具有内部构造安全保障体系系统解的分析
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摘要
冗余机器系统的可靠性分析是可靠性数学理论中所要研究的对象之一,所谓可靠性数学理论是指运用概率统计和运筹学等理论和方法对产品(单元或系统)的可靠性作定量的研究。它是可靠性理论的基础之一。有关这类问题,国内外上许多学者作出了大量研究,一般是运用补充变量的方法建立修复时间服从任意分布的可修复系统的数学模型,再运用Laplace等方法对系统的渐进稳定性给出证明,但对系统可靠性及系统特征值的相关问题没有很好的证明,本文主要研究这一问题。
本文研究的系统为具有内部构造安全保障体系的冗余机器系统。由两个同样的机器并联工作和一个安全保障体系独立运行构成随机数学模型。当应用时,被修机器及安全保障体系都与新的一样,而在两台机器故障时,整个系统故障。本文先将系统转化为Banach空间下的Volterra算子方程,得到系统解的存在唯一性,经过泛函分析处理,采用算子半群方法确立一正压缩C_0-半群,得出系统解存在且唯一,进而证明了该系统的稳定性,后又对系统对该系统的谱的特性进行了分析,知系统算子的谱点均位于复平面的左半平面,且虚轴上除了0点外无别的谱点.并通过一个具体实例来说明系统非零特征值是存在的,且特征值与特征向量是一一对应的,最后证明了该系统解的可靠性,即瞬态可靠性不小于牢固可靠性。
The reliability analysis on the Redundant Robot Configuration is one of the study objects in the reliability Mathemetical Theory,which makes quantitative study on the reliability of the products(units or systems) using theories and methods of probability,statistics and operations research and so on.It is one of reliable theory foundations.Many scholars,both domestic or abroad,have made massive researches concerning this kind of questions.Generally,they have set up a repairable system's mathematical model of which can repair time to be obedient to distributes,using the supplement variable method,then using Laplace method gives the proof to system's evolution stability,but to such related questions as system reliablility and system characteristic value,they haven't given good proof,yet.This article mainly studies this question.The system studied in this article is the redundant machine system which has the internal structure safety control system's,in which two same machines' parallel operating and an independently-running safety control system constitute a stochastic mathematical model.When being applied,the machine that is being repaired and the safety control system are almost new,but when these two machines both break down,then the whole system breaks down.In this article,the system solution existence uniqueness was firstly obtained by transforming the system into Banach under the spatial Volterra operator equation,then through functional analysis processing,the system solution existence,and also the only one, was obtained using the operator semi-group method to establish one is compressing C_0-semigroup, thus this system's stability has been proven.Based on the above,the analysis to this system's spectrum's characteristic has been carried on,so that the system operator spectrum that was located at the left half plane of the complex planes was known,and also there was no other spectrum spot besides 0 spots on the axis of imaginaries.And through a concrete example,it is explained that the system's non-vanishing characteristic value is in existence, and the characteristic value and the feature vector are in correspondences,finally it is shown that this system solution's reliability,namely the transient state reliability is not smaller than the reliable reliability had.
本文研究的系统为具有内部构造安全保障体系的冗余机器系统。由两个同样的机器并联工作和一个安全保障体系独立运行构成随机数学模型。当应用时,被修机器及安全保障体系都与新的一样,而在两台机器故障时,整个系统故障。本文先将系统转化为Banach空间下的Volterra算子方程,得到系统解的存在唯一性,经过泛函分析处理,采用算子半群方法确立一正压缩C_0-半群,得出系统解存在且唯一,进而证明了该系统的稳定性,后又对系统对该系统的谱的特性进行了分析,知系统算子的谱点均位于复平面的左半平面,且虚轴上除了0点外无别的谱点.并通过一个具体实例来说明系统非零特征值是存在的,且特征值与特征向量是一一对应的,最后证明了该系统解的可靠性,即瞬态可靠性不小于牢固可靠性。
The reliability analysis on the Redundant Robot Configuration is one of the study objects in the reliability Mathemetical Theory,which makes quantitative study on the reliability of the products(units or systems) using theories and methods of probability,statistics and operations research and so on.It is one of reliable theory foundations.Many scholars,both domestic or abroad,have made massive researches concerning this kind of questions.Generally,they have set up a repairable system's mathematical model of which can repair time to be obedient to distributes,using the supplement variable method,then using Laplace method gives the proof to system's evolution stability,but to such related questions as system reliablility and system characteristic value,they haven't given good proof,yet.This article mainly studies this question.The system studied in this article is the redundant machine system which has the internal structure safety control system's,in which two same machines' parallel operating and an independently-running safety control system constitute a stochastic mathematical model.When being applied,the machine that is being repaired and the safety control system are almost new,but when these two machines both break down,then the whole system breaks down.In this article,the system solution existence uniqueness was firstly obtained by transforming the system into Banach under the spatial Volterra operator equation,then through functional analysis processing,the system solution existence,and also the only one, was obtained using the operator semi-group method to establish one is compressing C_0-semigroup, thus this system's stability has been proven.Based on the above,the analysis to this system's spectrum's characteristic has been carried on,so that the system operator spectrum that was located at the left half plane of the complex planes was known,and also there was no other spectrum spot besides 0 spots on the axis of imaginaries.And through a concrete example,it is explained that the system's non-vanishing characteristic value is in existence, and the characteristic value and the feature vector are in correspondences,finally it is shown that this system solution's reliability,namely the transient state reliability is not smaller than the reliable reliability had.
引文
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[2]#12
[3]Campbell N.R.The replacement of perishable members of a continually operating system.J Roy.Statist.Soc.1941,7:110-130
[4]Lotka A.J.A contribution to the theory of self-renewing aggregates with special reference to industrial replacement.Ann.Math.Statist.1939,10:1-25
[5]何国伟.可靠性工程概论.国防工业出版社.1989
[6]程侃.寿命分布类与可靠性数学理论.科学出版社.1999
[7]Shapero A,Cooper J I.Human Engineering Testing and Mal-function Data Collection in Weapon System Test Programs.Wright Air Development Division,Wright-Patterson Air Force Base,OH,WADDTR 60-36,Feb.1960
[8]Christemsen J M,Howard J M etc.Human Detetion and Diagnosis of System Failure.New York:Plenum Press,1981.
[9]Dhillon B S,Rayapati S N.Reliability evaluation of human operators under stress.Microelectronic and Reliability,1985,25(4):729-752
[10]Dhillon B S,Rayapati S.Reliability analysis of standby systems with human errors.Proceedings of the IASTED International Conference,Applied Simulation and Modelling,ASM 86,Vancouver,Canada,June,1986,1-6
[11]Dhillon B S,Yang N.Availability of a man-machine system with critical and non-critical human error.microelectron.andReliability.1993,33(10):1511-1521.
[12]Pazy A.semigroups of Linear Operators.and Application to Partial Differential Equations.new York:Springgerverlag,1983.
[13]Dhillon B S.Yang N.Formulas for analyzing a redundant Robot configuration with a built-in safety system microelectron.Reliab.Vol.37.No.4pp.557-563.1997
[14]Dhillon B S.Robotreliability and Safety.Springer,New York,1991
[15]冯雪,张玉峰具有内部构造安全保障体系的冗余机器系统的解得半离散化分析[J]数学的实践与认识2006,36(3):199-207.
[16]于景元,郭宝珠,朱广田,半离散化人口发展系统的控制,系统科学与数学,1987 7(3):214-219
[17]Vagra R S.Matrix Iterative Analysis.Prentice-Hall.Inc,1962
[18]柳京爱郑福.具有四类故障可修复系统解的存在唯一性[J]。数学的实践与认识,2004,24(2).
[19]Adams RA.Sobolev空间[M]北京人民教育出版社1982
[20]周鸿兴,王连文线性算子半群理论及其应用[M]济南山东科学技术出版社1994
[21]夏道行,吴卓人,严绍宗,舒五昌.实变函数论与泛函分析.高等教育出版社.1982.
[22]张恭庆,郭懋正泛函分析讲义(F)[M]北京北京大学出版社1994
[23]Arendtw resolvent positive perators[J].Proclondon math Soc(3),1987,54:321-349.
[24]艾尼.吾甫尔一个可靠机器,一个不可靠机器和一个缓冲库构成的系统定性分析[J]数学的实践与认识2002。22(2):29-36
[25]Nagelr one-panameter semi-groups of positive operaters[M].Berlin:Springer-Verlag,1986.
[26]郭卫华.一类两个相同部件并联可修复系统解的存在和唯一性.数学的实践与认识.
[27]冯雪,张玉峰具有内部构造安全保障体系的冗余机器系统的特征值分布[J]数学的实践与认识2006,36(10):185-190.
[28]中国百科全书数学[M].北京、上海:中国百科全书出版社1988,11,155.
[29]秦化淑,林正国常微分方程及其应用[M]国防工业出版社1985,108-138.
[30]林武忠,汪志鸣,张九超常微分方程[M]北京科学出版社2003,71-85.
[2]#12
[3]Campbell N.R.The replacement of perishable members of a continually operating system.J Roy.Statist.Soc.1941,7:110-130
[4]Lotka A.J.A contribution to the theory of self-renewing aggregates with special reference to industrial replacement.Ann.Math.Statist.1939,10:1-25
[5]何国伟.可靠性工程概论.国防工业出版社.1989
[6]程侃.寿命分布类与可靠性数学理论.科学出版社.1999
[7]Shapero A,Cooper J I.Human Engineering Testing and Mal-function Data Collection in Weapon System Test Programs.Wright Air Development Division,Wright-Patterson Air Force Base,OH,WADDTR 60-36,Feb.1960
[8]Christemsen J M,Howard J M etc.Human Detetion and Diagnosis of System Failure.New York:Plenum Press,1981.
[9]Dhillon B S,Rayapati S N.Reliability evaluation of human operators under stress.Microelectronic and Reliability,1985,25(4):729-752
[10]Dhillon B S,Rayapati S.Reliability analysis of standby systems with human errors.Proceedings of the IASTED International Conference,Applied Simulation and Modelling,ASM 86,Vancouver,Canada,June,1986,1-6
[11]Dhillon B S,Yang N.Availability of a man-machine system with critical and non-critical human error.microelectron.andReliability.1993,33(10):1511-1521.
[12]Pazy A.semigroups of Linear Operators.and Application to Partial Differential Equations.new York:Springgerverlag,1983.
[13]Dhillon B S.Yang N.Formulas for analyzing a redundant Robot configuration with a built-in safety system microelectron.Reliab.Vol.37.No.4pp.557-563.1997
[14]Dhillon B S.Robotreliability and Safety.Springer,New York,1991
[15]冯雪,张玉峰具有内部构造安全保障体系的冗余机器系统的解得半离散化分析[J]数学的实践与认识2006,36(3):199-207.
[16]于景元,郭宝珠,朱广田,半离散化人口发展系统的控制,系统科学与数学,1987 7(3):214-219
[17]Vagra R S.Matrix Iterative Analysis.Prentice-Hall.Inc,1962
[18]柳京爱郑福.具有四类故障可修复系统解的存在唯一性[J]。数学的实践与认识,2004,24(2).
[19]Adams RA.Sobolev空间[M]北京人民教育出版社1982
[20]周鸿兴,王连文线性算子半群理论及其应用[M]济南山东科学技术出版社1994
[21]夏道行,吴卓人,严绍宗,舒五昌.实变函数论与泛函分析.高等教育出版社.1982.
[22]张恭庆,郭懋正泛函分析讲义(F)[M]北京北京大学出版社1994
[23]Arendtw resolvent positive perators[J].Proclondon math Soc(3),1987,54:321-349.
[24]艾尼.吾甫尔一个可靠机器,一个不可靠机器和一个缓冲库构成的系统定性分析[J]数学的实践与认识2002。22(2):29-36
[25]Nagelr one-panameter semi-groups of positive operaters[M].Berlin:Springer-Verlag,1986.
[26]郭卫华.一类两个相同部件并联可修复系统解的存在和唯一性.数学的实践与认识.
[27]冯雪,张玉峰具有内部构造安全保障体系的冗余机器系统的特征值分布[J]数学的实践与认识2006,36(10):185-190.
[28]中国百科全书数学[M].北京、上海:中国百科全书出版社1988,11,155.
[29]秦化淑,林正国常微分方程及其应用[M]国防工业出版社1985,108-138.
[30]林武忠,汪志鸣,张九超常微分方程[M]北京科学出版社2003,71-85.