Chapter-7-6-Phase-Margin-and-Gain-Margin

[e-mail]7CHAPTER7FrequencyResponseFrequencyResponse2008-1-32PhaseMarginandGainMarginReviewofNyquist’sStabilityCriterion9由Nyquist稳定判据可知：若已知系统的开环函数G(s)H(s),即可知开环的不稳定极点数（位于S的右半平面）PR，在画出该开环传递函数的极坐标图（Nyquist图）之后，闭环系统的稳定性则由Nyquist图包围点（-1,j0）的圈数N决定。闭环系统稳定的充要条件是：位于S右半平面的极点数ZR为0：ZR＝PR－N。9许多情况下，开环传递函数的某些系数发生变化时，Nyquist图也随之发生改变，闭环稳定性也会发生变化。9当Nyquist图穿过（－1，j0）点时，闭环系统临界稳定。9稳定性研究中，将（－1，j0）点称为临界点。Nyquist图相对于该点的位置即偏离临界点的程度，反映了系统的相对稳定性。9如果稳定性不够？？－－校正。2008-1-33PhaseMarginandGainMarginOutlineofChapter7Maple9Introduction9BodePlots(Logarithmicplots)9DirectPolarPlots9Nyquist’sStabilityCriterion-Part19Nyquist’sStabilityCriterion-Part29PhaseMarginandGainMarginandTheirRelationtoStability9StabilityFromtheNicholsPlot9Compensation9………2008-1-34PhaseMarginandGainMarginFrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStabilityThestabilityandapproximatedegreeofstabilitycanbedeterminedfromtheLmandphasediagram.Thestabilitycharacteristicisspecifiedintermsofthefollowingquantities.Gaincrossover（幅值穿越频率－－增益临界点）ThisisthepointontheplotofthetransferfunctionatwhichthemagnitudeofG(jω)isunity[LmG(jω)=0dB].Thefrequencyatgaincrossoveriscalledthephase-marginfrequencyωΦ.(有中文教材称此为截止频率ωC)Phasemarginangle（相角裕度）Thisis180°plusthenegativetrigonometricallyconsiderangleofthetransferfunctionatthegain-crossoverpoint.Itisdesignatedastheangleγ,whichcanbeexpressedasγ=180°+Φ,where∠G(jωΦ)=Φisnegative.2008-1-35PhaseMarginandGainMargin-1ωΦThepolarplotofG(jω)ωG(jω)LogmagnitudeandphasediagramofG(jω)γ(+)Φ-90°-135°-180°-225°-270°ω→LmG(jω)0dBPhasemarginangle,γ(+)ωΦForstablesystemγ=180°+Φ0FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-36PhaseMarginandGainMarginLogmagnitudeandphasediagramofG(jω)ω-1ThepolarplotofG(jω)G(jω)Φ-90°-135°-180°-225°-270°ω→LmG(jω)Phasemarginangle,γ(–)ωΦωΦγ(–)Forunstablesystemγ=180°+Φ0FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-37PhaseMarginandGainMarginThephasemarginangleistheamountofshiftatthefrequencyωΦthatwouldjustproduceinstability.Thisanglewouldmakethepolarplotgothroughthe–1point.Thephasemarginangleforminimum-phase(m.p.)systemsmustbepositiveforastablesystem,whereasanegativephasemarginmeansthatthesystemisunstable.Thephasemarginangleisrelatedtotheeffectivedampingratioζofthesystem.Satisfactoryresponseisusuallyobtainedwithaphasemarginof45°to60°.FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-38PhaseMarginandGainMarginPhasecrossover（相位穿越频率－－相位临界点）Thisisapointontheplotofthetransferfunctionatwhichthephaseangleis-180°.Thefrequencyatwhichphasecrossoveroccursiscalledthegain-marginfrequencyωc.(有中文教材称此为穿越频率ωx)Gainmargin（幅值裕度）Thegainmarginisthefactorabywhichthegainmustbechangedinordertoproduceinstability.Expressedintermsofthetransferfunctionatthefrequencyωc,itis1)(=⋅ajGcωOnthepolarplotofG(jω)thevalueatωcisajGc1)(=ωIntermsoftheLm,indecibels,thisis)(cjGLmLmaω−=FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-39PhaseMarginandGainMarginω-1ωΦγ(+)ΦThepolarplotofG(jω)G(jω)LogmagnitudeandphasediagramofG(jω)Forstablesystem-90°-135°-180°-225°-270°ω→ωΦPhasemarginangle,γ(+)LmG(jω)ωc1/aGainmargin,Lma(+)ωc1/a1/a1,a11,a10)(−=cjGLmLmaωFrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-310PhaseMarginandGainMarginω-1ωΦγ(–)Φ-90°-135°-180°-225°-270°ω→ωΦPhasemarginangle,γ(–)ThepolarplotofG(jω)G(jω)LogmagnitudeandphasediagramofG(jω)ForunstablesystemLmG(jω)1/aωcωcGainmargin,Lma(–)1/a1/a1,a11,a10)(−=cjGLmLmaωFrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-311PhaseMarginandGainMargin相角裕度和幅值裕度的求解方法通常有三种求解系统相角裕度和幅值裕度的方法，即解析法、极坐标图法和伯德图法。下面通过实例进行说明。（一）解析法根据系统的开环频率特性，由00180)()180()(+=−−=ΦΦωφωφγ和)0(1)()(+∞≤≤=ΦΦΦωωωjHjG求出相角裕度。由)0(180)()(0+∞≤≤−=∠cccjHjGωωω)()(1ccjHjGaωω=)()(lg20lg20ccjHjGaωω−=求出幅值裕度或FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-312PhaseMarginandGainMargin相角裕度和幅值裕度的求解方法)252(40)()(2++=ssssHsG试求出该系统的幅值裕度和相角裕度。例7-20已知最小相位系统的开环传递函数为解：系统的开环频率特性为)225(40)()(2ωωωωωjjjHjG+−=其幅频特性和相频特性分别是2224)25(401)()(ωωωωω+−=jHjG2025290)()(ωωωω−−−=∠arctgjHjGccFrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-313PhaseMarginandGainMargin相角裕度和幅值裕度的求解方法令，得1)()(=ωωjHjG82.1=Φω令，得0180)()(−=∠ωωjHjG5=cω°=−×−°=∠+°=ΦΦ5.8082.12582.1290)()(1802arctgjHjGωωγ)(94.125.1lg20)(dBdBa==或25.1)()(1==ccjHjGaωω即：该系统具有1.94分贝的幅值裕度，80.5度的相位裕度。FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-314PhaseMarginandGainMargin相角裕度和幅值裕度的求解方法（二）极坐标图法在GH平面上作出系统的开环频率特性的极坐标图，并作一单位圆，由单位圆与开环频率特性的交点A与坐标原点的连线与负实轴的夹角求出相角裕度γγ；由开环频率特性与负轴交点处的幅值的倒数得到幅值裕度aa。)()(ccjHjGωωa1mIeR[]GH+∞=ω0cωΦω1−γ+=0ω1jj−A例7-20的极坐标图FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationtoStability2008-1-315PhaseMarginandGainMargina1mIeR[]GH+∞=ω0cωΦω1−γ+=0ω1jj−A例7-20的极坐标图在例7-20中，先作出系统的开环频率特性曲线如图所示，作单位圆交开环频率特性曲线于A点，连接OA，射线OA与负实轴的夹角即为系统的相角裕度。开环频率特性曲线与负实轴的交点坐标为由此得到系统的幅值裕度：080≈γ)08.0(j，25.18.01==a相角裕度和幅值裕度的求解方法FrequencyResponse5.PhaseMarginandGainMarginandTheirRelationt