考研专业课-自动控制原理

根轨001100101010110100010100根轨2012-8- 第4第4 00110010 00101001011 2基本要0011001010101101000101001011返回 001100101010110100010100101140011递函数之间的关系，直接由开环传递函数零54－1根轨迹与根轨迹方程0011001010101101000101001011返回 例0011001如图所示二阶系统，系统的开环传递函数为GG(s)Ks(0.5s7开环传递函数有两个极

p10,p2 0011001010101101000101001011闭环传递函数(s)C(s)s22s闭环特征方程D(ss22s2Ks1s1 12K,s2 180011001010101101000101001011

s10,s2s11,s2s11j,s21s112j,s212js11j,s21j90011001010101101000101001010011001010101101000101001011(s)(s)G(s)1G(s)H0011001010101101000101001011

s

2s22sG(s)

ff(szi K i (spii

22T2sK2K2 GGTT12G10011001010101101

H(s)H(s)KH(szjj(spjjlhK* K*

(s(szi)(szjflG(s)H(s)K*Kj (spi)(spjql(szi)(szjflKj(spi)(spjqhj0011001010101101000101001011(s)(s)*G(szkf (spkknzk001100011101 01增益等于系统前向道的根轨迹增零点以及开环根轨迹增益K 根根轨迹法的任务是在已知开0011001010101101000101001011 （4-G(s)H(s)=- （4-001100111100000

1G(s)HG(s)H(s)K* 1(szim(spinKK|s|0011 1010110100101001011|in 1n|spi

(szi)(spi)(2k k0,1,2,0011001•10 000 01在实际应用中，用相角方程绘制根轨迹，而K*值。00110011010111000001011G(s)H(s)2K/(s

s12j4,s22证明：该系统的开环极

p12,p2s1,001100图图 p) 90D图00110010110101101000(2k 以s2为试验点，可

(k p) p)90

(2k

0011001010101101000101001011

例已知系统开环传递函数G(s)H(sK/(s00110010100111时其根轨迹如图4-5

s10.5j0.5所对应的K |0.5j0.51|422001|01KK400对应的K*值。一、根轨一、根轨分支数＝开环极＝开环特征方程的阶

4－2绘制根轨迹 返回 0011001010101101000101001011m(szi*i 1*ni

(spi 1010110110101101

000101001011K*0→spi0→ K*→szi

s(有(有nm个开环零点在无穷远处0011 在实轴上任取一试验点s1代入相角方 11(szi)(spi (sz1)(sz2(sp1(2k所以相角方程成立，即s1是根轨迹上的00110010110100101001011 (szi)(spi)(l (lh)(2k例001101G(s)=K(s+1)/[s(0.5s+1)],求K0G(s)2K(ss(s按绘制根规迹法则逐步进0011001010101101000101001011③法则四，在负实轴上，0到－1区间和到负无穷区间是根最最后绘制出根轨迹如图4－7所示0011001010101101000101001011图0011001010101101000101001011 (2kan z znmiajn0011001001011000101001G(s)H(s)

K*(ss(s4)(s22s z m极点p10,p24,p31j1p41j1,n40011001010101101000101001011(2k1)(2k1)(2k

n00

4 (k0 00 03

(k(k 5n 以下是几种开环传递函数的根轨KKs(sp1Ks(sp1)(sp2KKs2(sp)(sp)(sp123K0010a010110(0)10 s(sp1K

G(s)H(s)G(s)H(s)

s(sp1)(sp2Ks(sp1)(sp2)(sp3K

G(s)H(s)s2(sp)(sp)(sp 0011001010101101000101001011根轨迹的终止角是指终止于某根轨迹的起始角是指根轨迹在点处的切线与水平正方向001100110101101001100101计算公k(2k1)(pkzj)(pkpimnj 终止角计算公式zk(2k1)in p)kijjm zkj例00110011010110 10G(s)H(s)

K*(s2j)(s2j)(s1j2)(s1j2) K*K*(s2j)(s2j)(s1j2)(s1j2)①n=2，有两条根0011001010101101000101001011终于开环零(-2-j图00110010101011010001图00110010101010100001010 离点坐标定义：几条（两条或两条以上）根轨在s平面上相遇又分离点的坐标d可由下面方程0011001010101101000101001011n1m1i1dj1dz式中：zj为各开环零点的数例001100100101010001010101G(s)H(s)

K*(ss23s解：根据系统开环传递函数求出开环极001100101010 1.5 0011001010101101000101001011离开复平面极点的初始角 180D z1

p2180D116.57D90D 001100101010101j11.5j11

2 (2k1) 2 d1.5 d1.5 dd12.12d2 （舍去00110010101011010001图00110010101011010001010010111[(2k1)mdl(dz)nj(dsiildd0011001010101101000101001011sikkdl个重极点外，nl个非重根。会合角计算0011001010101101000101001011 1[(2k1)ndl(dp)ni(dsiiilsikkd时l个重极点外，nl个非重根。分离角与会合角不必经公式计算，可以用下列简0000100l条根轨迹进入d点，必有l条根轨迹离开l条进入dl条离开d点的根轨迹相间迹方向之间的夹角为 附近根轨迹方向的方法可根据法则2、法则4或取试验点用0011001010101101000101001011的K*和 的sj定，也可闭环特征的sj例0011001010101101000101001011

s(s3)(s22s解：按步骤画0011001010101101000101001011①有40，-3，-1+j1,1-j1③实轴上的根轨迹在0到-3④渐近线

(2k4

450, 031j11j1 ⑤确定分离点001100101010110100101001011 01d解方程d12.3,d2,30.92解方程（舍去00110010 10001100101 1800

p1

p2

p41800135026.570 ⑦确定根轨迹与虚轴0011001010101101000101001011D(s)s(s3)(s22s2)K*ss1.095,K*00110010图

例4－7根轨001100101010 10 n(sp)*im(sz)jn(ssijsnasn1asn2"12s

。若(nmK*00110010011010011001011

((例0011001010101101000101001011已知单位负反馈系统开环传递函G(s)

s(0.05s1)(0.05s20.2s试画

K0

时的闭环系统概略根轨迹KK递函数及闭环极

时的闭环解；根据根轨迹 则，按步计算0011001010101101000101001011②起始于开环极点0，-20，-2-j4, ③实轴上的根轨迹在（0，-20）400110010101000101001011466

i202j42j4 (2k1)(2k n k0,a 45D;k k 135D;k a⑤根轨迹的起始p p13 pp p2 4 3 180p p13 pp p2 4 3300110010101011010001010010111800116.5012.50900 3904⑥分离点坐标d1 d d2 d2 ⑦根轨迹与虚轴交00110010100 1011DD(s)s(s20)(s24s20)400K s 代解得

4.1K此时特征方程0011001010101101000101001临D(s)s(s20)(s24s 临D(s)s424s3100s2 .9ss34.2,s4001100101010110100010100图例4－8根轨迹00110000110100014－30011001010101101000101001011

G(s)HG(s)H(s)10

APAP(s)1Q(s)返回 令001100101010

(s)H(s)AP(s) Q(s)

11110KG(s)H(s)

s(s2)(sp1p10GG1(s)H1(s) p(s2s32s2K 例4-0011001010101101000101001011G(s)H(s)

s(s1)(Tas2试绘制当开环增益K为1,1,2 间常数Ta0变化时的根轨迹。2001100101010 等效开环传递函Tis2(sG1(s)H1(s) s2s等效开环传递函数有3个零点，即-1；2个极点，不同K值可计算出不按照常规根轨迹的则可绘制出义根轨迹如图4-210011001010101101000101001011图4-001100101010110100010100101100110010101

研究内回GsHs从而相角方程及模值方程0011001010101101000101001011 szispim msznK 1ns0011001010101101000101001011Ana计算公式根轨迹的起始角与0011001010101101000101001011 z p 2k pj

ij1j

pj

zz z

ji

jp分离角与会除除上述四个法则外，其他法则不00100100110100010100101

G(s)

K*(s(s3)(s22s

,H(s)K*0解：0011001010101101000101001011K*0变化时的根轨迹是零度根轨迹。实轴根轨迹在(3)和(2区间内。p13,p21j1,p31

001图4-例4-11根轨迹G(s)H *求闭环极点 0011001010101101000101001011返回 001100100101101000101001011K(sＮ阶系统K(s(s)

C(s) bsmbsm1"bR(s) asnasn1"

* j* mn

(ssiizj为闭环传si为闭环传递函数的极设输入为单位阶跃：r(t)=1(t),0011001010101101000101001011C(s)(s)R(s)

K

(szjs(ssii假设(s)中无重极点，上式分解为C(s)

A0

"

sAn Ak1s sAn Ak1s000101001011*000101001011*m(s)j(ssiinA0

K*K*(s mmKKm(s js(ssiiin

j1ns

sk(sksiii将C(s)表达式进行拉式反变换0011001010101101000101001011nkC(t)(0)k

k

Aesk

C(t)(0)C(t)(0)knA kk0011001010101101000101001011面的左半部；001100101010110100010100101

mK m jA小，闭环极点之间间 k

i1i

CC(t)(0)knA kk0011001010101101000101001011：位的极点，一般是离虚轴最近的极如果有两个极点：sk jsiij4sik0011010001一对靠得很近的闭环零、极点当当sks0.1时，就可以认sk与zi是一对偶极k在对系统进行分析时，就可以将其忽略不计00110010101010100010100011例4-0011001 00

试近似计算系统的动态性能指标%,ts s11.5,s2,34s10011001010101101s1

0011001010101101000101001011(s)(s)10.67s1Ts一阶系统无超调调节时间ts3T30.67s例4-0011001010101101000101001011(s)

(0.67s1)(0.01s20.08s0011001011010100010100100110010101011010001010010111(s)0.01s20.08sn 10sn1%e1

100%e0.43.14

3.5 s 0.4例4-0011001010101101000101001011G(s)

s(s1)(0.5s图4-图4-解0011110101010010s(s1)(s Ks(s1)(sK*001100101010110100010100101100K在平面上画出 s1s10.33j0.58之共轭的复数极点为s20.33