自動(dòng)控制原理第四章習(xí)題課

4-1已知系統(tǒng)的零、極點(diǎn)分布如圖，大致繪制出系統(tǒng)的根軌跡。章習(xí)題課(4-1)jσ0ω(1)(2)jσ0ω(3)jσ0ω(4)jσ0ω600900600第四章習(xí)題課(4-1)(5)jσ0ω(6)jσ0ωjσ0ω(7)(8)jσ0ω600450135036010804-2已知開環(huán)傳遞函數(shù)，試用解析法繪制出系統(tǒng)的根軌跡，并判斷點(diǎn)(-2+j0),(0+j1),(-3+j2)是否在根軌跡上。第四章習(xí)題課(4-2)解:Kr(s+1)G(s)=KrΦ(s)=s+1+KrKr=0s=-1-Krjσ0ω系統(tǒng)的根軌跡

s=-1-1Kr=→∞s=-∞s=-2+j0-2s=0+j10+j1-3+j2s=-3+j2σjω04-3已知系統(tǒng)的開環(huán)傳遞函數(shù)，試?yán)L制出根軌跡圖。第四章習(xí)題課(4-3)解:s(s+1)(s+5)(1)G(s)=Kr(s+1.5)(s+5.5)1）開環(huán)零、極點(diǎn)p1

p1=0p2p2=-1p3

p3=-52）實(shí)軸上根軌跡段p1~p2z1

z1=-1.5z2

z2=-5.5z1~p3z2~-∞3）根軌跡的漸近線n-m=1θ=+180o4）分離點(diǎn)和會(huì)合點(diǎn)A(s)B'(s)=A'(s)B(s)A(s)=s3+6s2+5sB(s)=s2+7s+8.25A(s)'=3s2+12s+5B(s)'=2s+7解得s1=-0.63s2=-2.5s3=-3.6s4=-7.28第四章習(xí)題課(4-3)s(s+1)(s+4)(2)G(s)=Kr(s+1.5)1）開環(huán)零、極點(diǎn)p1

p1=0p2p2=-1p3

p3=-4σjω02）實(shí)軸上根軌跡段p1~p2z1

z1=-1.5p3~z13）根軌跡的漸近線n-m=

2θ=

+90o2σ=-1-4+1.5

=-1.75-1.754）分離點(diǎn)和會(huì)合點(diǎn)A(s)=s3+5s2+4sB(s)=s+1.5A(s)'=3s2+10s+4B(s)'=1解得s=-0.625）系統(tǒng)根軌跡第四章習(xí)題課(4-3)s(s+1)2(3)G(s)=Kr1）開環(huán)零、極點(diǎn)p1=0p2=-1p3

p3=-12）實(shí)軸上根軌跡段p1~p2p3~-∞3）根軌跡的漸近線n-m=3σjω0p1

p23σ=-1-1

=-0.67-0.67

4）根軌跡與虛軸的交點(diǎn)θ=

+180o+60o,閉環(huán)特征方程為s3+2s2+s+Kr=0Kr=0Kr=2

ω2,3=±1ω1=01-15）分離點(diǎn)和會(huì)合點(diǎn)A(s)=s3+2s2+sB(s)=1A(s)'=3s2+4s+1B(s)'=0解得s=-0.336）系統(tǒng)根軌跡第四章習(xí)題課(4-3)1）開環(huán)零、極點(diǎn)p1=0p2=-3p3=-7σjω02）實(shí)軸上根軌跡段p1~p2p4

p4=-15z1

z1=-8p3~z1p4~-∞3）根軌跡的漸近線n-m=3s(s+3)(s+7)(s+15)(4)G(s)=Kr(s+8)p1

p2p3

3σ=

-3-7-15+8

=-5.67θ=

+180o+60o,-5.67

4）根軌跡與虛軸的交點(diǎn)閉環(huán)特征方程為s4+25s3+171s2+323s+8Kr=0Kr=0ω1=0Kr=638ω2,3=±6.26.2-6.25）分離點(diǎn)和會(huì)合點(diǎn)A(s)=s4+25s3+171s2+315sB(s)=s+8A(s)'=4s3+75s2+342s+315B(s)'=2s+7解得s=-1.46）系統(tǒng)根軌跡4-5已知系統(tǒng)的開環(huán)傳遞函數(shù)。(1)試?yán)L制出根軌跡圖。(2)增益Kr為何值時(shí)，復(fù)數(shù)特征根的實(shí)部為-2。第四章習(xí)題課(4-5)解:，s(s+1)Kr(s+2)G(s)=jσ0ωp1=0p1

p2=-1p2

z1=-2z1

p1~p2z1~-∞分離點(diǎn)和會(huì)合點(diǎn)s2+4s+2=0s1=-3.41s2=-0.59s2+s+Krs+2Kr=0閉環(huán)特征方程式系統(tǒng)根軌跡s=-2+jω(-2+jωω)2+(-2+j)(1+Kr)+2Kr=0ω-4+(1+Kr)ω=0ω4-2-2(1+Kr)+2Kr=0Kr=3ω=±1.414-6已知系統(tǒng)的開環(huán)傳遞函數(shù)，試確定閉環(huán)極點(diǎn)ζ=0.5時(shí)的Kr值。第四章習(xí)題課(4-6)解:p1=0p2=-1p3=-3p1~p2p3~-8+60oθ=+180o,3σ=-1-3=-1.3根軌跡的分離點(diǎn)：A(s)B'(s)=A'(s)B(s)3s2+8s+3=0s1=-0.45s2=-2.2s2沒有位于根軌跡上，舍去。σjωs(s+1)(s+3)KrG(s)H(s)=與虛軸交點(diǎn)s3+4s2+3s+Kr=02=0ωKr-4ω3+3ω=0-Kr=0Kr=12

ω2,3=±1.7ω1=01.7-1.7p3-30p1

p2-1系統(tǒng)根軌跡s1ζ=0.5得s1=-0.37+j0.8Kr=|s3||s3+1||s3+3|s3=-4+0.37×2=-3.26=3.26×2.26×0.26=1.9s3(1)s(s+3)(s2+2s+2)G(s)H(s)=Kr(2)解：p3.4=-1±jp2=-3p1=0p1~p24σ=-3-1-1=-1.25θ=+135o+45o,根軌跡的出射角θ3=+πθ1-θ2-θ4-=+π

-135o-90o-26.6o=-71.6o與虛軸的交點(diǎn)s(s+3)(s2+2s+2)+Kr=0s4+5s3+8s2+6s+Kr=0(jωω)4+5(jω)3+8(jω)2+j6+Kr=0ω4-8ω2+Kr=0-5ω3+6ω=0Kr=0ω1=0Kr=8.16

ω2,3=±1.1σjω0-1.2p1

p2

p3p4

13526.690-71.6