基于模糊PID控制的电阻炉炉温系统的研究Word格式.docx

基于模糊PID控制的电阻炉炉温系统的研究Word格式.docx

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学院：

自动化学院学号：

24

班级：

0710031104姓名：

杨京伟

一、外文原文

StabilityofLinearControlSystems

1.1INTRODUCTION

FromthestudiesoflineardifferentialequationswithconstantcoefficientsofSISOsystems,welearnedthatthehomogeneoussolutionthatcorrespondstothetransientresponseofthesystemisgovernedbytherootsofthecharacteristicequation.Basically,thedesignoflinearcontrolsystemsmayberegardedasaproblemofarrangingthelocationofthepolesandzerosofthesystemtransferfunctionsuchthatthesystemwellperformaccordingtotheprescribedspecifications.

Amongthemanyformsofperformancespecificationsusedindesign,themostimportantrequirementisthatthesystemmustbestable.Anunstablesystemisgenerallyconsideredtobeuseless.

Whenalltypesofsystemsareconsidered-linear,nonlinear,time-invariant,andtime-varying-thedefinitionofstabilitycanbegiveninmanydifferentforms.WeshalldealonlywiththestabilityoflinearSISOtime-invariantsystemsinthefollowingdiscussions.

Foranalysisanddesignpurposes,wecanclassifystabilityasabsolutestabilityandrelativestability.Absolutestabilityreferstotheconditionwhetherthesystemisstableorunstable;

itisayesornoanswer.Oncethesystemisfoundtobestable,itisofinteresttodeterminehowstableitis,andthisdegreeofstabilityisameasureofrelativestability.

Inpreparationforthedefinitionofstability,wedefinethetwofollowingtypesofresponsesforlineartime-invariantsystems:

1.Zero-stateresponse.Thezero-stateresponseisduetotheinputonly;

alltheinitialconditionsofthesystemarezero.

2.Zero-inputresponse.Thezero-inputresponseisduetotheinitialconditionsonly;

alltheinputsarezero.

Fromtheprincipleofsuperposition,whenasystemissubjecttobothinputsandinitialconditions,thetotalresponseiswritten

Totalresponse=zero-stateresponse+zero-inputresponse

Thedefinitionsjustgivenapplytocontinuous-dataaswellasdiscrete-datasystems.

1.2BOUNDED-INPUT,BOUNDED-OUTPUT（BIBO）,STABILITY-CONTIN

UOUS-DATASYSTEMS

Letu（t）,y（t）,andg（t）betheinput,output,andtheimpulseresponseofalineartime-invariantsystem,respectively.Withzeroinitialconditions,thesystemissaidtobeBIBO（bounded-inputbounded-output）stable,orsimplystable,ifitsoutputy（t）isboundedtoaboundedinputu（t）.

Theconvolutionintegralrelatingu（t）,y（t）,andg（t）is

y（t）=∫0∞u（t-τ）g（τ）dτ（1-1）

Takingtheabsolutevalueofbothsidesoftheequation,weget

│y（t）│=│∫0∞u（t-τ）g（τ）dτ│（1-2）

or

│y（t）│≤∫0∞│u（t-τ）││g（τ）│dτ（1-3）

Ifu（t）isbounded,

│u（t）│≤M（1-4）

whereMisafinitepositivenumber.Then,

│y（t）│≤M∫0∞│g（τ）│dτ（1-5）

Thus,ify（t）istobebounded,or

│y（t）│≤N<

∞（1-6）

whereNisafinitepositivenumber,thefollowingconditionmusthold:

M∫0∞│g（τ）│dτ≤N<

∞（1-7）

OrforanyfinitepositiveQ,

∫0∞│g（τ）│dτ≤Q<

∞（1-8）

TheconditiongiveninEq.（6-8）impliesthattheareaunderthe│g（τ）│-versus-τ-curvemust

befinite.

1.2.1RelationshipbetweenCharacteristicEquationRootsandStability

ToshowtherelationbetweentherootsofthecharacteristicequationandtheconditioninEq.（6-8）wewritethetransferfunctionG（s）,accordingtotheLaplacetransformdefinition,asG（s）=└[g（τ）]=∫0∞g（τ）e-stdt（1-9）

Takingtheabsolutevalueonbothsidesofthelastequation,wehave

│G（s）│=│∫0∞g（t）e-st│dt≤∫0∞│g（t）││e-st│dt（1-10）

Since│e-st│=│e-σt│,whereσistherealpartofs.WhensassumesavalueofapoleofG（s）,G（s）=∞,Eq.（1-10）becomes

∞≤∫0∞│g（t）││e-σt│dt（1-11）

Ifoneormorerootsofthecharacteristicequationareintheright-halfs-planeoronthejω-axis,σ≥0,then

e-σt≤M=1（1-12）

Equation（1-11）becomes

∞≤∫0∞M│g（t）│dt=∫0∞│g（t）│dt（1-13）

whichviolatestheBIBOstabilityrequirement.Thus,forBIBOstability,therootsofthecharacteristicequation,orthepolesofG（s）,cannotbelocatedintheright-halfs-planeoronthejω-axis,inotherwords,theymustalllieintheleft-halfs-plane.AsystemissaidtobeunstableifitisnotBIBOstable.Whenasystemhasrootsonthejω-axis,say,ats=jω0ands=-jω0,iftheinputisasinusoid,sinω0t,whichisunbounded,andthesystemisunstable.

1.3ZERO-INPUTANDASYMPTOTICSTABILITYOFCONTINUOUS-DATASYSTEMS

Inthissectionweshalldefinezero-inputstabilityandasymptoticstability,andestablishtheirrelationswithBIBOstability.

Zero-inputstabilityreferstothestabilityconditionwhentheinputiszero,andthesystemisdrivenonlybyitsinitialconditions.Weshallshowthatthezero-inputstabilityalsodependsontherootsofthecharacteristicequation.

Lettheinputofannth-ordersystembezero,andtheoutputduetotheinitialconditionsbey（t）.Then,y（t）canbeexpressedas

（1-14）

Where