10??p11??0?p11??02??p21????p21? 当?1??1时，3??????????12?7?6????p31???p31??
?p11??1?????解得： p21?p31??p11 令p11?1 得 P1??p21????1? ??p31?????1???p11???1?????（或令p11??1，得P1??p21???1?） ??p31????1??
10??p12??0?p12??02??p22???2?p22? 当?1??2时，3??????????12?7?6????p32???p32???p12??2?1????解得： p22??2p12,p32?p12 令p12?2 得 P2?p22??4
????2??p32????1?????p12??1???（或令p12?1，得P2?p22???2?） ???1???p32?????2?10??p13??0?p13??02??p23???3?p23? 当?1??3时，3??????????12?7?6????p33???p33???p13??1?????解得： p23??3p13,p33?3p13 令p13?1 得 P3?p23??3 ??????p33????3??
1-9将下列状态空间表达式化成约旦标准型（并联分解）
?1??41?2??x1??31??x?x?2???102??x2???27?u?????????3???x???1?13????x3????53??（2）
?x1??y1??120????y???011??x2?????2???x3?2????4?1???2??(??1)(??3)2?0 解：A的特征方程 ?I?A??1???1??3???1??1,2?3,?3?1
?41?2??p11??p11??2??p21??3?p21? 当?1?3时，10?????????1?13????p31???p31???p11??1?????解之得 p21?p31?p11 令p11?1 得 P1??p21???1?
?当?3时，?41?2??10???p11???p11??1?2?2p21??3p21???1? ??????1?13????p31???????p31????1??解之得 p12?p22?1,p22?p32 令p12?1 ?41?当?1时，?2???10?p13??p13?3?2?p23???p23? ?13???????1????p33????p33??解之得
p13?0,p23?2p33 令p33?1
?110??0?12?T???102? T???1???11?2?
?101?????01?1??
?0?12??31??8?1?T?1B???11?2??27????52?
????????01?1???53?????34??
CT???120???110???011???314???102 ??101?????203??
约旦标准型
?310??8?1~?x????030?~??x????52??u?001?????34?? y???314?~?203??x
1-10 已知两系统的传递函数分别为W1(s)和W2(s)
??p31????1???p12??1得 P2???p22????????0?? ?p32???0???p13??0? 得 P3???p23??p????2???33????1??
?1?W1(s)??s?1?0?1??1s?2? W(s)??s?32?1s?1???s?2??s?11?s?4?
?0??试求两子系统串联联结和并联连接时，系统的传递函数阵，并讨论所得结果 解：（1）串联联结
?1?W(s)?W2(s)W1(s)??s?31??s?1?1?(s?1)(s?3)??1?2??(s?1)
（2）并联联结
1??1s?4??s?1??0??0??21?s?2?s?1??s?2??s?5s?7(s?2)(s?3)(s?4)??1?(s?1)(s?2)??
?1?W(s)?W1(s)?W1(s)??s?1?0?1??1s?2???s?3s?1??1??s?2??s?11?s?4?
?0??1-11 （第3版教材）已知如图1-22所示的系统，其中子系统1、2的传递函数阵分别为
?1?W1(s)??s?1?0?1?s? W(s)??10?
2?01?1????s?2??求系统的闭环传递函数 解：
?1?s?1W1(s)W21(s)???0?1??1s??10???s?1??1??01???0?s?2???1?s? 1??s?2???1?s?1I?W1(s)W(s)?I???0??s?3s?1?s?2??s?3?0?1??s?2s??10???s?1??1??01??0??s?2???1??s?1s?2s?????s?2??0?s?1??1?s? s?3??s?2???I?W1(s)W2(s)??1s?1?s(s?3)?
?s?2??s?3?1??1?s?3s?1?s?2?1s??s?1W(s)??I?W1(s)W2(s)?W1(s)??s?2??s?3?0??ss?1???s?31??1s?1??????s?2?s?1?(s?2)(s?1)ss(s?3)??????1??1s?3??00???s?1??s?3???
1?s?1??s?2??
1-11（第2版教材） 已知如图1-22所示的系统，其中子系统1、2的传递函数阵分别为
?1?s?1W1(s)???2?1?s? W(s)??10?
2?01?1????s?2??求系统的闭环传递函数 解：
?1?W1(s)W1(s)??s?1?2?1?1??1?s??10???s?1s?
?1??1?01?????2?s?2?s?2??1?1??1?s?2???s??10???s?1s? I?W1(s)W1(s)??s?1???1??s?301??2?2???s?2?s?2???1??s?3s?2s? ?I?W1(s)W1(s)??1?2s(s?1)?s?2?s?5s?2???2?s?1???1??1?s?3??s?2s(s?1)?s?2?1sW(s)??I?W1(s)W1(s)?W1(s)?2???s?5s?2??2s?2??ss?1???s?32s?31??????s(s?1)?(s?2)2ss(s?2)s(s?2)?2??22(s?2)21s?5s?2???????s?2?s?1ss?1???(s?1)2(3s?8)?22(s?2)(s?5s?2)??s3?6s2?6s??(s?2)(s2?5s?2)?
1-12 已知差分方程为
1?s?1??s?2??
s?1??s2?5s?2?s?2??2s?5s?2???y(k?2)?3y(k?1)?2y(k)?2u(k?1)?3u(k)
试将其用离散状态空间表达式表示，并使驱动函数u的系数b(即控制列阵)为 （1）b??? 解法1：
?1?
?1?
W(z)?2z?311??
z2?3z?2z?1z?2??10??1?x(k?1)??x(k)?u(k) ????0?2??1?y(k)??11?x(k)
解法2：
x1(k?1)?x2(k)x2(k?1)??2x1(k)?3x2(k)?u y(k)?3x1(k)?2x2(k)1??0?0?x(k?1)???x(k)??1?u(k)
?2?3????y(k)??32?x(k)?1??11??1?1??1求T,使得TB??? 得T??? 所以 T??01?
101???????1
1??1?1???40??11??0T?1AT?????2?3??01????5?1?
01?????????1?1?CT??32????3?1? ??01?所以，状态空间表达式为
??40??1?z(k?1)??z(k)?u(k)??? ??5?1??1?y(k)??3?1?z(k)
第二章习题答案
2-4 用三种方法计算以下矩阵指数函数eAt。
?11?
（2） A=??
41??
解：第一种方法： 令
?I?A?0
则
??1?4?12?0 ，即???1??4?0。 ??1求解得到?1?3，?2??1 当?1?3时，特征矢量p1???p11? ??p21??11??p11??3p11?由 Ap1??1p1，得???p???3p?
41???21??21?即??p11?p21?3p11?1?，可令p1???
?2??4p11?p21?3p21?p12?当?2??1时，特征矢量p2???
p?22?
《现代控制理论》第3版(刘豹_唐万生)课后习题答案
