﻿ 线性熵的系统故障熵模型及其时变研究
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 智能系统学报  2021, Vol. 16 Issue (6): 1136-1142  DOI: 10.11992/tis.202006034 0

### 引用本文

CUI Tiejun, LI Shasha. System fault entropy model and its time-varying based on linear entropy[J]. CAAI Transactions on Intelligent Systems, 2021, 16(6): 1136-1142. DOI: 10.11992/tis.202006034.

### 文章历史

1. 辽宁工程技术大学 安全科学与工程学院，辽宁 阜新 123000;
2. 辽宁工程技术大学 工商管理学院，辽宁 葫芦岛 125105

System fault entropy model and its time-varying based on linear entropy
CUI Tiejun 1, LI Shasha 2
1. College of Safety Science and Engineering, Liaoning Technical University, Fuxin 123000, China;
Abstract: The concept of system fault entropy is proposed to study the general rule, fault variation degree, and the fault information of the system fault under different superposed factors. A linear entropy model with a multifactor phase divided into two states is derived based on the linear uniformity of the linear entropy. The linear entropy can represent the system fault entropy; thus, the time-varying characteristics of the system fault entropy can be studied. The system fault probability distribution is obtained by counting the faults under the superposition of different factor states in continuous time intervals. Then, the time-varying curve of the system fault entropy is drawn. The results show that at least three tasks can be completed: 1) Obtaining the change of the system fault entropy under different factors from the change law. 2) Obtaining the general change rule of the system fault entropy. 3) Studying the stability of the system reliability. The research results can be applied to fault and data analyses in various fields in similar cases.
Key words: intelligent science    safety science    safety system engineering    space fault tree    factor space    system fault entropy    linear entropy    time-varying analysis

1 系统故障熵

2 线性熵

 $H(R) = \frac{pH(P) + qH(Q) + H(p,q)}{2(p + q)}$ (1)

 $\begin{array}{l} J\left( {{P^{(k)}}} \right) = (({p^{(k)}_0}{\grave{U}} {p^{\left( k \right)}_1}) + ({p^{(k)}_0}J({P^{(k - {1)}}_0}) + \\ \;\;\;\;\;\;\;\;{p^{(k)}_1}J({P^{(k - 1)}_1}))/2\left( {{p^{(k)}_0} + {p^{(k)}_1}} \right) \end{array}$ (2)

 $J\left( {{P_{1/0XX}}} \right) = \frac{{p_{0X}}\hat {\;}{p_{1X}} + {p_{00}}\hat {\;}{p_{01}} + {p_{10}}\hat {\;}{p_{11}}}{{p_{XX}}}$ (3)

k=3时，线性熵如式(4)所示:

 \begin{aligned} J\left( {{P_{1/0XXX}}} \right) = [{p_{0XX}}\hat {\;}{p_{1XX}} + 1/2\left( {p_{0XX}}J\left( {{P_{0XX}}} \right) + \right.\\ \left. {p_{1XX}}J\left( {{P_{1XX}}} \right) \right)]/{p_{XXX}}\quad\quad\quad\quad\quad \end{aligned} (4)

k=4时，线性熵如式(5)所示：

 \begin{aligned} J\left( {{P_{1/0XXXX}}} \right) = [{p_{0XXX}}\hat {\;}{p_{1XXX}} + 1/2\left( {p_{0XXX}}J\left( {{P_{0XXX}}} \right) + \right.\\ \left. {p_{1XXX}}J\left( {{P_{1XXX}}} \right) \right)]/{p_{XXXX}}\quad\quad\quad\quad\quad\quad \end{aligned} (5)

k=n时，线性熵如式(6)所示：

 \begin{aligned} J\left( {{P_{1/0X}}^n} \right) = \Big[{p_{0X}}{^{n - 1}}\hat {\;}{p_{1X}}{^{ n- 1}} + \quad\quad\\ 1/2\left( {p_{0X}}{^{n - 1}}J ({{P_{0X}}{^{n - 1}}}) + {p_{1X}}{^{n - 1}}J ({{P_{1X}}{^{n - 1}}}) \right)\Big]/{p_X}^n \end{aligned} (6)

3 系统故障熵时变分析

4 实例分析

 \begin{aligned} J\left( {{P_{00XX}}} \right) = ({p^{(2)}_{0X}}\Lambda {p^{\left( 2 \right)}_{1X}}) + {p^{\left( 1 \right)}_{00}}\Lambda {p^{\left( 1 \right)}_{01}} + {p^{(1)}_{10}}\Lambda {p^{\left( 1 \right)}_{11}})/\quad\quad\\ \left( {{p^{(2)}_{0X}} + {p^{(2)}_{1X}}} \right) = [\left( {{p_{0000}} + {p_{0001}}} \right)\Lambda \left( {{p_{0010}} + {p_{0011}}} \right) + \quad\quad\\ {p_{0000}}\Lambda {p_{0001}} + {p_{0010}}\Lambda {p_{0011}}] /\left( {{p_{0000}} + {p_{0001}} + {p_{0010}} + {p_{0011}}} \right) =\\ [\left( {{\rm{ }}0.045\;6 + 0.046\;7} \right)\Lambda \left( {{\rm{ }}0.057\;2 + 0.069\;3} \right)+{\rm{ }} \quad\quad\quad\quad\\ 0.045\;6\Lambda 0.046\;7 +{\rm{ }}0.057\;2\Lambda 0.069\;3]/\quad\quad\quad\quad\\ \left( {{\rm{ }}0.045\;6 + 0.046\;7 + 0.057\;2 + 0.069\;3} \right){\rm{ }} = 0.891\;7 \quad\quad\quad\end{aligned}

 \begin{aligned} J\left( {{P_{XXXX}}} \right) =\quad\quad\quad\quad\quad\quad\quad\quad\\ [({\rm{sum}}( {p_{0000}},{p_{0001}},{p_{0010}},{p_{0011}}, {p_{0100}},{p_{0101}},{p_{0110,}}{p_{0111}} )\Lambda\\ {\rm{sum}}({p_{1000}},{p_{1001}},{p_{1010}}, {p_{1011}},{p_{1100}},{p_{1101}},{p_{1110}},{p_{1111}})) +\\ 1/2({\rm{sum}}({p_{0000}},{p_{0001}},{p_{0010}},{p_{0011}},{p_{0100}},{p_{0101}},{p_{0111}},{p_{0111}} )\times\\ 0.911\;8 + {\rm{ sum}}({p_{1000}},{p_{1001}},{p_{1010}},{p_{1011}},{p_{1100}}, {p_{1101}},\quad\\ {p_{1110}},{p_{1111}} ) \times 0.852\;0)]/1 = 0.900\;4\quad\quad\quad\quad \end{aligned}

 Download: 图 1 不同状态系统故障熵的时变规律 Fig. 1 Time-varying law of the system fault entropy in different states

1)不同因素影响下系统故障熵的变化不同。图1中曲线可成对分析，00XX与01XX、10XX与11XX、0XXX与1XXX。00XX与01XX在图中距离较大，说明湿度变化对温度不变的电压磁场状态叠加时系统故障熵影响较大。10XX与11XX在图中距离很小，说明湿度变化对温度不变的电压磁场状态叠加时系统故障熵影响较小。0XXX与1XXX表明温度变化对其余3个因素状态叠加时系统故障熵影响较大。同理，可横向对比，00XX与10XX表明温度变化对湿度不变电压磁场状态叠加时系统故障熵影响较大。通过计算两条曲线的距离平均值获得影响因素的影响程度排序。该计算较为简单，这里不再详述。进一步可通过这些影响的对比和排序有的放矢地采取措施方式故障发生。

2)系统故障熵的总体变化规律。图1中7条曲线给出了所有情况下系统故障熵随时间的变化规律。可见，无论何种情况，虽然局部可能递减，但系统故障熵总体上都是递增的。根据熵的基本含义，熵值增加说明系统变得更加混乱。考虑哲学意义，该电气系统是人造系统，以完成预定功能。对该系统而言，在系统制造完成时系统故障熵为0(如果可靠性是100%)。自然对系统(人造)的影响是使系统失去功能，变得杂乱。不加维护的长时间使用，系统可靠性逐渐降低为0，这时系统故障熵为1。因此在不维护时使用系统必将导致系统故障熵的持续升高。

3)判断系统可靠性的稳定性。系统可靠性与故障发生是互补关系。可靠性稳定证明在运行过程中故障发生也是稳定的，反之亦然。图1表明在这7种4个因素状态叠加时系统故障熵曲线都是近似连续的，具有较小且稳定的斜率。这说明，系统故障熵是稳定的，系统可靠性是稳定的，没有跳跃式变化。如果在连续时间间隔上，系统故障熵在某种条件下出现大幅变化，可能是由于系统修缮，或系统失效将要出现重大故障。

5 结束语

1)定义了系统故障熵。系统故障熵是基于系统故障概率分布曲面得到的。可研究系统故障变化的混乱程度和信息量。其变化可衡量不同因素状态下的系统故障变化情况，得到系统故障变化总体规律及系统可靠性的稳定性。

2)定义了线性熵。与传统熵相比，线性熵满足它的前3个条件。熵并非线性均匀度而是对数均匀度，线性熵才是线性均匀度，即线性熵具有的第4条件。给出了线性熵在不同因素数量时的模型。认为线性熵可表征和计算系统故障熵。

3)对系统故障熵进行了时变分析。通过实例研究得到了不同时间和不同因素状态叠加时系统故障熵及其变化规律。得到了考虑不同因素状态叠加时系统故障熵的变化不同；系统故障熵总体随时间增长而增长；可应用于判断系统故障稳定性。

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