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System identification and control for large aperture fast-steering mirror driven by PZT
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摘要:
本文提出了一种基于随机梯度优化算法的倾斜镜模型辨识方法，实现对大口径压电倾斜镜的复杂频率响应规律的辨识与控制带宽提高。文章介绍了压电倾斜镜原理和数学模型，描述了随机梯度优化算法在模型辨识的应用过程，并通过实验验证的方式检验了算法辨识模型的准确性以及在提高系统控制带宽方面的能力；最后，利用随机梯度下降算法本文还开展了对抖动输入频谱的辨识，结合倾斜镜模型的辨识结果，获得了对特定频谱区域更高抑制能力的控制效果。
Abstract:Novel system identification for large aperture fast-steering mirror (FSM) is presented in this paper. Using the stochastic parallel gradient descent method (SPGD), the new system identification method is able to identify the complex piezoelectric fast-steering mirror (PZT-FSM) model exactly and greatly improve the correction effect. The principle and mathematical model of the PZT-FSM are stated briefly in the paper firstly. Then the use process of the SPGD algorithm in the system identification for the large aperture PZT-FSM is presented. By using the identified model, the validity and feasibility of the proposed approach is confirmed by our close-loop experiments. To expand the usage of the new method, the input jitter spectrum is also identified using the similar method, which enables us to get a higher correction effect for the special frequency region.
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Key words:
- fast steering mirror /
- structural resonance /
- PZT /
- large aperture
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Overview: Fast steering mirror driven by piezoelectric material has been used widely for control of opto-axis stabilization. The system identification for such fast steering mirror is very important. It has decided whether we are able to reach the mirror's full correction potential. However, the system identification is difficult for mirror with large aperture, because we found that the larger aperture, the more complex of the mirror responding.
Novel system identification method based on the stochastic parallel gradient descent (SPGD) algorithm is presented for the large aperture fast-steering mirror (FSM). The proposed method can identify the complicated frequency response of the large aperture FSM accurately and improve the correcting effect of the system. The principle and mathematical model of the piezoelectric fast-steering mirror (PZT-FSM) are stated briefly in the paper firstly. Then the use process of the SPGD algorithm in the system identification for the large aperture PZT-FSM is presented. A PZT-FSM with the diameter of 250 mm is taken as an example to test the effectiveness of the proposed method. Compared with the actual frequency response curves, the frequency response curves of the two kinds of models identified by different order (7th-order and 9th-order) are consistent with the actual curves. Especially for the 9th-order identified model, both the overall distribution of the curves and the local details are highly consistent with the actual curves. As a contrast, the results using the Levy method for identification are also presented. Levy method cannot accurately identify complex models. Even if the order of identification is increased, there is no significant improvement in performance.
In order to verify the accuracy of the identified model, we conduct two confirmatory experiments. First, we explore the resonance elimination of PZT-FSM by the identified model. After the resonance elimination, the frequency response is close to the frequency response of a pure-delay system. The amplitude response of the system is close to the ideal case, and the fluctuation of the amplitude response is within ± 2 dB in the range of 0 Hz to 2000 Hz. The closed-loop control effects of PZT-FSM before and after the resonance elimination are presented in the next section, where we found that the error bandwidth of the PZT-FSM has been significantly improved after the resonance elimination. Under the same condition of overshoot, the closed-loop error bandwidth increases from 105 Hz to 210 Hz. We compared the correction capability of beam jitter before and after the resonance elimination. After the correction without resonance elimination, the root mean square (RMS) of the jitter amplitude declined from 11.6 μrad to 7.9μrad. However, after the resonance elimination and correction, the RMS of the jitter amplitude decreased to 3.5 μrad. The experimental results show that the proposed method can not only eliminate the resonance, but also improve the closed-loop error bandwidth.
To expand the usage of the new method, the input jitter spectrum is also identified using the similar method, which enables us to get a higher correction effect for the special frequency region. Accurate model identification for the large aperture FSM is also meaningful to advanced control methods, such as LQG control and adaptive filters control method. Through the combination of these advanced control methods, on-line identification and correction of specific resonance frequency can be realized.
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图 1 用于辨识压电倾斜镜系统模型光路结构示意图
Figure 1. Optical structure for piezoelectric fast-steering mirror system identification
图 2 250 mm口径压电倾斜镜系统对不同频率的响应曲线
Figure 2. Frequency response of piezoelectric fast-steering mirror with a 250 mm aperture
图 3 辨识模型与实测模型对不同频率的响应曲线。(a) 7阶模型辨识结果；(b) 9阶模型辨识结果；(c) 7阶模型辨识局部细节；(d) 9阶模型辨识局部细节
Figure 3. Frequency responses of identified model and detected models. (a) Identification of a 7th order model; (b) Identifi-cation of a 9th order model; (c) Local detail for the 7th order identification; (d) Local detail for the 9th order identification
图 4 同样阶次条件下利用Levy方法辨识结果
Figure 4. Identification result using Levy method with the same identified order
图 5 250 mm压电倾斜镜系统中消谐振后，系统的实测响应曲线
Figure 5. Actual detected frequency response after resonant depression for the 250 mm piezoelectric fast-steering mirror
图 6 消谐振前后闭环误差控制曲线。(a)未消谐振闭环误差曲线；(b)消谐振后闭环误差曲线
Figure 6. Close loop of the FSM before and after resonant depression. (a) Close loop of the FSM without resonant depression; (b) Close loop of the FSM with resonant depression
图 8 消谐振前后倾斜镜校正残差分布
Figure 8. Residual jitter error with and without resonant depression
图 9 (a) 输入抖动频谱分布和校正后残余抖动频谱分布；(b)积分功率谱曲线
Figure 9. (a) Power spectrums of input jitter and residual jitter error; (b) Power spectrum integral of (a)
图 10 含输入抖动频谱成分的系统误差控制曲线
Figure 10. The close loop of the FSM considering input jitter spectrum
图 11 将入射抖动频谱进行辨识后结合PZT倾斜镜模型进行抖动抑制效果。(a)输入抖动频谱分布和校正后残余抖动频谱分布；(b)积分功率谱曲线
Figure 11. The effect of jitter control combining identification of PZT-FSM and input jitter spectrum. (a) Power spectrum of input jitter and residual jitter error; (b) Corresponding power spectrum integral of (a)
表 1 9阶参数模型辨识结果
Table 1. Identification results for the 9th order model
序号 极点频率 零点频率 极点振荡因子 零点振荡因子 1 1261.6 2243.2 0.021 0.384 2 715.2 1337.4 0.192 0.464 3 710.0 1186.8 0.020 0.030 4 661.6 664.5 0.018 0.055 5 625.9 649.3 0.019 0.117 6 576.2 643.2 0.021 0.023 7 528.2 577 0.822 0.047 8 527.6 577 0.046 0.063 9 298.6 304 0.016 0.020 
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