Piezoelectric system hysteresis modeling method based on symmetric madelung model

Abstract

The piezoelectric system hysteresis modeling method based on the symmetrical Madelung model solves the problem of poor real-time performance of the piezoelectric system caused by the P-I model, and belongs to the field of piezoelectric intelligent material hysteresis nonlinear modeling and compensation. The P-I model in the piezoelectric system hysteresis modeling is replaced, the output of the P-I model is obtained, and specifically comprises the following steps: S1, an improved Madelung model is established, the improved Madelung model is a segmented function, and the segmented function is obtained by searching; S2, the current input voltage x i is determined, the corresponding segmented function is selected in the improved Madelung model according to the position on the hysteresis trajectory, the value y i is calculated according to the corresponding segmented function, and the output of the P-I model is efficiently calculated.

Description

Technical Field

[0001] This invention relates to a hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model, belonging to the field of hysteresis nonlinear modeling and compensation for piezoelectric smart materials. Background Technology

[0002] Piezoelectric ceramic actuators based on the inverse piezoelectric effect possess nanometer-level positioning accuracy and show great promise in high-end manufacturing and opto-mechatronics fields. However, the application of piezoelectric ceramic actuators in practical engineering is severely limited due to the influence of hysteresis nonlinearity, significantly reducing system performance. To improve hysteresis linearity and even eliminate its effects, establishing a model describing hysteresis nonlinearity is crucial. Among operator-based models, the Prandtl-Ishlinskii (PI) model is the most widely used in practice. Its basic idea is to linearly weight and superimpose multiple simple backlash operators with different thresholds to describe complex hysteresis phenomena. It is a single integral, with relatively low computational complexity, making it suitable for designing hysteresis compensation control methods in control systems. Furthermore, the PI model not only has a mathematical analytical inverse, but the inverse PI model is also of the PI type; therefore, the parameters of the inverse PI model can be derived from the parameters of the PI model. This characteristic is beneficial for the design of feedforward controllers. However, as an operator-based hysteresis model, its modeling accuracy is related to the number of superimposed operators. The number of superimposed backlash operators directly affects the accuracy of the PI model. Although the modeling accuracy will increase with the increase in the number of operators, the computation time required by the model will also increase. When the PI model is applied to engineering practice to realize composite control, the real-time performance of the piezoelectric system is challenged. Summary of the Invention

[0003] To address the problem of high computational complexity and poor real-time performance of piezoelectric ceramic systems in the PI model modeling process, this invention provides a hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model.

[0004] This invention provides a hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model, used to replace the PI model in hysteresis modeling of piezoelectric systems and obtain the output of the PI model. Specifically, it includes:

[0005] S1. Establish an improved Madelung model, which is a piecewise function:

[0006]

[0007] Where, x i y represents the input voltage of the piezoelectric ceramic at time i. i Indicates with x iThe corresponding output displacement of the piezoelectric ceramic, x0 represents the input voltage corresponding to the starting point A0 of the piezoelectric ceramic hysteresis trajectory, x k ,y k Indicates x on the hysteresis trajectory i The input voltage and output displacement corresponding to the previous inflection point, x k-1 x represents k The input voltage corresponding to the previous inflection point, f0(·) represents the main rising curve function, which is a known quantity and is obtained through search; k represents the inflection point sorting number according to the starting order of the trajectory, k=1,2,…;

[0008] S2. Determine the current input voltage x i At the position on the hysteresis trajectory, select the corresponding piecewise function in the improved Madelung model, and calculate y based on the corresponding piecewise function. i value.

[0009] Preferably, the main ascending curve function f0(x) i ):

[0010]

[0011]

[0012] Among them, w n r is the weight value. n For the threshold, n PI To represent the number of operators as integers, the main ascending curve function f0(x) i Discretize the data according to the resolution of the data acquisition system and store it as a known quantity.

[0013] Preferably, in S2, the current input voltage x is determined. i Position on the hysteresis trajectory:

[0014] S21, Move stack X L X R Y L and Y R Initialization, stack X L Y L Store the x and y coordinates of the left turning point respectively, X R and Y R The x and y coordinates of the right turning point are stored in the middle.

[0015] S22, when (x i -x pre1 (x) pre1 -x pre2 ) < 0 and (x i -x pre1 When x > 0, xpre1 ,y pre1 Stored into stack X respectively L Y L And set TP_flag = 0;

[0016] When (x) i -x pre1 (x) pre1 -x pre2 ) < 0 and (x i -x pre1 When x ≤ 0, pre1 ,y pre1 Stored into stack X respectively R Y R And set TP_flag = 1;

[0017] (x i -x pre1 (x) pre1 -x pre2 When )≥0, check if the current TP_flag is equal to 0. If it is, on stack X R Mid-position x i The location, if not, on stack X L Mid-position x i Location;

[0018] x pre1 x represents i The input voltage at the previous moment, x pre2 x represents pre1 The input voltage at the previous moment, y pre1 For x pre1 The corresponding output displacement.

[0019] Preferably, S2 further includes:

[0020] S23, i = i + 1, if the input voltage x at the current moment i Satisfy x i ≠x pre1 When, x pre1 =x i-1 y pre1 =y i-1 x pre2 =x pre1 Then proceed to S22.

[0021] As a preferred method, a binary search algorithm is used in stack X. R Mid-position x i The position is determined using a binary search algorithm on stack X. L Mid-position x i The location.

[0022] As a preferred option, yk via f k-1 (x k (Get)

[0023] The beneficial effects of this invention are that it establishes the Madelung model based on the central symmetry mechanism, which significantly reduces the amount of computation while obtaining the same output as the PI model, and efficiently calculates the output of the PI model. Attached Figure Description

[0024] Figure 1 This is a flowchart illustrating the modeling method of the present invention;

[0025] Figure 2 This is a schematic diagram illustrating the central symmetry relationship of the present invention;

[0026] Figure 3 Output response plots for the PI model and the Madelung model. Detailed Implementation

[0027] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0028] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0029] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0030] This embodiment addresses the problem that increasing the number of operators in the classical PI model describing hysteresis nonlinearity in the prior art leads to a significant increase in computational load when improving modeling accuracy. The present invention establishes the Madelung model algorithm based on the central symmetry mechanism. This modeling approach is clear and easy to understand, and while obtaining the same output as the PI model, it significantly reduces the computational load and efficiently calculates the output of the PI model.

[0031] This implementation uses an improved Madelung model to calculate the output of the PI model. The improved Madelung model is based on the principle of centrosymmetry to describe the current hysteresis curve (CHC), thereby obtaining the corresponding output displacement y of the piezoelectric ceramic. i ;

[0032] like Figure 2 As shown, curves A1A0 and A0A1 are centrally symmetric. Therefore, points (x, y) on curve A1A0 are centrally symmetric.i ,f1(x i The point (x) on its corresponding curve A0A1 t ,f0(x t The expression satisfies the following:

[0033]

[0034] Where, x i y represents the current input voltage of the piezoelectric ceramic. i Indicates with x i The corresponding output displacement of the piezoelectric ceramic, x0 represents the input voltage corresponding to the starting point A0 of the piezoelectric ceramic hysteresis trajectory, coordinates The center point of the hysteresis loop A0A1-A1A0. The function f0(x) of the main ascending curve of the Madelung model. i It is described by the following polynomial:

[0035] f0(x i ) = c n (x i ) n +c n-1 (x i ) n-1 +…+c1x i +c0 (2)

[0036] Where c i (i = 0, 1, ..., n) are the coefficients of the polynomial, and n is the degree of the polynomial.

[0037] In this implementation, the starting point A0 of the hysteresis trajectory is set as the origin, at which point f0(x0) = 0. After multiple iterations, the current expression of the hysteresis curve can be obtained as follows:

[0038]

[0039] Where, x i y represents the input voltage of the piezoelectric ceramic at time i. i Indicates with x i The corresponding output displacement of the piezoelectric ceramic, x0 represents the input voltage corresponding to the starting point A0 of the piezoelectric ceramic hysteresis trajectory, x k ,y k Indicates x on the hysteresis trajectory i The input voltage and output displacement corresponding to the previous inflection point, x k-1 x represents k The input voltage corresponding to the previous inflection point, f0(·) represents the main rising curve function, which is a known quantity and is obtained through search; k represents the inflection point sorting number according to the starting order of the trajectory, k=1,2,…;

[0040] When k is odd, the curve described by formula (3) is a descending curve, while when k is even, it is an ascending curve. Therefore, the parity of the value of k determines the direction of the current curve.

[0041] Formula (3) is used as an improved Madelung model, and the output of the PI model is calculated using the Madelung model. In formula (3), f0(x0+x) k -x i ) and f0(x0-x k +x i All are based on f0(x) i f0(x) obtained in this embodiment; i Using a piecewise function representation, we determine f0(x) below. i The expression for the PI model is as follows: The PI model is formed by superimposing a set of Backlash operators with weights. Considering that the input signal of the piezoelectric ceramic actuator is positive, the PI model obtained by weighting based on the Backlash operators can be expressed as follows:

[0042]

[0043] In the formula H n [t] represents the one-sided Backlash operator, w n r is the weight value. n n is the threshold. PI T is an integer representing the number of operators. s The sampling period is [value]. The main ascending curve function of the PI model is expressed as a piecewise function as follows:

[0044]

[0045]

[0046] Assumption Because the PI model satisfies central symmetry and follows the Madelung rule, to calculate the output of the PI model using the symmetric hysteresis Madelung model, one only needs to set f0(x) to the maximum value of f0(x). i ) represents In other words, the output of the symmetric hysteresis Madelung model is equivalent to the output of the PI model. This equivalence is not accidental, but because the symmetric hysteresis Madelung model captures the inherent symmetry of the PI model and guarantees equivalence in terms of mechanism.

[0047] The process for calculating the PI model output in this implementation is as follows: First, the position of x needs to be determined. Then, by selecting the corresponding piecewise function in the improved Madelung model, the y value is calculated based on the corresponding piecewise function. The specific process is as follows: Figure 1 As shown, the specific process is as follows: Figure 1 As shown, it includes:

[0048] First, initialization is performed. At the start of the program, stack X is first loaded. L X R Y L and Y R Initialize the main ascending curve function f0(x) i The data is discretized according to the resolution of the data acquisition system and then stored as a known quantity. Secondly, for a quantity with n... s A piezoelectric ceramic actuator system with a maximum input resolution of 1 bit exists at most There are several potential inflection points, and since inflection points occur in pairs and are erased in pairs, the length of these stacks is set to... Finally, using the two input values ​​x before x, pre1 and x pre2 and y pre1 Used as an auxiliary variable to identify inflection points.

[0049] Secondly, determine (x) pre1 ,y pre1 Is it an inflection point? Clearly, when the inequality (x) i -y pre1 (x) pre1 -x pre2 When x < 0, it means x pre1 It is an extreme point of the input, using the inequality (x i -x pre1 ) > 0 can further determine (x) pre1 ,y pre1 The type of ) is then determined, and (x) is assigned according to its type. pre1 ,y pre1 Points are stored in the corresponding stack.

[0050] Next, the erase mechanism is implemented. The input value x is used in this part. i Need to work with X L or X R All elements are compared to determine how many inflection points need to be surpassed. A binary search algorithm is used to speed up the algorithm's execution; given the high efficiency of binary search, the input value x... i At stack position p x It can pass through at most n s The second comparison found it.

[0051] Finally, the output value y is calculated based on the CHC description method. i The function is to update the variable x here. pre1 x pre2 and y pre1It should be noted that because consecutive identical input values ​​can affect the identification of inflection points, leading to incorrect results and wasting computational resources, the program only considers x. i ≠x pre1 Updated regularly.

[0052] The complexity of the modeling method in this embodiment is calculated, and the time and space complexity of the algorithm are verified here:

[0053] (1) Calculate the time complexity of the binary search method. Since x i The position can be determined through multiple comparisons, so the number of numerical comparisons p during algorithm execution directly affects the computation time of binary search. According to the properties of binary search, there is a logarithmic relationship between p and m, specifically expressed as:

[0054] p = (log₂m + 1) (7)

[0055] Where m is the array X LTP or array X RTP The number of non-empty elements in the middle.

[0056] (2) Calculate the time complexity of the Madelung modeling algorithm. Assume that the AD / DA boards of the analog-to-digital converter and the digital-to-analog converter have n... s Bit resolution, the maximum number of distinct input values ​​in a data acquisition system during HIL simulation. At this point, the maximum value of the number of comparisons p is:

[0057]

[0058] (3) Calculate the space complexity of the algorithm. To study the space complexity of the algorithm in the extreme case where all inflection points are recorded, we set the maximum number of inflection points that can be generated and stored during the entire algorithm operation to N. u This number represents the maximum number of distinct input values ​​for the entire hysteresis system. It has n. s Digital data acquisition systems with bit-to-analog (DA) and analog-to-digital (AD) converters have a maximum of There are 10 different input values, corresponding to 10 different values. There are several potential inflection points, therefore the number of input signals for the algorithm can be represented by resolution n. s The corresponding space complexity is expressed as

[0059] Example 1:

[0060] To verify the equivalence of the two models, a PI model is constructed using 10 operators. A series of signals of different types are input into both models. Figure 3By plotting the corresponding output signals, it can be clearly seen that the outputs of the PI model and the symmetric hysteresis Madelung model are equivalent.

[0061] Further analysis shows that for any different value of k, formula (3) consists of three terms, the first of which is f. k-1 (x k This value represents the current inflection point; the other two values ​​can be derived from the main loop curve f0(x). i The calculation is obtained. The Madelung modeling algorithm operates because f0(x) is used to calculate... i The value of f0(x) has been stored in the computer; a simple search is all that is needed to obtain it. i The method outputs the value without performing the complex weight superposition calculations found in the PI model, thus making it more computationally efficient than the PI model.

[0062] Example 2:

[0063] Next, we calculate the algorithm complexity of the Madelung modeling algorithm at 16-bit AD / DA resolution, starting with the time complexity. We assume the AD / DA boards of the analog-to-digital converter and digital-to-analog converter have 16-bit resolution. As shown in equation (8), the maximum value of the number of comparisons p is 16. The relationship between the computation time and the number of comparisons in binary search can be fitted by a straight line 0.19p + 0.57. Therefore, each comparison takes about 0.19 μs. At this time, the computation time corresponding to the value of p is 3.61 μs, which is the maximum computation time.

[0064] Next, we calculate the space complexity. Since inflection points have the characteristics of alternating generation and paired erasure, we can simply assume that the number of left and right inflection points is the same, and the maximum value of the input array length L is... To store all generated inflection points, stack X L X R Y L and Y R The lengths of all of them are set to 2. 16-1 =2 15 Furthermore, since each element in the stack is 16 bits, or two bytes, the memory resources required to run the modeling algorithm program are approximately 2... 15 ×4×2Bytes=256KB.

[0065] In summary, this application addresses the problem that increasing the number of operators in the classical PI model describing hysteresis nonlinearity in the prior art leads to a significant increase in computational load when improving modeling accuracy. This invention establishes the Madelung model algorithm based on the central symmetry mechanism. This modeling approach is clear and easy to understand, and while obtaining the same output as the PI model, it significantly reduces the computational load and efficiently calculates the output of the PI model.

[0066] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model, characterized in that, The method is used to replace the PI model in the hysteresis modeling of piezoelectric systems and obtain the output of the PI model, specifically including: S1. Establish an improved Madelung model, which is a piecewise function: Where, x i y represents the input voltage of the piezoelectric ceramic at time i. i Indicates with x i The corresponding output displacement of the piezoelectric ceramic, x0 represents the input voltage corresponding to the starting point A0 of the piezoelectric ceramic hysteresis trajectory, x k ,y k Indicates x on the hysteresis trajectory i The input voltage and output displacement corresponding to the previous inflection point, x k-1 x represents k The input voltage corresponding to the previous inflection point, f0(·) represents the main rising curve function, which is a known quantity and is obtained through search; k represents the inflection point sorting number according to the starting order of the trajectory, k=1,2,…; S2. Determine the current input voltage x i At the position on the hysteresis trajectory, select the corresponding piecewise function in the improved Madelung model, and calculate y based on the corresponding piecewise function. i value.

2. The hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model according to claim 1, characterized in that, The main ascending curve function f0(x) i ): Among them, w n r is the weight value. n For the threshold, n PI To represent the number of operators as integers, the main ascending curve function f0(x) i Discretize the data according to the resolution of the data acquisition system and store it as a known quantity.

3. The hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model according to claim 1, characterized in that, In S2, the current input voltage x is determined. i Position on the hysteresis trajectory: S21, Move stack X L X R Y L and Y R Initialization, stack X L Y L Store the x and y coordinates of the left turning point respectively, X R and Y R The x and y coordinates of the right turning point are stored in the middle. S22, when (x) i -x pre1 (x) pre1 -x pre2 ) < 0 and (x i -x pre1 When x > 0, x pre1 ,y pre1 Stored into stack X respectively L Y L And set TP_flag = 0; When (x) i -y pre1 (x) pre1 -x pre2 ) < 0 and (x i -x pre1 When x ≤ 0, pre1 ,y pre1 Stored into stack X respectively R Y R And set TP_flag = 1; (x i -x pre1 (x) pre1 -x pre2 When )≥0, check if the current TP_flag is equal to 0. If it is, on stack X R Mid-position x i The location, if not, on stack X L Mid-position x i Location; x pre1 x represents i The input voltage at the previous moment, x pre2 x represents pre1 The input voltage at the previous moment, y pre1 For x pre1 The corresponding output displacement.

4. The hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model according to claim 3, characterized in that, S2 further includes: S23, i = i + 1, if the input voltage x at the current moment i Satisfy x i ≠x pre1 When, x pre1 =x i-1 y pre1 =y i-1 x pre2 =x pre1 Then proceed to S22.

5. The hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model according to claim 3, characterized in that, Using binary search on stack X R Mid-position x i The position is determined using a binary search algorithm on stack X. L Mid-position x i The location.

6. The hysteresis modeling method for piezoelectric systems based on the symmetric Madelung model according to claim 3, characterized in that, y k via f k-1 (x k (Get) 7. A computer-readable storage device storing a computer program, characterized in that, When the computer program is executed, it implements the piezoelectric system hysteresis modeling method based on the symmetric Madelung model as described in any one of claims 1 to 6.

8. A piezoelectric system hysteresis modeling device based on the symmetric Madelung model, comprising a storage device, a processor, and a computer program stored in the storage device and executable on the processor, characterized in that, The processor executes the computer program to implement the piezoelectric system hysteresis modeling method based on the symmetric Madelung model as described in any one of claims 1 to 6.

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