﻿ 基于机器学习算法的谐振点漂移的伺服系统最优谐振抑制

# 基于机器学习算法的谐振点漂移的伺服系统最优谐振抑制 Optimal Resonance Suppression of Servo System with Resonance Point Drift Based on Machine Learning Algorithm

Abstract: In order to suppress the resonance of the servo system when the resonance point drifts, a notch filter is connected in the servo system to suppress the mechanical resonance of the two mass system. Firstly, the mathematical model of two mass system is established, the resonance mechanism is analyzed, and the causes of mechanical resonance are described; Secondly, the online search of drift resonance point is carried out by using the trisection method; Thirdly, the principle of notch filter is analyzed, and the width and depth parameters of notch filter are determined by simulated annealing algorithm; Then, on the basis of determining the optimal parameters, the robustness of the system is analyzed; Finally, based on the determination of the optimal parameters, experiments and system robustness analysis are carried out. The experimental results show that this method can not only quickly search the drift resonance point, effectively suppress the resonance and maintain the stability of the system, but also avoid the defect of time-consuming manual adjustment of parameters, and can accurately and quickly suppress the drift resonance point.

1. 引言

1) 利用三分法在线搜索谐振频率的效率高，搜索速度是现有文献中提出算法的搜索速度的10至20倍；

2) 利用机器学习算法中的模拟退火算法求解到陷波滤波器的最优参数，可以达到最优谐振抑制的目的；

3) 本文研究的谐振抑制方法既能抑制机械谐振，又能保持系统的稳定性；

4) 本文提出的最优谐振抑制方法可以快速、准确抑制谐振，避免手动调节参数耗时长的缺陷，提高谐振抑制的效率。

2. 二质量系统模型建立

Figure 1. Simplified model of two-mass system

$\left\{\begin{array}{l}{J}_{m}\frac{\text{d}{\omega }_{m}}{\text{d}t}={T}_{e}-{T}_{s}\\ {J}_{l}\frac{\text{d}{\omega }_{l}}{\text{d}t}={T}_{s}-{T}_{l}\\ {T}_{s}={K}_{s}\left({\theta }_{m}-{\theta }_{l}\right)+{K}_{W}\left({\omega }_{m}-{\omega }_{l}\right)\\ {\stackrel{˙}{\theta }}_{m}={\omega }_{m}\\ {\stackrel{˙}{\theta }}_{l}={\omega }_{l}\end{array}$ (1)

$G\left(s\right)=\frac{1}{{J}_{m}s}×{G}_{r}\left(s\right)=\frac{1}{{J}_{m}s}\frac{{s}^{2}+2p{\xi }_{r}{\omega }_{n}s+p{\omega }_{n}^{2}}{{s}^{2}+2{\xi }_{r}{\omega }_{n}s+{\omega }_{n}^{2}}$ (2)

${G}_{r}\left(s\right)=\frac{{s}^{2}+2p{\xi }_{r}{\omega }_{n}s+p{\omega }_{n}^{2}}{{s}^{2}+2{\xi }_{r}{\omega }_{n}s+{\omega }_{n}^{2}}$

$\left\{\begin{array}{l}{\xi }_{r}=\sqrt{\frac{\left({J}_{m}+{J}_{l}\right){K}_{W}^{2}}{4{K}_{s}{J}_{m}{J}_{l}}}\\ {\omega }_{n}=\sqrt{\frac{{K}_{s}\left({J}_{m}+{J}_{l}\right)}{{J}_{m}{J}_{l}}}\\ {\omega }_{an}=\sqrt{\frac{{K}_{s}}{{J}_{l}}}\\ p=\frac{{J}_{m}}{{J}_{m}+{J}_{l}}\end{array}$ (3)

3. 漂移谐振点的在线搜索

1) 如果 $h\left({x}_{1}\right)>h\left({x}_{2}\right)$，则谐振点位于区间 $\left[a,{x}_{2}\right]$，计算区间长度l；

2) 如果 $h\left({x}_{1}\right)=h\left({x}_{2}\right)$，则谐振点位于区间 $\left[{x}_{1},{x}_{2}\right]$，计算区间长度l；

3) 如果 $h\left({x}_{1}\right)，则谐振点位于区间 $\left[{x}_{1},b\right]$，计算区间长度l。

4. 陷波滤波器最优谐振抑制

4.1. 陷波滤波器原理

${G}_{N}\left(s\right)=\frac{{s}^{2}+2\pi \xi ks+{\omega }_{0}^{2}}{{s}^{2}+2\pi ks+{\omega }_{0}^{2}}$ (4)

${G}_{0}\left(s\right)={G}_{r}\left(s\right){G}_{N}\left(s\right)$ (5)

4.2. 陷波滤波器的参数确定

${G}_{o}\left(s\right)={G}_{r}\left(s\right){G}_{N}\left(s\right)=\frac{{s}^{2}+2p{\xi }_{r}{\omega }_{n}s+p{\omega }_{n}^{2}}{{s}^{2}+2{\xi }_{r}{\omega }_{n}s+{\omega }_{n}^{2}}\cdot \frac{{s}^{2}+2\pi \xi ks+{\omega }_{0}^{2}}{{s}^{2}+2\pi ks+{\omega }_{0}^{2}}$ (6)

$s=j\omega$，则级联陷波器在任意频率处的幅值为：

${H}_{0}\left(\omega \right)=|{G}_{0}\left(j\omega \right)|=\sqrt{\frac{{\left(j\omega \right)}^{2}+2p{\xi }_{r}{\omega }_{n}\left(j\omega \right)+p{\omega }_{n}^{2}}{{\left(j\omega \right)}^{2}+2{\xi }_{r}{\omega }_{n}\left(j\omega \right)+{\omega }_{n}^{2}}}\sqrt{\frac{{\left(j\omega \right)}^{2}+2\pi \xi k\left(j\omega \right)+{\omega }_{0}^{2}}{{\left(j\omega \right)}^{2}+2\pi k\left(j\omega \right)+{\omega }_{0}^{2}}}$ (7)

Figure 2. Bode diagram of a two mass system with a given reference threshold

$\underset{{\omega }_{i}\in \left[{\omega }_{1},{\omega }_{2}\right]}{{H}_{0}\left({\omega }_{i}\right)}\le {H}_{th}$ (8)

$\varphi =\text{arctan}\frac{2\pi \left(\xi -1\right)k{\omega }_{c}\left({\omega }_{0}^{2}-{\omega }_{c}^{2}\right)}{4{\pi }^{2}\xi k{\omega }_{c}^{2}+{\left({\omega }_{0}^{2}-{\omega }_{c}^{2}\right)}^{2}}$ (9)

$\underset{\underset{{\omega }_{i}\in \left[{\omega }_{1},{\omega }_{2}\right]}{{H}_{0}\left({\omega }_{i}\right)\le {H}_{th}}}{\mathrm{min}}\varphi =\mathrm{arctan}\frac{2\pi \left(\xi -1\right)k{\omega }_{c}\left({\omega }_{0}^{2}-{\omega }_{c}^{2}\right)}{4{\pi }^{2}\xi k{\omega }_{c}^{2}+{\left({\omega }_{0}^{2}-{\omega }_{c}^{2}\right)}^{2}}$ (10)

Step 1 选取最大迭代次数 ${K}_{\mathrm{max}}$，可行域内任意收敛方向 ${v}_{r}$ 和最优收敛方向 ${v}_{b}$，以及可行域内任意初始状态 ${x}_{0}=\left({\xi }_{0},{k}_{0}\right)$

Step 2 令 $T\left(i\right)=1-i/\left({K}_{\mathrm{max}}+1\right)$，则T单调递减趋于0，随机选择正参数 $\lambda$，令 $v\left(i\right)=T\left(i\right){v}_{r}+\left(1-T\right){v}_{b}$${x}_{i}={x}_{0}+\lambda \left(i\right)v\left(i\right)$

Step 3 计算相角裕度 $\phi \left({x}_{0}\right)$$\phi \left({x}_{i}\right)$ 并进行比较，如果满足 $\phi \left({x}_{0}\right)>\phi \left({x}_{i}\right)$，就将 ${x}_{i}$ 接收为新的初始状态；否则以概率 $p=\mathrm{exp}\left\{-\left(f\left({x}_{i}\right)-f\left({x}_{0}\right)\right)/T\left(i\right)\right\}$ 接收 ${x}_{i}$ 为新的状态；

Step 4 重复Step 3，直至得到对可行域内任意的 ${x}_{j}$，都有 $\phi \left({x}_{j}\right)<\phi \left({x}^{\ast }\right)$ 或达到最大迭代次数时的相角裕度最小值所对应的 ${x}^{\ast }$，则得到的最优解为 ${x}^{\ast }$

5. 实验

5.1. 谐振抑制实验

Table 1. Parameters of permanent magnet synchronous motor

Figure 3. Experimental setup

Figure 4. Current response

Figure 5. Speed response

(a)(b)

Figure 6. Spectrum; (a) Spectrum before suppression; (b) Spectrum after suppression

5.2. 鲁棒性分析

Figure 7. Robustness analysis results

6. 总结

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