CN113311712A - Identification method for hysteresis characteristic of rapid tilting mirror - Google Patents
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Abstract
A method for identifying the hysteresis characteristic of a fast tilting mirror solves the problem of low precision existing in the existing modeling method by regarding the nonlinear hysteresis phenomenon of the fast tilting mirror as the hysteresis phenomenon of single piezoelectric ceramic, and belongs to the field of hysteresis nonlinear model parameter identification. The method comprises the steps of establishing a symmetrical hysteresis model and a nonlinear function h of single piezoelectric ceramic of the rapid tilting mirror, taking the output of the symmetrical hysteresis model as the input of the nonlinear function h, and obtaining the nonlinear functions h of two pieces of diametrically opposite piezoelectric ceramic1And h2Nonlinear function of difference f ═ h1‑h2(ii) a Calculating the parameter f by using the output angle of the fast tilting mirror and the output of the symmetric hysteresis model; calculating the parameters of the nonlinear function h according to the f for identifying the parameters; and obtaining the lengths of the two pieces of piezoelectric ceramics in the diameter alignment by using the symmetric hysteresis model and the nonlinear function h, which calculate the parameters, and obtaining the inclination angles of the two pieces of piezoelectric ceramics in the diameter alignment according to the difference value of the lengths of the two pieces of piezoelectric ceramics.
Description
Technical Field
The invention relates to an identification method for a hysteresis characteristic of a fast tilting mirror, and belongs to the field of hysteresis nonlinear model parameter identification.
Background
The fast tilting mirror is a parallel mechanism consisting of a plurality of piezoelectric ceramics. As shown in figure 1, four piezoelectric ceramic columns are symmetrically distributed in a cross manner in the rapid tilting mirror, the mirror surface platform on each piezoelectric ceramic column can be pushed to deflect by stretching of each piezoelectric ceramic column, and the mirror surface platform is connected with the shell through a flexible hinge. Although the fast tilting mirror internally comprises four piezoelectric ceramic actuators, the piezoelectric ceramics do not work independently, but work simultaneously for two radial piezoelectric ceramics, and the mirror platform is driven to rotate along the direction vertical to the two piezoelectric ceramics. As shown in FIG. 2, the electrical interfaces of the No. 1 and No. 2 ceramics are connected in series, and share the fixed 100V voltage input, and the expansion and contraction of the two ceramics can be controlled simultaneously by changing the control voltage, so as to complete the angle control in one direction.
Because the piezoelectric ceramic of the driving element of the rapid tilting mirror has a nonlinear hysteresis effect, the input voltage and the deflection angle of the rapid tilting mirror also have a hysteresis nonlinear relationship. This nonlinear hysteresis is manifested in that when a triangular wave input voltage as shown in fig. 3 is input to the fast-tilting mirror, the output angle of the fast-tilting mirror with respect to time is as shown in fig. 4. Further, the relationship between the output angle and the input voltage of the fast tilting mirror is shown in fig. 5. In order to effectively compensate the nonlinear hysteresis effect, the nonlinear hysteresis must be modeled, and the relationship between the established model and the actual approximation degree reaches the quality of the next compensation effect (compensation purpose: linear relationship between input voltage and output displacement is realized, and further control is facilitated).
At present, although the hysteresis model of a single piezoelectric ceramic can be established more accurately, the hysteresis model of the fast tilting mirror is more complex compared with the hysteresis model of a single piezoelectric ceramic because the fast tilting mirror is driven by a plurality of piezoelectric ceramics. However, in the prior art, the internal structure of the fast tilting mirror is ignored, the hysteresis nonlinearity of the fast tilting mirror is directly regarded as the hysteresis nonlinearity relationship of a single piezoelectric ceramic, that is, the hysteresis characteristics of the fast tilting mirror and the single piezoelectric ceramic are considered to be the same, and then the modeling compensation is performed on the hysteresis phenomenon of the fast tilting mirror. Through the use of the rapid tilting mirror, the hysteresis characteristic of the rapid tilting mirror is not the same as that of a single piezoelectric ceramic, so that the problem of low precision exists in the conventional modeling of regarding the nonlinear hysteresis phenomenon of the rapid tilting mirror as the hysteresis phenomenon of the single piezoelectric ceramic.
Disclosure of Invention
Aiming at the problem that the accuracy is not high when the existing modeling is carried out by regarding the nonlinear hysteresis of the rapid tilting mirror as the hysteresis of single piezoelectric ceramic, the invention provides an identification method for the hysteresis characteristic of the rapid tilting mirror.
The invention relates to a method for identifying hysteresis characteristics of a fast tilting mirror, which comprises the following steps:
s1, establishing a symmetrical hysteresis model of the single piezoelectric ceramic of the rapid tilting mirror, wherein the input of the symmetrical hysteresis model is the input voltage of the single piezoelectric ceramic, the output of the symmetrical hysteresis model is the length of the single piezoelectric ceramic, and the input and the output are used for identifying the parameters of the symmetrical hysteresis model;
s2, inputting a random signal into the fast tilting mirror to obtain an output angle of the fast tilting mirror, converting the random signal into input voltages of two diametrically opposite piezoelectric ceramics in the fast tilting mirror, and respectively inputting the input voltages into a symmetric hysteresis model with identified parameters to obtain output of the symmetric hysteresis model of the two diametrically opposite piezoelectric ceramics;
s3, establishing a nonlinear function h of the single piezoelectric ceramic, wherein the input of the nonlinear function h is the output of a symmetrical hysteresis model of the single piezoelectric ceramic in S2, the output of the nonlinear function h is the length output of the single piezoelectric ceramic, and the parameter of the nonlinear function h is A;
s4, obtaining two radial nonlinear functions h of the piezoelectric ceramics according to S31And h2Obtaining a non-linear function f ═ h1-h2The parameter in the nonlinear function f is B;
s5, identifying and calculating the parameter B in the nonlinear function f by using the output angle of the fast tilting mirror obtained in S2 and the output of the symmetric hysteresis model;
s6, calculating a parameter A of the nonlinear function h according to the nonlinear function f of the identified parameter;
and S7, obtaining the lengths of the two pieces of piezoelectric ceramics in the diameter alignment by using the symmetric hysteresis model and the nonlinear function h of the calculated parameters, and obtaining the inclination angles of the two pieces of piezoelectric ceramics in the diameter alignment according to the difference value of the lengths of the two pieces of piezoelectric ceramics.
Preferably, the symmetric hysteresis model is a PI model.
Preferably, the S1 includes:
s11, obtaining nonlinear hysteresis main loop data of the fast tilting mirror, comprising:
input voltage vector X ═ X (1), X (2), …, X (n)]TAnd outputting the angle vector Y1＝[y1(1),y1(2),…,y1(n)]T
S12, outputting angle vector Y1Processing is carried out to obtain the length output vector Z of the single piezoelectric ceramic, wherein the length output vector Z is [ Z (1), Z (2), …, Z (n)]T；
S13, allocating operator threshold values to the PI model, wherein if m operators are used, the operator threshold values are respectively
The weight corresponding to each operator threshold is rhoiThen, construct weight vector ρ ═ ρ1,ρ2,…,ρm]T；
S14, using input voltage vector X ═ X (1), X (2), …, X (n)]TAnd output vector Z ═ Z (1), Z (2), …, Z (n)]TRho ═ rho in symmetric hysteresis model for single piezoelectric ceramic1,ρ2,…,ρm]TAnd (5) performing identification.
Preferably, S14 includes:
using the input voltage vector X ═ X (1), X (2), …, X (n)]TConstructing matrices
Weight vector in PI model
And completing the identification of parameters in the symmetrical hysteresis model of the single piezoelectric ceramic.
Preferably, the nonlinear function h and the nonlinear function f are polynomial functions.
Preferably, the nonlinear function h of a single piezoelectric ceramic in S3 is h (z) ═ a1·z+a2·z2+a3·z3+a4·z4+a5·z5Z is input, h (z) is output, a1、a2、a3、a4、a5As the parameter(s) is (are),
preferably, f is b1·zp+b2·zp 2+b3·zp 3+b4·zp 4+b5·zp 5，b1、b2、b3、b4·、b5As the parameter(s) is (are),
zpis the output of the symmetrical hysteresis model of the single piezoelectric ceramic, and f is the output.
Preferably, in S5, the input matrix Φ is constructed using the outputs of the symmetric hysteresis model obtained in S2, and the output matrix Y is constructed using the output angles of the fast-tilting mirrors obtained in S22Identifying B ═ phi by using least square methodT·φ)-1·φT·Y2And completing the identification of the parameter B in the nonlinear function f.
Preferably, the method further comprises S7, the S7 comprising:
inputting the same signals to the fast tilting mirror, the symmetric hysteresis model with the identified parameters and the nonlinear function h respectively, observing the difference value between the output of the fast tilting mirror and the inclination angle obtained according to S7, and obtaining the prediction error of the symmetric hysteresis model in series with the nonlinear function h by using the difference value.
The method has the advantages that the modeling is carried out on the single piezoelectric ceramic in the rapid tilting mirror from the internal structure of the rapid tilting mirror, and then the final angle output is deduced through the internal structure of the rapid tilting mirror, so that the modeling precision of the rapid tilting mirror is effectively improved. In addition, compared with the existing modeling method of the rapid tilting mirror, the invention provides a new modeling approach, namely, the hysteresis characteristic of a single driving element in the rapid tilting mirror structure is established, and finally the output of the rapid tilting mirror is assembled into a whole.
Drawings
FIG. 1 is a schematic diagram of the internal structure of a fast tilting mirror;
FIG. 2 is a schematic diagram of the input voltage of a fast tilting mirror;
FIG. 3 is an input voltage for a fast tilting mirror, where the abscissa is the sampling time t and the ordinate is the input voltage;
FIG. 4 is an output angle of the fast tilting mirror, with the abscissa being the sampling time t and the ordinate being the output angle;
FIG. 5 is a graph of the relationship between the output angle of a fast tilting mirror and the input voltage;
FIG. 6 is a graph showing the relationship between the output angle of a fast tilting mirror and the input voltage when a complex signal is input;
FIG. 7 is a schematic diagram of the principles of the present invention;
FIG. 8 is a solid line of the predicted output of the fast tilting mirror nonlinear hysteresis model constructed using the method of the present invention, and a dashed line of the output of the true fast tilting mirror;
FIG. 9 is a model prediction error for the non-linear hysteresis of the fast tilting mirror constructed in accordance with the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
The invention is further described with reference to the following drawings and specific examples, which are not intended to be limiting.
In one embodiment, a method for identifying a hysteresis characteristic of a fast tilting mirror includes:
step one, establishing a symmetrical hysteresis model of a single piezoelectric ceramic of the rapid tilting mirror, wherein the input of the symmetrical hysteresis model is the input voltage of the single piezoelectric ceramic, the output of the symmetrical hysteresis model is the length of the single piezoelectric ceramic, and the input and the output are used for identifying the parameters of the symmetrical hysteresis model;
inputting a random signal into the rapid tilting mirror to obtain an output angle of the rapid tilting mirror, converting the random signal into input voltages of two diametrically opposite piezoelectric ceramics in the rapid tilting mirror, and respectively inputting the input voltages into a symmetrical hysteresis model with identified parameters to obtain output of the symmetrical hysteresis model of the two diametrically opposite piezoelectric ceramics;
as shown in fig. 2, the number 1 piezoelectric ceramic and the number 2 piezoelectric ceramic are two radial piezoelectric ceramics in the rapid tilting mirror, the random signal refers to a control voltage, after the control voltage is input, the input voltage of the number 1 piezoelectric ceramic is the control voltage, and the input voltage of the number 2 piezoelectric ceramic is 100v minus the control voltage;
step three, establishing a nonlinear function h of the single piezoelectric ceramic, wherein the input of the nonlinear function h is the output of the symmetrical hysteresis model of the single piezoelectric ceramic in the step two, the output of the nonlinear function h is the length output of the single piezoelectric ceramic, and the parameter of the nonlinear function h is A;
step four, obtaining two radial nonlinear functions h of the piezoelectric ceramics according to the step three1And h2Obtaining a non-linear function f ═ h1-h2The parameter in the nonlinear function f is B;
step five, identifying and calculating the parameter B in the nonlinear function f by using the output angle of the fast tilting mirror obtained in the step two and the output of the symmetric hysteresis model;
step six, identifying and calculating the parameter A of the nonlinear function h according to the nonlinear function f of the identified parameter;
and seventhly, acquiring the lengths of the two pieces of diametrically opposite piezoelectric ceramics by using the symmetric hysteresis model with the identified parameters and the nonlinear function h, and acquiring the inclination angles of the two pieces of diametrically opposite piezoelectric ceramics according to the difference value of the lengths of the two pieces of piezoelectric ceramics.
As shown in FIG. 7, the input voltages of the two diametrically aligned piezoelectric ceramics are v (t) and u (t), respectively, and the outputs of the two symmetric hysteresis models are z (t), respectively1(t) and z2(t) and then input to the nonlinear function h1And h2In (h)1And h2Middle parameter equal, non-linear function h1And h2Respectively output h1(t) and h2And (t), subtracting to obtain angle information.
The modeling method provided by the embodiment is to model a single piezoelectric ceramic in the fast tilting mirror, and then to deduce the final angle output through the internal structure of the fast tilting mirror. Furthermore, the hysteresis output of the fast tilting mirror is not directly modeled, but the angle output data of the fast tilting mirror is used for carrying out certain transformation (the transformation method is determined according to the internal structure of the fast tilting mirror), and then the transformed data is used for carrying out hysteresis modeling on the single piezoelectric ceramic of the driving element in the fast tilting mirror. As shown in fig. 7, the fast tilting mirror nonlinear hysteresis model of the present embodiment includes a symmetric hysteresis model and a nonlinear function for compensation, which are respectively established for two pieces of diametrically opposite piezoelectric ceramics, and obtains the tilt angle by calculating the difference between the output results of the two nonlinear functions; in one direction, the difference value of the two models of the piezoelectric ceramics after modeling can be regarded as the output of the rapid tilting mirror, and then the modeling of the rapid tilting mirror is realized.
The method adopted by the modeling of the single piezoelectric ceramic model in the embodiment is that a symmetric hysteresis model is connected with a nonlinear function in series, in a preferred embodiment, the symmetric hysteresis model in the embodiment is a PI model, and the nonlinear function is a polynomial function.
Identifying a symmetric hysteresis model, namely a PI model, in a single piezoelectric ceramic model, wherein parameters of the PI model mainly comprise a threshold value of an operator and weight of the operator; in a preferred embodiment, the first step of this embodiment includes:
the method comprises the following steps of firstly, obtaining nonlinear hysteresis main loop data of a fast tilting mirror, and comprising the following steps:
step one, applying a triangular wave voltage signal with a full amplitude (100V) to the fast tilting mirror, as shown in fig. 2, to obtain an angle output signal shown in fig. 3, wherein the input signal and the output signal are data after normalization. Discretizing the input and output signals to obtain an input voltage vector X ═ X (1), X (2), …, X (n)]TAnd outputting the angle vector Y1＝[y1(1),y1(2),…,y1(n)]TWherein x (1), x (2), …, x (n), y1(1),y1(2),…,y1And (n) is the corresponding input voltage and output angle when the sampling time is 1, 2, … …, n.
Input voltage vector X ═ X (1), X (2), …, X (n)]TAnd outputting the angle vector Y1＝[y1(1),y1(2),…,y1(n)]T
Step one and two, output angle vector Y1Processing is carried out to obtain the length output vector Z of the single piezoelectric ceramic, wherein the length output vector Z is [ Z (1), Z (2), …, Z (n)]T，
Step one, the processing of two pairs of experimental data is obtained according to the analysis of the specific internal structure of the fast tilting mirror, namely the processing conditions are as follows: first the fast tilting mirror must be of this internal construction, see fig. 1; secondly, the piezoelectric ceramics No. 1 and No. 2 in fig. 2 have similar hysteresis characteristics.
Step three, distributing operator threshold values to the PI model, wherein if m operators are used, the operator threshold values are respectively
The weight corresponding to each operator threshold is rhoiThen, construct weight vector ρ ═ ρ1,ρ2,…,ρm]T；
Step four, using input voltage vector X ═ X (1), X (2), …, X (n)]TAnd output vector Z ═ Z (1), Z (2), …, Z (n)]TRho ═ rho in symmetric hysteresis model for single piezoelectric ceramic1,ρ2,…,ρm]TAnd (5) performing identification.
In a preferred embodiment, the first step and the fourth step of the present embodiment include:
using the input voltage vector X ═ X (1), X (2), …, X (n)]TConstructing matrices
Identifying weight vectors in PI models
And completing the identification of parameters in the symmetrical hysteresis model of the single piezoelectric ceramic.
In step two of this embodiment, a random signal is applied to the fast tilting mirror to obtain inner loop data of its nonlinear hysteresis, and the output angle vector of the fast tilting mirror is Y2＝[y2(1),y2(2),…,y2(n)]T. Meanwhile, inputting random signals into the identified symmetric model to obtain model output data ZPI＝[zp(1),zp(2),…,zp(n)]TAnd p is 1 or 2.
In a preferred embodiment, the nonlinear function h of a single piezoelectric ceramic in step three of this embodiment is h (z) ═ a1·z+a2·z2+a3·z3+a4·z4+a5·z5Z is input, h (z) is output, a1、a2、a3、a4、a5As the parameter(s) is (are),
for the symmetric hysteresis model of No. 1 piezoelectric ceramic, the input is v (t), and the output is z1，
h1(z1)＝a1·z1+a2·z1 2+a3·z1 3+a4·z1 4+a5·z1 5
For the symmetric hysteresis model of No. 2 piezoelectric ceramic, the input is u (t), and the output is z2，
h2(z2)＝a1·z2+a2·z2 2+a3·z2 3+a4·z2 4+a5·z2 5
In a preferred embodiment, in step three of this embodiment, f ═ b1·zp+b2·zp 2+b3·zp 3+b4·zp 4+b5·zp 5，b1、b2、b3、b4·、b5As the parameter(s) is (are),
zpis the output of the symmetrical hysteresis model of the single piezoelectric ceramic, and f is the output. Z of the present embodimentpIs z1Or z2(ii) a In zpIs z1For example, the following steps are carried out:
since v (t) + u (t) is 100v, and both pass through the same symmetric hysteresis model, z is1+z2＝1；
f＝h1(z1)-h2(z2)
＝(a1z1+…a5z5)-[(a1(1-z1)+…a5(1-z5)]
After simplification, f ═ b1·z1+b2·z1 2+b3·z1 3+b4·z1 4+b5·z1 5。
In a preferred embodiment, in step five of this embodiment, the output of the symmetric hysteresis model obtained in step two is used to construct an input matrix Φ, and the output angle of the fast tilting mirror obtained in step two is used to construct an output matrix Y2Identifying B ═ phi by using least square methodT·φ)-1·φT·Y2And completing the identification of the parameter B in the nonlinear function f.
At the moment, the construction of a single piezoelectric ceramic nonlinear hysteresis model is completed, and the modeling of the nonlinear hysteresis of the integral input voltage and the output angle of the rapid tilting mirror is also completed.
In a preferred embodiment, the present embodiment further includes a seventh step, where the seventh step includes:
the same input signals are respectively given to the fast tilting mirror and the identified model, the respective outputs are observed, the prediction error of the model is obtained by using the difference value, and the identification result of the model is observed, as shown in fig. 8 and 9.
Although the invention herein has been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that features described in different dependent claims and herein may be combined in ways different from those described in the original claims. It is also to be understood that features described in connection with individual embodiments may be used in other described embodiments.
Claims (9)
1. A method of identifying a hysteresis characteristic for a fast tilting mirror, the method comprising:
s1, establishing a symmetrical hysteresis model of the single piezoelectric ceramic of the rapid tilting mirror, wherein the input of the symmetrical hysteresis model is the input voltage of the single piezoelectric ceramic, the output of the symmetrical hysteresis model is the length of the single piezoelectric ceramic, and the input and the output are used for identifying the parameters of the symmetrical hysteresis model;
s2, inputting a random signal into the fast tilting mirror to obtain an output angle of the fast tilting mirror, converting the random signal into input voltages of two diametrically opposite piezoelectric ceramics in the fast tilting mirror, and respectively inputting the input voltages into a symmetric hysteresis model with identified parameters to obtain output of the symmetric hysteresis model of the two diametrically opposite piezoelectric ceramics;
s3, establishing a nonlinear function h of the single piezoelectric ceramic, wherein the input of the nonlinear function h is the output of a symmetrical hysteresis model of the single piezoelectric ceramic in S2, the output of the nonlinear function h is the length output of the single piezoelectric ceramic, and the parameter of the nonlinear function h is A;
s4, obtaining two radial nonlinear functions h of the piezoelectric ceramics according to S31And h2Obtaining a non-linear function f ═ h1-h2The parameter in the nonlinear function f is B;
s5, identifying and calculating the parameter B in the nonlinear function f by using the output angle of the fast tilting mirror obtained in S2 and the output of the symmetric hysteresis model;
s6, calculating a parameter A of the nonlinear function h according to the nonlinear function f of the identified parameter;
and S7, obtaining the lengths of the two pieces of piezoelectric ceramics in the diameter alignment by using the symmetric hysteresis model and the nonlinear function h of the calculated parameters, and obtaining the inclination angles of the two pieces of piezoelectric ceramics in the diameter alignment according to the difference value of the lengths of the two pieces of piezoelectric ceramics.
2. An identification method for the hysteresis characteristic of a fast tilting mirror according to claim 1, characterized in that the symmetric hysteresis model is a PI model.
3. A method for identifying a hysteresis characteristic of a fast tilting mirror as claimed in claim 2, wherein the S1 comprises:
s11, obtaining nonlinear hysteresis main loop data of the fast tilting mirror, comprising:
input voltage vector X ═ X (1), X (2), …, X (n)]TAnd outputting the angle vector Y1＝[y1(1)，y1(2)，…，y1(n)]T
S12, outputting angle vector Y1Processing is carried out to obtain the length output vector Z of the single piezoelectric ceramic, wherein the length output vector Z is [ Z (1), Z (2), …, Z (n)]T；
S13, allocating operator threshold values to the PI model, wherein if m operators are used, the operator threshold values are respectively
i is 1, 2, …, m; the weight corresponding to each operator threshold is rhoiThen, construct weight vector ρ ═ ρ1，ρ2，…，ρm]T；
S14, using input voltage vector X ═ X (1), X (2), …, X (n)]TAnd output vector Z ═ Z (1), Z (2), …, Z (n)]TRho ═ rho in symmetric hysteresis model for single piezoelectric ceramic1，ρ2，…，ρm]TAnd (5) performing identification.
4. A method for identifying hysteresis characteristics of a fast tilting mirror as claimed in claim 3, wherein S14 comprises:
using the input voltage vector X ═ X (1), X (2), …, X (n)]TConstructing matrices
Weight vector in PI model
And completing the identification of parameters in the symmetrical hysteresis model of the single piezoelectric ceramic.
5. A method of identifying hysteresis characteristics for a fast tilting mirror as claimed in claim 4 wherein the non-linear function h and the non-linear function f are polynomial functions.
6. A method for identifying hysteresis characteristics of fast tilting mirrors according to claim 5, wherein the nonlinear function h of a single piezoelectric ceramic in S3 is h (z) a1·z+a2·z2+a3·z3+a4·z4+a5·z5Z is input, h (z) is output, a1、a2、a3、a4、a5As the parameter(s) is (are),
7. a method of identifying hysteresis characteristics for a fast tilting mirror as claimed in claim 5, characterized in that f ═ b1·zp+b2·zp 2+b3·zp 3+b4·zp 4+b5·zp 5，b1、b2、b3、b4·、b5As the parameter(s) is (are),
zpis the output of the symmetrical hysteresis model of the single piezoelectric ceramic, and f is the output.
8. According to claimThe method for identifying the hysteresis characteristic of a fast tilting mirror of claim 1 or 7, wherein in S5, the input matrix Φ is constructed using the output of the symmetric hysteresis model obtained in S2, and the output matrix Y is constructed using the output angle of the fast tilting mirror obtained in S22Identifying B ═ phi by using least square methodT·φ)-1·φT·Y2And completing the identification of the parameter B in the nonlinear function f.
9. A method of identifying hysteresis characteristics for fast tilting mirrors as claimed in claim 8, the method further comprising S7, the S7 comprising:
inputting the same signals to the fast tilting mirror, the symmetric hysteresis model with the identified parameters and the nonlinear function h respectively, observing the difference value between the output of the fast tilting mirror and the inclination angle obtained according to S7, and obtaining the prediction error of the symmetric hysteresis model in series with the nonlinear function h by using the difference value.
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