Madhelg model construction method based on hysteresis feature multi-stage grabbing strategy

Abstract

The invention provides a Madelung model construction method based on a hysteresis characteristic multi-level grabbing strategy, which is characterized in that a general expression of CHC is established by adopting a multi-level similarity scaling mode based on rich hysteresis characteristics in a main hysteresis loop, an improved CHC description method is obtained by combining a weight superposition strategy, and thus an improved Madelung model is established. Compared with a traditional method, the method has the advantages that the hysteresis characteristics in the main hysteresis loop are fully excavated, the proposed general expression can realize multiple CHC description methods, each CHC description method does not need to be independently designed, the implementation is easy, the expansibility is higher, and the CHC description precision is effectively improved. Finally, the improved Madelung model is connected with a linear dynamic link in series, and the effectiveness of the method is verified through a design experiment.

Description

Technical Field

[0001] The present invention belongs to the field of hysteresis models, and in particular relates to a method for constructing a Madelung model based on a multi-level grasping strategy of hysteresis characteristics. Background Art

[0002] Piezoelectric ceramic actuators offer advantages such as nanometer-level resolution, fast response, and immunity to electromagnetic interference, making them widely used in precision positioning and precision manipulation. However, piezoelectric ceramic actuators exhibit an inherent hysteresis effect between input voltage and output displacement. This means that the output depends not only on the current input but also on the historical state of the input. This nonlinear behavior can lead to control errors, reduce the system's positioning accuracy, and severely impact practical applications. Without understanding the system's hysteresis behavior, it is impossible to predict the system's output. By establishing a hysteresis nonlinear model, hysteresis characteristics can be more accurately predicted, laying the foundation for eliminating hysteresis.

[0003] Among established traditional hysteresis models, phenomenon-based hysteresis models are more widely used in practice. They are mainly categorized into micro-classification models, operator-based models, and models that directly construct hysteresis trajectories. Micro-classification hysteresis models, such as the Bouc-Wen model, have the advantage of requiring fewer parameters to be identified, but their modeling accuracy is heavily dependent on parameter identification, and they cannot be inverted analytically. Operator-based hysteresis models, such as the Prandtl-Ishlinskii (PI) model, can be inverted analytically. However, since the model's accuracy is determined by the number of operator superpositions, when sufficient model accuracy is required, the model requires more parameters, which makes the model more complex and computationally demanding.

[0004] Directly constructing hysteresis trajectory models has the advantages of clear modeling ideas and easy implementation. The recently proposed Madelung model solves the problem of reduced modeling accuracy caused by the loss of inflection point information representing the historical information of hysteresis characteristics in polynomial models. The established model contains all inflection point information.

[0005] Although the existing Madelung model contains all the inflection point information, the description accuracy of its current hysteresis curve (CHC) is insufficient and still needs to be further improved. The essential reason is that the existing Madelung model fails to fully explore the characteristic information of the hysteresis curve. For example, in the previously proposed CHC description strategy based on weight superposition, since each CHC method needs to be designed separately and the accuracy of each proposed CHC description method depends on the designer's experience, the scalability is relatively poor. In addition, although the main hysteresis loop curve contains rich hysteresis features, the existing Madelung model fails to deeply explore the hysteresis features in the main hysteresis loop, which is also the reason for the insufficient description accuracy of CHC. Summary of the Invention

[0006] In response to the problems existing in the prior art, the present invention provides a Madelung model construction method based on a multi-level capture strategy of hysteresis features. Based on the rich hysteresis features in the main hysteresis loop, a universal expression of CHC is established by adopting a multi-level similarity scaling method, and then an improved CHC description method is obtained by combining the weight superposition strategy, thereby establishing an improved Madelung model. Compared with the traditional method, not only the hysteresis features in the main hysteresis loop are fully exploited, but the proposed universal expression can also realize multiple CHC description methods, without the need to design each CHC description method separately. It is easy to implement and has stronger scalability, effectively improving the description accuracy of CHC. Finally, the improved Madelung model is connected in series with the linear dynamic link, and the effectiveness of the proposed method is verified by design experiments.

[0007] The technical solutions of the present invention are as follows:

[0008] The Madelung model construction method based on the hysteresis feature multi-level grasping strategy includes the following steps:

[0009] Step 1: Establish the expression of the main hysteresis loop

[0010] The main hysteresis loop expression can be established using a polynomial function or a piecewise linear function. Taking the polynomial function as an example, the main hysteresis loop is divided into the main rising curve f0(x) and the main falling curve g0(x);

[0011] f0(x)=c n x n +c n-1 x n-1 +...+c1x 1 +c0

[0012]

[0013] in are the coefficients of the main ascending curve and main descending curve polynomials respectively;

[0014] Step 2: Establish a general hysteresis curve expression based on the expression of the main hysteresis loop

[0015] Assume that the rising hysteresis curve expression obtained by the mth hysteresis construction scheme is recorded as f mk (x), x is the current input signal, and the universal rising hysteresis curve expression is obtained through multi-level scaling.

[0016]

[0017] where x∈(x k ,x k-1 ), and s=x k-1 -xk , h=y k-1 -y k , S=x frm -x flm ,

[0018] Similarly, assuming that the expression of the descending hysteresis curve obtained by the mth hysteresis construction scheme is g mk (x), after multi-level similarity scaling, the descending curve g is obtained mk (x) function is:

[0019]

[0020] where x∈(x k-1 ,x k ), x glm <x grm , is the scaled intermediate curve function;

[0021] Step 3: Superimpose the weights of different hysteresis curve description methods to obtain the improved hysteresis curve expression

[0022] The improved rising hysteresis curve f obtained by weight superposition k (x) expression

[0023]

[0024] Improved descending hysteresis curve g obtained by weight superposition k (x) expression

[0025]

[0026] α m is the weight coefficient, where 0≤α m ≤1, m=1,2,3...N.

[0027] Step 4: Incorporate the improved hysteresis curve expression into the erasure mechanism algorithm to establish an improved Madelung model;

[0028] Step 5: Based on the Hammerstein structure, the improved Madelung model is connected in series with the linear dynamic link to establish a dynamic hysteresis model.

[0029] The part between the improved Madelung model output y and the rate-dependent dynamic mode output H is regarded as a linear dynamic link G(s);

[0030] When the input signal satisfies the increasing relationship, the rate-dependent dynamic hysteresis model H f (x) is

[0031] H f (x) = f k (x)G(s);

[0032] When the input signal satisfies the decreasing relationship, the rate-dependent dynamic hysteresis model H g (x) is

[0033] H g (x) = g k (x)G(s);

[0034] The parameters of the linear dynamic link G(s) can be obtained by identification.

[0035] Preferably, in the Madelung model construction method based on the hysteresis feature multi-level grasping strategy,

[0036] The minimum input voltage of the piezoelectric actuator is x min , the maximum input voltage is x max , the corresponding output displacement is y min 、y max ;

[0037] The main rising curve f0(x) refers to the piezoelectric actuator input voltage from x min Load to x max The input-output displacement trajectory of the piezoelectric actuator during the process, and the main descending curve g0(x) refers to the piezoelectric actuator voltage from x max Load to x min Output displacement trajectory of the piezoelectric actuator during the process.

[0038] Preferably, in the Madelung model construction method based on the multi-level capture strategy of hysteresis characteristics, a universal rising hysteresis curve expression is obtained by multi-level scaling, comprising the following steps:

[0039] First, select the endpoints x on the main rising hysteresis curve f0(x) flm and x frm The corresponding curve is used as the curve to be scaled, where x flm <x frm .

[0040] Secondly, keep the horizontal coordinate of the curve to be scaled unchanged and scale the vertical coordinate with a scaling factor of η (η≠0). The intermediate curve obtained after scaling is Satisfy x∈(x flm ,x frm );

[0041] Again, continue with the middle curve Perform similarity transformation, and the scaling ratio in the X and Y directions is and In this link, both the horizontal and vertical coordinates need to be scaled;

[0042] Finally, after two similarity scalings, the hysteresis curve A is obtained. k A k-1 The function f mk (x) is

[0043]

[0044] where x∈(x k ,x k-1 ), and s=x k-1 -x k , h=y k-1 -y k , S=x frm -x flm ,

[0045] Preferably, in the Madelung model construction method based on the hysteresis feature multi-level grasping strategy, the improved hysteresis curve expression is integrated into the erasure mechanism algorithm to establish the improved Madelung model, comprising the following steps:

[0046] Initialize the variables used;

[0047] Determine whether the input signal is an inflection point. If it is an inflection point, store it in the corresponding stack according to the inflection point type.

[0048] Determine how many inflection points need to be surpassed, erase the surpassed inflection points and their corresponding inflection points, and thereby determine the movement trajectory interval of the CHC;

[0049] Based on the improved rising hysteresis curve f k (x) and improved descending hysteresis curve g k (x) Calculate the output value y.

[0050] The function of the erasing mechanism is to achieve the erasure of useless inflection points, and then determine the movement trajectory range of CHC.

[0051] The present invention has the following beneficial effects:

[0052] Based on the rich hysteresis features in the main hysteresis loop, the present invention establishes a universal expression of CHC by adopting a multi-level similarity scaling method, and obtains an improved CHC description method by combining a weight superposition strategy, thereby establishing an improved Madelung model.

[0053] Compared with traditional methods, the improved Madelung model established in the present invention deeply explores the hysteresis characteristics in the main hysteresis loop, effectively improves the modeling accuracy of hysteresis nonlinearity, and solves the problem of relatively poor modeling accuracy and scalability caused by the need for each CHC method in the existing Madelung model to rely on the designer's own experience to design it individually.

[0054] The application of the present invention can improve the prediction accuracy of the hysteresis characteristics of the piezoelectric ceramic actuator, and lay a solid foundation for eliminating the inherent hysteresis effect of the piezoelectric ceramic actuator.

[0055] Other advantages, objectives and features of the present invention will be reflected in part from the following description and will be understood by those skilled in the art through study and practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 The CHC rising curve f obtained by the mth hysteresis construction scheme in one embodiment of the Madelung model improvement method based on the hysteresis feature multi-level capture strategy proposed by the present invention is mk (x);

[0057] Figure 2 The pseudo code of the erasure mechanism algorithm in one embodiment of the improved Madelung model method based on the hysteresis feature multi-level grasping strategy proposed by the present invention;

[0058] Figure 3 A dynamic hysteresis model in an embodiment of the Madelung model improvement method based on the hysteresis feature multi-stage grasping strategy proposed by the present invention;

[0059] Figure 4 A table of endpoint values ​​and scaling coefficients of three weighted superposition strategies in one embodiment of the improved Madelung model method based on the hysteresis feature multi-level grasping strategy proposed by the present invention;

[0060] Figure 5 This is a diagram of model prediction results in one embodiment of the Madelung model improvement method based on the hysteresis feature multi-level grasping strategy proposed by the present invention;

[0061] Figure 6 This is a prediction error diagram in one embodiment of the Madelung model improvement method based on the hysteresis feature multi-stage grasping strategy proposed by the present invention. DETAILED DESCRIPTION

[0062] The present invention will be described in further detail below in conjunction with the accompanying drawings so that those skilled in the art can implement the invention with reference to the description.

[0063] This paper proposes an improved Madelung model method based on a multi-stage grasping strategy based on hysteresis characteristics. The following examples will be introduced using a piezoelectric ceramic actuator platform as the research object. The specific steps are as follows:

[0064] Step 1: Establish the expression of the main hysteresis loop. The minimum input voltage of the piezoelectric actuator is x min , the maximum input voltage is x max , the corresponding output displacement is y min 、y max The main hysteresis loop is divided into the main rising curve f0(x) and the main falling curve g0(x). The main rising curve f0(x) refers to the piezoelectric actuator input voltage from x min Load to x max The input-output displacement trajectory of the piezoelectric actuator during the process, and the main descending curve g0(x) refers to the piezoelectric actuator voltage from x max Load to x min The output displacement trajectory of the piezoelectric actuator during the process. All rising CHC description methods are based on the hysteresis characteristics of the main rising curve f0(x), and all falling CHC description methods are based on the hysteresis characteristics of the main falling curve g0(x). The present invention uses a polynomial form to establish the main hysteresis curve expression:

[0065] f0(x)=c n x n +c n-1 x n-1 +...+c1x 1 +c0

[0066]

[0067] in are the coefficients of the main ascending curve and main descending curve polynomials respectively.

[0068] Step 2: This step is based on the established expression of the main hysteresis loop to establish a general CHC expression formula. Assume that the CHC rising curve obtained by the mth hysteresis construction scheme is denoted as f mk (x), the following two scaling schemes are used to establish f mk (x). First, select the endpoints on the main rising curve f0(x) as x flm and x frm The corresponding curve is used as the curve to be scaled, where x flm <x frm Secondly, keep the horizontal coordinate of the curve to be scaled unchanged and scale the vertical coordinate with a scaling factor of η (η≠0). The intermediate curve obtained after scaling is Again, continue with the middle curve Perform similarity transformation, and the scaling ratio in the X and Y directions is and In this step, both the horizontal and vertical coordinates need to be scaled. Finally, after two similarity scalings, f is established. mk The expression of (x) is

[0069]

[0070] where x∈(x k ,x k-1 ), and s=x k-1 -x k , h=y k-1 -y k , S=x frm -x flm ,

[0071] Similarly, assuming that the CHC decline curve obtained by the mth hysteresis construction scheme is denoted as g mk (x), after two similarity scalings, we can get the descending curve g mk (x) function is:

[0072]

[0073] where x∈(x k-1 ,x k ), x glm <x grm , is the scaled intermediate curve function.

[0074] Step 3: Superimpose the weights of different CHC description methods to obtain the final improved CHC expression. As shown in the following formula, the improved CHC rising curve f obtained by weight superposition is k (x) expression

[0075]

[0076] Improved CHC decline curve g obtained by weighted superposition k (x) expression

[0077]

[0078] α m is the weight coefficient, where 0≤α m ≤1, m=1,2,3...N,because f k (x)∈[y k ,y k-1 ], g k (x)∈[yk-1 ,y k ], so the endpoint of the curve obtained after weighted superposition is still A k and A k-1 In addition, due to A k-1 In A k Generated before, so x k-1 and y k-1 is considered to be a predefined value, ie, a known value.

[0079] Step 4: Incorporate the improved CHC description method established in step 3 into the erasure mechanism algorithm to establish the improved Madelung model. The erasure mechanism is used to erase useless inflection points and thus determine the CHC trajectory range. The pseudo code of the erasure mechanism algorithm is as follows: Figure 2 As shown, it is divided into four parts.

[0080] The first part is to initialize the variables used. L 、X R 、Y L and Y R Initialize and use the two input values ​​x before x pre1 and x pre2 and y pre1 As an auxiliary variable to identify inflection points.

[0081] The second part is to determine the latest point of the input signal (x pre1 ,y pre1 ) is an inflection point, when the inequality (xy pre1 )(x pre1 -x pre2 )<0 means x pre1 is an extreme point of the input. At this time, if the inequality (xx pre1 )>0, then x pre1 and y pre1 Recorded in X L and Y L and set Flag=0; if the inequality (xx pre1 )<0, then x pre1 and y pre1 Recorded in X R and Y R and set Flag = 1.

[0082] The third part is to compare the input value x with X L or X R All elements in are compared to determine how many inflection points need to be surpassed. By setting the public pointer P = P x The exceeded inflection point and its corresponding inflection point are erased, and the movement trajectory interval of the CHC is determined accordingly.

[0083] The fourth part is based on the improved CHC rising curve f proposed in step 3 k (x) and improved CHC decline curve g k (x) Calculate the output value y and update the variable x here pre1 、x pre2 and y pre1 Since continuous identical input values ​​will affect the identification of inflection points and lead to erroneous results, and waste computing resources, the program only runs when x≠x pre1 Updates are made when needed.

[0084] Step 5: In order to better characterize the rate-dependent dynamic hysteresis characteristics of the piezoelectric ceramic actuator, based on the Hammerstein structure, the improved Madelung model is connected in series with the linear dynamic link to establish a dynamic hysteresis model. Figure 3 As shown in the figure, the improved Madelung model is regarded as a static nonlinear hysteresis link. The influence of the dynamic link does not need to be considered in this part. The part between the output y of the improved Madelung model and the output H of the rate-dependent dynamic mode is regarded as a linear dynamic link G(s). Therefore, when the input signal satisfies the increasing relationship, the rate-dependent dynamic hysteresis model H f (x) is

[0085] H f (x) = f k (x)G(s);

[0086] When the input signal satisfies the decreasing relationship, the rate-dependent dynamic hysteresis model H g (x) is

[0087] H g (x) = g k (x)G(s);

[0088] The parameters of the linear dynamic link G(s) can be obtained by identification.

[0089] Step 6: Perform parameter identification. In the present invention, the least squares method is used to identify the main hysteresis loop curve expression. The parameters of the main rising curve f0(x) and the main falling curve g0(x) are identified separately.

[0090] When identifying the parameters of the main rising curve f0(x), the input voltage of the piezoelectric actuator is changed from x min Load to x max In the process, the input signal-output signal is loaded into the least squares algorithm, and the polynomial order is set to 5. The expression of the main rising curve f0(x) can be obtained as follows:

[0091] f0(x)=0.943x 5 -2.895x4 +3.639x 3 -2.208x 2 +1.522x

[0092] Similarly, the expression of the main descent curve g0(x) can be obtained as:

[0093] g0(x)=0.445x 5 -0.489x 4 +0.056x 3 +0.375x 2 +0.61x

[0094] The improved Madelung hysteresis model proposed in this invention adopts three weight superposition strategies, namely N=3, and the endpoint value x of each strategy is flm 、x frm 、x glm 、x grm , the scaling factor η is set as Figure 4 On this basis, the weight coefficients of the three superposition strategies are obtained by particle swarm optimization identification, namely, α1 = 0.432, α2 = 0.487, and α3 = 0.081.

[0095] Since the hysteresis nonlinearity of the piezoelectric ceramic actuator is small under the action of small amplitude signals, in order to minimize the adverse effects of the hysteresis nonlinear part on the identification of the linear part parameters, a small signal is used to excite the piezoelectric ceramic actuator to perform the parameter identification of the linear dynamic part. The transfer function expression of the linear part G(s) is obtained by the least squares method:

[0096]

[0097] Step 7: Perform experimental verification. To verify the effectiveness of the improved Madelung model proposed in this patent, a sinusoidal signal with superimposed frequencies of 10 Hz, 30 Hz, and 50 Hz was input into the piezoelectric ceramic actuator. Figure 5 The actual experimental output and the rate-related model output proposed in this invention are shown. The output of the model is represented by a solid line, and the actual output is represented by a dotted line. It can be found that the model output predicts the actual experimental output very well. The corresponding modeling error is given by Figure 6 The results show that the normalized root mean square error is 0.67%, which not only verifies the effectiveness of the proposed improved Madelung model algorithm, but also shows that after the proposed improved Madelung model is connected in series with the linear dynamic link, high-precision modeling of the hysteresis rate-related hysteresis nonlinearity of the piezoelectric ceramic actuator can be achieved.

[0098] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the description and implementation methods. They can be fully applied to various fields suitable for the present invention. For those familiar with the art, additional modifications can be easily implemented. Therefore, without departing from the general concept defined by the claims and the scope of equivalents, the present invention is not limited to specific details.

Claims

1. A Madelung model construction method based on a multi-level grasping strategy based on hysteresis characteristics, characterized in that: The following steps are involved: Step 1: Establish the expression of the main hysteresis loop The main hysteresis loop is divided into the main rising curve f0(x) and the main falling curve g0(x). The main hysteresis loop expression is established using polynomial function or piecewise linear function. Take the polynomial function as an example: f0(x)=c n x n +c n-1 x n-1 +...+c1x 1 +c0 where c0,...,c n , are the coefficients of the main ascending curve and main descending curve polynomials respectively; Step 2: Establish a general hysteresis curve expression based on the expression of the main hysteresis loop Assume that the rising hysteresis curve expression obtained by the mth hysteresis construction scheme is recorded as f mk (x), x is the current input signal, and the universal rising hysteresis curve expression is obtained through multi-level scaling. where x∈(x k ,x k-1 ), and s=x k-1 -x k , h=y k-1 -y k , S=x frm -x flm , Similarly, assuming that the expression of the descending hysteresis curve obtained by the mth hysteresis construction scheme is g mk (x), after multi-level similarity scaling, the descending curve g is obtained mk (x) function is: where x∈(x k-1 ,x k ), x glm <x grm , is the scaled intermediate curve function; Step 3: Superimpose the weights of different hysteresis curve description methods to obtain the improved hysteresis curve expression The improved rising hysteresis curve f obtained by weight superposition k (x) expression Improved descending hysteresis curve g obtained by weight superposition k (x) expression α m is the weight coefficient, where 0≤α m ≤1, m=1,2,3...N Step 4: Incorporate the improved hysteresis curve expression into the erasure mechanism algorithm to establish an improved Madelung model; Step 5: Based on the Hammerstein structure, the improved Madelung model is connected in series with the linear dynamic link to establish a dynamic hysteresis model. The part between the improved Madelung model output y and the rate-dependent dynamic mode output H is regarded as a linear dynamic link G(s); When the input signal satisfies the increasing relationship, the rate-dependent dynamic hysteresis model H f (x) is H f (x)=f k (x)G(s); When the input signal satisfies the decreasing relationship, the rate-dependent dynamic hysteresis model H g (x) is H g (x)=g k (x)G(s); The parameters of the linear dynamic link G(s) can be obtained by identification.

2. The method for constructing a Madelung model based on a multi-level grasping strategy based on hysteresis characteristics according to claim 1, characterized in that: The minimum input voltage of the piezoelectric actuator is x min , the maximum input voltage is x max , the corresponding output displacement is y min 、y max ; The main rising curve f0(x) refers to the piezoelectric actuator input voltage from x min Load to x max The input-output displacement trajectory of the piezoelectric actuator during the process, and the main descending curve g0(x) refers to the piezoelectric actuator voltage from x max Load to x min Output displacement trajectory of the piezoelectric actuator during the process.

3. The method for constructing a Madelung model based on a multi-level grasping strategy based on hysteresis characteristics according to claim 2, characterized in that: The general rising hysteresis curve expression is obtained through multi-level scaling, including the following steps: First, select the endpoints x on the main rising hysteresis curve f0(x) flm and x frm The corresponding curve is used as the curve to be scaled, where x flm <x frm ; Secondly, keep the horizontal coordinate of the curve to be scaled unchanged and scale the vertical coordinate with a scaling factor of η (η≠0). The intermediate curve obtained after scaling is Satisfy x∈(x flm ,x frm ); Again, continue with the middle curve Perform similarity transformation, and the scaling ratio in the X and Y directions is and In this link, both the horizontal and vertical coordinates need to be scaled; Finally, after two similarity scalings, the hysteresis curve A is obtained. k A k-1 The function f mk (x) is where x∈(x k ,x k-1 ), and s=x k-1 -x k , h=y k-1 -y k , S=x frm -x flm , 4. The method for constructing a Madelung model based on a multi-level grasping strategy based on hysteresis characteristics according to claim 3, characterized in that: The improved hysteresis curve expression is incorporated into the erasure mechanism algorithm to establish an improved Madelung model, which includes the following steps: Initialize the variables used; Determine whether the input signal is an inflection point. If it is an inflection point, store it in the corresponding stack according to the inflection point type. Determine how many inflection points need to be surpassed, erase the surpassed inflection points and their corresponding inflection points, and thereby determine the movement trajectory interval of the CHC; Based on the improved rising hysteresis curve f k (x) and improved descending hysteresis curve g k (x) Calculate the output value y.

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