1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 9Lecture 9StabilityStability2Course roadmapCourse roadmapLaplace transformLaplace transformTransfer functionTransfer functionModels for systemsModels for systems••electricalelectrical••mechanicalmechanical••electromechanicalelectromechanicalBlock diagramsBlock diagramsLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime responseTime response••TransientTransient••Steady stateSteady stateFrequency responseFrequency response••Bode plotBode plotStabilityStability••RouthRouth--HurwitzHurwitz••((NyquistNyquist))Design specsDesign specsRoot locusRoot locusFrequency domainFrequency domainPID & LeadPID & Lead--laglagDesign examplesDesign examples((MatlabMatlabsimulations &) laboratoriessimulations &) laboratories3Simple mechanical examplesSimple mechanical examplesWe want mass to stay at x=0, but wind gave some We want mass to stay at x=0, but wind gave some initial speed (initial speed (F(tF(t)=0). What will happen?)=0). What will happen?How to characterize different behaviors with TF?How to characterize different behaviors with TF?MMx(tx(t))f(tf(t))MMx(tx(t))f(tf(t))KKMMx(tx(t))f(tf(t))BBMMx(tx(t))f(tf(t))BBKK4StabilityStabilityUtmost important specification in control design!Utmost important specification in control design!Unstable systems have to be stabilized by Unstable systems have to be stabilized by feedback.feedback.Unstable closedUnstable closed--loop systems are useless.loop systems are useless.What happens if a system is unstable?What happens if a system is unstable?••may hit mechanical/electrical may hit mechanical/electrical ““stopsstops””(saturation)(saturation)••may break down or burn outmay break down or burn out5What happens if a system is unstable?What happens if a system is unstable?Tacoma Narrows Bridge (July 1Tacoma Narrows Bridge (July 1--Nov.7, 1940)Nov.7, 1940)20082008……WindWind--induced vibrationinduced vibrationCollapsed!Collapsed!6Mathematical definitions of stabilityMathematical definitions of stabilityBIBOBIBO(Bounded(Bounded--InputInput--BoundedBounded--Output) Output) stabilitystability: : Any bounded input generates a bounded output.Any bounded input generates a bounded output.Asymptotic stability Asymptotic stability ::Any ICs generates Any ICs generates y(ty(t) converging to zero.) converging to zero.BIBO stable BIBO stable systemsystemu(tu(t))y(ty(t))ICs=0ICs=0AsympAsymp. stable . stable systemsystemu(tu(t)=0)=0y(ty(t))ICsICs7Some terminologiesSome terminologiesZeroZero: roots of : roots of n(sn(s))PolePole: roots of : roots of d(sd(s))Characteristic polynomialCharacteristic polynomial: : d(sd(s))Characteristic equationCharacteristic equation: : d(sd(s)=0)=0Ex.Ex.8Stability condition in sStability condition in s--domain domain (Proof omitted, and not required)(Proof omitted, and not required)For a system represented by a transfer For a system represented by a transfer function function G(sG(s),),system is BIBO stablesystem is BIBO stablesystem is asymptotically stablesystem is asymptotically stableAll the poles of All the poles of G(sG(s) are in the open left ) are in the open left half of the complex plane.half of the complex plane.9““IdeaIdea””of stability conditionof stability conditionAsymAsym. Stability: . Stability: ((U(sU(s)=0))=0)BIBO Stability: BIBO Stability: (y(0)=0)(y(0)=0)ExampleExampleBounded if Bounded if ReRe(α(α)>)>0010Remarks on stabilityRemarks on stabilityFor a general system (nonlinear etc.), BIBO For a general system (nonlinear etc.), BIBO stability condition and asymptotic stability stability condition and asymptotic stability condition are different.condition are different.For For linear timelinear time--invariant (LTI) systemsinvariant (LTI) systems(to which (to which we can use Laplace transform and we can we can use Laplace transform and we can obtain a transfer function), the conditions obtain a transfer function), the conditions happen to be the same.happen to be the same.In this course, we are interested in only LTI In this course, we are interested in only LTI systems, we use simply systems, we use simply ““stablestable””to mean both to mean both BIBO and asymptotic stability.BIBO and asymptotic stability.11Remarks on stability (contRemarks on stability (cont’’d)d)Marginally stableMarginally stableififG(sG(s) has no pole in the open RHP (Right Half Plane), &) has no pole in the open RHP (Right Half Plane), &G(sG(s) has at least one simple pole on ) has at least one simple pole on --axis, &axis, &G(sG(s) has no multiple poles on ) has no multiple poles on --axis.axis.UnstableUnstableif a system is neither stable nor if a system is neither stable nor marginally stable.marginally stable.Marginally stableMarginally stableNOT marginally stableNOT marginally stable12ExamplesExamplesRepeated polesRepeated polesDoes marginal stability imply BIBO stability?Does marginal stability imply BIBO stability?TF:TF:PickPickOutputOutput13Stability summaryStability summary(BIBO, asymptotically) stable(BIBO, asymptotically) stableififRe(sRe(sii)<0 for all i.)<0 for all i.marginally stablemarginally stableififRe(sRe(sii)<=0 for all i, and)<=0 for all i, andsimple root for simple root for Re(sRe(sii)=0)=0unstableunstableififit is neither stable nor it is neither stable nor marginally stable.marginally stable.Let Let ssiibe be polespolesof G. of G. Then, G is Then, G is ……14Mechanical examples: revisitedMechanical examples: revisitedMMx(tx(t))f(tf(t))MMx(tx(t))f(tf(t))KKMMx(tx(t))f(tf(t))BBMMx(tx(t))f(tf(t))BBKKPoles= Poles= stable?stable?Poles= Poles= stable?stable?Poles= Poles= stable?stable?Poles= Poles= stable?stable?15ExamplesExamplesStable/marginally stableStable/marginally stable/unstable/unstable????????????????16Summary and ExercisesSummary and ExercisesStability for LTI systemsStability for LTI systems(BIBO and asymptotically) stable, marginally stable, (BIBO and asymptotically) stable, marginally stable, unstableunstableStability for Stability for G(sG(s) is determined by poles of G.) is determined by poles of
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