《现代控制理论参考答案》
第一章答案
1-1 试求图1-27系统的模拟结构图，并建立其状态空间表达式。
U(s)+-K1Kps?K1+-Kps?K1s+-1J1sKbJ2s2?(s)Kns图1-27系统方块结构图
解：系统的模拟结构图如下：
Kp+U(s)-K1Kp+-K1Kp?x6+-K1x5++?+-1J1?x4x3KbJ2?x2??(s)x1Kn?图1-30双输入--双输出系统模拟结构图
系统的状态方程如下：
x1?x2x2????Kbx3J2KpKpKn1x3??x3?x4?x5?x6J1J1J1J1 x4?x3x5??K1x3?K1X6x6?????K1KKx1?1x6?1uKpKpKp令?(s)?y，则y?x1
所以，系统的状态空间表达式及输出方程表达式为
?????1??x???x????2???x3????????x4???????x5????????x6?????00000K1Kp10Kb0J2Kp0?J10100?K1000?KnJ1000001J0000???0?0??x1???0???x??Kp??2??0??x?J1??3???0?u?x4??0????0???xK1??5??K1???K1??x6???K?p?? ???Kp???x1??x??2??x?y??100000??3??x4??x5????x6???
1-2有电路如图1-28所示。以电压u(t)为输入量，求以电感中的电流和电容上的电压作为状态变量的状态方程，和以电阻R2上的电压作为输出量的输出方程。
R1L1L2i1CUi2---------Uc---------R2图1-28 电路图
解：由图，令i1?x1,i2?x2,uc?x3，输出量y?R2x2
R1x1?L1x1?x3?u有电路原理可知：L2x2?R2x2?x3??x1??x2??x3?????R111x1?x3?uL1L1L1R21x2?x3L2L2
既得
x1?x2?Cx3?11x1?x2CCy?R2x2写成矢量矩阵形式为：
?R1????L1??x?1。?x???02?。???x3??1???????C。0?R2L21?C?1??1?L1??x????1L1????1?x2?0uL2????????x3???0??? 0?????y??0R2
?x1??0??x2????x3??1-4 两输入u1，u2，两输出y1，y2的系统，其模拟结构图如图1-30所示，试求其状态空间表达式和传递函数阵。
u1b1+---??y1a1a5a6?++a21u2b2+---?a3y2
a4图1-30双输入--双输出系统模拟结构图
解：系统的状态空间表达式如下所示：
?1??0?x?x?2???a2?????x?3??1????4??0?x1?a10?a5000?a40??x1??0?x??b?a6???2???11??x3??0?????a3??x4??00?0??u0??b2??x1??x?y??1010??2??x3????x4??1?s?as?a1(sI?A)??2??10?a5?000sa40?a6?? ?1??a3?
?1?s?as?a1?1Wux(s)?(sI?A)B??2??10?a5?000sa40?a6???1??a3??1?0?b?1?0??00?0?? 0??b2??1?s?as?a1Wuy(s)?C(sI?A)?1B??1010??2??10?a5?0
1-5系统的动态特性由下列微分方程描述
00sa40?a6???1??a3??1?0?b?1?0??00?0?? 0??b2?(2)y?5y?7y?3y?u?3u?2u
列写其相应的状态空间表达式，并画出相应的模拟结构图。 解：令x1?y，x2?y，x3?y，则有
...?......?。?1?10??x1??0??0?x。??x???0?u?x???0012??2????。???x3????3?7?5????x3????1?? ?????x1??y??231??x2????x3??相应的模拟结构图如下：
13u+---?5x3?x2?x12++y73
1-6 （2）已知系统传递函数W(s)?6(s?1),试求出系统的约旦标准型的实现，并画出相应的模拟结构图 2s(s?2)(s?3)?1016(s?1)?43?3?3 ??解：W(s)?s(s?2)(s?3)2(s?3)2s?3s?2s?1???310?x?x?2??0?30?????3??0?x0?2????00?x4??00??x1??0?0??x2??1???????u0??x3??1??????0??x4??1?
?x1???101??x2??y???4?333????x3????x4?1-7 给定下列状态空间表达式
?1??010??x1??0??x?x?2????2?30??x2???1?u?????????3???x????11?3????x3????2??‘
?x1?y??001??x2?????x3??（1） 画出其模拟结构图
（2） 求系统的传递函数 解：
0??s?1?0? （2）W(s)?(sI?A)?2s?3????1?1s?3??sI?A?s(s?3)2?2(s?3)?(s?3)(s?2)(s?1)
??s?3?2?s?30??1(sI?A)?1??2(s?3)s(s?3)0?
(s?3)(s?2)(s?1)???s?5s?1(s?1)(s?2)?????s?3?2??0?s?30????1Wux(s)?(sI?A)?1B??2(s?3)s(s?3)01????(s?3)(s?2)(s?1)???s?1(s?1)(s?2)???s?5??2??(s?3)?1?s(s?3)???(s?3)(s?2)(s?1)???(2s?1)(s?3)???(s?3)?1?Wuy(s)?C(sI?A)?1B??001??s(s?3)??(s?3)(s?2)(s?1)? ?(2s?1)(s?3)??(2s?1)?(s?2)(s?1)
1-8 求下列矩阵的特征矢量
10??0?02? （3）A?3?????12?7?6??0????1??2???3?6?2?11??6?0 解：A的特征方程 ?I?A??3?????127??6??解之得：?1??1,?2??2,?3??3
《现代控制理论》第3版(刘豹_唐万生)课后习题答案
