一类含hysteresis的自控系统模型
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摘要
很多自控系统都存在时滞效应,描述有害物质反应扩散过程的数学模型就是某种含时滞的微分方程。由于迟滞因子的非线性特性,如函数的许多优良性质被破坏,使得许多具有迟滞特性的模型几乎无法进行分析,因此在一般自控系统的建模中,人们都略去时滞因素。首先根据空气净化系统的功能原理引进了一类时滞算子,通过一系列细致分析给出了时滞算子的若干重要性质。然后主要利用不动点定理证明了含时滞的微分方程整体古典解存在唯一性,最后提出了精细地测算净化设备投资与效益的决策模型。
There exists hysteresis effects in many automatic control systems.Models of mathematical describing harmful matter to react-diffuse process are some kind of differential equations with hysteresis in air cleaning system.Due to the nonlinearity of hysteresis factors,such as the destruction of many excellent properties of functions,many models with hysteresis characteristics hardly can be analyzed.Therefore,in the modeling of general automatic control systems,hysteresis factors are omitted.First,a class of hysteresis operators are introduced according to the functional principle of air purification system,and some important properties of hysteresis operators are given through a series of detailed analysis.Then the existence and uniqueness of the global classical solution of differential equation with hysteresis are proved by using the fixed point theorem.Finally,a decision model for calculating the investment and benefit of purification equipment is proposed.
There exists hysteresis effects in many automatic control systems.Models of mathematical describing harmful matter to react-diffuse process are some kind of differential equations with hysteresis in air cleaning system.Due to the nonlinearity of hysteresis factors,such as the destruction of many excellent properties of functions,many models with hysteresis characteristics hardly can be analyzed.Therefore,in the modeling of general automatic control systems,hysteresis factors are omitted.First,a class of hysteresis operators are introduced according to the functional principle of air purification system,and some important properties of hysteresis operators are given through a series of detailed analysis.Then the existence and uniqueness of the global classical solution of differential equation with hysteresis are proved by using the fixed point theorem.Finally,a decision model for calculating the investment and benefit of purification equipment is proposed.
引文
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[2]盘川,侯畅,邬依林.海洋柔性立管二维振动的输出反馈控制[J].中山大学学报(自然科学版),2016,55(4):18-25.PAN C,HOU C,WU Y L.Output feedback control for two dimensional vibration of a flexible marine riser[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2016,55(4):18-25.
[3]赵亢,汪祖陆,邓达.室内空气净化器净化效果评价方法探讨[J].环境与健康杂志,2000,17(6):338-340.ZHAO K,WANG Z L,DENG D.Discussion on evaluation method of indoor air purifier purification effect[J].Journal of Environment and Health,2000,17(6):338-340.
[4]张琴,范海明,陈兆文.空气净化器对甲醛、苯和氨气的净化效果研究[J].舰船科学技术,2008,30(6):239-242.ZHANG Q,FAN H M,CHEN Z W.Study on purifying effect of air purifier on formaldehyde,benzene and ammonia[J].Ship Science and Technology,2008,30(6):239-242.
[5]宋静,孙立,闻真,等.空气净化器甲醛净化能力的测试方法[J].上海计量测试,2015(3):43-47.SONG J,SUN L,WEN Z,et al.Test method for formaldehyde purification capacity of air purifiers[J].Shanghai Measurement Test,2015(3):43-47.
[6]张言,杨宏伟,王娟.一种空气净化器在真实室内环境中的空气净化效果研究[J].绿色科技,2012,12:77-49.ZHANG Y,YANG H W,WANG J.Study on the effect of a kind of air purifier in real indoor environment[J].Green Science and Technology,2012,12:77-49.
[7]厉龙,朱兰,暴海霞,等.空气净化器甲醛洁净空气量检测探讨[J].工业计量,2017,27(2):23-28.LI L,ZHU L,BAO H X,et al.Discussion on the measurement of formaldehyde clean air volume in air purifier[J].Industrial Metering,2017,27(2):23-28.
[8]何锦强.空气净化器净化室内TVCO效率实验研究[J].福建分析测试,2017,26(1):52.HE J Q.Experimental study on the efficiency of indoor TVCO purification by air purifier[J].Fujian Analysis Test,2017,26(1):52.
[9]鲁建国.解读GB/T 18801-2015空气净化器国家标准[J].质量与认证,2015(12):65-66.LU J G.Interpretation of the national standard of GB/T18801-2015 air purifier[J].Quality and Certification,2015(12):65-66.
[10]KAN H D,WU T C.Ambient air pollution and human health in China:the past and future[J].Academic Journal of Second Military Medical University,2013,34(7):697-699.
[11]CHEN R,ZHAO Z,KAN H.Heavy smog and hospital visits in Beijing,China[J].Am J Respir Crit Care Med,2013,188(9):1170-1171.
[12]NORBCK D,NORDSTRM K,ZHAO Z.Carbon dioxide(CO2)demand-controlled ventilation in university computer classrooms and possible effects on headache,fatigue and perceived indoor environment:an intervention study[J].Int Arch Occup Environ Health,2013,86(2):199-209.
[13]CHEN R,ZHAO A,CHEN H,et al.Cardiopulmonary benefits of reducing indoor particles of outdoor origin:a randomized,double-blind crossover trial of air purifiers[J].Am Coll Cardiol,2015,65(21):2279-2287.
[14]BRUNER E V,FORCHHAMMER L,MLLER P,et al.Indoor particles affect vascular function in the aged:an air filtration-based intervention study[J].Am J Respir Crit Care Med,2008,177:419-425.
[15]VISINTIN A.A model for hysteresis of distributed systems[J].Annali Di Matematica Pura Ed Applicata(IV),CXXI,1982,131(1):203-231.
[16]KREJCI P.Hysteresis and periodic solutions of semilinear and quaslinear wave equations[J].Math Z,1986,193(2):247-264.
[17]FRIEDMAN A.Partial Differential Equations of parabolic type[M].New York:Prentice-Hall,1964:71-85.
[18]CLADYZENSKAAJA A O.Linear and quasi-linear equations of parabolic type[M].American:The American Mathematical society,1968:204-210.
[19]JIANG L S,CHEN Y Z.Equations of mathematical physics[M].Beijing:Higher Education Publishing House,1986.(in Chinese)