Verify the fitting accuracy; Calculated from the final extended vector: in, The root mean square error, For the first The measured output displacement of the pillow-type piezoelectric ceramic actuator unit 1 at any given time. For the first The calculated output displacement of pillow-type piezoelectric ceramic actuator unit 1 at any given time, where N is the number of iterations. Let be the regression vector at the convergence time.

[0008] Thus, a high-precision method for calculating broadband hysteresis and identifying adaptive parameters of piezoelectric ceramics has been completed.

[0009] The beneficial effects of this invention are as follows: A high-precision method for wideband hysteresis calculation and adaptive parameter identification of piezoelectric ceramics is proposed, along with a supporting high-precision wideband hysteresis characteristic testing platform for piezoelectric ceramics. This provides high-precision, high-stability measured data support and hardware assurance for hysteresis calculation and parameter identification. The proposed method decomposes the output displacement of the piezoelectric ceramic into linear displacement components and nonlinear hysteresis displacement components, while introducing a voltage-rate cross-correction term. This achieves accurate characterization of asymmetric hysteresis characteristics under wideband operating conditions, significantly reducing the problem of excessive high-frequency fitting errors in traditional methods and greatly improving calculation accuracy. The testing platform, through the precise assembly and coordinated operation of components such as a pillow-shaped piezoelectric ceramic actuator unit, a controllable preload-dissipating helical spring, and a low-dielectric-loss piezoelectric transmission line, enables high-precision acquisition of displacement and electrical signals under wideband excitation of piezoelectric ceramics, effective suppression of electromagnetic interference, and accurate elimination of vibration interference. This provides a reliable hardware foundation for model verification of hysteresis calculation and acquisition of measured data for parameter identification, effectively avoiding the impact of low measurement accuracy, large signal interference, and poor adaptability of traditional testing devices on the calculation results. Meanwhile, this method contains only 5 parameter vectors to be identified. Combined with a recursive least squares method with a forgetting factor, it achieves rapid parameter convergence through a time-by-time recursive process of initialization, residual calculation, gain update, parameter iteration, and covariance matrix correction. This avoids the defects of parameter redundancy and computational time consumption in traditional methods, significantly improving the real-time performance and convenience of engineering applications. The rapid assembly and accurate calibration characteristics of the supporting test platform further reduce the engineering complexity of method implementation and improve the efficiency of field applications. In addition, the method proposed in this invention does not require separate calibration for specific frequencies and can adaptively adapt to the wideband dynamic variation scenarios of aircraft vibration measurement. The supporting test platform can also accurately adapt to the testing requirements of 0~2000Hz wideband vibration conditions. The two work together to effectively overcome the limitation of weak generalization ability of existing technologies across the entire frequency band. The method of this invention complements the high-precision piezoelectric ceramic broadband hysteresis characteristic testing platform, which not only has a solid theoretical foundation and reliable hardware support, but also demonstrates high universality and practicality in engineering practices such as high-precision vibration measurement in aviation. It provides a precise and efficient calculation method and measurement tool for hysteresis error compensation of piezoelectric ceramics, which is of great significance for promoting the engineering application of piezoelectric ceramics in the field of high-precision vibration measurement. Attached Figure Description

[0010] Figure 1 This is a flowchart of a method for constructing a high-precision piezoelectric ceramic broadband hysteresis model and identifying adaptive parameters; Figure 2 This is a schematic diagram of a piezoelectric ceramic hysteresis characteristic testing platform; Figure 3 This is a comparison chart of hysteresis fitting of piezoelectric ceramics; Figure 4 yes Figure 3A magnified view of a portion of the image; Figure 5 This is a convergence curve of the piezoelectric ceramic hysteresis model parameters; where (a) is... (Linear static gain) convergence curve, (b) is (Linear dynamic gain) convergence curve, (c) is (Hysteresis strength coefficient) convergence curve, (d) is (Hysteresis coefficient) convergence curve, (e) is (d) is (Voltage-rate cross correction coefficient) convergence curve.

[0011] Figure 2 In the middle, 1-pillow-shaped piezoelectric ceramic actuator unit, 2-precision polypropylene insulating sheet, 3-pillow-shaped piezoelectric ceramic actuator unit semi-covered limit cap, 4-piezoelectric ceramic mounting reference platform end face cover plate, 5-piezoelectric ceramic mounting reference platform end face cover plate limit screw, 6-piezoelectric ceramic mounting reference platform, 7-trapezoidal piezoelectric ceramic limit insert plate, 8-controllable pre-tightening and dissipation helical spring, 9-low dielectric loss piezoelectric transmission line, 10-wideband piezoelectric sensing signal preamplifier unit. Detailed Implementation

[0012] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0013] The test was conducted using a high-precision wideband hysteresis characteristic testing platform for piezoelectric ceramics, with the following parameters: Pillow-type piezoelectric ceramic actuator unit 1: displacement stroke 0~100μm, positioning accuracy 1nm, piezoelectric strain constant d33≥300 pm / V; excitation signal: adopts 500Hz sinusoidal composite voltage under aviation broadband vibration conditions, the expression is u(t)=10sin(2π×500t)+10 V; data acquisition: sampling period Ts=0.001 s, acquisition time 5s.

[0014] A high-precision method for calculating broadband hysteresis and identifying adaptive parameters of piezoelectric ceramics, comprising the following steps: First, define the forgetting factor. Initial expansion vector of parameters Initial covariance matrix .

[0015] Table 1 Data from the first 5 typical sampling times

[0016] by Taking 2 as an example, we can get (V / s). Definition ,in, Linear static gain, unit The value ranges from 0.2 to 0.3; Linear dynamic gain, unit The value range is 0 to 0.0002; Hysteresis strength coefficient, unit 1 / V, value range 0.01~0.03; : Lag memory coefficient, unit The value ranges from 0.01 to 0.02. Voltage-rate cross-correction factor, unit The value range is 0 to 0.001.

[0017] Based on u(2) = 13.090 V, u(2) = 3090.000 V / s, and Ts = 0.001 s, the following values ​​were calculated: (V / s) , Therefore Parameter extension vector [0, 0, 0, 0, 0, 0] T .

[0018] Then, recursive calculations are performed to fit the residuals. Gain vector =[0.0072, 0, 5.5×10⁻⁶, -0.0017, 0.0223, 0.0005], thus obtaining the updated extended parameter vector. [0, 0, 0, 0, 0, 0] T +[0.0072, 0, 5.5 × 10 -6 [ -0.0017, 0.0223, 0.0005] T =[-0.0059, 0, -4.5×10 -6 [0.0014, -0.0183, -0.0004] T The updated covariance matrix = Repeating the above steps multiple times will yield the parameter identification results.

[0019] Table 2 Comparison of Core Parameter Identification Results and Physical Meaning

[0020] Table 3 Convergence Time of Core Parameters

[0021] The average convergence time is 8.7ms, which meets the stringent real-time requirements of aviation systems and verifies the efficiency of the algorithm and the simplified parameter structure.

[0022] Calculated from the final extended vector: =0.08 Within the excitation frequency range of 0~2000Hz, the fitting error is less than 0.8%, verifying the accurate adaptation capability of the voltage-rate cross correction term to asymmetric hysteresis.

[0023] This high-precision piezoelectric ceramic broadband hysteresis calculation and adaptive parameter identification method decomposes the output displacement into linear and nonlinear hysteresis components and introduces a voltage-rate cross-correction term. This accurately adapts to the asymmetric hysteresis characteristics under broadband operating conditions, significantly reducing the problem of excessive high-frequency fitting errors in traditional models. The simplified five high-precision piezoelectric ceramic broadband hysteresis calculation and adaptive parameter identification methods, combined with a recursive least squares method with a forgetting factor, achieve rapid parameter convergence, avoiding the redundancy and computationally intensive drawbacks of traditional models, and effectively improving the real-time performance of engineering applications. The combination of non-contact high-precision measurement and time-by-time recursive identification reduces environmental interference and human error. The vibration isolation structure is adapted to aerospace vibration scenarios, further enhancing the system's measurement stability and engineering reliability, and flexibly meeting the stringent requirements of high-precision aerospace vibration measurement.

[0024] Compared to traditional hysteresis modeling methods, this method can efficiently and accurately solve for the hysteresis parameters of piezoelectric ceramics under wide-frequency (0~2000Hz) conditions, avoiding the limitations of traditional algorithms that require separate calibration for specific frequencies and have weak generalization ability, effectively reducing the modeling cycle and computational cost. This method also has high universality, applicable to modeling the hysteresis characteristics of piezoelectric ceramic components of different specifications, providing reliable theoretical support and engineering tools for hysteresis error compensation of piezoelectric ceramics in aerospace, precision manufacturing, and other fields. It is a highly practical method for hysteresis modeling and parameter identification.

Claims

1. A high-precision method for calculating broadband hysteresis and identifying adaptive parameters of piezoelectric ceramics, characterized in that, The steps are as follows: The first step is to build a high-precision test platform for the broadband hysteresis characteristics of piezoelectric ceramics; The second step is to read the output displacement signal from the broadband piezoelectric sensing signal preamplifier unit (10) and decompose it into linear displacement components and nonlinear hysteresis displacement components. in, The output displacement of the pillow-type piezoelectric ceramic actuator unit (1) at time t is a function of time t; The linear displacement component of the pillow-type piezoelectric ceramic actuator unit (1) at time t is a function of time t; The nonlinear hysteresis displacement component of the pillow-type piezoelectric ceramic actuator unit (1) at time t is a function of time t; The third step is to introduce a voltage-rate cross-correction term. Define the parameter vector to be identified: in, for The derivative with respect to time t; Let be the input driving voltage applied to the pillow-type piezoelectric ceramic actuator unit (1) at time t, which is a function of time t; for The derivative with respect to time t; Linear static gain, unit ; Linear dynamic gain, unit ; The hysteresis strength coefficient is expressed in units of 1 / V. The unit is the lag memory coefficient. ; Voltage-rate cross-correction factor, unit ; The fourth step is to identify parameters based on the recursive least squares method with a forgetting factor. Step 5: Verify the fitting accuracy; Calculated from the final extended vector: in, The root mean square error, For the first The measured output displacement of the pillow-type piezoelectric ceramic actuator unit (1) at any given time. For the first The calculated output displacement of the pillow-type piezoelectric ceramic actuator unit (1) at any given time, where N is the number of iterations. Linear static gain The final convergence value, Linear dynamic gain The final convergence value, Let be the regression vector at the convergence time. This is the final parameter extension vector.

2. The high-precision piezoelectric ceramic broadband hysteresis calculation and adaptive parameter identification method according to claim 1, characterized in that, The first step is to build a high-precision test platform for the broadband hysteresis characteristics of piezoelectric ceramics, as detailed below: The high-precision piezoelectric ceramic broadband hysteresis characteristic test platform includes a pillow-shaped piezoelectric ceramic actuator unit (1), a precision polypropylene insulating sheet (2), a pillow-shaped piezoelectric ceramic actuator unit semi-covered limit cap (3), a piezoelectric ceramic mounting reference platform end face cover plate (4), a piezoelectric ceramic mounting reference platform end face cover plate limit screw (5), a piezoelectric ceramic mounting reference platform (6), a trapezoidal piezoelectric ceramic limit insert plate (7), a controllable pre-tightening and dissipation helical spring (8), two low dielectric loss piezoelectric transmission lines (9), and a broadband piezoelectric sensing signal preamplifier unit (10). The piezoelectric ceramic mounting reference platform (6) is horizontally calibrated in both directions to ensure that the flatness of the piezoelectric ceramic mounting reference platform (6) meets the test accuracy requirements. Then, the piezoelectric ceramic mounting reference platform end face cover plate (4) is symmetrically locked on both sides of the piezoelectric ceramic mounting reference platform (6) by the limit screws (5). Two pillow-shaped piezoelectric ceramic actuator units (1) are placed on the left and right sides of the piezoelectric ceramic mounting reference platform (6). The pillow-shaped piezoelectric ceramic actuator unit semi-covered limit caps (3) are covered on the pillow-shaped piezoelectric ceramic actuator units (1) to ensure that the half-covered limit caps (3) of the pillow-shaped piezoelectric ceramic actuator units on both sides are in contact with the piezoelectric ceramic mounting reference platform end face cover plate (4). Then, the limit is set in the middle of the piezoelectric ceramic mounting reference platform (6). A trapezoidal piezoelectric ceramic limiting plate (7) is inserted into the slot to achieve lateral constraint and positioning of the pillow-type piezoelectric ceramic actuator unit (1); a low dielectric loss piezoelectric transmission line (9) is used to connect the four in series in the order of input signal, two pillow-type piezoelectric ceramic actuator units (1) and grounding terminal; then a precision polypropylene insulating sheet (2) is attached and laid on the half-covering limiting cap (3) of the pillow-type piezoelectric ceramic actuator units on both sides, and a controllable pre-tightening relief helical spring (8) is installed between the precision polypropylene insulating sheet (2) and the piezoelectric ceramic mounting reference platform (6); after connecting another low dielectric loss piezoelectric transmission line (9) to the electrode of the longitudinal displacement sensor on the precision polypropylene insulating sheet (2), it is then connected to the broadband piezoelectric sensing signal preamplifier unit (10).

3. The high-precision piezoelectric ceramic broadband hysteresis calculation and adaptive parameter identification method according to claim 2, characterized in that, The fourth step involves parameter identification based on the recursive least squares method with a forgetting factor. The specific implementation process is as follows: First, set the sampling period to... , at t= Discretize at any time. ; Discretization of voltage change rate: Discretization of hysteresis displacement rate: Combining the above equations, we get: in, It is the regression vector at time t=k0. It is the parameter vector to be identified The parameter extension vector of the hysteresis displacement at the previous moment; Secondly, define the forgetting factor. Initial expansion vector Initial covariance matrix , It is a 6th order identity matrix; Then, the fitting residual of the recursive least squares method with forgetting factor at time t=k0 is calculated. and parameter gain vector : Using the fitting residual at time t=k0 and parameter gain vector Update the covariance matrix of the recursive least squares method with forgetting factor at time k0. and parameter extension vector : The final parameter extension vector is obtained when the iterations converge. and the convergence time k final : in, Expand the vector for the final parameters. These are linear static gains. Linear dynamic gain Hysteresis strength coefficient Delayed memory coefficient Voltage-rate cross correction factor The final convergence value.

Images