3-14 已知单位反馈系统的闭环传递函数为G(s)?函数输入和抛物线函数输入时的稳态误差。
解：将闭环传递函数化为单位反馈形式
an?1s?an求斜坡nn?1s?a1s???an?1s?anan?1s?anG(s)an?1s?ansn?a1sn?1???an?2s2 ?(s)?n ??n?1as?as?a1s???an?1s?an1?1?G(s)n?1nnn?12s?a1s??an?2s 所以开环传递函数为 G(s)?an?1s?anan?1s? nn?122n?2n?3s?a1s???an?2ss(s?a1s???an?2)则其静态误差系数为：
静态位置误差系数 Kp?limG(s)H(s)??
s?0静态误差 ess?1?0
1?Kps?0静态速度误差系数 KV?limSG(s)H(s)?? 静态误差 ess?1?0 KV2静态加速度误差系数 Ka?limSG(s)H(s)?s?0an an?2静态误差 ess?1a?n?2 Kaan故当斜坡输入时，系统静态误差系数为无穷大，其静态误差为零。
4-1 某单位反馈系统的开环传递函数为G(s)?表达式。
(1)xi(t)?sin(t?30?) (2) xi(t)?3cos(2t?60?)
5，试求下列输入时，输出的稳态响应s?1解：上述控制系统的闭环传递函数为
?(s)?其频率特性为
X0(s)G(s)55 ???Xi(s)1?G(s)s?1?5s?6?(j?)?X0(j?)5 ?Xi(j?)j??656??22则 |?(j?)|??536??2
??(j?)??tg?1??6
当输入信号为 xi(t)?sin(t?30) 时 可令 t?t?30 即 xi(t)?sint 此时
'''???1
555???0.822
36?1376.08?1则 |?(j?)|? ??(j?)??tg'1??tg?10.167??9.46???9?28' 6'?即 xi(t)?|?(j?)|sin(t?9.46) 将变量t换成t
xi(t)?|?(j?)|sin(t?30?9.46) ?0.822sin(t?20.54) ?0.822sin(t?2032) 当输入信号为xi(t)?3cos(2t?60)时 可令 t?t?30
即 xi(t)?3cos2t?3sin(90?2t) 此时
''?'???'??''???2 R?3
536?22?55??0.791 406.32则 |?(j?)|? ??(j?)??tg?1'2??tg?10.333??18.43???18?26' 6?'?即 xi(t)?R|?(j?)|sin[(90?2t)?18.43] ?3?0.791cos(2t?18.43) ?2.373cos(2t?1826) 将变量t换成t
xi(t)?2.373cos(2t?60?18.43) ?2.373cos(2t?78.43) ?2.373cos(2t?7826)
4-2 试画出具有下列传递函数的极坐标图。
11(1) G(s)? (2) G(s)?
s(0.1s?1)0.01s?12(0.3s?1)7.5(0.3s?1)(s?1)(3) G(s)?2 (4) G(s)?
s(s2?12s?100)s(5s?1)(0.2s?1)(0.025s?1)(5) G(s)?2 (6) G(s)?5e?0.1s
s(0.005s?1)(0.001s?1)解：（1）G(s)??'???'?'?''1
0.01s?111?e?j?arctan?T1?
1?jT?1?T2?2Im0 G?j??? 1 （2）G(s)?1
s(0.1s?1)1e??j?90??arctTa?n ?G?j????1??T22 当??0时，G?j??????90?
???
???2T?j???t?22??2
当???时，G?j???0??180?
Re?G?j?????T??0.1
lim???
Im-0.1 0 ???
（3）G(s)?2(0.3s?1)
s2(5s?1) G?j???21?T12?2?21?T22?2e?j?180??arctanT1??arctanT2??21?T1T2?2?j2?T1?T2?? ???21?T22?2???? 当??0时，G?j??????180? 当???时，G?j???0??180?
Im?? ? 0
(4) G(s)?7.5(0.3s?1)(s?1)
s(s2?12s?100)G(j?)?7.5(j?T2?1)(j?T3?1) 22?[(1??T1)?j2??T1]当??0时，G?j??????180? 当???时，G?j???0??180?
Im?? ? 0
（5）G(s)?(0.2s?1)(0.025s?1) 2s(0.005s?1)(0.001s?1)G(j?)?(j?T3?1)(j?T4?1) 2?(j?T1?1)(j?T2?1)当??0时，G?j??????180? 当???时，G?j???0??360?
Im ??0 ? 0???Re
（6）G(s)?5e?0.1s
G?j???5e?j0.1?
Im ???0 Re