Modeling method of piezoelectric fast-tilting mirror
By establishing a series structure of system delay, rate-dependent hysteresis and linear dynamic models, the modeling problem of piezoelectric fast tilt mirrors in a wide frequency range is solved, and high-precision dynamic characteristic description and performance improvement are achieved.
Patent Information
- Application Number
- CN202411106694.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-08-13
- Publication Date
- 2025-08-12
- Estimated Expiration
- 2044-08-13
AI Technical Summary
The existing piezoelectric fast tilt mirror modeling methods are difficult to accurately describe their dynamic characteristics over a wide frequency range, especially because of the large positioning accuracy error caused by hysteresis nonlinearity.
A series structure consisting of the system delay model, rate-dependent hysteresis model and linear dynamic model is adopted. By identifying each model parameter, an FSM system model is established, and the rate-dependent hysteresis inverse model is combined with the rate-dependent hysteresis inverse compensation to reduce the influence of the system dynamics.
High-precision modeling of piezoelectric fast tilting mirrors over a wide frequency range is realized, and the positioning accuracy and overall performance of the system are improved.
Smart Images
Figure CN118862511B_ABST
Abstract
Description
Technical Field
[0001] The invention relates to a Hammerstein-like modeling method for a piezoelectric fast-tilt mirror, and belongs to the field of wide-frequency domain modeling of piezoelectric fast-tilt mirrors. Background Art
[0002] In satellite laser communication systems, the stable tracking capability of the beam control system is crucial to maintaining the stability of the communication link. As a key actuator in the beam control system, the piezoelectric fast tilt mirror (FSM) features high closed-loop bandwidth and fast response speed, enabling precise aiming and stable tracking of the communication laser beam. However, the piezoelectric ceramic actuators within the FSM exhibit hysteresis and nonlinearities such as multi-value mapping, nonlocal characteristics, and memory effects, which can result in full-scale errors of up to 15%, severely impacting the FSM's positioning accuracy.
[0003] To study the effects of hysteresis nonlinearity, various modeling approaches have been proposed. Among them, phenomenological hysteresis models have attracted widespread attention due to their relatively simple mathematical structure. Most hysteresis models have analytical inverse models that can effectively counteract hysteresis effects through open-loop inverse compensation. However, these hysteresis models are primarily used to describe rate-independent hysteresis. As signal frequency increases, effectively describing dynamic hysteresis becomes challenging. To address this issue, two common modeling approaches have been proposed: rate-dependent models and Hammerstein models. Rate-dependent models use linear functions to express the relationship between the slope of the hysteresis loading curve and the control input rate, fully accounting for the impact of input signal rate variations. These rate-dependent models describe dynamic hysteresis by incorporating correction terms related to the derivative of the input signal. Hammerstein models separate the system into rate-independent hysteresis and linear dynamics, and connect these two components in series to describe the input-output relationship of the entire system. These models capture the overall dynamic behavior of the system. Rate-dependent models can only effectively describe the system's dynamic performance within a certain frequency range (a few hundred Hz). At higher frequencies, other system characteristics come into play, making description difficult using correction terms related to the derivative alone. Although the Hammerstein model can describe the dynamic characteristics of the system within a wide frequency range, its ability to describe nonlinear systems is still limited. For more complex nonlinear systems, model establishment, parameter identification, and subsequent control strategy design may pose greater challenges. Summary of the Invention
[0004] In response to the problem of how to accurately model a piezoelectric fast-tilt mirror over a wide frequency range, the present invention provides a method for modeling a piezoelectric fast-tilt mirror with high accuracy over a wide frequency range by using a series structure consisting of delay, rate-dependent hysteresis model and linear dynamics.
[0005] A piezoelectric fast-tilt mirror modeling method of the present invention comprises:
[0006] S1. Establish the FSM system model, including the system delay model, the rate-dependent hysteresis model and the linear dynamics model G(s). The input of the system delay model is u(t) and the output is v(t). The input of the rate-dependent hysteresis model is v(t) and the output is H D (v(t)), the input of the linear dynamic model is H D (v(t)), the output is y(t);
[0007] S2. determining a system delay model according to the delay of the piezoelectric fast-tilt mirror;
[0008] S3. Use the low-frequency sinusoidal input signal u(t) and the corresponding output signal y(t) of the piezoelectric fast tilt mirror to identify the parameters of the main rise and main fall curves. Based on the parameters, a rate-independent hysteresis model is established using the hysteresis curve description method. The output H of the rate-dependent hysteresis model is D (v(t)) is the output H of the rate-independent hysteresis model S The sum of (v(t)) and the rate-related term Ψ(v(t));
[0009] S4. Input the low-intermediate-frequency sinusoidal input signal u(t) into the FSM system model, and use the output v(t) of the system delay model and the output signal y(t) corresponding to the low-intermediate-frequency sinusoidal input signal u(t) of the piezoelectric fast tilt mirror to identify the rate-dependent hysteresis model and identify the parameters of the rate-dependent hysteresis model;
[0010] S5. Input the high-frequency sinusoidal input signal u(t) into the FSM system model and use the output H of the rate-dependent hysteresis model. D The linear dynamic model G(s) is identified by the output signal y(t) corresponding to the high-frequency sinusoidal input signal u(t) of the piezoelectric fast tilt mirror, and the parameters of the linear dynamic model G(s) are obtained.
[0011] Preferably, in S2, the step input signal u(t) and the corresponding output signal y(t) are used to identify the delay time of the piezoelectric fast tilt mirror, and then determine the system delay model.
[0012] As a preference, in S4, the output H of the rate-independent hysteresis model SThe rate-dependent term Ψ(v(t)) is obtained by performing a differential operation on the output signal y(t) corresponding to the low-intermediate-frequency sinusoidal input signal u(t) of the piezoelectric fast-tilt mirror. The parameters of the rate-dependent term Ψ(v(t)) are obtained based on the input and output of the rate-dependent term Ψ(v(t)) and the least squares method.
[0013] Preferably, in S5, the least squares method is used to identify the parameters of the linear dynamic model G(s).
[0014] The beneficial effects of the present invention are as follows: the present invention is a unique improved Hammerstein-like modeling method, which ultimately enables the piezoelectric fast-tilt mirror to have higher modeling accuracy within a wider frequency range. On the one hand, the model adopts a series structure composed of delay, rate-dependent hysteresis model and linear dynamics, comprehensively considering the system delay and hysteresis nonlinearity, and can accurately describe the dynamic characteristics of the piezoelectric fast-tilt mirror within a wider frequency range; on the other hand, the FSM system model of the present invention can describe more system dynamic characteristics that can be accurately modeled due to its rate-related terms, and can effectively reduce the influence of system dynamics by constructing a feedforward inverse compensation method based on the rate-dependent hysteresis inverse model, thereby improving the overall performance of the system. The method of the present invention can be implemented in both software and hardware in actual engineering, and has strong practical value and engineering significance. BRIEF DESCRIPTION OF THE DRAWINGS
[0015] Figure 1 Schematic diagram of the structure of the FSM system model of the present invention;
[0016] Figure 2 Schematic diagram of rate-dependent hysteresis model;
[0017] Figure 3 It is the open-loop system response diagram of the FSM system model;
[0018] Figure 4 Comparison of the prediction and experimental results of the FSM system model of the present invention under multi-frequency sinusoidal signals
[0019] Figure 5 Comparison of the model predictions and experimental results of the FSM system of the present invention under sinusoidal excitation of different frequencies, where (a) the intermediate frequency f = 500 Hz, (b) the intermediate frequency f = 1000 Hz, (c) the intermediate frequency f = 1500 Hz, (d) the intermediate frequency f = 2000 Hz, (e) the intermediate frequency f = 3000 Hz, and (f) the intermediate frequency f = 4000 Hz. DETAILED DESCRIPTION
[0020] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.
[0021] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.
[0022] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.
[0023] This embodiment provides a method for modeling a piezoelectric fast-tilt mirror, comprising:
[0024] Step 1: Establish the FSM system model, such as Figure 1 As shown, it includes the system delay model, the rate-dependent hysteresis model and the linear dynamics model G(s), where the input of the system delay model is u(t) and the output is v(t), the input of the rate-dependent hysteresis model is v(t) and the output is H D (v(t)), the input of the linear dynamic model is H D (v(t)), the output is y(t);
[0025] Step 2: Determine the system delay model based on the delay of the piezoelectric fast-tilt mirror:
[0026] Input a step signal u(t) and apply it to the piezoelectric fast tilt mirror to be modeled to obtain the output signal y(t);
[0027] The delay time τ of the piezoelectric fast tilt mirror is determined by using the step input signal u(t) and the corresponding output signal y(t), and the system delay model e is determined based on -τs ;
[0028] Step 3: Apply the low-frequency sinusoidal input signal u(t) to the piezoelectric fast-tilt mirror to be modeled to obtain the output signal y(t);
[0029] The low-frequency sinusoidal input signal u(t) and the corresponding output signal y(t) of a piezoelectric fast-tilting mirror are used to identify the parameters of the main rise and main fall curves. Based on these parameters, a rate-independent hysteresis model is established using the hysteresis curve description method.
[0030] At low-frequency inputs, the effects of the system delay model, rate-dependent terms, and linear dynamics model G(s) can be ignored. Therefore, the rate-independent hysteresis model identified by the low-frequency sinusoidal input signal u(t) and the corresponding output signal y(t) is sufficient to reflect the rate-independent hysteresis characteristics of the piezoelectric fast-tilt mirror.
[0031] Step 4: Apply the low-intermediate frequency sinusoidal input signal u(t) to the piezoelectric fast tilt mirror to be modeled to obtain the output signal y(t); then input the low-intermediate frequency sinusoidal input signal u(t) into the FSM system model, delay the model output v(t), and use the v(t) and the output signal y(t) corresponding to the low-intermediate frequency sinusoidal input signal u(t) to identify the rate-dependent hysteresis model, identify the parameters of the rate-dependent hysteresis model, and the output H of the rate-dependent hysteresis model. D (v(t)) is the output H of the rate-independent hysteresis model S The sum of (v(t)) and the rate-dependent term Ψ(v(t)). The form of the rate-dependent term is not fixed, except Figure 2 Other rate-related models are also applicable.
[0032] The low-intermediate-frequency sinusoidal input signal u(t) only affects the delay model and the rate-dependent hysteresis model, and has a very small effect on the linear dynamic model G(s) and can be ignored. Therefore, the rate-dependent hysteresis model is identified by using the low-intermediate-frequency sinusoidal input signal u(t) and the corresponding output signal y(t), which can accurately describe the rate-dependent hysteresis characteristics of the piezoelectric fast-tilt mirror.
[0033] The output G of the rate-independent hysteresis model S The rate-dependent term Ψ(v(t)) is obtained by performing a differential operation on the output signal y(t) corresponding to the low-intermediate frequency sinusoidal input signal u(t). The parameters of the rate-dependent term Ψ(v(t)) are obtained based on the input and output of the rate-dependent term Ψ(v(t)) and the least squares method.
[0034] Step 5: Apply the high-frequency sinusoidal input signal u(t) to the piezoelectric fast tilt mirror to be modeled to obtain the output signal y(t); then input the high-frequency sinusoidal input signal u(t) into the FSM system model, and use the output H of the rate-dependent hysteresis model. D The linear dynamic model G(s) is identified by the output signal y(t) corresponding to the high-frequency sinusoidal input signal u(t), and the parameters of the linear dynamic model G(s) are obtained using the least squares method.
[0035] This embodiment combines the advantages of rate-dependent models and Hammerstein-like models to propose an improved Hammerstein-like model: the FSM system model, which adopts a series structure consisting of delay, rate-dependent hysteresis model, and linear dynamics. Due to its rate-dependent terms, the FSM system model can describe more system dynamic characteristics that can be accurately modeled, and can effectively mitigate the impact of system dynamics by constructing a feedforward inverse compensation method based on the rate-dependent hysteresis inverse model, thereby improving the overall performance of the system. The FSM system model of this embodiment comprehensively considers system delay and hysteresis nonlinearity, and can accurately describe the dynamic characteristics of the FSM system over a wide frequency range. Figure 1 The rate-dependent model in Figure 2 As shown,
[0036] use Figure 1 and Figure 2 Schematic diagram of the model structure, the present invention has completed the establishment of a wide frequency range model of the piezoelectric fast tilt mirror through the above steps. Figure 3 The comparison between the actual open-loop response of the FSM system and the model output response is shown. In order to verify that the method of the present invention has high modeling accuracy for the piezoelectric fast tilt mirror in a wide frequency range, a multi-frequency sinusoidal signal containing 15Hz, 40Hz, 75Hz and 100Hz and multiple high-frequency sinusoidal signals are sequentially input into the system. The results are shown in Figure 2. Figure 4 and Figure 5 It can be observed that the method of the present invention has a good modeling effect in a wide frequency domain, which fully verifies the effectiveness and feasibility of the method of the present invention.
[0037] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the invention. It should be understood that many modifications may be made to the illustrative embodiments, and that other arrangements may be devised, without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in ways other than those described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be employed in conjunction with other described embodiments.
Claims
1. A piezoelectric fast tilt mirror modeling method, characterized in that: The method comprises: S1. Establish the FSM system model, including the system delay model, the rate-dependent hysteresis model and the linear dynamics model G(s). The input of the system delay model is u(t) and the output is v(t). The input of the rate-dependent hysteresis model is v(t) and the output is H D (v(t)), the input of the linear dynamic model is H D (v(t)), the output is y(t); S2. determining a system delay model according to the delay of the piezoelectric fast-tilt mirror; S3. Use the low-frequency sinusoidal input signal u(t) and the corresponding output signal y(t) of the piezoelectric fast tilt mirror to identify the parameters of the main rise and main fall curves. Based on the parameters, a rate-independent hysteresis model is established using the hysteresis curve description method. The output H of the rate-dependent hysteresis model is D (v(t)) is the output H of the rate-independent hysteresis model s The sum of (v(t)) and the rate-related term Ψ(v(t)); S4. Input the low-intermediate-frequency sinusoidal input signal u(t) into the FSM system model, and use the output v(t) of the system delay model and the output signal y(t) corresponding to the low-intermediate-frequency sinusoidal input signal u(t) of the piezoelectric fast tilt mirror to identify the rate-dependent hysteresis model and identify the parameters of the rate-dependent hysteresis model; S5. Input the high-frequency sinusoidal input signal u(t) into the FSM system model and use the output H of the rate-dependent hysteresis model. D The linear dynamic model G(s) is identified by the output signal y(t) corresponding to the high-frequency sinusoidal input signal u(t) of the piezoelectric fast tilt mirror, and the parameters of the linear dynamic model G(s) are obtained.
2. The piezoelectric fast tilt mirror modeling method according to claim 1, characterized in that: In S2 , the delay time of the piezoelectric fast tilt mirror is identified using the step input signal u(t) and the corresponding output signal y(t), thereby determining the system delay model.
3. The piezoelectric fast tilt mirror modeling method according to claim 1, characterized in that: In S4, the output H of the rate-independent hysteresis model s The rate-dependent term Ψ(v(t)) is obtained by performing a differential operation on the output signal y(t) corresponding to the low-intermediate-frequency sinusoidal input signal u(t) of the piezoelectric fast-tilt mirror. The parameters of the rate-dependent term Ψ(v(t)) are obtained based on the input and output of the rate-dependent term Ψ(v(t)) and the least squares method.
4. The piezoelectric fast tilt mirror modeling method according to claim 1, characterized in that: In S5, the least squares method is used to identify the parameters of the linear dynamic model G(s).
5. A computer-readable storage device storing a computer program, characterized in that: When the computer program is executed by a processor, the steps of the piezoelectric fast-tilt mirror modeling method according to any one of claims 1 to 4 are implemented.
6. A piezoelectric fast tilt mirror modeling device, 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 steps of the piezoelectric fast-tilt mirror modeling method according to any one of claims 1 to 4.
7. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method for modeling a piezoelectric fast-tilt mirror according to any one of claims 1 to 4 are implemented.