自動控制原理第四章根軌跡法

1、)15 . 0( ssK)(sR)(sC0222 KssKssKs222)(2 )15 . 0()( ssKsGkKs2112, 1 0 K1 K5 K5 . 0 K011 2 1 nj Ks2112,1 0 K1 K5 K5 . 0 K011 2 1 nj )(sR)(sC)(sG)(sH)()(1)()(sHsGsGs 0)()(1 sHsG1)()( sHsG)()(sHsG1)()( sHsG0)()(1 sHsG )12()()( ksHsG2,1,0 k njjmiigpszsKsHsG11)()()()(。；：為為開開環(huán)環(huán)零零極極點點，稱稱為為根根軌軌跡跡增增益益式式中中jigp

2、zK 1| )( | )( |)()(1111 njjmiignjjmiigpszsKpszsK njjmiigpszsKsHsG11)()()()(。；：為為開開環(huán)環(huán)零零極極點點，稱稱為為根根軌軌跡跡增增益益式式中中jigpzk 2,1,0,)12()()(11 kkpszsnjjmii 2,1,0,)12()()(11 kkpszsnjjmii 1| )( | )( |)()(1111 njjmiignjjmiigpszsKpszsK njjmiigpszsKsHsG11)()()()(gjnjmiiKpszs1)()(11 0 gK gK)1(njpsj 0 gK0)(1lim)()(l

3、im111 mnjjsjnjmiispspszsiz )1(mizsi gK s2s3s1s1z3p4p2p1p2z3412)10)(5)(1()2()(2 sssssKsGgk 012510ksks minjjiknmpszs11)12()()(), 1 , 0( ,)12( kmnk a mnzpmiinjja 11)()( 開開環(huán)環(huán)零零點點數(shù)數(shù)開開環(huán)環(huán)極極點點數(shù)數(shù)開開環(huán)環(huán)零零點點的的實實部部之之和和開開環(huán)環(huán)極極點點的的實實部部之之和和 a )5)(1()( sssKsGgk5, 1, 0321 ppp20351)()( mnzpiia 180,60)12( mnk 60601800125

4、AB1p2pz0gK0gKgKgK)()()()()(11sNsMKpszsKsGgnjjmiigk )()(sMsNKg 2)()()()()(sMsMsNsMsNdsdKg 0 dsdKg1)( sGk0 dsdKg)5)(1()( sssKsGgk0 , 15,( 和和13.1312845. 15275. 34725. 03722, 1 ggKKs 1 5 0 0)5)(1(1)( sssKsGgk1)56()5)(1(23ssssssKg 0)5123(2 ssdsdKg nliiilmjjlplppzpk11)()() 12( mjjlzp1)( nliiilpp1)( mljjjl

5、niilzlzzpzk11)()() 12( niilpz1)( mljjjlzz1)(23011114321 zppjpjp，2 4 3 1p 1p 2p 4p 3p z 11;45, 1 tg)(6 .266 .2626 .26180) 12(6 .261359045) 12(11考慮到周期性 cpkkk 6 .262 p 2p ;902 ;1353 6 .26, 5 . 044 tggcKgcK js gcK)5)(1()( sssKsGgkgcK05623 gKsssgggKKKssss6306510123 030 gK30 gcK062 gcKs52, 1js js 05623 gK

6、jj 30,05,0050632ggKK 0 gK0 30 gcK5 5j 05623 gKsss0)(1 sGk0)()(11 mjjgniizsKps niiss1)(nnnnasasas 111. niinniinssss111.)( njjniipsa111gKgK)1()1()()(2 TsssKsHsG )()()1()1()()(22psszsKTsssTKsHsGg 04 T j0 0 。T1 1 0902)12( k傾傾角角為為 232)/1(/1 Ta與實軸交點為與實軸交點為 23 )1()1()()(2TsssKsHsGg )22)(3()2()()(2 sssssKsH

7、sGg)22)(3()2()()(2 sssssKsHsGg00180,603)12( k傾傾角角為為13)2(113 a 與與實實軸軸交交點點為為)22)(3()2()()(2 sssssKsHsGg0)2(685234 sKssssg00 gK 761. 1 gK 61. 1 j )22)(3()2()()(2 sssssKsHsGg)2()1(10)()( sssKsHsGf)2(10 ss)(sR)(sCsKf 10101022 sKssf除特征方程得除特征方程得以以1022 ss01021012 sssKf102102 sssKGfkeq102102 sssKGfkeqj3j3 j0

8、 0。1 2 3 3 3 得得由由0 dsdKf0)22(1022 ssss0102 s16. 3102, 1 s解得解得j3處的出射角處的出射角 o90 opk4 .198)12(1 ootg4 .1831901 o4 .198op4 .1982 j0 0。1 2 3 3 3 102102 sssKGfkeq102102 sssKGfkeq j0 0。1 2 3 3 3 o4 .198102102 sssKGfkeq j0 0。1 2 3 3 3 5 . 0 o605 . 0cos o60 AB)(pssKg )(sR)(sC12 gKssp1 gK4 gK9 gK16 gK1)(1 pss

9、Kg0 p2 p4 p6 psKssKsEsCsG11)2()()()( 0)2(11 KsKss 0)2(111 KsssK )(sC)(sR)(sE)2(1 ssKs -9-8-7-6-5-4-3-2-101-8-6-4-202468Real AxisImag Axis101 K201 K501 K) 42)(6 . 0()()(211 sssKsHsGf4212 ss6 . 0 sKf)(sC)(sR cK1 . 0s-2-1.5-1-0.500.51-3-2-10123Real AxisImag Axis0) 42)(6 . 0(2 fKsss83. 15 . 083. 15 . 06

10、 . 1321jpjpp ) 6 . 3)(6 . 1() 1 . 0)(6 . 0()()(2 sssssKsHsGc-2-1.5-1-0.500.51-2-1.5-1-0.500.511.52Real AxisImag Axis)()(1)()()(1sHsGsGsRsC 0)()(1 sHsG)(sG)(sH)(sC)(sR)(sGc1)()( sHsG1111 njjmiipszsK.3 , 2 , 12)()(11 kkpszsnjjmii 1)()( sHsG k2 、2102 kmnka nliiilmjjlplppzpk11)()(2 mljjjlniilzlzzpzk11)(

11、)(2 )3)(1()()(011 sssKsHsG1,3,0 )1(0 ssK31 s)(sC)(sR 2 sKc10120,01802、 kmnka 3403)3()1(0)()(11 mnzpmiinjja 034023 Ksss038320 sssdKd舍舍去去）(45. 022. 221 ss-4-3-2-1012-5-4-3-2-1012345Real AxisImag Axisse )(sC)(sR )(sG)(1)()()(sGesGesRsCss 0)(1 sGes 1)( sGes eeeeejjs)( 3 .57)(180 )()()(111inijmjpszsKsG 1

12、)()()(111 sinijmjsepszsKsGe 1111 epszsKinijmj niimiikpszs11) 12(3 .57)()( ,2,1,0 k1)()()(111 sinijmjsepszsKsGe 1111 epszsKinijmj1111Kepszsinijmj ,ips ,jzs 3 .57180oN ,2,1,0,1,2 N01 dsdK0)( dssGeds nliiilmjjlplppzpk11)()() 12(3 .57 mljjjlniilzlzzpzk11)()() 12(3 .57 niimiikpszs11) 12(3 .57)()( 求求解解0)(

13、1 sGes 代代入入特特征征方方程程令令 js 1)1 seKesGss （0111 seKs ) 12() 1(3 .57 ks、210 k 3 .57180 N ,2,1,0,1,2 N ) 12() 1(3 .57 ks、210 k 3 .57180) 1( s1j11 p1s13 .57180 22! 21111sseess ses 11ses 12/12/1sses 221111 nnn 1)2)(1()()( sssKsHsGg5 . 0 系系統(tǒng)統(tǒng)不不穩(wěn)穩(wěn)定定。具具有有負負實實部部的的共共軛軛復(fù)復(fù)根根個個負負實實根根和和一一對對時時系系統(tǒng)統(tǒng)有有當當軛軛復(fù)復(fù)根根；和和一一對對具具有

14、有負負實實部部的的共共個個負負實實根根時時系系統(tǒng)統(tǒng)有有當當個個負負實實根根；時時系系統(tǒng)統(tǒng)有有當當,1616385. 03385. 00 gggKKK385. 0 gK423. 0 s385. 0 gK2js 6 gK6 gK385. 0 gK6 gK5 . 0 o605 . 0cos o60 58. 033. 02, 1js 06. 1 gK34. 2)(21213 sss為為共共軛軛的的主主導(dǎo)導(dǎo)極極點點。2,1509.733.034.2s 06. 1 gK%5 .16%10021 ep58. 033. 02, 1js 0)(21 ssss0445. 066. 02 ss即即67. 0445.

15、 0 n )(94.1196. 843秒秒 st5 . 0 )(180)12(mnka )()(mnzpiia )22()()(21 sssKsHsG3210zzzzpszsKsGff )()(180)12(mnka )()(mnzpiia 1cp2cpccniimiipzpzKK 1101. 0 cp1 . 0 cz niimjjcnnnnmmmmpszsKasasasbsbsbsbsRsCs111111110)()()()()()0(! 21)0()0()( )( )()(210210eeesssssrCCCtrCtrCtrCte )(1)(11)(ssGse nmeab 1)0(1)0(211)0()0(nmnnmeabaab mjjcmzKb1 niinpa1)1(11 nii