5-4 奈奎斯特稳定判据

1Nyquist)()(ωωjHjG)()(1)(sHsGsF+=2)()(1)(sHsGsF+=)()(sHsG——)())(()()(210111nmmmmpspspsbsbsbsbsHsG−……−−++……++=−−)())(()())(()(2121nnpspspszszszsksF−……−−−……−−=)(sF),,2,1(niZi……=0)()(1=+sHsG)(sF3)(sF)(sF)(sF)1(1)()(+=sssHsG)1(1)()(1)(2+++=+=sssssHsGsF0=′s1−=′′s)(sF211js+=15.095.0)121)(21(1)21()21()(21jjjjjsF−=+++++++=4—)(sF15.095.0)(1jsF−=211js+=)(1sF1s)(sFωj2j1s's0σ1−''s1[]SmI0eR)(1sF()[]SF15.0−95.055—38S1s)(sFFΓsΓsΓ)(sF)(sF)(sFωj[]Sσ1P2P1S2S3S3P1Z2Z3Z0)(amI[])(SFeR)(1SF)(2SF)(3SF0)(bsΓFΓ6ΓsΓsΓsF(s)ΓFsΓsΓFπΓFF(s)ΓsF(s)PZΓFF(s)ΓFF(s)ΓFF(s)—ΓsΓsF(s)ΓFF(s)ΓsP-Z)(sF)(sF)(sF)(sF7ΓFNSΓsZZ=P-N5-109ΓsΓF5-110S5—39SΓs,ΓsΓs,(5-111)))()(())()(()(321321pspspszszszssF−−−−−−=)(sF)()(sHsG)(sF1S(1sF∑∑==−∠−−∠=∠3131111)()()(jiijpszssF)8sΓss1Γss10Γss1———π2)(1=∠sF)(sFπ2)(1−=∠sF)(1sF)(sF)(sF)(1sF0)(1=∠sFNZPsFππ2)(2)(=−=∠)(1sFπ2)(101)(1−=∠−===SFNPZamI()[]sFeR)(1sF1−=NΓF1Zωj1p11PS−2p11zs−21ps−31ps−3p3z31zs−1s2z21zs−0Γsσ9σ[]Sωj1S1Z1P02Z2P3P3ZsΓ0)(1SFIm[])(SFRe1=NFΓπ2)(110)(1=∠===SFNPZb0Im[])(SF)(1SFRe0=NFΓ3Z3P2P1S2Z01P[]SωjσsΓ0)(011)(1=∠===SFNPZcΓs—F(s))(sF−∞=ω+∞=ω+∞=ω−∞=ω∞→Rωjσ[]SΓFN=P5-114)(sF)(sF)(sF)(sFΓs)(sFsΓsΓF)(sF)(sF)(sF)()(sHsG-GHΓF)()(1)(sHsGsF+=1)()()(−=sFsHsG)(sF)1(οj−GHΓ)1(οj−)(sF(-1,j0)00)(sF[GH][F]154112)()(sHsG)1(οj−)1(οj−)()(sHsGGHΓ)()(sHsGGHΓ1S5—421s2ss)()(sHsG)()(ωωjHjG)()(ωωjHjG)()(sHsGωjs=)()(ωωjHjG)()(sHsG0≤∞−ω+∞≤ω0φjRs−∞→=Relimπφ=ω−∞→∞+−∞=ω+∞=ω[]S∞→F)1()2()3(ωjσΓs1sS)()()()()()()()(ωωωωωωωjHjGjjsejHjGjHjGsHsG∠−−==−−=ωjs−=[GH]5—43114[]GHmIeR)2(0)1()3(−∞=+∞=ωω0=ωKmnb)(GHΓmI[]GHeR)2(0)1()3(−∞=+∞=ωω0=ωKkmna=)(GHΓΓs2s)()()()()()()()(ωωωωωωωjHjGjjsejHjGjHjGsHsG∠===ωjs=-SGH12ΓsGHSGHKK3sΓsοjs=KjHjGsHsGjs===)()()()(οοοφjRs−∞→=Relimφφφ)(Relim01110111Relim)1lim()()(mnjmnnmRsnnnnmmmmseRabasasasabsbsbsbsHsGjRjR−−∞→=−−−−=⋅=++……++++……++=−∞→−∞→kabsHsGnmsjR==−∞→=φRelim)()(—5-121Γs—Γsφοφ)(Relim)()(mnjsesHsGjR−=⋅=−∞→(5-120)(5-119)16Γs123—)()(sHsGΓsΓsθjrres0lim→=)22(πθπ≤≤−+−→οοω)()(sHsG)())(()())(()()(2121vnvmpspspsszszszsksHsG−−……−−−……−−=θjrres0lim→=θθθθjvjvvrrenvmreseerKpspspsszszszsksHsGjrjr−−→=∞==−……−−−……−−=→→0lim2121limlim)())(()())(()()(00(5-123)17Γs—ΓsνπmI−=0ω+=0ω0−∞=+∞=ωω∞→R0=ω1=veR)(a[]GHωj+∞=ω+=0ω−=0ω0−∞=ω)1()2(∞→R)3()4(0→rσ[]SΓs−=0ω+=0ω1−0∞→R+∞=−∞=ωω0=ω[]GH2=veR)(bmI0≠v18GH)()(ωωjHjG)()(ωωjHjG)()(ωωjHjG∞+∞−ωGHΓGHΓ),1(οj−),1(οj−GHΓ19SN=PZ=P-N=0(iii)N0Z=P-N0GHSGH),1(οj−),1(οj−)()(sHsG)()(sHsGGHΓGHΓGHΓGHΓGHΓGHΓ),1(οj−),1(οj−),1(οj−),1(οj−20——5-46SSP=05-46N=0SZ=PN=0)()1)(1()()(2121TTsTsTKsHsG++=)1)(1()()(21++=ωωωωjTjTKjHjG∞−ω∞+11T−21T−GHΓ),1(οj−21——)0(1=KP2)0(PK=∞→K21T−11T−ωj[]S0σ∞→K+∞=−∞=ωω1−0K0=ωeRmI[]GH22——5—48v=15—48)10()12()()(22++=ζζTssTsKsHsGv)21()()(22ωζωωωωTjTjKjHjGv+−=∞−ω∞++−00ω23SP=0N=05-48a,,5-48bζ2TKv−12ζTKv),1(οj−12ζTKv),1(οj−GHΓGHΓ2−=N0−∞=+∞=ωωξ2TKV−=0ω+=0ωeR[]GHmI012)(=NTKaVξ1−1−−∞=+∞=ωωξ2TKV−=0ω+=0ωeR[]GHmI212)(−=NTKbVζ024—τT)1()1()()(2++=TsssKsHsGττττ=TTT)1()1()()(2ωωτωωωjTjKjHjG+−+=2222211)()(ωωωτωωTKjHjG++=τωωωωarctgarctgTjHjG+−−=∠0180)()(,0+0-180oIII-180o5-50aGHN=0SP=0τωωarctgarctgT)()(ωωjHjG)()(ωωjHjG∠),1(οj−τT25ωτ=T0180)()(−=∠ωωjHjGτωωarctgarctgT=)()(ωωjHjG∞+0ω∞),1(οj−τ=Tb05-50b5-51bSτTτωωarctgarctgTω∞+)()(ωωjHjG∞0180)()(−∠ωωjHjG),1(οj−τTτTc00II-1800—26)(a+−==00ωωmIeR[]GHτT1−0+∞=−∞=ωω1−0+∞=−∞=ωωmIeRτ=T)(b[]GH+−==00ωω+=0ω−=0ω1−0[]GHmIeRτT)(c−∞=ω+∞=ω21PP(τ=T)(b0ωj[]Sσ021PP3PT1−τ1−ωj[]Sσ(τT)(a21PP3p0T1−τ1−ωjσ)(τT)(c[]S27ss0dB0dB0dB)0,1(j−)0,1(j−π−−+−=NNN02=−=NPZ28°−180ωω)(ωL)(ωϕ−+−=NNNNPZ2−=