Hysteresis modeling and feedforward control method of piezoelectric actuator

Abstract

The invention discloses a hysteresis modeling and feedforward control method of a piezoelectric actuator, which effectively solves the problem of memory characteristic deficiency of piezoelectric ceramics in polynomial modeling and improves the modeling precision. The modeling method comprises the following steps: S1, describing a full-scale hysteresis model of the piezoelectric ceramic actuator through a polynomial; s2, constructing a hysteresis scheme by using a Madelung rule, and performing weighted stacking on the full-scale hysteresis model in the S1 based on the hysteresis scheme to obtain a weighted model; and S3, identifying parameters of the weighted model to obtain a complete hysteresis model. According to the feed-forward control method, on the basis of the modeling method, input and output are exchanged, a hysteresis inverse model is constructed, feed-forward compensation is provided for a control system, nonlinear hysteresis of the control system is eliminated, and the control precision is improved.

Description

Technical Field

[0001] The present invention belongs to the field of micro-nano driving and control, and in particular relates to a hysteresis modeling and compensation control method for a piezoelectric actuator. Background Art

[0002] In the fields of high-end manufacturing, high-precision orientation, etc., piezoelectric actuators have great development prospects due to their fast response speed, wide operating frequency band and nanometer-level positioning accuracy. Piezoelectric actuators use piezoelectric ceramics as driving components and utilize the inverse piezoelectric effect of piezoelectric ceramic materials to directly convert electrical energy into mechanical energy, generate tiny displacements, and generate corresponding driving forces according to control signals.

[0003] However, the inherent nonlinear hysteresis characteristics of piezoelectric materials greatly limit their application. In actual systems, hysteresis is not a simple lag phenomenon between input and output. First, there is multi-valued between its input and output; second, the output of the system is not only related to the current input, but also to the historical input of the system, and has a certain memory. The existing polynomial hysteresis modeling is difficult to describe its historical memory, resulting in low accuracy in nonlinear hysteresis description, which has hindered the application of piezoelectric actuators.

[0004] Therefore, it is urgent to establish a new model to describe the nonlinear hysteresis of piezoelectric actuators, solve the problem that it is difficult to describe the historical memory of the piezoelectric actuator description system in the existing technology, improve the accuracy of hysteresis description, and eliminate the influence of hysteresis in the control system. Summary of the invention

[0005] In order to solve the problem in the prior art that hysteresis modeling is difficult to describe the historical memory of the piezoelectric actuator description system, resulting in low hysteresis description accuracy, the present invention provides a hysteresis modeling and compensation control method for a piezoelectric actuator.

[0006] Technical solution: The present invention discloses a hysteresis modeling method for a piezoelectric actuator, comprising:

[0007] S1.1, describing the full-scale hysteresis model of the piezoelectric ceramic actuator by polynomials;

[0008] S1.2, constructing a hysteresis scheme using the Madelung rule, and performing weighted superposition on the full-scale hysteresis model of S1.1 based on the hysteresis scheme to obtain a weighted model;

[0009] S1.3. Identify the parameters of the weighted model to obtain complete hysteresis modeling.

[0010] Furthermore, the full-scale hysteresis model uses y to represent the displacement of the piezoelectric actuator. For the displacement hysteresis loop function F(u)=y, it is described as the superposition of the hysteresis loop center line X(u) and the hysteresis component H(u), u represents the voltage, and a polynomial is used to fit the center line of the hysteresis loop. At the same time, a polynomial is used to describe the distance of the upper and lower hysteresis loops from the center line as the hysteresis component, which is then multiplied by the voltage derivative. The sign function of the hysteresis loop is used to express the full-scale hysteresis model as follows:

[0011]

[0012] Where N and M are positive integers, indicating the order, k ui and k wj are the coefficients of the polynomials f1 and f2 respectively.

[0013] Furthermore, the left and right inflection point information is recorded in S1.2, which is used to construct the left and right inflection point voltages and displacements of the full-scale hysteresis function, which are recorded as initial values ​​U L1 , Y L1 , U R1 and Y R1 , the subsequent left and right turning point information is stored in U L , Y L Store the horizontal and vertical coordinates of the left turning point, U R and Y R Store the horizontal and vertical coordinates of the right turning point, and record the latest turning point data as U L(end) .

[0014] Furthermore, the method for determining the inflection point is as follows: let ui be the current input voltage, u1 be the input voltage at the moment before ui, and u2 be the input voltage at the moment before u1. When (ui-u1)(u1-u2)<0, and (ui-u1)>0, store u1 and y1 in the stack U. L , Y L , otherwise it is stored in U R , Y R If the current voltage ui L(end) orui>U R(end) , erase U L(end) , Y L(end) , U R(end) , Y R(end) , back to the previous hysteresis loop.

[0015] Furthermore, the construction of the hysteresis scheme includes:

[0016] Solution 1 F1(u):

[0017]

[0018] Option 2 F2(u): ​

[0019]

[0020] Solution 3 F3(u):

[0021]

[0022] Where U l is a point selected to satisfy U R1 -U l =U R(end) -U L(end) ;

[0023] Scheme 4 F4(u):

[0024]

[0025] Where U r is a point selected to satisfy U r -U L1 =U R(end) -U L(end) .

[0026] Furthermore, the full-scale hysteresis model of S1.1 is weightedly superimposed as described in S1.2, and is expressed as:

[0027]

[0028] Among them, F m (u) represents the mth hysteresis scheme constructed using the Madelung rule; a m represents the weight of the mth solution, and M(u) represents the output of the weighted model.

[0029] Furthermore, S1.3 identifies the coefficients of the polynomial hysteresis model F(u) by the least squares method and identifies the weights a by the differential evolution algorithm. m .

[0030] The present invention also discloses a hysteresis feedforward control method of a piezoelectric actuator, comprising:

[0031] S2.1. The hysteresis modeling method according to any one of claims 1 to 6, establishing a hysteresis model;

[0032] S2.2, swap the input-output relationship, repeat S2.1, and obtain the inverse model of the hysteresis model described in S2.1;

[0033] S2.3. Perform feedforward compensation on the control system based on the inverse model of the hysteresis model.

[0034] Furthermore, the hysteresis inverse model of the piezoelectric actuator in S2.2 is:

[0035]

[0036] Where u = g(y) represents voltage, N, M are positive integers representing the order, k xi and k yj are the coefficients of the polynomials g1 and g2, respectively, and the required voltage is calculated by the expected displacement.

[0037] Beneficial effects: The present invention discloses a hysteresis modeling method for a piezoelectric actuator, which is approximated by polynomials and uses third-order, fourth-order or higher-order fitting hysteresis characteristics; the Madelung rule is used to combine the erasure mechanism with the polynomial model to describe the hysteresis displacement under various circumstances, and four modeling schemes are proposed. The accuracy of the secondary ring hysteresis description is improved by weighting the four schemes.

[0038] The present invention also discloses a hysteresis feedforward control method for a piezoelectric actuator, which swaps the input-output relationship of the piezoelectric actuator hysteresis model in the aforementioned modeling method to obtain a hysteresis inverse model, and connects the hysteresis inverse model in series to the front of the piezoelectric actuator to eliminate nonlinear hysteresis in the control system and improve the control accuracy of the piezoelectric actuator. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 A flowchart of the Madelung rule hysteresis modeling algorithm of the present invention;

[0040] Figure 2 is the input voltage of the hysteresis model of Embodiment 1 of the present invention;

[0041] Figure 3 It is a schematic diagram of the actual output of the hysteresis model and the model predicted output in Example 1 of the present invention;

[0042] Figure 4 It is a schematic diagram of the actual output of the hysteresis inverse model and the model predicted output of Example 2 of the present invention;

[0043] Figure 5 This is a schematic diagram of the output voltage predicted by the inverse model of the variable amplitude expected displacement according to the second embodiment of the present invention;

[0044] Figure 6 Schematic diagram of the inverse model feedforward compensation structure of embodiment 2 of the present invention;

[0045] Figure 7 Schematic diagram of the relationship between the expected displacement and the actual displacement of the inverse model feedforward compensation in Example 2 of the present invention. DETAILED DESCRIPTION

[0046] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0047] The present invention discloses a hysteresis modeling method for a piezoelectric actuator, a preferred embodiment 1, and the specific steps are as follows:

[0048] Step (1.1) describes the full-scale hysteresis model of the piezoelectric actuator using a polynomial.

[0049] The hysteresis loop function F(u) is described as the superposition of the hysteresis loop center line X(u) and the hysteresis component H(u). u represents the voltage. A polynomial is used to fit the center line of the hysteresis loop. At the same time, a polynomial is used to describe the distance between the upper and lower hysteresis loops and the center line as the hysteresis component, which is then multiplied by the voltage derivative. The sign function of is used to express the hysteresis loop. The hysteresis model of the piezoelectric actuator is:

[0050]

[0051] Where N and M are positive integers, indicating the order, k ui and k wj are the coefficients of polynomials f1 and f2 respectively. Choose the appropriate order according to the hysteresis curve of the actual material.

[0052] In this embodiment, a third-order polynomial is used to describe the full-scale hysteresis model of the piezoelectric actuator, which is expressed as:

[0053]

[0054] Step (1.2) uses the Madelung rule to construct four hysteresis schemes and perform weighted superposition to obtain the complete model.

[0055] The four hysteresis schemes based on the Madelung rule consist of two parts: the inflection point recording and erasure mechanism and the use of inflection point data to describe the current hysteresis curve. First, the inflection point recording and erasure mechanism is realized, such as Figure 1 As shown, its function is to accurately describe the current hysteresis loop.

[0056] The steps of the inflection point recording and erasing mechanism are as follows:

[0057] First, perform initialization. The left and right inflection point voltages and displacements used to construct the full-scale hysteresis function are recorded as initial values ​​U L1 , Y L1 , U R1 and Y R1 , the subsequent left and right turning point information is stored in UL , Y L Store the horizontal and vertical coordinates of the left turning point, U R and Y R Store the horizontal and vertical coordinates of the right turning point, and record the latest turning point data as U L(end) wait.

[0058] Secondly, determine whether (u1, y1) is an inflection point. Let ui be the current input voltage, u1 be the input voltage at the moment before ui, and u2 be the input voltage at the moment before u1. When (ui-u1)(u1-u2)<0, and (ui-u1)>0, store u1 and y1 in the stack U. L , Y L , otherwise it is stored in U R , Y R .

[0059] Next, the erase mechanism is implemented. If the current voltage ui L(end) orui>U R(end) , erase U L(end) , Y L(end) , U R(end) , Y R(end) , back to the previous hysteresis loop.

[0060] The hysteresis model was constructed according to four hysteresis schemes based on Madelung's rule and by weighted superposition.

[0061] The first piezoelectric actuator hysteresis construction scheme outputs F1(u):

[0062]

[0063] The second piezoelectric actuator hysteresis construction scheme outputs F2(u):

[0064]

[0065] The third piezoelectric actuator hysteresis construction scheme outputs F3(u):

[0066]

[0067] Where U l is a point selected to satisfy U R1 -U l =U R(end) -U L(end) ;

[0068] The fourth piezoelectric actuator hysteresis construction scheme outputs F4(u):

[0069]

[0070] Where U​r is a point selected to satisfy U r -U L1 =U R(end) -U L(end) ;

[0071] Finally, the recorded inflection point information and four different hysteresis curve construction schemes are used to describe the current curve, and the model is made more accurate through weighted form. The weighted model is expressed as:

[0072]

[0073] Among them, F m (u) represents the mth hysteresis scheme based on Madelung rule; a m represents the weight of the mth solution, and M(u) is obtained by scaling the curves of different intervals on the hysteresis model F(u) of the piezoelectric actuator using the similarity method, and weighted superposition is used to more accurately describe the complex hysteresis characteristics.

[0074] Step (1.3) identifies the parameters of the hysteresis model, including k ui , k wj and a m .

[0075] In this embodiment, the least square method is used to identify the parameters and obtain the parameter k ui and k wj The values ​​of are shown in Table 1; at the same time, the differential evolution algorithm is used to obtain the weight a m .

[0076] Table 1 Parameter k in Example 1 ui , k wj Value

[0077]

[0078] Specifically, this embodiment uses the above parameters to carry out experiments, such as Figure 2 As shown, a set of voltages with an initial range of 0-120V and subsequent voltage amplitudes gradually decreasing are taken as input. Figure 3 The actual output of the piezoelectric actuator and the model output are presented. Figure 2 It can be seen that the hysteresis model disclosed in the present invention has a high accuracy in predicting piezoelectric hysteresis. It is proved that the hysteresis model proposed in the present invention has good accuracy in describing the nonlinear hysteresis of the control system.

[0079] In a second aspect, the present invention discloses a piezoelectric actuator feedforward control method based on a hysteresis inverse model, an embodiment 2 as a preferred specific implementation method, and the specific steps are as follows:

[0080] Step (2.1) uses the hysteresis modeling method of the above steps (1.1) to (1.3) to establish a full-scale hysteresis model describing the piezoelectric actuator.

[0081] Step (2.2) swaps the input-output relationship and remodels using the aforementioned method to obtain the inverse model of the hysteresis model.

[0082] The hysteresis inverse model of the piezoelectric actuator is:

[0083]

[0084] Where u = g(y) represents voltage, N and M are positive integers representing the order, and k xi and k yj are the coefficients of the polynomials g1 and g2, respectively, and the required voltage is calculated by the expected displacement.

[0085] In this embodiment, a third-order polynomial is used to describe the full-scale inverse hysteresis model of the piezoelectric actuator:

[0086] g(y)=g1+g2sign(y),

[0087] Where k xi and k yj are the coefficients of polynomials g1 and g2 respectively. The least squares method is used to identify the parameters and obtain k xi and k xj The parameters obtained are shown in Table 2; at the same time, the differential evolution algorithm is used to obtain the weight a m .

[0088] Table 2 Parameter k in Example 2 xi , k yj Value

[0089]

[0090] Using the above parameters to carry out experiments, we first verify the accuracy of the inverse model. The relationship between the predicted voltage output by the inverse model and the actual voltage required for the expected displacement is as follows: Figure 4 As shown, the two have a high degree of matching. The expected displacement of the variable amplitude is then used to verify the memory of the inverse model. The relationship between the expected displacement and the output voltage is shown in Figure 5 shown.

[0091] Step (2.3) performs feedforward compensation on the control system according to the inverse model of the hysteresis model.

[0092] The feedforward control compensation structure based on the inverse model is as follows: Figure 6As shown, the inverse model is connected in series in front of the piezoelectric actuator to complete the feedforward control, so that the output voltage u(t) of the inverse model of the feedforward control tracks the input expected displacement y of the upper inverse model. d (t) Actual voltage required. Figure 7 The inverse model feedforward compensation effect is demonstrated, and the expected displacement is approximately linearly related to the actual displacement, proving that the inverse model feedforward compensation proposed in the present invention has good accuracy.

Claims

1. A hysteresis modeling method for a piezoelectric actuator, characterized in that: include: S1.1, describing the full-scale hysteresis model of the piezoelectric ceramic actuator by polynomials; S1.2, constructing a hysteresis scheme using the Madelung rule, and performing weighted superposition on the full-scale hysteresis model of S1.1 based on the hysteresis scheme to obtain a weighted model; S1.

3. Identify the parameters of the weighted model and obtain the complete hysteresis model.

2. The hysteresis modeling method according to claim 1, characterized in that: The full-scale hysteresis model uses y to represent the displacement of the piezoelectric actuator. For the displacement hysteresis loop function F(u)=y, it is described as the superposition of the hysteresis loop center line X(u) and the hysteresis component H(u). u represents the voltage. A polynomial is used to fit the center line of the hysteresis loop. The distance between the upper and lower hysteresis loops and the center line is described by a polynomial as the hysteresis component, which is multiplied by the voltage derivative. The sign function of the full-scale hysteresis model is expressed as: Where N and M are positive integers, indicating the order, k ui and k wj are the coefficients of polynomials f1 and f2 respectively.

3. The hysteresis modeling method according to claim 1, characterized in that: The left and right inflection point information is recorded in S1.2, which is used to construct the left and right inflection point voltages and displacements of the full-scale hysteresis function, respectively recorded as the initial values ​​U L1 , Y L1 , U R1 and Y R1 , the subsequent left and right turning point information is stored in U L , Y L Store the horizontal and vertical coordinates of the left turning point, U R and Y R Store the horizontal and vertical coordinates of the right turning point, and record the latest turning point data as U L(end) .

4. The hysteresis modeling method according to claim 3, characterized in that: The method for determining the inflection point is: let ui be the current input voltage, u1 be the input voltage at the moment before ui, and u2 be the input voltage at the moment before u1. When (ui-u1)(u1-u2)<0, and (ui-u1)>0, store u1 and y1 in the stack U. L , Y L , otherwise it is stored in U R , Y R If the current voltage ui L(end) orui>U R(end) , erase U L(end) , Y L(end) , U R(end) , Y R(end) , back to the previous hysteresis loop.​ 5. The hysteresis modeling method according to claim 4, characterized in that: The hysteresis construction scheme comprises: Solution 1 F1(u): Option 2 F2(u): Solution 3 F3(u): Where U l is a point selected to satisfy U R1 -U l =U R(end) -U L(end) ; Scheme 4 F4(u): Where U r is a point selected to satisfy U r -U L1 =U R(end) -U L(end) .

6. The hysteresis modeling method according to claim 5, characterized in that: S1.2 performs weighted superposition of the full-scale hysteresis models F1(u), F2(u), F3(u), and F4(u) of S1.1, expressed as: Among them, F m (u) represents the mth hysteresis scheme constructed using the Madelung rule; a m represents the weight of the mth solution, and M(u) represents the output of the weighted model.

7. The hysteresis modeling method according to claim 1, characterized in that: S1.3 identifies the coefficients of the polynomial hysteresis model F(u) by the least squares method and the weights a by the differential evolution algorithm. m .

8. A hysteresis feedforward control method for a piezoelectric actuator, characterized in that: include: S2.

1. The hysteresis modeling method according to any one of claims 1 to 6, establishing a hysteresis model; S2.2, swap the input-output relationship, repeat S2.1, and obtain the inverse model of the hysteresis model described in S2.1; S2.

3. Perform feedforward compensation on the control system based on the inverse model of the hysteresis model.

9. The hysteresis feedforward control method according to claim 8, characterized in that: The hysteresis inverse model of the piezoelectric actuator in S2.2 is: Where u = g(y) represents voltage, N and M are positive integers representing the order, and k xi and k yj are the coefficients of the polynomials g1 and g2, respectively, and the required voltage is calculated by the expected displacement.

10. The hysteresis feedforward control method according to claim 9, characterized in that: The feedforward compensation described in S2.3 is to connect the hysteresis inverse model obtained in S2.2 in series in front of the piezoelectric actuator to offset the nonlinear hysteresis of the piezoelectric actuator control system.

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