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1、ProblemsandChallengesinControlTheory(forMechanicalSystems)AndrewD.Lewis¤16/03/2005¤AssociateProfessor,DepartmentofMathematicsandStatistics,Queen'sUniversity,Kingston,ONK7L3N6,CanadaEmail:andrew@mast.queensu.ca,URL:http://penelope.mast.queensu.ca/~andr
2、ew/Whatiscontroltheory?²Controltheoryisthestudyofthemanipulationofdynamicalprocessestoachievedesiredobjectives.²Therearemanysortsofmodelsconsideredincontroltheory.Letus¯xone:x_(t)=f(x(t);u(t));y(t)=g(x(t);u(t)):1.Thevariablex2Rnisthestatewhosebehaviou
3、risbeingcontrolled.2.Thevariableu2Rmisthecontrolorinputwhichwecanspecifyaswelike.3.Thevariabley2Rpistheoutput,andmightrepresentwhatcanbemeasured.AndrewD.LewisQueen'sUniversityMathematics&StatisticsSimpleexamples²Servo-motor:!(t)+V(t)motor¡1.State:Angu
4、larpositionandvelocityof°ywheel(andpossiblyarmaturecurrentinmotor,foraccuratemodel).2.Input:VoltageV(t).3.Output:Angularvelocityof°ywheel.4.Typicalcontrolobjective:DesignV(t)sothatadesired°ywheelangularvelocity!(t)isachievedandmaintained.²Coupledtanks
5、:FinFin,1Fout,1Fin,2Fout,21.States:Heightsh1(t)andh2(t)ofwaterintanks.2.Input:Input°owFin(t).3.Output:Oneorbothtankheights.4.Typicalcontrolobjective:DesignFin(t)sothatdesiredheightsh1(t)andh2(t)areachievedandmaintained.AndrewD.LewisQueen'sUniversityMa
6、thematics&StatisticsTypicalcontrolproblems1.Stabilisation:Designthecontroludes(asafunctionofstate,ortime,orpossiblyboth)sothatadesiredstatex0isastableequilibriumforthedi®erentialequationx_(t)=f(x(t);udes(x;t)):
7、{z}orudes(t)orudes(x)2.Outputtracking:Ob
8、tainudessothattheoutputfollowsaspeci¯edtrajectoryydes(t).3.Motionplanning:Steerthestatefromx1tox2.4.Optimalcontrol:Doanyoftheabovewhileminimisingsomecostfunction,e.g.,time,orcontrolenergy."Typicalapproaches²Linearsystems:x_(t)=Ax(t)+Bu(t);y(t)=Cx(t)+
9、Du(t):1.Much
10、ahugeamount
11、hasbeendoneusingtoolsfromlinearalgebraandfunctionalanalysis.2.Thisiswhatistypicallyusedinpractice.²Nonlinearsystems:1.Linearise
12、thisworkssometimes.2.Ingeneral,nonlinearcontroltheoryisverydi±cult.3.Needtofocusonstructur
