?(s?3)?Ws)?C(sI?A)?1B??001???s(s?3)?1uy(???(2s?1)(s?3)??(s?3)(s?2)(s?1) ?(2s?1)(s?2)(s?1)
1-8 求下列矩阵的特征矢量
?（3）A??010??302? ??12?7?6?????解：A的特征方程 ?I?A????10???3??2???3?6?2?11??6?0 ???6??127??解之得：?1??1,?2??2,?3??3
?当??010??p11???p11?1??1时，?302??p21???p21? ??12?7?6??????????p31????p31??
?p11??1解得： p21?p31??p11 令p?????11?1 得 P1??p21????1? ??p31?????1???p11???（或令p??1，得P???1??111??p21????1?） ?p31????1??
?当??010??p12??p12?1??2时，?302??p22???2?p22? ?12?7?6???????????p32????p32???p12??2?解得： p??2p1????2212,p32?2p12 令p12?2 得 P2??p22?????4???p32???1???p??12?p?1?（或令??12?1，得P2??p22???2?） ???p????1?32?2??欢迎下载
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?当??010??p13???p13?1??3时，?302???12?7?6????p23???3??p23?? ????p33????p33???p13??1?解得： p23??3p13,p33?3p13 令p13?1 得 P3???p23????3? ?p????33????3??
1-9将下列状态空间表达式化成约旦标准型（并联分解）
???x1??41?2??x1??31??x?2?（2）?????102???x2???27?u?x??x???3????1?13????3????53????y??x
1?1?2?????120???y?011???x2???x3???解：A的特征方程 ?I?A????4?12???1??2??(??1)(??3)2?0 ??1??1??3???1,2?3,?3?1
?41?2?当??3时，??10???p11??p11?2p21??3?p21?1 ?13??????1????p31?????p31???p11??1解之得 p21?p31?p11 令p?????11?1 得 P1??p21p???1? ??31????1??
?当??41?2?????p11??p11??1?2?3时，102p21??3?p21???1? ??????1?13????p31???????p31????1???解之得 p12?p22?1,p22?p32 令p??p12??p22???1??12?1 得 P2????0? ?p32????0???当?3?1时，?41?2??p13??p13??102??p23???p23? ???????1?13????p33????p33??欢迎下载
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?p?2p13??0?解之得
p13?0,p2333 令p33?1 得 P3???p23??p????2??
?33????1??
?110??0?12?T???102? T?1??11?2?
?????101????01?1??
?0?12??31??8?1?T?1B???11?2??27????5?1?1?????2??
?0???53?????34??
?1CT???120??10???011???102?????314??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)
?11??11?W??1(s)?s?1ss??2?1? W2(s)???s??13s?4?
?0s?2????s?10???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
?11??11?W(s)?W)W?2(s1(s)??s?s?4?s?13?s?10????s?1s??2?1????0s?2???s2?5s?7?
??1?(s?1)(s?3)(s?2)(s?3)(s?4)??11???(s?1)2(s?1)(s?2)???欢迎下载
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（2）并联联结
?11??11?W(s)?W?W?1(s)1(s)??s?1ss?4?
?s??2?1???s?3?0s?2??1???s?10???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?1?W(s)??1?s?1s??? W(s)??10?12?01?
?0s?2????求系统的闭环传递函数 解：
?1?1??1?1?W)W?1(s21(s)??s?1s??1s????10?01?1? ?0s?2???????s?1??0s?2????1??1?I?W?1?10??s?21(s)W(s)?I??s?11s????s?s?3? ?0s?2???01?????s?1??0s?2???s?31??s?1s?1??I?W(s)W?1?s?1?s?2s?12(s)?s?2????s?2s(s?3)?s?3??s?2?
?0s?1?????0s?3????s?31??1?1?W(s)??I?W??1Ws?1?1(s)W2(s)1(s)?s?2s??s?1s?s?3??1??0s?2??s?1????ss?2???s?31??1s?1
?s?1?s?3?(s?2)(s?1)?s??s?2??s(s?3)??1??????0s?1??????01s?3???
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?1?W?1(s)??s?11s?? W(s)??10?2??01?
?2s?2????求系统的闭环传递函数 解：
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?1?1??11?W?1(s)W1(s)??s?11s???10????s??2s?2????01????1s?
?1??2s?2???11??s1?I?W?1(s)W1(s)??s?1??2s??10?????s?1?s? ?21??0?s?3?s?2???1???2s?2???s?31??I?W)W?s(s?1)??s?2s?1(s1(s)??1s2?5s?2?s?2? ??2s?1???s?311W(s)??I?W?1s(s?1)???1??1(s)W1(s)?W1(s)?s?s2s?2s??s?2?5s?2??1???2s?2??s?1????ss?2???s?32s?31??s(s?1)?(s?2)2?s?s(s?2)?s(s?2)?s2?5s?2??22(s?2)21????s?2?s?1?s?s?1?????(s?1)2(3s?8)?s?1???(s?2)2(s2?5s?2)s2?5s?2??s3?6s2?6s??(s?2)(s2?5s?2)?s?2??s2?5s?2??
1-12 已知差分方程为
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???1??
?1?
解法1：
W(z)?2z?31z2?3z?2?z?1?1z?2 x(k?1)???10???0?2??x(k)??1???1??u(k) y(k)??11?x(k)
解法2：
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《现代控制理论》第3版课后习题答案
