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1. 第二炮兵工程大学控制科学与工程系, 西安 710025;
2. 北京航空航天大学 能源与动力工程学院, 北京 100083

Design of dynamic total variance and its fast algorithm
WANG Lixin1 , LI Can1 , JIANG Zhou2 , ZHU Zhanhui1 , TIAN Ying1
1. Department of Control Science and Engineering, The Second Artillery Engineering University, Xi'an 710025, China ;
2. School of Energy and Power Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-07-20; Accepted: 2015-10-30; Published online: 2015-12-17 10:41
Foundation item: Equipment Technical Infrastructure Projects of the Second Artillery(EP114054)
Corresponding author. E-mail:wlxxian@sina.com
Abstract: There exist two shortcomings in Allan variance approach when it is used to analyze random error. First, the vibration of estimated value is large in long-term correlation. Second it can't follow the dynamic change of signal.This paper integrates the thought and advantage of total variance and dynamic allan variance, and proposes the dynamic total variance approach. Firstly, original data is truncated by window function. Secondly, we continue the data in the window. Then the data after continuation is analyzed by total variance, so that local random characteristic is acquired. With the window sliding, the change of random characteristic of original signal with time is obtained. Through verification, the dynamic total variance can solve the two shortcomings of Allan variance approach. Eventually, linear vibration trial of hemispherical resonator gyro(HRG) is designed to clarify this new algorithm, which indicates that dynamic total variance possesses advantages in analysis accuracy and data quantity that it used. However, this new algorithm has the problem of excessive calculation and low speed of analysis, so we derive recursive formula of it, based on which fast algorithm is provided.
Key words: Allan variance approach     total variance algorithm     dynamic total variance algorithm     recursive formula     fast algorithm

1 Allan方差及陀螺随机误差分析 1.1 Allan方差法原理

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1.2 陀螺随机误差

1)角度随机游走

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2)零偏不稳定性

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3)速率随机游走

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4)速率斜坡

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5)量化噪声

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1.3 Allan方差法的缺陷

 图 1 平稳白噪声和方差渐大白噪声 Fig. 1 Stationary white noise and white noise with gradually bigger variance
 图 2 平稳白噪声和方差渐大白噪声Allan方差分析 Fig. 2 Allan variance analysis of stationary white noise and white noise with gradually bigger variance

1)在长相关时间下，由于Allan方差估计的自由度比较小，导致Allan方差估计值置信度比较低，相对误差比较大，震荡非常严重。

2)平稳白噪声和方差渐大白噪声分析结果相同，说明Allan方差无法跟踪随机序列的动态变化(即无法跟踪信号的方差变化)。

2 总方差法原理分析

Allan方差法存在的第1个缺陷是由于估计值的自由度比较小，那么很自然的一个想法就是通过数据延拓的方法增加数据的自由度。总方差(total variance)法就是通过镜像映射将原始数据进行延长，达到增加估计值置信度、稳定估计值的效果。

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 随机误差组分 b c F(τ) 零偏不稳定性 3/2 0 3.000 角度随机游走 24(ln 2)2/π2 0.222 2.097 量化噪声 140/151 0.358 1.514

 图 3 Allan方差与总方差相对误差 Fig. 3 Relative error of Allan variance and total variance

 图 4 随机序列的Allan方差和总方差分析 Fig. 4 Allan variance and total variance analysis of random sequence
3 动态Allan方差法分析

Allan方差法的第2个缺陷是无法跟踪信号的非平稳变化，那么可以用窗函数的方法，逐段截取数据对其进行Allan方差分析，窗函数就像显微镜一般，显微出原始数据的局部特性，然后将这些特性在时间轴上展开就可以得到随机特性随时间的变化，这就是动态Allan方差法的理论基础。动态Allan方差法的具体实现步骤[14-15]

1)确定原始数据的分析时间起点t1和窗函数宽度W，要求

2)用中心点为t1、宽度为W的窗函数截断原始信号Ω(t)。

3)计算窗口内数据的Allan方差值σA2(t1, τ)，并使用最小二乘拟合的方法得到此时的随机误差系数。

4)将窗口滑动到下一个时刻t2，要求以t2为中心的窗要与以t1为中心的窗互相重叠。重复步骤2)和步骤3)，得到此时窗口内的Allan方差值σA2(t2, τ)和随机误差系数。

5)以此类推，重复上述过程，得到Allan方差值的集合{σA2(ti, τ)}(i=1, 2, …, n)和随机误差系数序列。以tτ为变量，将Allan方差值的变化展现在三维分布图上，就得到了动态Allan分析的结果；将各时刻的随机误差系数在时间轴上展开，得到随机误差特性的动态变化过程。

 图 5 平稳白噪声和方差渐大白噪声动态Allan方差分析 Fig. 5 Dynamic Allan variance analysis of stationary white noise and white noise with gradually bigger variance
4 动态总方差法设计

1)确定原始数据的分析起点t1和窗口宽度L

2)用中心点为t1、宽度为L窗函数截取原始信号Ω(t)，得到窗口截断信号ΩT(t1)，支撑变量描述窗内渐逝的时间，记为t′，则

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3)将窗内信号ΩT(t1, t′)按式(15)给出的方式进行镜像映射，得到延拓序列ΩT*(t1, t′)。

4)按照式(16)给出的总方差法计算式计算得到t1时刻的总方差值σtol2(t1, τ)，同时使用最小二乘拟合得到t1时刻的随机误差系数。

5)将窗口滑动到t2，重复步骤2)~步骤4)得到t2时刻的动态总方差值σtol2(t2, τ)和随机误差系数。以此类推，得到动态总方差法序列{σtol2(ti, τ)}(i=1, 2, …, N)和各时刻的随机误差系数。

6)将动态总方差值展示在σtol-τ-t三维图中来分析信号的非平稳特征；将随机误差系数在时间轴上展开得到随机误差随时间的变化规律。

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5 动态试验验证

 图 6 原始随机信号 Fig. 6 Original random signal
 图 7 陀螺输出的动态总方差分析 Fig. 7 Dynamic total variance analysis of gyro output

 图 8 动态Allan方差分析结果 Fig. 8 Results of dynamic Allan variance analysis
 图 9 动态总方差法分析结果 Fig. 9 Results of dynamic total variance analysis

1)从数量级上来讲，动态总方差法分析精度比动态Allan方差法至少高一个数量级。

2)角度随机游走和零偏不稳定性系数在振动过程中增大，而量化噪声系数在振动过程中减小，说明周期性振动弥补了量化噪声的周期性误差。

 处理算法 Q N B V R Allan方差(1 600) 3.8×10-3 1.8×10-4 3.8×10-3 2.8×10-2 6.7×10-2 动态总方差(401) 3.2×10-3 1.9×10-4 3.9×10-3 2.9×10-2 7.5×10-2 动态Allan(401) 4.8×10-2 2.5×10-3 3.7×10-2 0.4 1.2

6 动态总方差快速算法

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 数据长度 运行时间/s 传统算法 快速算法 102 3.0×10-3 3.1×10-4 103 0.30 4.5×10-3 104 118 0.25

7 结论

1)本文设计了动态总方差法，当使用相同的数据量进行分析时，动态总方差法相对于动态Allan方差法的分析精度至少提高了一个数量级；当分析精度一致时，动态总方差法需要的数据量是Allan方差法的1/4。

2)本文设计了动态总方差法的快速算法，对比快速算法和定义式算法的运行时间可以看出，快速算法极大地减少了算法的运行时间，而且原始数据量越大，则快速算法的优势越明显。

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#### 文章信息

WANG Lixin, LI Can, JIANG Zhou, ZHU Zhanhui, TIAN Ying

Design of dynamic total variance and its fast algorithm

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(7): 1352-1360
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0488