华科电气自动控制理论m13-2

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1、13.2Applicationtosystemstability:NyquiststabilitycriterionSupposeF(s)tobetheCEofasystem:F(s)=1+GH(s)mm−1bs+bs+...+bmm−10F(s)=1+G(s)H(s)=1+(s−p)(s−p)...(s−p)12nK(s−z)(s−z)...(s−z)12n=(s−p)(s−p)...(s−p)12nNote:polesofF(s)=polesofopen-loopTFG(s)H(s)ZerosofF(s)=Clo

2、sed-looppolesNumberofopen–looppolesinΓsN=Z–PNumberofclosed–looppolesinΓs•DefineaCWcontourгins-planethatencirclesthewholeRHP.s•ObservethenumberofCWencirclementoforiginintheF(s)plane.Question1:whatdoesthes-planecontourlooklike?гconsistsof3segments:sΓs1.0≤s<+j∞2.-

3、j∞

4、-shiftedby1.3.InfinitesemicircleImage:F(s)=1(formostcases)Thinkwhy?Question2:HowtoobtainthecontourofF(s)?Infact,wedon’tbothertodrawthecontourofF(s)directly,since:EncirclementoftheImoriginbyF(s)=1+GH(s).GH(s)1+GH(s)ReEncirclementofpoint-1(-1,0)byGH(s).mQuestion3

5、:HowtoplotгGHfromгs?KΠ(s−zi)GH(s)=i=1n1.For0≤s<+j∞Π(s−p)jj=1GH(s)=GH(jω)=

6、GH

7、∠φThisistheNyquistdiagramoftheopen-loopsystem.2.For-j∞

8、GH

9、∠-φThisistheMirroredNyquistdiagramoftheopen-loopsystem.3.Fors=Rejθ(R=∞,θ:π/2~-π/2)⎧0(forn>m)GH(s)=⎨⎩RootLocu

10、sGain(forn=m)Ineithercase,thisisafixedpointontherealaxis!Nyquiststabilitycriterion:ForasystemwithPopenlooppolesintheright-halfplanetobestable,theopenloopfrequencyresponsemustencirclethe(-1+j0)pointPtimecounter-clockwiseintheGHplane.Z=N+PZ:thenumberofclosed-loop

11、polesintheright-halfplane.N:thenumberofclockwiseencirclementsofthecriticalpointforfrequencyresponse.P:thenumberofopen-looppolesofGH(s)intheright-halfplane.Steps:(1)DrawNyquistcurve(2)DetermineN⇒Z=N+PExample-1:KN=0j1.G(s)H(s)=Ts+11P=0（-1，j0）Ｋ2.Kω=∞ω=0G(s)H(s)=N=

12、0(Ts+1)(Ts+1)012P=01K3.G(s)H(s)=N=22(Ts+1)(Ts+1)(Ts+1)1233P=01、2:stable，3:unstableDiscussiononP(numberofpolesofF(s)inRHP):Note:polesofF(s)arealsopolesofGH(s).Formostsystems,