华科电气自动控制理论m16

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1、Module16BodeAnalysis,Stability,andGainandPhaseMarginsObjectives:Attheendofthislecturestudentsshouldbeableto:&GainmarginandphasemarginonBodeplots.&Analyzethesystemstabilityusingbodeplots&Analyzethesystemtimeresponseusingbodeplots.16.1GainandPhaseMargins¾NyquistdiagramtoBodeDiagramsM(ω)[GH]db1（-1，j

2、0）−ωhcωωxωx00PM=γωcGMdb-hϕ(ω)ω0°γhGM==db−20lg

3、(GjHjωx)(ωx)

4、-180°GainCrossoverPoint0dbPhaseCrossoverPoint-180ºSample:ω=6(rad/s)ϕ11GM==Im

5、GH(jω)

6、0.3ϕ1GM=20lgGM=20lgdb0.3ω=6rad/sRe=10.46(db)0.30ω135°MG()

7、()()

8、1ωcc=jωωHjc=ω=2rad/sω=2(rads/)cPM=180°+−(135)°=45°Mdb16.2conditionalstabilityωKGH(s)=Ks(1+2

9、s)(1+3s)3KcKφ2K1-90-180-270Bodeplotscanbeusedtodeterminethesystemstability.StabilityfromBodePlots:GainandPhaseMarginsonBodeplotsEx1:NoFrequencyCrossover100K(1+s)50GH(s)=2s(1+s/10)0-50GM:indeterminate-100-120PM:about45º-140-160PM-180-2-10123101010101010Frequency(rad/sec)WeshouldtrustPM.Thesystemis

10、unconditionallystable.(seeFig.16.9)Ex2:MultipleFrequencyCrossoverGM(db)>0,GM(db)>0,butPM<0.Againweshould12relyonPManddiscardGM.Thesystemisunstable.(1+s)(1+s/10)G(s)H(s)=3s(1+s/100)(1+s/1000)ForK=1,Z=2,thesystemisunstable.PM<0,GM>0Ex3:SingleFrequencyCrossoverwithIncreasingPhase150100250(1+s/2)GM=K

11、1(db)GH(s)=03s-50-100-50-100-150-200PM-250-300-2-10123101010101010GM(db)>0,butPM<0.WeshouldrelyonPManddiscardGM.Thesystemisunstable(seeFig.16.13,Fig.16.14).¾ThesystemisstableifithasPositivePhaseMargin.¾Thegainmargincanbeusedtoestablishconditionalstability.•Althoughbothgainandphasemarginsareutiliz

12、edtodeterminesystemstability,itisimportanttorealizethatphasemarginistheonlyreliableparameteronwhichtobasesuchdeterminations(seep.326).Sample:10G(s)=dBs(s+1)(s+5)４0-20dB/dec-４0dB/dec20ω010-1001012110-20GM=9.5db-60dB/decdb0º-90º-180ºPM=25.4deg-270ºω-360º10-10010121ω（rad/sec）10Sample:100G(s)=K

13、nditionalstabledBs(s+1)(s+5)-20dB/dec４0-４0dB/dec20ωGM=-10.5dB010-100101-60dB/dec2110-200º-90º-180ºPM=-23.7deg-270ºω-360º10-10010121ω（rad/sec）102KMdb=20lgK−20lgω−20lg(Tω)+1SP.1GH(s)=s(Ts+1)−1φ=−90°−tgTωNophasecrossoverp